$\dot{x}$ becomes null. More explicitly, consider a parameter interval $\delta
t$ on $\mathpzc{q}$, short enough that the metric, $V$, and $\dot{x}$ can be
considered constant. Let $\mathpzc{q}^{\prime}$ connect the endpoints of this
interval by way of two timelike segments with tangent covectors
$\dot{x}^{\prime}_{\pm}$, each of proper time $\delta t^{\prime}/2$, so that
$\delta x^{\prime}_{+}+\delta x^{\prime}_{-}=\dot{x}\delta t$, where $\delta
x^{\prime}_{\pm}=\dot{x}^{\prime}_{\pm}\delta t/2$. Then, on
$\mathpzc{q}^{\prime}$, using (14), $|V_{\perp}|=|\dot{x}^{\prime}_{\pm}\wedge
V|$, so the contribution of these segments to the left-hand side of (70) is
$|\delta x^{\prime}_{+}\wedge V|+|\delta x^{\prime}_{-}\wedge V|$. In the
limit that the segments become null, again using (14), $|\delta
x^{\prime}_{\pm}\wedge V|$ goes to $|\delta x^{\prime}_{\pm}\cdot V|$. Since
$\delta x^{\prime}_{+}\cdot V$ and $\delta x^{\prime}_{+}\cdot V$ have the
same sign, the sum of these terms is $|(\dot{x}\delta t)\cdot V|$. By (16),
and using the fact that $V$ is timelike, this equals
$|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|\,\delta t$, which is the
contribution of the original segment of $\mathpzc{q}$ to the left-hand side of
(68). Adding up such segments, we have
$\lim_{\mathpzc{q}^{\prime}\to\text{null}}\int_{\mathpzc{q}^{\prime}}dt^{\prime}|V_{\perp}|=\int{q}dt|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|$.  Thus, (70), when imposed on all timelike curves, is
equivalent to (68) imposed on all timelike curves and therefore, by
continuity, on all causal curves.

So far we have shown that (56) implies that (68) holds for all causal curves
(or equivalently that (70) holds for all timelike curves). Remarkably, as
shown in theorem 4.7 (subsection 4.4.3), the converse holds. This theorem is
proved by constructing, given $V$, a function $\phi\in\mathscr{S}_{\rm c}$
such that $d\phi\pm V\in\mathfrak{j}^{+}$. We can therefore choose to take
(68) (or (70)) as the defining norm bound for a V-flow, and dispense with the
function $\phi$. In other words, we can define $\mathscr{F}$ as the set of
1-forms $V$ such that

$d{*V}=0\,,\qquad*V|_{\mathcal{I}}=0$ (71)
$\forall\mathpzc{q}\in\mathscr{Q}\,,\,\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\leq 1\,,$ (72)

where $\mathscr{Q}$ is the set of inextendible causal curves in $\mathcal{M}$.
From this viewpoint, the function $\phi$ can be thought of as a certificate,
or witness, for (68), so that one does not have to check it for every causal
curve.

##### Time-sheet bound:

The second bound is a bound on the flux of a V-flow through an arbitrary time-
sheet $\tau$ (not necessarily homologous to $D(A)$, although that will be an
important special case). As we show in subsection 4.4.1 (see lemma 4.2), given
an orthonormal pair of covectors $(T,N)$ with $T\in\mathfrak{i}^{+}$, the
constraint $d\phi\pm V\in\mathfrak{j}^{+}$ implies424242 In lemma 4.2, we also
show the converse: if $(W,V)\in\mathfrak{j}^{+}\times T^{*}$ obeys $N\cdot
V\leq-T\cdot W$ for any orthonormal pair $(T,N)$ with $T\in\mathfrak{i}^{+}$,
then $W\pm V\in\mathfrak{j}^{+}$. Therefore, if we wish to use a scalar
function and a local condition to guarantee (75), then $d\phi\pm
V\in\mathfrak{j}^{+}$ is the weakest condition we can impose.

$N\cdot V\leq-T\cdot d\phi\,.$ (73)

The level sets of $\phi$ on $\tau$ can be written in terms of its level sets
$\sigma_{t}$ on $\bar{\mathcal{M}}$:

$\gamma_{t}:=\sigma_{t}\cap\tau\,.$ (74)

We know from the fact that $d\phi\in\mathfrak{j}^{+}$ that $\sigma_{t}$, and
hence $\gamma_{t}$, is achronal. Assume temporarily that $\gamma_{t}$ is
spacelike (or empty) for all $t$; we will use continuity to address the null
or partly null case below. Let $N$ be the unit normal covector field on $\tau$
and $T$ the unit future-directed covector tangent to $\tau$ and normal to
$\gamma_{\tau}$. The flux density of $V$ on $\tau$ is $N\cdot V$. The proper
time between nearby surfaces $\gamma_{t}$ and $\gamma_{t+dt}$ is $dt/(-T\cdot
d\phi)$, so the flux per unit spatial area on $\gamma_{t}$ in the strip of
$\tau$ between $\gamma_{t}$ and $\gamma_{t+dt}$ is $dt(N\cdot V)/(-T\cdot
d\phi)$, which by (73) is bounded above by $dt$. Integrating over $\gamma_{t}$
and over $t$, we find

$\int_{\tau}*V\leq\int_{-1/2}^{1/2}dt\,\operatorname{area}(\gamma_{t})\,.$
(75)

By continuity, (75) holds even when $\gamma_{t}$, for some values of $t$, is
null or partly null (but still achronal). The right-hand side of (75) is
bounded above by the maximum area in $\Gamma_{\tau}$, so we have

$\int_{\tau}*V\leq\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,.$
(76)

Again by continuity, (76) also holds when $\tau$ is null or partly null. The
bound (76) is the analogue of the bound
$\int_{\gamma}*v\leq\operatorname{area}(\gamma)$ on the flux of a Riemannian
flow through a surface $\gamma$. It implies for example that an observer who
carries a window of area $a$ along his worldline will see a total flux of $V$
through the window over his lifetime that is bounded above by $a$.

When applied to a time-sheet $\tau\in\mathscr{T}$ (i.e. a time-sheet
homologous to $D(A)$ relative to $\mathcal{I}$), (76) bounds the flux of $V$
through $D(A)$:

$\int_{D(A)}*V=\int_{\tau}*V\leq\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,,$
(77)

where in the first equality we used the constraints $d{*V}=0$,
$*V|_{\mathcal{I}}=0$. Maximizing the left-hand side of (77) over V-flows and
minimizing the right-hand side over time-sheets yields the inequality $S_{\rm
c}\leq S_{+}$, for which we sketched a different proof in subsection 3.3.

###  4.2 U-flows

In the previous subsection we started from the first, maximin formula for
$S_{\rm c}$ in (49) and dualized on $\psi$ for fixed $\phi$. We can also start
from the second, minimax formula, and dualize on $\phi$ for fixed $\psi$. The
dualization is carried out in subsection 4.4.2. The result is a _minimization_
convex program:434343 It is possible to dualize (78) on $\psi$, to obtain a
minimax formula in terms of 1-forms $U$, $V$. However, this formula does not
tell us anything new. See subsection 4.4.2 for details.

$S_{\rm c}=\inf_{U\in\mathscr{G}}\int_{\mathcal{I}^{+}}*U\,,$ (78)

where $\mathscr{G}$ is the set of U-flows, and a U-flow is a 1-form $U$ on
$\bar{\mathcal{M}}$ satisfying

$d{*U}=0\,,\qquad*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0\,,$ (79)
$\exists\,\psi\in\mathscr{T}_{\rm c}\text{ s.t. }U\pm
d\psi\in\mathfrak{j}^{+}$ (80)

Here the condition $U\pm d\psi\in\mathfrak{j}^{+}$ appearing in the norm bound
(80) imposes a _lower_ bound on the flux, requiring $U\in\mathfrak{j}^{+}$ and
$|U|\geq|d\psi|$; see subsection 2.3.1 (especially figure 5) with $U$ taking
place of $W$ and $d\psi$ taking place of $X$.

The symmetry of $S_{\rm c}$ can easily be understood in the language of
U-flows. Under exchange of $A$ and $B$, whereas a V-flow and its scalar
witness were mapped as $(V,\phi)\to(-V,\phi)$, preserving the objective and
constraints, the corresponding map for a U-flow is $(U,\psi)\to(U,-\psi)$.

The U-flow formula (78) is related to the V-flow one (54) via the minimax
formulas (49). These two formulas are also directly related by Lagrange
duality on both variables, exchanging $(V,\phi$) for $(U,\psi)$; the
dualization can be found in subsection 4.4.2. The relationships among these
formulas is summarized in the following diagram:

$\minCDarrowwidth 50pt\CD\boxed{\sup_{\phi\in\mathscr{S}_{\rm c}}\
\operatorname*{inf\vphantom{p}}_{\psi\in\mathscr{T}_{\rm c}}\
\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|}@Z\text{minimax}Z\text{theorem}Z\boxed{\operatorname*{inf\vphantom{p}}_{\psi\in\mathscr{T}_{\rm
c}}\ \sup_{\phi\in\mathscr{S}_{\rm c}}\
\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|}\\\
@X{\scriptsize\begin{array}[]{r}\psi\leftrightarrow V\\\
\text{duality}\end{array}}XX@XX{\scriptsize\begin{array}[]{r}\phi\leftrightarrow
U\\\ \text{duality}\end{array}}X\\\
\boxed{\sup_{V\in\mathscr{F}}\int_{D(A)}*V}@Z{(V,\phi)\leftrightarrow(U,\psi)}Z\text{duality}Z\boxed{\inf_{U\in\mathscr{G}}\int_{\mathcal{I}^{+}}*U}$
(81)

####  4.2.1 Time-sheet-flows

Analogously to the slice-flows discussed in subsection 4.1.1, we can define a
_time-sheet-flow_ as a U-flow such that, for some time-sheet
$\tau\in\mathscr{T}$, $\psi=-1/2$ on the spacetime homology region
interpolating between $\tau$ and $D(A)$, and $\psi=1/2$ on the complement. The
flux of a time-sheet-flow is bounded below by the area of any surface
$\gamma\in\Gamma_{\tau}$ on $\tau$ (see (98)), and in particular by the
maximal-area surface:

$\int_{\mathcal{I}^{+}}*U\geq\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,.$
(82)

In fact, as a consequence of the strong duality between the $U$ and $\phi$
programs, this inequality is saturated. Given a time-sheet
$\tau\in\mathscr{T}$ and the corresponding function $\phi$, we have, for any
$\psi\in\mathscr{S}_{c}$,

$\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|=\int_{\mathcal{M}}\sqrt{g}\,|d\phi\wedge
d\psi|=\int_{-1/2}^{1/2}dt\,\operatorname{area}(\tau\cap\sigma_{t})\leq\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,,$
(83)

where $\sigma_{t}$ is the level set of $\phi$ with $\phi=t$ and in the first
equality we used the fact that $\tau$ is everywhere timelike so its normal
covector $d\psi$ is everywhere spacelike. Strong duality of the $U$ and $\phi$
programs says that the inf of the $U$ objective equals the sup of the $\phi$
objective (for fixed $\psi$), so

$\inf_{U}\int_{\mathcal{I}^{+}}*U=\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,,$
(84)

where the inf is over time-sheet-flows with fixed time-sheet $\tau$.
Minimizing over time-sheets, we have, analogously to (58),

$S_{+}=\inf_{U\text{ time-sheet-flow}}\int_{\mathcal{I}^{+}}*U\,.$ (85)

So in a spacetime like that of figure 6 where $S_{c}<S_{+}$, the flux cannot
attain $S_{c}$. As with the slice-flows, this is a case where the convex
relaxation (going from time-sheet-flows to general U-flows) changes the
optimal value.

Just as slice-flows are directly related via (57) to Riemannian flows, time-
sheet-flows are related to Lorentzian flows, although the relation is a bit
more complicated. Recall that a Lorentzian flow, as defined in
Headrick:2017ucz , is a 1-form $u$ on a Lorentzian manifold-with-boundary
obeying

$u\in\mathfrak{i}^{+}\,,\qquad|u|\geq 1\,,\qquad d*u=0\,.$ (86)

(No-flux boundary conditions may also be imposed on part of the manifold’s
boundary.) A time-sheet-flow involves a piece that is delta-function localized
on the time-sheet, $\delta(x^{1})u_{a}dx^{a}$, where $\\{u^{1},u^{a}\\}$ is a
set of Gaussian normal coordinates about $\tau$ and $u$ is a 1-form on $\tau$
obeying $u\in\mathfrak{i}^{+}$, $|u|\geq 1$ (as well as the no-flux boundary
condition $*u|_{\tau\cap\mathcal{I}^{0}}=0$). However, the flux of $U$ may
enter or leave the time-sheet, so $u$ is not necessarily divergenceless, nor
does $U$ necessarily vanish off of $\tau$. In fact, if $\tau$ has spacelike
seams at which timelike pieces meet, then the flux must continue past the
seam, so $U$ cannot vanish outside $\tau$.

####  4.2.2 Worked example

Figure 8:  The U-flow $\hat{U}=U_{{}_{|\\!|\\!|}}+U_{{}_{\\!/\\!/}}$ on the
spacetime of figure 6. The vertical lines represent the component
$U_{{}_{|\\!|\\!|}}$ (62), while the diagonal lines in the stripe represent
the component $U_{{}_{\\!/\\!/}}$ (63). The thick line in the upper right
represents a delta-function carrying the flux from $U_{{}_{\\!/\\!/}}$ along
$D(B)$ toward $\mathcal{I}^{+}$; similarly for the thick line in the lower
left.

To see the effect of relaxation, we return again to the spacetime of figure 6.
The minimum flux of any time-sheet-flow is $S_{+}=a_{+}$. Instead of a single
time-sheet, we can consider a smeared time-sheet, represented for example by
the function $\hat{\psi}=x^{1}/L$. In the $a_{-}$ region, the smallest
covector $U$ obeying $U\pm d\hat{\psi}\in\mathfrak{j}^{+}$ is $dx^{0}/L$. This
can be continued into the $a_{+}$ region to make a divergenceless 1-form:

$U_{{}_{|\\!|\\!|}}=\begin{cases}\frac{1}{L}\,dx^{0}\quad&(a_{-}\text{
region})\\\ \frac{a_{-}}{La_{+}}\,dx^{0}\quad&(a_{+}\text{
region})\end{cases}\,.$ (87)

$U_{{}_{|\\!|\\!|}}$ has flux $a_{-}$ through $\mathcal{I}^{+}$. However, it
does not obey $U\pm d\hat{\psi}\in\mathfrak{j}^{+}$ in the $a_{+}$ region. We
must add extra flux through the stripe. The minimal extra 1-form that we can
add to satisfy this constraint is

$U_{{}_{\\!/\\!/}}=\begin{cases}0\quad&(a_{-}\text{ region})\\\
\left(1-\frac{a_{-}}{a_{+}}\right)\frac{1}{L}\frac{1}{1-\beta}\,(dx^{0}-\beta\,dx^{1})\quad&(a_{+}\text{
region})\end{cases}\,.$ (88)

Since the dual vector field $U_{{}_{\\!/\\!/}}^{\mu}$ is parallel to the edges
of the stripe, $U_{{}_{\\!/\\!/}}$ is divergenceless. However, it does not
obey the no-flux boundary condition on $\mathcal{N}$. We therefore also have
to add a delta-function 1-form in the $x^{0}$-direction on $D(A)$ from
$\mathcal{I}^{-}$ to the top edge of the stripe, and on $D(B)$ from the bottom
edge of the stripe to $\mathcal{I}^{+}$. The total flux of
$\hat{U}=U_{{}_{|\\!|\\!|}}+U_{{}_{\\!/\\!/}}$ is

$\int_{\mathcal{I}^{+}}*\hat{U}=a_{-}+\frac{T^{\prime}}{L}\frac{\beta}{1-\beta}\,(a_{+}-a_{-})\,.$
(89)

By virtue of (61), this is smaller than $a_{+}$. On the other hand, it is
larger than the flux (66) for the V-flow of the previous subsection. In fact,
this holds for any pair $(V,\phi)$ and $(U,\psi)$ of feasible configurations,
and is a consequence of the duality between the V-flow and U-flow programs. We
can say more: as a consequence of the vertical dualities in the diagram (81),
the functional appearing in the minimax formulas (49) always sits between
these two values:

$\int_{D(A)}*V\leq\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|\leq\int_{\mathcal{I}^{+}}*U\,.$ (90)

For the functions $\hat{\phi}$, $\hat{\psi}$ we used in the previous
subsection and this one, we have

$\int_{\mathcal{M}}\sqrt{g}\,|d\hat{\phi}\mathclap{\hskip
6.75pt\cdot}{\wedge}d\hat{\psi}|=a_{-}+\frac{T^{\prime}}{T}\,(a_{+}-a_{-})\,,$
(91)

which indeed (using (60)) obeys (90).

This configuration $(\hat{U},\hat{\psi})$, like $(\hat{V},\hat{\phi})$, is not
optimal. Indeed, we do not know what the optimal configuration is, nor the
value of $S_{\rm c}$ for this spacetime.444444 One natural generalization of
the scalar fields $\phi$ and $\psi$ being linear is for them to be piecewise
linear, with different constant gradient inside and outside the stripe.
Setting the outside parts to be the same as before and optimizing over the
gradients inside, we can see that the extremized fluxes of $V$ and $U$
approach each other further (but not enough to coincide), while maintaining
the requisite nesting. The detailed results are presented in appendix B.  We
only know that it lies somewhere between (66) and (89).

####  4.2.3 Bounds

U-flows obey non-local bounds analogous to those obeyed by V-flows. The
condition $U\pm d\psi\in\mathfrak{j}^{+}$ implies that $U\in\mathfrak{j}^{+}$
and also that, for any covector $Y$,

$|U\mathclap{\hskip 6.75pt\cdot}{\wedge}Y|\geq Y\cdot d\psi$ (92)

(see lemma 4.4 in subsection 4.4.1). Let $\mathpzc{p}$ be a curve in
$\mathcal{M}$ starting in $D(A)$ and ending in $D(B)$, parametrized by $s$,
and $\dot{x}$ be the covector dual to its tangent vector $dx^{\mu}/ds$. The
bound (92), with $Y=\dot{x}$, together with the boundary conditions on $\psi$,
imply

$\int{p}ds\,|U\mathclap{\hskip 6.75pt\cdot}{\wedge}\dot{x}|\geq 1\,.$ (93)

(Note that, since the pairing $|\ \mathclap{\hskip 6.75pt\cdot}{\wedge}\ |$ is
homogeneous in the second argument, the left-hand side of (93) is independent
of the parametrization of $\mathpzc{p}$.) For spacelike $\mathpzc{p}$, with
$s$ the proper-distance parameter, (93) is equivalent to

$\int{p}ds\,|U_{\perp}|\geq 1\,,$ (94)

where $U_{\perp}$ is the projection of $U$ perpendicular to $\dot{x}$.

According to theorem 4.8 (subsection 4.4.3), the converse is also true: if
$U\in\mathfrak{j}^{+}$ everywhere and (93) holds for all curves $\mathpzc{p}$
from $D(A)$ to $D(B)$, then (80) holds. Thus we can take (93) as the defining
norm bound for $U$ and dispense with the function $\psi$, defining a U-flow
for $A$ as a 1-form $U$ obeying

$d{*U}=0\,,\qquad*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0\,,$ (95)
$U\in\mathfrak{j}^{+}\,,\qquad\forall\,\mathpzc{p}\in\mathscr{P}\,,\,\int{p}ds\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\geq 1\,,$ (96)

where $\mathscr{P}$ is the set of all curves from $D(A)$ to $D(B)$. From this
viewpoint, $\psi$ can be thought of as a witness for (93).

We also have an analogue of the bound (77) for the flux of $U$, proved
similarly. From lemma 4.2, we know that, for any orthonormal pair of covectors
$(T,N)$ with $T\in\mathfrak{i}^{+}$, the constraint $U\pm
d\psi\in\mathfrak{j}^{+}$ implies

$-T\cdot U\geq N\cdot d\psi\,.$ (97)

Therefore, for any everywhere-spacelike slice $\sigma$,

$\int_{\mathcal{I}^{+}}*U=\int_{\sigma}*U=\int_{\sigma}\sqrt{h}\,(-T\cdot
U)\geq\int_{\sigma}\sqrt{h}\,N\cdot
d\psi=\int_{\sigma}\sqrt{h}\,|d_{\sigma}\psi|=\int_{-1/2}^{1/2}ds\,\operatorname{area}(\gamma_{s})\,,$
(98)

where $d_{\sigma}$ is the gradient on $\sigma$; in going from $N\cdot d\psi$
to $|d_{\sigma}\psi|$ we chose $N$ to be the normalized projection of $d\psi$
onto $\sigma$; and $\gamma_{s}:=\tau_{s}\cap\sigma$ is the level set of $\psi$
on $\sigma$. Hence

$\int_{\mathcal{I}^{+}}*U\geq\inf_{\gamma\in\Gamma_{\sigma}}\operatorname{area}(\gamma)\,.$
(99)

By continuity, (99) holds for general slices. Minimizing the left-hand side
over U-flows and maximizing the right-hand side over slices yields the
inequality $S_{\rm c}\geq S_{-}$.

###  4.3 Multiple regions & subadditivity

So far, we have considered just two boundary regions $A,B$. But it is also
interesting to consider more regions $A,B,C,\ldots$, allowing us to explore
increasingly fine relational properties of the entanglement structure. As
explained in subsection 2.1, any entangling surfaces have been excised in
passing to the regulated spacetime $\mathcal{M}$, so we have $D(AB)=D(A)\cup
D(B)$, etc., and the conformal boundary of $\mathcal{M}$ is
$\mathcal{N}=D(A)\cup D(B)\cup D(C)\cup\cdots$. The quantity $S_{\rm
c}(\mathcal{A}:\mathcal{A}^{c})$ is then defined for any subset $\mathcal{A}$
of the regions.

It is important to understand where the choice of region enters in the max
V-flow and min U-flow programs. The constraints (55) and (56) (or equivalently
(72)) that define a V-flow do not depend on the choice of region, which
instead enters in the objective, which is the flux of $V$ through
$D(\mathcal{A})$. The situation is the opposite for the U-flows: the objective
(78), the flux of $U$ through the future boundary, is independent of the
choice of region, whereas the constraints depend on it; more precisely, the
conditions (79) do not depend on it, but the norm bound (80) does, through the
boundary conditions on $\psi$: $\psi|_{D(\mathcal{A})}=-1/2$,
$\psi|_{D(\mathcal{A}^{c})}=1/2$; the equivalent norm bound (96) also does
through the set $\mathscr{P}$ of curves from $D(\mathcal{A})$ to
$D(\mathcal{A}^{c})$.454545 It’s important to note that this discussion of the
dependence of the set of V- and U-flows on the choice of boundary region is in
the context of a _fixed_ regulated spacetime, of which the regions being
considered are unions of entire boundary connected components. As we will see
when we return to the original unregulated spacetime in section 7.1, the set
of V-flows there _does_ depend on the choice of boundary region.

Let us now specialize to the case of three boundary regions $A,B,C$. We then
have three different $S_{\rm c}$ quantities: $S_{\rm c}(A:BC)$, $S_{\rm
c}(B:AC)$, and $S_{\rm c}(C:AB)$, which we will write as $S_{\rm c}(A)$,
$S_{\rm c}(B)$, and $S_{\rm c}(AB)$. We will show that they obey the
subadditivity, or triangle, inequality:

$S_{\rm c}(AB)\leq S_{\rm c}(A)+S_{\rm c}(B)\,.$ (100)

Interestingly, it is possible to prove (100) using either V-flows or U-flows.
The proofs are quite different, illustrating the different points of view
afforded by the two types of flows.

###### Proof via V-flows:.

Let $V$ be a maximal V-flow for $AB$. We then have

$S_{\rm c}(AB)=\int_{D(AB)}*V=\int_{D(A)}*V+\int_{D(B)}*V\leq S_{\rm
c}(A)+S_{\rm c}(B)\,,$ (101)

where in the inequality we used the fact that $V$ is an allowed (though not
necessarily maximal) V-flow for $A$ and for $B$, since the constraints (55),
(56) (or (72)) in the definition of a V-flow do not depend on the choice of
boundary region. (This is is essentially the same proof as in the Riemannian
setting Freedman:2016zud .) ∎

###### Proof via U-flows:.

For the proof of (100) using U-flows, on the other hand, we have to contend
with the fact that the definition of a U-flow _does_ depend on the choice of
boundary region. Specifically, the boundary conditions for the function $\psi$
appearing in the norm bound (80) depend on the boundary region; equivalently,
the curves $\mathpzc{p}$ enforcing the norm bound in the form (96) connect the
given boundary region to its complement. Let $U_{A}$ be a minimal U-flow for
$A$ and $U_{B}$ a minimal U-flow for $B$. Then we claim that
$U_{AB}:=U_{A}+U_{B}$ is an allowed (though not necessarily minimal) U-flow
for $AB$. First, it clearly obeys the constraints (79), which are linear. We
can show that it also obeys the norm bound (80) or equivalently (96).

For (80), let $\psi_{A}$, $\psi_{B}$ be witness functions for $U_{A}$, $U_{B}$
respectively. These obey

$\left.\psi_{A}\right|_{D(A)}=-\frac{1}{2}\,,\quad\left.\psi_{A}\right|_{D(BC)}=\frac{1}{2}\,,\qquad\left.\psi_{B}\right|_{D(B)}=-\frac{1}{2}\,,\quad\left.\psi_{B}\right|_{D(AC)}=\frac{1}{2}\,.$
(102)

We then define

$\psi_{AB}:=\psi_{A}+\psi_{B}-\frac{1}{2}\,.$ (103)

This obeys

$\left.\psi_{AB}\right|_{D(AB)}=-\frac{1}{2}\,,\quad\psi_{AB}|_{D(C)}=\frac{1}{2}\,.$
(104)

Also, since $U_{A}\pm d\psi_{A}\in\mathfrak{j}^{+}$ and $U_{B}\pm
d\psi_{B}\in\mathfrak{j}^{+}$, we have $U_{AB}\pm
d\psi_{AB}\in\mathfrak{j}^{+}$, so $U_{AB}$ obeys (80).

To show that $U_{AB}$ obeys (96), first note that clearly
$U_{AB}\in\mathfrak{j}^{+}$. Let $\mathpzc{p}$ be a curve connecting $D(A)$
and $D(C)$. We then have

$\int{p}ds\,|U_{AB}\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\geq\int{p}ds\left(|U_{A}\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|+|U_{B}\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\right)=\int{p}ds\,|U_{A}\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|+\int{p}ds\,|U_{B}\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\geq 1\,,$ (105)

where in the first inequality we used the fact that $|\ \mathclap{\hskip
6.75pt\cdot}{\wedge}\ |$ is homogeneous and concave, hence superadditive, in
the first argument; and in the second inequality we used the fact that
$\mathpzc{p}$ connects $D(A)$ to its complement $D(BC)$ (since $D(C)$ is
contained in $D(BC)$) and $U_{A}$ is a U-flow for $A$, so the first integral
is at least 1, and the second integral is clearly at least 0. Similarly for a
curve connecting $D(B)$ and $D(C)$. Hence $U_{AB}$ obeys (96).

We now have

$S_{\rm
c}(AB)\leq\int_{\mathcal{I}^{+}}*U_{AB}=\int_{\mathcal{I}^{+}}*U_{A}+\int_{\mathcal{I}^{+}}*U_{B}=S_{\rm
c}(A)+S_{\rm c}(B)\,.$ (106)

∎

The fact that the quantity $S_{\rm c}$ is subadditive immediately raises
several other questions. First, are the minimax and maximin quantities
$S_{\pm}$ also subadditive? We are not aware of either a proof or a
counterexample, but we suspect that they are not. It is also natural to ask
whether $S_{\rm c}$ obeys higher entropy inequalities such as strong
subadditivity and MMI. In the Riemannian setting, flow-based proofs of these
inequalities use the nesting property Freedman:2016zud ; Headrick:2017ucz  and
the existence of a max multiflow Cui:2018dyq , respectively. The proofs of
these properties do not carry over straightforwardly to the current Lorentzian
setting.464646 Those proofs use strong duality between the max flow and
relaxed min cuts programs. On the cut side, they require a smeared cut, or
barrier function $\psi$, for a (nested or disjoint) set of boundary regions to
be decomposed into a corresponding set of barrier functions. Due to the non-
local nature of the constraints, we have not found a way to effect such a
decomposition in the Lorentzian setting.  We do not know whether these
properties hold for U- and V-flows in a general spacetime, but we suspect that
they do not. In subsection 6.3, we will return to this issue in the more
specific setting of standard holographic spacetimes.

###  4.4 Proofs

In this subsection we provide proofs and derivations for many of the
statements made in the rest of this section. At a technical level, this
subsection is the heart of the paper. The reader who is mainly interested in
the results of the paper is nonetheless permitted to skip ahead to section 5.

####  4.4.1 Covector-pair lemmas

The following set of covector pairs, at a given point $x\in\mathcal{M}$, has
played an important role throughout this section:

$\mathscr{K}:=\left\\{(W,X)\in T^{*}\times T^{*}\quad\big{|}\quad W\pm
X\in\mathfrak{j}^{+}\right\\}$ (107)

Here we prove four lemmas providing alternative characterizations of
$\mathscr{K}$ that have been used in various parts of this section. We also
give two formulas for the pairing $|\ \mathclap{\hskip 6.75pt\cdot}{\wedge}\
|$ in terms of $\mathscr{K}$ that will be used in the dualizations of the next
subsection.

######  Lemma 4.1.

$\mathscr{K}$ is the convex hull in $T^{*}\times T^{*}$ of

$\mathscr{K}^{\prime}:=\left\\{(W,X)\in\mathscr{K}\quad\big{|}\quad W\cdot
X=0\right\\}.$ (108)

###### Proof.

Suppose $(W,X)\in\mathscr{K}\setminus\mathscr{K}^{\prime}$. Consider first the
case $W\cdot X<0$; then $W+X\in\mathfrak{i}^{+}$ (since we know
$W+X\in\mathfrak{j}^{+}$ and $(W+X)^{2}=(W-X)^{2}+4W\cdot X<0$). Choose a
covector $Y$ such that $Y\cdot(W+X)=0$, $Y^{2}=-W\cdot X$. The reader can
check that the two points $(W_{\pm},X_{\pm}):=(W\pm Y,X\pm Y)$, whose average
is $(W,X)$, are both elements of $\mathscr{K}^{\prime}$. The case $W\cdot X>0$
is the same, but with $Y$ chosen such that $Y\cdot(W-X)=0$, $Y^{2}=W\cdot X$.
∎

Now define a set of orthonormal pairs of covectors, one of which is timelike:

$\mathscr{L}:=\left\\{(T,N)\in\mathfrak{i}^{+}\times T^{*}\quad\big{|}\quad
T\cdot N=0,\ |T|=|N|=1\right\\}.$ (109)

We can rewrite the condition on $(W,X)\in\mathscr{K}$ as $W$ being future-
directed causal with $X$ lying in its causal diamond, which we can
conveniently express in terms of $T$ and $N$ as follows:

######  Lemma 4.2.

$(W,X)\in\mathscr{K}$ if and only if

$\forall\,(T,N)\in\mathscr{L}\,,\ \,X\cdot N\leq-W\cdot T\,.$ (110)

###### Proof.

Given covectors $(T,N)\in T^{*}\times T^{*}$, we can null-decompose them by
defining ${U^{\pm}:=(T\pm N)/2}$. Then $(T,N)\in\mathscr{L}$ if and only if
$U^{\pm}$ are future-directed null covectors and $U^{+}\cdot U^{-}=-1/2$. For
$(W,X)\in T^{*}\times T^{*}$, we have

$X\cdot N+W\cdot T=(W+X)\cdot U^{+}+(W-X)\cdot U^{-}\,.$ (111)

If $(W,X)\in\mathscr{K}$ then $(W\pm X)\cdot U^{\pm}\leq 0$, so $X\cdot
N\leq-W\cdot T$.

For the converse, suppose $W+X\not\in\mathfrak{j}^{+}$. Then there exists a
future-directed null covector $U^{+}$ such that $(W+X)\cdot U^{+}>0$. Let
$U^{-}$ be a future-directed null covector such that $U^{+}\cdot U^{-}=-1/2$.
By rescaling $U^{+}$ by a positive factor $\alpha$ and $U^{-}$ by $1/\alpha$,
(111) can be made positive. Similarly for $W-X\not\in\mathfrak{j}^{+}$. ∎

In fact, using the $|\ \mathclap{\hskip 6.75pt\cdot}{\wedge}\ |$ product, we
can re-express the condition $(W,X)\in\mathscr{K}$ in terms of a single
arbitrary covector $Y$, in two different ways.

######  Lemma 4.3.

$(W,X)\in\mathscr{K}$ if and only if

$\forall\,Y\in\mathfrak{j}^{+}\,,\ \,|Y\mathclap{\hskip
6.75pt\cdot}{\wedge}X|\leq-Y\cdot W\,.$ (112)

###### Proof.

If (112) holds, then for any $Y\in\mathfrak{j}^{+}$,

$Y\cdot(W\pm X)\leq Y\cdot W+|Y\cdot X|\leq Y\cdot W+|Y\mathclap{\hskip
6.75pt\cdot}{\wedge}X|\leq 0\,,$ (113)

implying $(W,X)\in\mathscr{K}$.

Conversely, if $(W,X)\in\mathscr{K}$, then $W\in\mathfrak{j}^{+}$ and, for any
$Y\in\mathfrak{j}^{+}$, $|Y\cdot X|\leq-Y\cdot W$. Furthermore, for any
$Y\in\mathfrak{i}^{+}$, $|Y\wedge X|=|Y||X_{\perp}|$, where $X_{\perp}$ is the
projection of $X$ perpendicular to $Y$. The condition $(W,X)\in\mathscr{K}$
implies $|Y||W_{\perp}\pm X_{\perp}|\leq|Y||W_{\parallel}\pm
X_{\parallel}|=-Y\cdot(W\pm X)$; taking the average of these two inequalities
and applying the triangle inequality in the (spacelike) hyperplane orthogonal
to $Y$ yields $|Y||X_{\perp}|\leq-Y\cdot W$. Hence $|Y\mathclap{\hskip
6.75pt\cdot}{\wedge}X|\leq-Y\cdot W$. By continuity, the inequality holds for
any $Y\in\mathfrak{j}^{+}$. ∎

######  Lemma 4.4.

$(W,X)\in\mathscr{K}$ if and only if

$W\in\mathfrak{j}^{+}\text{ and }\forall\,Y\in T^{*}\,,\ \,X\cdot
Y\leq|W\mathclap{\hskip 6.75pt\cdot}{\wedge}Y|\,.$ (114)

###### Proof.

If (114) holds then, for any $Y\in\mathfrak{j}^{+}$, $(W+X)\cdot
Y=-|W\mathclap{\hskip 6.75pt\cdot}{\wedge}Y|+X\cdot Y\leq 0$, so
$W+X\in\mathfrak{j}^{+}$. Similarly, for any $Y$ such that
$-Y\in\mathfrak{j}^{+}$, $(W-X)\cdot Y=|W\mathclap{\hskip
6.75pt\cdot}{\wedge}Y|-X\cdot Y\geq 0$, so $W-X\in\mathfrak{j}^{+}$.

Conversely, if $(W,X)\in\mathscr{K}$, then for any $Y$ such that
$Y\in\mathfrak{j}^{+}$ or $-Y\in\mathfrak{j}^{+}$, $X\cdot
Y\leq|W\mathclap{\hskip 6.75pt\cdot}{\wedge}Y|$. For spacelike $Y$, assume
first that $W\in\mathfrak{i}^{+}$. Decomposing $X$ and $Y$ into components
perpendicular and parallel to $W$, we have
$|X_{\perp}|+|X_{\parallel}|\leq|W|$ and $|Y_{\parallel}|<|Y_{\perp}|$, so

$X\cdot Y=X_{\perp}\cdot Y_{\perp}+X_{\parallel}\cdot
Y_{\parallel}\leq|X_{\perp}||Y_{\perp}|+|X_{\parallel}||Y_{\parallel}|<\left(|X_{\perp}|+|X_{\parallel}|\right)\,|Y_{\perp}|\leq|W||Y_{\perp}|=|W\mathclap{\hskip
6.75pt\cdot}{\wedge}Y|\,,$ (115)

where in the first inequality we used the Cauchy-Schwarz inequality in the
(spacelike) hyperplane orthogonal to $W$. By continuity the inequality holds
for $W\in\mathfrak{j}^{+}$. ∎

The preceding two lemmas give bounds on the $|\ \mathclap{\hskip
6.75pt\cdot}{\wedge}\ |$ product, which we now show are tight.

######  Lemma 4.5.

Given $X,Y\in T^{*}$,

$\sup_{\begin{subarray}{c}W:\\\ (W,X)\in\mathscr{K}\end{subarray}}Y\cdot
W=\begin{cases}+\infty\,,\quad&Y\not\in\mathfrak{j}^{+}\\\ -|Y\mathclap{\hskip
6.75pt\cdot}{\wedge}X|\,,\quad&Y\in\mathfrak{j}^{+}\end{cases}\,.$ (116)

###### Proof.

If $Y\not\in\mathfrak{j}^{+}$, then there exists a covector
$Z\in\mathfrak{i}^{+}$ such that $Y\cdot Z>0$. For sufficiently large
$\alpha$, $(\alpha Z,X)\in\mathscr{K}$. So $W\cdot Y$ can be made arbitrarily
large for $(W,X)\in\mathscr{K}$.

If $Y\in\mathfrak{j}^{+}$, then by lemma 4.3, for any $W$ such that
$(W,X)\in\mathscr{K}$, $Y\cdot W\leq-|Y\mathclap{\hskip
6.75pt\cdot}{\wedge}X|$. So the supremum on LHS of (116) likewise satisfies
this inequality. To show that it can be actually saturated, the following
constructs a $W$ such that $(W,X)\in\mathscr{K}$ and $W\cdot
Y=-|Y\mathclap{\hskip 6.75pt\cdot}{\wedge}X|$:

$W=\begin{cases}X\,\quad&X\in\mathfrak{j}^{+}\\\
-X\,\quad&-X\in\mathfrak{j}^{+}\\\
|X|\,\hat{Y}_{\perp}\,\quad&X^{2}>0\end{cases}\,,$ (117)

where $\hat{Y}_{\perp}:=Y_{\perp}/|Y_{\perp}|$ is the unit covector in the
direction of the projection of $Y$ orthogonal to $X$. ∎

######  Lemma 4.6.

Given $W\in\mathfrak{j}^{+}$, $Y\in T^{*}$,

$\sup_{\begin{subarray}{c}X:\\\ (W,X)\in\mathscr{K}\end{subarray}}X\cdot
Y\textbf{}=|W\mathclap{\hskip 6.75pt\cdot}{\wedge}Y|\,.$ (118)

###### Proof.

From lemma 4.4, we have that $X\cdot Y\leq|W\mathclap{\hskip
6.75pt\cdot}{\wedge}Y|$. For $\pm Y\in\mathfrak{j}^{+}$, the bound is achieved
by $X=\mp W$. For $Y$ spacelike and $W\in\mathfrak{i}^{+}$, the bound is
achieved by

$X=|W|\,\hat{Y}_{\perp}\,,$ (119)

where $\hat{Y}_{\perp}$ is the unit covector in the direction of the
projection of $Y$ orthogonal to $W$. By continuity, (118) then holds for
$W\in\mathfrak{j}^{+}$. ∎

####  4.4.2 Dualizations

In this subsection we carry out five Lagrange dualizations:

  * •

on $\psi$ in the first formula for $S_{\rm c}$ in (49), with $\phi$ fixed, to
obtain the V-flow program (54);

  * •

on $\phi$ in the second formula for $S_{\rm c}$ in (49), with fixed $\psi$, to
obtain the U-flow program (78);

  * •

on $(V,\phi)$ in the V-flow program (54), to obtain the U-flow program (78);

  * •

on $\phi$ in the V-flow program (54), with $V$ fixed, to obtain a maximin
formula involving $U$ and $V$;

  * •

and finally on $\psi$ in the U-flow program (78), with $U$ fixed, to obtain a
minimax formula involving $U$ and $V$.

We include the last two dualities for completeness. As we will explain, the
resulting maximin and minimax formulas do not tell us anything new.

For each duality, we will check whether Slater’s condition holds. Slater’s
condition, a sufficient condition for strong duality, is the existence of a
(not necessarily optimal) point in the relative interior of the feasible set
for the primal problem (the interior with respect to the lowest-dimensional
affine space containing the feasible set). For a review and examples of
Lagrange dualization as applied to this type of problem see Headrick:2017ucz .

For simplicity, when writing the programs to be dualized, we will drop the
condition $-1/2\leq\psi\leq 1/2$ in the definition of $\mathscr{T}_{\rm c}$
and the conditions $-1/2\leq\phi\leq 1/2$, $d\phi\in\mathfrak{j}^{+}$ in the
definition of $\mathscr{S}_{\rm c}$. As we will now show, it makes no
difference whether we include those conditions or not. In the $\psi$ case,
given a function $\psi$ that satisfies the boundary conditions
$\psi|_{D(A)}=-1/2$, $\psi|_{D(B)}=1/2$ but falls outside the range
$[-1/2,1/2]$ somewhere in $\bar{\mathcal{M}}$, we can define a new function
$\psi^{\prime}$ that is in the range $[-1/2,1/2]$, and therefore is in
$\mathscr{T}_{\rm c}$, by

$\psi^{\prime}=\begin{cases}-1/2\,,\quad&\psi<-1/2\\\
\psi\,,\quad&-1/2\leq\psi\leq 1/2\\\ 1/2\,,\quad&\psi>1/2\end{cases}\,.$ (120)

Furthermore, if $U\pm d\psi\in\mathfrak{j}^{+}$, then $U\pm
d\psi^{\prime}\in\mathfrak{j}^{+}$. So the set of feasible $U$s is the same,
whether or not the condition $-1/2\leq\psi\leq 1/2$ is imposed. For $\phi$,
the conditions $d\phi\in\mathfrak{j}^{+}$, $-1/2\leq\phi\leq 1/2$, are already
implied by the conditions $d\phi\pm V\in\mathfrak{j}^{+}$,
$\phi|_{\mathcal{I}^{\pm}}=\pm 1/2$, so there is no need to include them
explicitly.

##### From $\psi$ to $V$, with fixed $\phi$:

The starting program is

$\text{minimize}\int_{\mathcal{M}}\sqrt{g}\ |d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|\text{ over }\psi\text{ such that
}\psi|_{D(A)}=-\frac{1}{2}\,,\quad\psi|_{D(B)}=\frac{1}{2}$ (121)

(where the function $\phi$ is regarded as fixed and obeys
$d\phi\in\mathfrak{j}^{+}$). We rewrite this by introducing a 1-form $X$:

$\text{minimize}\int_{\mathcal{M}}\sqrt{g}\ |d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}X|\text{ over }\psi,X\text{ such that
}d\psi=X\,,\quad\psi|_{D(A)}=-\frac{1}{2}\,,\quad\psi|_{D(B)}=\frac{1}{2}\,.$
(122)

Slater’s condition merely amounts in this case to the existence of a feasible
configuration $\psi,X$.

We will impose the constraint $d\psi=X$ using a Lagrange multiplier 1-form
$V$. The boundary conditions will be imposed implicitly. The Lagrangian is

$\begin{split}L[\psi,X,V]&=\int_{\mathcal{M}}\sqrt{g}\
\left[|d\phi\mathclap{\hskip 6.75pt\cdot}{\wedge}X|-V\cdot(X-d\psi)\right]\\\
&=\int_{\mathcal{M}}\sqrt{g}\ \left[|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}X|-V\cdot
X\right]-\int_{\mathcal{M}}\psi\,d{*V}-\int_{\mathcal{I}}\psi\,(*V)+\frac{1}{2}\left(\int_{D(A)}*V-\int_{D(B)}*V\right),\end{split}$
(123)

where, in the second line, after integrating by parts and using Stokes’
theorem in the form (24), we imposed the boundary conditions on $\psi$.

We now minimize $L$ with respect to $\psi$ and $X$, with the locus of 1-forms
$V$ such that $L$ is bounded below defining the dual constraints. The two
terms in the bulk integrand involving $X$ are ${|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}X|-V\cdot X}$. This is homogeneous in $X$, so if it is
negative for any $X$ then it is unbounded below, and otherwise its minimum is
0. By lemma 4.4, this quantity is non-negative for all $X$ if and only if
$d\phi\pm V\in\mathfrak{j}^{+}$, which is therefore a dual constraint. The
bulk term involving $\psi$, $-\psi\,d{*V}$, is bounded below if and only if
$d{*V}=0$, i.e. $V$ is divergenceless. Similarly, the boundary term involving
$\psi$, $\int_{\mathcal{I}}\psi\,(*V)$, is bounded below if and only if
$*V|_{\mathcal{I}}=0$. We are left only with the two terms in the large
parentheses, which, given the last two conditions on $V$, are equal. We are
finally left with the following dual program:

$\text{maximize}\int_{D(A)}*V\text{ over }V\text{ such that }d\phi\pm
V\in\mathfrak{j}^{+}\,,\quad d{*V}=0\,,\quad*V|_{\mathcal{I}}=0\,.$ (124)

##### From $\phi$ to $U$, with fixed $\psi$:

The starting program is

$\text{maximize}\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|\text{ over }\phi\text{ such that
}d\phi\in\mathfrak{j}^{+}\,,\quad\phi|_{\mathcal{I}^{\pm}}=\pm\frac{1}{2}$
(125)

(where the function $\psi$ is regarded as fixed). We rewrite this by
introducing a 1-form $W$:

$\text{maximize}\int_{\mathcal{M}}\sqrt{g}\ |W\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|\text{ over }\phi,W\text{ such that }d\phi=W\,,\quad
W\in\mathfrak{j}^{+}\,,\quad\phi|_{\mathcal{I}^{\pm}}=\pm\frac{1}{2}\,.$ (126)

Slater’s condition amounts here to the existence of a function $\phi_{0}$
obeying $d\phi_{0}\in\mathfrak{i}^{+}$ everywhere and
$\phi_{0}|_{\mathcal{I}^{\pm}}=\pm 1/2$.474747 Such a function $\phi_{0}$ can
be constructed as follows. Let $\Omega$ be a continuous function on
$\bar{\mathcal{M}}$ that is positive on $\mathcal{M}$ and such that the metric
$\tilde{g}:=\Omega^{2}g$ on $\mathcal{M}$ has finite total spacetime volume.
Define $\phi_{\pm}(x)$ as the spacetime volume of $J^{\pm}(x)$ with respect to
$\tilde{g}$. Then $\phi_{\pm}\to 0$ on $\mathcal{I}^{\pm}$ and $\mp
d\phi_{+}\in\mathfrak{i}^{+}$ on $\mathcal{M}$ (with respect to both
$\tilde{g}$ and $g$). The function
$\phi_{0}:=\frac{1}{2}\tanh\left(\frac{1}{\phi_{+}}-\frac{1}{\phi_{-}}\right)$
(127) then has the claimed properties.

We will impose the constraint $d\phi=W$ using a Lagrange multiplier 1-form
$U$. The boundary conditions on $\phi$ and the constraint
$W\in\mathfrak{j}^{+}$ will be imposed implicitly. The Lagrangian is

$\begin{split}L[\phi,W,U]&=\int_{\mathcal{M}}\sqrt{g}\
\left[|W\mathclap{\hskip 6.75pt\cdot}{\wedge}d\psi|+(W-d\phi)\cdot U\right]\\\
&=\int_{\mathcal{M}}\sqrt{g}\ \left[|W\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|+W\cdot
U\right]+\int_{\mathcal{M}}\phi\,d{*U}+\int_{\mathcal{I}^{0}\cup\mathcal{N}}\phi\,(*U)+\frac{1}{2}\left(\int_{\mathcal{I}^{+}}*U-\int_{\mathcal{I}^{-}}*U\right),\end{split}$
(128)

where in the second line, after integrating by parts and using Stokes’ theorem
in the form (24), we imposed the boundary conditions on $\phi$.

We now maximize $L$ with respect to $\phi$ and $W$. The bulk integrand
$|W\mathclap{\hskip 6.75pt\cdot}{\wedge}d\psi|+W\cdot U$, by lemma 4.3 along
with homogeneity in $W$, is bounded above (as a function of $W$ on
$\mathfrak{j}^{+}$) if and only if $U\pm d\psi\in\mathfrak{j}^{+}$, in which
case the maximum is 0. The bulk integrand $\phi\,d{*U}$ is bounded above as a
function of $\phi$ if and only if $d{*U}=0$, i.e. $U$ is divergenceless. The
boundary integrand $\phi\,(*U)$ on $\mathcal{I}^{0}\cup\mathcal{N}$ is bounded
above if and only if $*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0$. The last two
conditions imply that the two terms in the big parentheses in (128) are equal.
We are finally left with the following program:

$\text{minimize}\int_{\mathcal{I}^{+}}*U\text{ over }U\text{ such that }U\pm
d\psi\in\mathfrak{j}^{+}\,,\qquad
d{*U}=0\,,\qquad*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0\,.$ (129)

##### From $(V,\phi)$ to $(U,\psi)$:

Here we start with the V-flow program (with local norm bound in terms of the
function $\phi$) and dualize it to obtain the U-flow program (with local norm
bound in terms of the function $\psi$).

The V-flow program is:

$\text{maximize}\int_{D(A)}*V\text{ over }(V,\phi)\text{ such that }\\\
d\phi\pm V\in\mathfrak{j}^{+}\,,\qquad
d{*V}=0\,,\qquad*V|_{\mathcal{I}}=0\,,\qquad\phi|_{\mathcal{I}^{\pm}}=\pm\frac{1}{2}\,.$
(130)

We rewrite this by introducing 1-forms $W^{\pm}:=(d\phi\pm V)/2$ and
eliminating $V$:

$\displaystyle\text{maximize}\int_{D(A)}*(W^{+}-W^{-})\text{ over
}(W^{+},W^{-},\phi)\text{ such that }$ (131) $\displaystyle W^{+}+W^{-}$
$\displaystyle=d\phi\,,$ (132) $\displaystyle d*(W^{+}-W^{-})$
$\displaystyle=0\,,$ (133) $\displaystyle
W^{\pm}\in\mathfrak{j}^{+}\,,\qquad*(W^{+}-W^{-})|_{\mathcal{I}}=0\,,\qquad\phi|_{\mathcal{I}^{\pm}}$
$\displaystyle=\pm\frac{1}{2}\,.$ (134)

To show that Slater’s condition holds, we set $W^{+}=W^{-}=d\phi/2$ and
$\phi=\phi_{0}$, where $\phi_{0}$ is the function on $\bar{\mathcal{M}}$
obeying $d\phi_{0}\in\mathfrak{i}^{+}$ everywhere in $\mathcal{M}$ and
$\phi_{0}|_{\mathcal{I}^{\pm}}=\pm 1/2$ (such as the one indicated in footnote
47).

We will treat (132), (133) as explicit constraints, introducing Lagrange
multipliers $U$ (a 1-form) and $\psi$ (a scalar) respectively to enforce them.
The constraints (134) will be treated implicitly. The Lagrangian functional is

$\begin{split}L[W^{+},W^{-},\phi,U,\psi]&=\int_{D(A)}*(W^{+}-W^{-})+\int_{\mathcal{M}}\left[\sqrt{g}\,U\cdot(W^{+}+W^{-}-d\phi)-\psi\,d*(W^{+}-W^{-})\right]\\\
&=\int_{D(A)}\left[(1+\psi)*\\!(W^{+}-W^{-})+(*U)\,\phi\right]+\int_{D(B)}\left[(*U)\,\phi+\psi*\\!(W^{+}-W^{-})\right]\\\
&\qquad\qquad{}+\frac{1}{2}\int_{\mathcal{I}^{+}}*U-\frac{1}{2}\int_{\mathcal{I}^{-}}*U+\int_{\mathcal{I}^{0}}(*U)\,\phi\\\
&\qquad\qquad{}+\int_{\mathcal{M}}\left[\sqrt{g}\,\left((U+d\psi)\cdot
W^{+}+(U-d\psi)\cdot W^{-}\right)+\phi\,(d*U)\right],\end{split}$ (135)

where in the second line we integrated by parts and used Stokes’ theorem in
the form (24) and the boundary conditions on $W^{\pm}$ and $\phi$ in (134).

Following the usual procedure, our task now is to (1) find the constraints on
$(U,\psi)$ that are necessary and sufficient for $L$ to be bounded above as a
functional of $(W^{+},W^{-},\phi)$; and (2) assuming those constraints are
satisfied, maximize $L$ with respect to $W^{\pm}$ and $\phi$. The results of
(1) and (2) are the constraints and objective respectively of the dual
program. The integrand on $D(A)$ is bounded above (as a function of $W^{\pm}$
and $\phi$) if and only if $\psi=-1$ and $*U|_{D(A)}=0$, in which case it
vanishes. Similarly, the integrand on $D(B)$ is bounded above if and only if
$\psi=0$ and $*U|_{D(B)}=0$, in which case it vanishes. On
$\mathcal{I}^{\pm}$, there is no constraint on $\psi$ or $U$. On
$\mathcal{I}^{0}$ the integrand is bounded above if and only if
$*U|_{\mathcal{I}^{0}}=0$, in which case it vanishes. On $\mathcal{M}$, the
first term is bounded above (given the implicit constraint
$W^{\pm}\in\mathfrak{j}^{+}$) if and only if $U\pm d\psi\in\mathfrak{j}^{+}$,
in which case the maximum is at $W^{\pm}=0$ and vanishes. The second term is
bounded above if and only if $d*U=0$, in which case it vanishes. All in all we
are left with the following program:

$\text{minimize
}\left(\frac{1}{2}\int_{\mathcal{I}^{+}}*U-\frac{1}{2}\int_{\mathcal{I}^{-}}*U\right)\text{
over }(U,\psi)\text{ such that}\\\ U\pm d\psi\in\mathfrak{j}^{+}\,,\qquad
d*U=0\,,\qquad\psi|_{D(A)}=-1\,,\qquad\psi|_{D(B)}=0\,,\qquad*U|_{\mathcal{N}\cup\mathcal{I}^{0}}=0\,.$
(136)

Given the no-flux boundary condition for $*U$ on
$\mathcal{N}\cup\mathcal{I}^{0}$, and the fact that $\mathcal{I}^{+}$ is
homologous to $\mathcal{I}^{-}$ with opposite orientation, relative to
$\mathcal{N}\cup\mathcal{I}^{0}$, the two terms in the objective are actually
equal, so we might as well combine them into one term. (In fact, we could have
gotten the same result by shifting the boundary condition for $\phi$ by $1/2$
in the primal program (130).) Finally, we can restore the symmetry between
$D(A)$ and $D(B)$ by shifting $\psi$ by $-1/2$. After these cosmetic
adjustments, we end up with the U-flow program:

$\text{minimize }\int_{\mathcal{I}^{+}}*U\text{ over }(U,\psi)\text{ such
that}\\\ U\pm d\psi\in\mathfrak{j}^{+}\,,\qquad
d*U=0\,,\qquad\psi|_{D(A)}=-\frac{1}{2}\,,\qquad\psi|_{D(B)}=\frac{1}{2}\,,\qquad*U|_{\mathcal{N}\cup\mathcal{I}^{0}}=0\,.$
(137)

##### From $\phi$ to $U$, with fixed $V$:

The primal program is

$\text{maximize}\int_{D(A)}*V\text{ over }\phi\text{ such that }d\phi\pm
V\in\mathfrak{j}^{+}\,,\quad\phi|_{\mathcal{I}^{\pm}}=\pm\frac{1}{2}\,.$ (138)

Notice that the objective here does not depend on the variable. In such a case
the program amounts to a feasibility test: if a feasible point exists, the
program returns the objective, and if not it returns $-\infty$ (the supremum
of the empty set). In the former case, Slater’s condition may or may not be
satisfied, and, in the latter case, it is definitely not satisfied. However,
we will see explicitly that the dual program has the same optimal value.

We introduce a 1-form $W$:

$\text{maximize}\int_{D(A)}*V\text{ over }\phi,W\text{ such that
}d\phi=W\,,\quad W\pm
V\in\mathfrak{j}^{+}\,,\quad\phi|_{\mathcal{I}^{\pm}}=\pm\frac{1}{2}\,.$ (139)

We will impose the constraint $d\phi=W$ using a Lagrange multiplier 1-form
$U$, and impose the other constraints implicitly. The Lagrangian is

$\begin{split}L[\phi,W,U]&=\int_{D(A)}*V+\int_{\mathcal{M}}\sqrt{g}\,(W-d\phi)\cdot
U\\\ &=\int_{D(A)}*V+\int_{\mathcal{M}}\sqrt{g}\,W\cdot
U+\int_{\mathcal{M}}\phi\,d{*U}+\int_{\mathcal{I}^{0}\cup\mathcal{N}}\phi(*U)+\frac{1}{2}\left(\int_{\mathcal{I}^{+}}*U-\int_{\mathcal{I}^{-}}*U\right).\end{split}$
(140)

We now maximize the Lagrangian over $\phi$ and $W$. Requiring the Lagrangian
to be bounded above with respect to $\phi$ implies $d{*U}=0$,
$*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0$, which makes the two terms in
parentheses equal. Meanwhile, $W\cdot U$ is bounded above with respect to $W$
(such that $W\pm V\in\mathfrak{j}^{+}$) if and only if $U\in\mathfrak{j}^{+}$,
in which case the maximum is $-|U\mathclap{\hskip 6.75pt\cdot}{\wedge}V|$ (see
lemma 4.5). All in all, we are left with the following dual program:

$\text{minimize}\int_{D(A)}*V+\int_{\mathcal{I}^{+}}*U-\int_{\mathcal{M}}\sqrt{g}\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\text{ over $U$ such that }U\in\mathfrak{j}^{+}\,,\quad
d{*U}=0\,,\quad*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0\,.$ (141)

We now wish to show that strong duality holds, which in this case means that
if the primal (138) is feasible then the dual objective equals the primal
objective, and if not then the dual objective equals $-\infty$. The first case
is established by weak duality (which says that the dual objective is bounded
below by the primal objective) and the fact that the last two terms in the
dual objective can be set to zero by setting $U=0$. It remains to be shown
that the dual objective attains $-\infty$ when the primal is infeasible. In
theorem 4.7, we show that infeasibility implies the existence of an
inextendible causal curve $\mathpzc{q}$ such that
$\int{q}dt\,|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|>1$. If $U$ is a
delta-function flux tube on $\mathpzc{q}$ with flux $\alpha$, then

$\int_{\mathcal{M}}\sqrt{g}\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}V|=\alpha\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|>\alpha=\int_{\mathcal{I}^{+}}*U\,.$ (142)

Hence the dual objective can be made arbitrarily large and negative.

##### From $\psi$ to $V$, with fixed $U$:

The primal here is

$\text{minimize}\int_{\mathcal{I}^{+}}*U\text{ over }\psi\text{ such that
}U\pm
d\psi\in\mathfrak{j}^{+}\,,\quad\psi|_{D(A)}=-\frac{1}{2}\,,\quad\psi|_{D(B)}=\frac{1}{2}\,.$
(143)

The story here is similar to the previous one; in particular, again the primal
objective is independent of the variable, so the program amounts to a
feasibility test. The dualization proceeds very similarly to the previous one,
this time using lemma 4.6; we leave the details as an exercise to the reader.
The dual program is

$\text{maximize}\int_{\mathcal{I}^{+}}*U+\int_{D(A)}*V-\int_{\mathcal{M}}\sqrt{g}\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\text{ over $V$ such that
}d{*V}=0\,,\quad*V|_{\mathcal{I}}=0\,.$ (144)

If the primal is infeasible, then by theorem 4.8 and letting $V$ be a flux
tube along a curve $\mathpzc{p}\in\mathscr{P}$ on which
$\int{p}ds\,|U\mathclap{\hskip 6.75pt\cdot}{\wedge}\dot{x}|<1$, the dual
objective is unbounded above, establishing strong duality.

##### Discussion:

From the last two dualities we find the following minimax pair of formulas for
$S_{\rm c}$:

$\displaystyle S_{\rm c}$
$\displaystyle=\sup_{V}\operatorname*{inf\vphantom{p}}_{U}\left(\int_{\mathcal{I}^{+}}*U+\int_{D(A)}*V-\int_{\mathcal{M}}\sqrt{g}\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\right)$ (145)
$\displaystyle=\operatorname*{inf\vphantom{p}}_{U}\sup_{V}\left(\int_{\mathcal{I}^{+}}*U+\int_{D(A)}*V-\int_{\mathcal{M}}\sqrt{g}\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\right)\,,$ (146)

where in both formulas $U$, $V$ obey

$U\in\mathfrak{j}^{+}\,,\quad
d{*U}=0\,,\quad*U|_{\mathcal{I}^{0}\cup\mathcal{N}}=0\,,\qquad
d{*V}=0\,,\quad*V|_{\mathcal{I}}=0\,.$ (147)

Note that, since the first two terms in the objective functional are linear
and $|U\mathclap{\hskip 6.75pt\cdot}{\wedge}V|$ is concave in $U$ and convex
in $V$, the objective functional is convex in $U$ and concave in $V$, as one
would expect for a minimax pair with equality. In (145), the inf on $U$ simply
forces $V$ to be feasible (otherwise the objective is unbounded below, as
explained below (141)), in which case the first and third terms cancel and the
objective reduces to the second term, returning us to the max V-flow formula
(54). Similarly, (146) returns us to the min U-flow formula (78). Thus, these
two formulas do not teach us anything new.

####  4.4.3 Equivalence of norm bounds

In this subsection we prove the following theorems:

######  Theorem 4.7.

Let $V$ be a 1-form on $\bar{\mathcal{M}}$. The following conditions are
equivalent:

$\exists\ \phi\in\mathscr{S}_{\rm c}\text{ s.t. }d\phi\pm
V\in\mathfrak{j}^{+}$ (148)
$\forall\,\mathpzc{q}\in\mathscr{Q}\,,\,\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\leq 1\,,$ (149)

where $\mathscr{Q}$ is the set of inextendible causal curves in $\mathcal{M}$,
$t$ is any parameter along the curve $\mathpzc{q}$, and $\dot{x}$ is the
covector dual to the tangent vector $dx^{\mu}/dt$ of the curve $\mathpzc{q}$.

######  Theorem 4.8.

Let $U$ be a 1-form on $\bar{\mathcal{M}}$. The following conditions are
equivalent:

$\exists\ \psi\in\mathscr{T}_{\rm c}\text{ s.t. }U\pm
d\psi\in\mathfrak{j}^{+}$ (150)
$U\in\mathfrak{j}^{+}\,,\qquad\forall\,\mathpzc{p}\in\mathscr{P}\,,\,\int{p}ds\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\geq 1\,,$ (151)

where $\mathscr{P}$ is the set of curves in $\mathcal{M}$ starting in $D(A)$
and ending in $D(B)$, $s$ is any parameter along the curve $\mathpzc{p}$, and
$\dot{x}$ is the covector dual to the tangent vector $dx^{\mu}/ds$ of the
curve $\mathpzc{p}$ .

Notice that the theorems do not require the conditions (55), (79) in the
definitions of V- and U-flows respectively.

We will need the following lemma, which provides a generalization of the usual
Hamilton-Jacobi formula from classical mechanics to a situation where the
Lagrangian is a convex but not necessarily differentiable function of the
velocity. The Hamilton-Jacobi formula states that the variation in the on-
shell action under a change in the final position is given by the final value
of the canonical momentum; here, the same formula applies, but the canonical
momentum is a subgradient, rather than the gradient, of the Lagrangian with
respect to the velocity.

######  Lemma 4.9.

Let $M$ be a manifold (or manifold with boundary) and
$L:TM\to\mathbf{R}\cup\\{\infty\\}$ a function on its tangent bundle; we write
$L(x,v)$ where $x\in M$ and $v\in T_{x}M$. Suppose $L$ is convex in $v$ for
fixed $x$; we write $\partial_{v}L(x,v)$ for its subdifferential with respect
to $v$ (this is a convex subset of the cotangent space at $x$, $T^{*}_{x}M$).
Fix real numbers $t_{0}<t_{1}$ and a set $I\subseteq M$ (the allowed initial
positions). Define the following function on $M$ (the “on-shell action”):

$S_{\rm min}(y):=\inf_{\begin{subarray}{c}X\\\ X(t_{0})\in I\\\
X(t_{1})=y\end{subarray}}\int_{t_{0}}^{t_{1}}dt\,L(X(t),\dot{X}(t))\,.$ (152)

Then, on the domain where $S_{\rm min}$ is finite, it is continuous and almost
everywhere differentiable, and at a differentiable point $y$,

$dS_{\rm min}(y)\in\partial_{v}L(y,\dot{X}(t_{1}))\,,$ (153)

where $X$ is any minimizing trajectory.484848 If, on the domain where it is
finite, $L$ is differentiable with respect to $x$ with gradient $F(x,v)$, then
a minimizing trajectory $X$ will admit a solution $\pi$ to the generalized
Euler-Lagrange equations
$\pi(t)\in\partial_{v}L(X(t),\dot{X}(t))\,,\qquad\dot{\pi}(t)=F(X(t),\dot{X}(t))\,,$
(154) and $dS_{\rm min}=\pi(t_{1})$ for some solution.

We will not prove this lemma. It can be proven either (1) directly by
augmenting the standard calculus of variations with the subdifferential
calculus494949 There exists a large mathematical literature on the calculus of
variations with non-differentiable Lagrangians. For introductions, see e.g.
Rockafellar2001 ; MR3026831 . or (2) by regularizing $L$ to make it a finite
and differentiable function of $v$, applying the standard results, and then
taking the limit as the regulator is removed; see HeadrickNonsmooth .

As a warm-up to proving theorems (4.7) and (4.8), we will first prove an
analogous result in the simpler Riemannian setting, which will also be useful
to us in subsection 5.1.

######  Theorem 4.10.

Let $M$ be a Riemannian manifold (or manifold with boundary), $A$, $B$
disjoint subsets of $M$, $\mathscr{P}$ the set of curves in $M$ connecting $A$
and $B$, and $\lambda$ a non-negative function on $M$. Then the following
conditions are equivalent:

$\exists\ \psi:M\to[-1/2,1/2]\text{ s.t.
}\psi|_{A}=-1/2\,,\quad\psi|_{B}=1/2\,,\quad|d\psi|\leq\lambda$ (155)
$\forall\,\mathpzc{p}\in\mathscr{P}\,,\,\int{p}ds\,\lambda\geq 1\,,$ (156)

where $s$ is the proper distance along the curve $\mathpzc{p}$.

###### Proof.

First assume (155). Then for any $\mathpzc{p}\in\mathscr{P}$,

$\int{p}ds\,\lambda\geq\int{p}ds\,|d\psi|\geq\int{p}d\psi=\psi|^{B}_{A}=1$
(157)

(where in the second inequality we used the Cauchy-Schwarz inequality),
implying (156).

Now assume (156). Rewrite the integral over $\mathpzc{p}$ in a
reparametrization-invariant way:
$\int{p}ds\,\lambda=\int{p}dt\,|\dot{x}|\lambda$. Define the following
functions on $M$:

$\psi_{-}(y):=\inf_{\begin{subarray}{c}\mathpzc{p}\\\ A\text{ to
}y\end{subarray}}\int{p}dt\,|\dot{x}|\lambda\,,\qquad\psi_{+}(y):=\inf_{\begin{subarray}{c}\mathpzc{p}\\\
B\text{ to }y\end{subarray}}\int{p}ds\,|\dot{x}|\lambda\,,$ (158)

where the infimum is over curves $\mathpzc{p}$ from $A$ to $y$ and from $B$ to
$y$ respectively. Clearly for all $y\in M$ we have

$\psi_{-}(y)+\psi_{+}(y)\geq 1\,.$ (159)

We also have

$\lim_{y\to A}\psi_{-}(y)=0\,,\qquad\lim_{y\to B}\psi_{+}(y)=0\,.$ (160)

We now calculate the gradient of $\psi_{\pm}$ using lemma 4.9. The
“Lagrangian” here, $|v|\lambda$, is a convex but not differentiable function
of $v$. Its subdifferential is

$\partial_{v}(|v|\lambda)=\begin{cases}\\{\lambda v_{\mu}/|v|\\}\,,&v\neq 0\\\
\\{w_{\mu}:|w|\leq\lambda\\}\,,&v=0\end{cases}$ (161)

(where on the right-hand side we switched to an index notation to represent
the covectors); hence $|d\psi_{\pm}|\leq\lambda$.505050 Use of lemma 4.9 is
actually overkill in this case, because by a reparametrization we can always
choose to make the final velocity non-zero, so that the usual canonical
momentum is well-defined and therefore $|d\psi_{\pm}|=\lambda$. Nonetheless we
persist, in order to gain practice in applying that lemma. By (159), the
regions where $\psi_{-}<1/2$ and where $\psi_{+}<1/2$ do not overlap. We
therefore define515151 Another function that would also work is
$\psi:=(\psi_{-}-\psi_{+})/(2(\psi_{-}+\psi_{+}))$. The argument is similar to
the one for $\phi$ in the proof of theorem 4.7.

$\psi(y):=\begin{cases}\psi_{-}(y)-1/2\,,\quad&\psi_{-}(y)<1/2\\\
1/2-\psi_{+}(y)\,,\quad&\psi_{+}(y)<1/2\\\
0\,,\quad&\text{otherwise}\end{cases}\,.$ (162)

This function is continuous and obeys the constraints of (155). ∎

The proofs of theorems 4.7 and 4.8 are essentially the same, just with more
complicated integrands along the curves and, in the first case, switching
maximization for minimization. We start with the closer analogue, theorem 4.8.

###### Proof of theorem 4.8.

The proof that (150) implies (151) was given in subsection 4.2.3, but we
repeat it here for completeness. $U\pm d\psi\in\mathfrak{j}^{+}$ implies
$U\in\mathfrak{j}^{+}$. By lemma 4.4 it also implies $|U\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\geq\dot{x}\cdot d\psi$. Integrating over $s$ and
using the boundary conditions $\psi|_{D(A)}=-1/2$, $\psi|_{D(B)}=1/2$ yields
the inequality in (151).

For the converse, given $U$ satisfying (151), define the following functions
on $\mathcal{M}$:

$\psi_{-}(y):=\inf_{\begin{subarray}{c}\mathpzc{p}\\\ D(A)\text{ to
}y\end{subarray}}\int{p}ds\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\,,\qquad\psi_{+}(y):=\inf_{\begin{subarray}{c}\mathpzc{p}\\\
D(B)\text{ to }y\end{subarray}}\int{p}ds\,|U\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|\,,$ (163)

where the infimum is over curves $\mathpzc{p}$ from $D(A)$ to $y$ and from
$D(B)$ to $y$ respectively. Clearly for all $y\in\mathcal{M}$ we have

$\psi_{-}(y)+\psi_{+}(y)\geq 1\,.$ (164)

We also have

$\lim_{y\to D(A)}\psi_{-}(y)=0\,,\qquad\lim_{y\to D(B)}\psi_{+}(y)=0\,.$ (165)

We now calculate the gradient of $\psi_{\pm}$ using lemma 4.9. Assume first
that $U$ is timelike at $y$. If, for the minimizing trajectory, $\dot{x}$ is
timelike or spacelike at $y$, we have:

$|U\mathclap{\hskip 6.75pt\cdot}{\wedge}\dot{x}|=\begin{cases}U\cdot\dot{x}\\\
-U\cdot\dot{x}\\\
|U||\dot{x}_{\perp}|\end{cases}\qquad\Rightarrow\qquad\pi=\begin{cases}U\,,\quad&-\dot{x}\in\mathfrak{i}^{+}\\\
-U\,,\quad&\dot{x}\in\mathfrak{i}^{+}\\\
|U|\,\dot{x}_{\perp}/|\dot{x}_{\perp}|\,,\quad&\dot{x}\text{
spacelike}\end{cases}$ (166)

(where $\dot{x}_{\perp}$ is the projection of $\dot{x}$ orthogonal to $U$). In
all three cases, $U\pm\pi\in\mathfrak{j}^{+}$. What if $\dot{x}$ is null? The
gradient of $|U\mathclap{\hskip 6.75pt\cdot}{\wedge}\dot{x}|$ with respect to
$\dot{x}$ is discontinuous when $\dot{x}$ is null. The subgradient set (or
subdifferential) is the convex hull of the gradients on either side of the
discontinuity, which we have already computed in (166). Since the set of
covectors $X$ obeying $U\pm X\in\mathfrak{j}^{+}$ is convex (for fixed $U$),
all the subgradients obey that condition, and therefore $\pi$ does. The case
where $U$ is null can be treated as a limit of the timelike case, or directly
using the formula $|U\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{x}|=|U\cdot\dot{x}|$. We conclude that $U\pm
d\psi_{-}\in\mathfrak{j}^{+}$ and $U\pm d\psi_{+}\in\mathfrak{j}^{+}$
everywhere in $\mathcal{M}$.

By (164), the regions where $\psi_{-}<1/2$ and where $\psi_{+}<1/2$ do not
overlap. We therefore define525252 Another function that would also work is
$\psi:=(\psi_{-}-\psi_{+})/(2(\psi_{-}+\psi_{+}))$. The argument is similar to
the one for $\phi$ below.

$\psi(y):=\begin{cases}\psi_{-}(y)-1/2\,,\quad&\psi_{-}(y)<1/2\\\
1/2-\psi_{+}(y)\,,\quad&\psi_{+}(y)<1/2\\\
0\,,\quad&\text{otherwise}\end{cases}\,.$ (167)

This function is continuous and obeys the constraints of (150). ∎

###### Proof of theorem 4.7.

For the $V$-case we proceed similarly. The proof that (148) implies (149) was
given in subsection 4.1.3, but we repeat it here for completeness. By lemma
4.3, with $Y=-\dot{x}$, the condition $d\phi\pm V\in\mathfrak{j}^{+}$ implies
$|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|\leq\dot{x}\cdot d\phi$.
Integrating over $t$ and using the boundary conditions
$\phi|_{\mathcal{I}^{\pm}}=\pm 1/2$ yields the inequality in (149).

For the converse, given a 1-form $V$ obeying (149), we define the following
functions on $\mathcal{M}$:

$\phi_{-}(y)=\sup_{\begin{subarray}{c}\mathpzc{q}\text{ causal}\\\
\mathcal{I}^{-}\text{ to
}y\end{subarray}}\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\,,\qquad\phi_{+}(y)=\sup_{\begin{subarray}{c}\mathpzc{q}\text{
causal}\\\ y\text{ to
}\mathcal{I}^{+}\end{subarray}}\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\,.$ (168)

Clearly for all $y\in\mathcal{M}$ we have

$\phi_{-}(y)+\phi_{+}(y)\leq 1\,.$ (169)

We also have

$\lim_{y\to\mathcal{I}^{-}}\phi_{-}(y)=0\,,\qquad\lim_{y\to\mathcal{I}^{+}}\phi_{+}(y)=0\,.$
(170)

We now calculate the gradient of $\phi_{-}$. In this case, the integrand is
differentiable as a function of the velocity $\dot{x}^{\mu}$ on its domain,
namely the set of future-directed causal vectors. However, having a
constrained velocity again leads to a subtlety in the application of the
Hamilton-Jacobi formula, since the velocity on the maximizing trajectory may
be on the boundary of the domain (i.e. may be null), in which case the
canonical momentum may differ from its naive value.535353 In the familiar case
of the action for a massive particle, $-m\int dt\sqrt{1-\dot{x}^{2}}$, the
velocity is similarly constrained to the future light cone. However, the
classical trajectory is never null, so the constraint is never active and the
canonical momentum is indeed given by its “naive” value. To deal with this
issue, we enlarge the domain to be the whole tangent space, and implement the
constraint by defining the integrand to equal $-\infty$ whenever the velocity
is outside the future light cone. Thus we write

$\begin{split}\phi_{-}(y)&=\sup_{\begin{subarray}{c}\mathpzc{q}\\\
\mathcal{I}^{-}\text{ to
}y\end{subarray}}\int{q}dt\,\begin{cases}|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\,,\quad&-\dot{x}\in\mathfrak{j}^{+}\\\
-\infty\,,\quad&\text{otherwise}\end{cases}\,,\\\
\phi_{+}(y)&=\sup_{\begin{subarray}{c}\mathpzc{q}\\\ y\text{ to
}\mathcal{I}^{+}\end{subarray}}\int{q}dt\,\begin{cases}|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\,,\quad&-\dot{x}\in\mathfrak{j}^{+}\\\
-\infty\,,\quad&\text{otherwise}\end{cases}\,.\end{split}$ (171)

(Recall that the velocity vector $\dot{x}^{\mu}$ is future-directed causal if
and only if the dual covector $\dot{x}$ obeys $-\dot{x}\in\mathfrak{j}^{+}$.)
We now deal with the various cases in turn. When $V$ is spacelike and
$\dot{x}^{\mu}$ is timelike, $|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|=|V||\dot{x}_{\perp}|$, so the gradient with respect to
$\dot{x}^{\mu}$ is $\pi_{\mu}=-|V|\dot{x}_{\perp\mu}/|\dot{x}_{\perp}|$, which
satisfies $\pi\pm V\in\mathfrak{j}^{+}$. In the limit that $\dot{x}^{\mu}$
becomes null, $\pi$ goes to infinity in the direction of $-\dot{x}$; the
subgradient set has only this one element, so again $\pi\pm
V\in\mathfrak{j}^{+}$. When $V$ is future-directed timelike or null,
$|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|=V\cdot\dot{x}$, so for
timelike $\dot{x}^{\mu}$, $\pi=V$, and again $\pi\pm V\in\mathfrak{j}^{+}$.
For null $\dot{x}^{\mu}$, however, the integrand is discontinuous so the
gradient is undefined; the subgradient set consists of all covectors $\pi$ of
the form $V-\alpha\dot{x}$ for $\alpha\geq 0$; again $\pi\pm
V\in\mathfrak{j}^{+}$. Similarly for the case where $V$ is past-directed
timelike or null. We conclude that $d\phi_{-}\pm V\in\mathfrak{j}^{+}$.

By the same argument, but referring to a change in the initial rather than
final position (and noting that $\phi_{+}$ decreases to the future), we have
$-d\phi_{+}\pm V\in\mathfrak{j}^{+}$.

We now define the function

$\phi:=\frac{\phi_{-}-\phi_{+}}{2(\phi_{-}+\phi_{+})}\,.$ (172)

We have

$d\phi=\frac{\phi_{+}}{(\phi_{-}+\phi_{+})^{2}}\
d\phi_{-}+\frac{\phi_{-}}{(\phi_{-}+\phi_{+})^{2}}\,(-d\phi_{+})\,.$ (173)

Since $d\phi_{-}$ and $-d\phi_{+}$ both satisfy $X\pm V\in\mathfrak{j}^{+}$,
and since the set of such covectors is closed under linear combinations of the
form $\alpha_{1}X_{1}+\alpha_{2}X_{2}$ with $\alpha_{1},\alpha_{2}\geq 0$,
$\alpha_{1}+\alpha_{2}\geq 1$, we find (using (169)) that $d\phi$ is also in
the set. This function thus obeys the constraints of (148). ∎

##  5 Thread distributions

A bit thread is a bulk curve connecting two boundary points in a holographic
spacetime, that represents a unit of entanglement between boundary regions.
One way to define a configuration of bit threads is as the field lines (or
integral curves) of a flow. However, such a configuration can also be defined
simply as a set of curves obeying some density bound. The two definitions are
related but not equivalent, the main difference being that, whereas the field
lines of a flow cannot intersect, there is a priori no such constraint for a
set of curves. Both descriptions have been used in the literature, and the
relationship between them has been discussed to some extent (e.g. Cui:2018dyq
; Headrick:2020gyq ).545454 A similar distinction between a vector field and
its field lines arises for the electic and magnetic fields. In that case, as
textbooks often emphasize, the vector fields are the “real” physical objects,
while the field lines are merely a device for visualizing the fields. One way
to see this is that the vector fields obey the superposition principle: the
solution for a combination of sources is the sum of the individual solutions,
while it would be incorrect to combine the respective sets of field lines. In
the case at hand, it remains to be seen whether the flows or the thread
distributions — if either one — reflect the underlying physics of holographic
entanglement. In this section we will develop the second description more
systematically, showing how to apply the technology of convex optimization and
prove max flow-min cut type theorems directly in terms of threads, without
appealing to flows. We start in subsection 5.1 in the Riemannian setting. The
proofs there are new as far as we know, and of some interest in their own
right. In subsection 5.2, we then apply the same ideas to define thread
versions of the V-flows and U-flows of the previous section. The method can
also be applied in other settings, such as the graph max flow-min cut theorem
and the Lorentzian min flow-max cut theorem Headrick:2017ucz .555555
Riemannian threads have recently been generalized to so-called hyperthreads
Harper:2021uuq ; Harper:2022sky . We leave it to the interested reader to work
out the details in those cases.

###  5.1 Riemannian thread distributions

In this subsection we will define a thread distribution on a Riemannian
manifold, prove analogues of the max flow-min cut and max multiflow theorems,
and explain how to map a flow or multiflow to a thread distribution and back.

Throughout this subsection we fix a compact oriented $d$-dimensional
Riemannian manifold-with-boundary $M$.565656 Note that we purposefully denote
the manifold and its dimensionality differently from the previous section, in
order to make an easier contact to the Lorentzian case of the next subsection
wherein the present Riemannian manifold may be viewed as a slice $\sigma$.
Correspondingly, the regions $A$, $B$ are now codimension-0 within the
boundary $\partial M$.  Let $\mathscr{P}$ be a set of (unoriented575757
Threads can be defined to be either oriented or unoriented; in fact in the
original definition they were oriented Freedman:2016zud . However, making them
unoriented as we do here simplifies the analysis a bit, especially in the
multiflow case, and also reflects more closely the physics of entanglement,
which is a symmetric concept (in the sense that $A$ is entangled with $B$ to
the same degree that $B$ is entangled with $A$).) curves in $M$. (Below we
will specialize to the case where $\mathscr{P}$ is the set of curves
connecting regions $A$ and $B$ on the boundary of $M$, but for now we can be
general.) Let $\delta(x,y)$ be the delta function on $M\times M$ supported on
the diagonal $y=x$ and normalized such that, for any $x$,
$\int_{M}d^{d}y\sqrt{g(y)}\,\delta(x,y)=1$. We then define the following
function on $M\times\mathscr{P}$:

$\Delta(x,\mathpzc{p}):=\int{p}ds\,\delta(x,y(s))\,,$ (174)

where $s$ is a proper-distance parameter along $\mathpzc{p}$ and $y(s)$ is the
corresponding point in $M$. For fixed $x$, $\Delta(x,\mathpzc{p})$ is a delta-
function on $\mathscr{P}$ supported on the codimension-$(d-1)$ locus of curves
passing through $x$. Note also that $\Delta(x,\mathpzc{p})$ reflects the
multiplicity of times $\mathpzc{p}$ passes through $x$. Given a measure
$\upmu$ on $\mathscr{P}$,585858 We remind the reader that a measure $\upmu$ on
$\mathscr{P}$ assigns to each subset $P\subseteq\mathscr{P}$ a non-negative
number $\upmu(P)$ obeying certain properties (such as $\upmu(P_{1}\cup
P_{2})=\upmu(P_{1})+\upmu(P_{2})$ when $P_{1}\cap P_{2}=\emptyset$), and can
be used to define the Lebesgue integral of a function on $\mathscr{P}$. We
will be fairly sloppy about the analysis and measure-theory here, ignoring
issues such as what regularity conditions are imposed on threads, what
topology is put on the set of threads $\mathscr{P}$, what conditions are put
on the measure $\upmu$, what constitutes a measurable set, and so on.  we
define the _thread density_ at a given point $x\in M$ by
$\int_{\mathscr{P}}d\upmu(\mathpzc{p})\Delta(x,\mathpzc{p})$. It is a density
in the sense that its integral over any region $r\subseteq M$ equals the
length of $\mathpzc{p}\cap r$, integrated over $\mathscr{P}$ with respect to
$\upmu$:

$\int_{r}d^{d}\\!x\sqrt{g}\int_{\mathscr{P}}d\upmu(\mathpzc{p})\Delta(x,\mathpzc{p})=\int_{\mathscr{P}}d\upmu(\mathpzc{p})\int_{\mathpzc{p}\cap
r}ds\,.$ (175)

A _thread distribution_ is a measure $\upmu$ such that the thread density
nowhere exceeds 1:595959 When threads are allowed to intersect, as they are
here or in the case of a multiflow (but not in the case of a flow), there
exist several distinct natural density bounds, which were extensively explored
in Headrick:2020gyq . The bound (176) corresponds in the language of
Headrick:2020gyq  to the so-called $\nu_{v}$-bound and is the most amenable to
dualization and therefore the easiest to analyze.

$\forall\,x\in
M\,,\quad\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(x,\mathpzc{p})\leq
1\,.$ (176)

Since the measures on $\mathscr{P}$ form a convex cone, and (176) is a set of
convex (in fact linear) constraints, the thread distributions also form a
convex set. Therefore the problem of maximizing the total measure
$\upmu(\mathscr{P})$ subject to (176) defines a linear program:

$\text{Maximize}\quad\upmu(\mathscr{P})\quad\text{over measure $\upmu$ on
$\mathscr{P}$ subject to:}\quad\forall\,x\in
M\,,\quad\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(x,\mathpzc{p})\leq
1\,.$ (177)

Dualizing this linear program is straightforward. The Lagrange multiplier for
the constraint is a function $\lambda$ on $M$, constrained to be non-negative
since it is enforcing an inequality constraint. The Lagrangian is

$\begin{split}L[\upmu,\lambda]&=\upmu(\mathscr{P})+\int_{M}d^{d}\\!x\sqrt{g}\,\lambda(x)\left(1-\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(x,\mathpzc{p})\right)\\\
&=\int_{M}d^{d}\\!x\sqrt{g}\,\lambda(x)+\int_{\mathscr{P}}d\upmu(\mathpzc{p})\left[1-\int{p}ds\,\lambda(y(s))\right],\end{split}$
(178)

where in the second line we rearranged terms, switched the order of the
integrations over $\mathpzc{p}$ and $x$, applied the definition (174) of
$\Delta$, and performed the integration over $x$ in the last term. The
Lagrangian is bounded above as a function of $\upmu$ if and only if, for all
$\mathpzc{p}$, the quantity in square brackets is non-positive; this becomes
the constraint for the dual program, and when it is satisfied the second term
vanishes on the maximum. The dual program is thus:

$\text{Minimize}\quad\int_{M}d^{d}\\!x\sqrt{g}\,\lambda(x)\quad\text{over
function $\lambda:M\to\mathbf{R}^{+}$ subject
to:}\quad\forall\,\mathpzc{p}\in\mathscr{P}\,,\quad\int{p}ds\,\lambda(y(s))\geq
1\,.$ (179)

Strong duality is guaranteed for linear programs by the existence of a
feasible configuration for the primal; for (177), such a feasible
configuration is the zero measure $\upmu=0$.

We will now apply the duality between (177) and (179) to prove an analogue, in
the language of thread distributions, of the Riemannian max flow-min cut
theorem. Given non-overlapping regions $A,B$ of $\partial M$, let
$\mathscr{P}$ be the set of curves in $M$ with one endpoint in $A$ and the
other in $B$. Given a hypersurface $\gamma$ in $M$ homologous to $A$ relative
to $\gamma^{0}:=\partial M\setminus AB$, every curve
$\mathpzc{p}\in\mathscr{P}$ intersects $\gamma$. Therefore a delta function
supported on $\gamma$ and normalized in the obvious way (i.e. equal to
$\delta(x^{1})$ in Gaussian normal coordinates with $x^{1}$ the coordinate
transverse to $\gamma$) is a feasible function $\lambda$ for (179). The
objective evaluated on this function equals the area of $\gamma$. So by the
weak duality between (177) and (179) we have, for any thread distribution
$\upmu$,

$\upmu(\mathscr{P})\leq\operatorname{area}(\gamma)\,.$ (180)

The max flow-min cut theorem asserts that this inequality is tight, namely

$\sup\upmu(\mathscr{P})=S(A):=\inf_{\gamma\sim
A}\operatorname{area}(\gamma)\,,$ (181)

where the supremum is over thread distributions and the infimum over surfaces
homologous to $A$ relative to $\gamma^{0}$. Given strong duality, to prove the
theorem it is sufficient to show that the optimal value of (179) equals
$S(A)$, in other words that the integral of any feasible $\lambda$ is bounded
below by $S(A)$. By theorem 4.10, given a feasible $\lambda$, there exists a
function $\psi:M\to[-1/2,1/2]$ such that $\psi|_{A}=-1/2$, $\psi|_{B}=1/2$,
$|d\psi|\leq\lambda$. We then have

$\int_{M}\sqrt{g}\,\lambda\geq\int_{M}\sqrt{g}\,|d\psi|\geq S(A)\,,$ (182)

where in the second inequality we used the coarea formula and the fact that
the level sets of $\psi$ are homologous to $A$ (relative to $\gamma^{0}$).

The generalization to multiple boundary regions is straightforward, yielding a
thread-distribution version of the max multiflow theorem Cui:2018dyq . Let
$\\{A_{i}\\}$ be a set of non-overlapping regions of $\partial M$, let
$\mathscr{P}_{i}$ be the set of threads connecting $A_{i}$ to $\cup_{j\neq
i}A_{j}$, and set $\mathscr{P}:=\cup_{i}\mathscr{P}_{i}$. We will now show
that a thread distribution exists such that $\upmu(\mathscr{P}_{i})=S(A_{i})$
for all $i$. We have $\upmu(\mathscr{P})=\sum_{i}\upmu(\mathscr{P}_{i})/2$,
since each curve in $\mathscr{P}$ contributes to two terms in the sum. Hence
the primal objective is bounded above by $\sum_{i}S(A_{i})/2$, and achieves
this bound if and only if $\upmu(\mathscr{P}_{i})=S(A_{i})$ for all $i$. We
will now show that the dual objective is bounded below by the same quantity,
implying that both bounds are achieved. This part of the proof is essentially
the same as that of the multiflow-based theorem, so we will be brief; details
can be found in Cui:2018dyq . Given a feasible function $\lambda$ for the dual
program, we define $\psi_{i}(x)$ as the minimal integral of $\lambda$ over any
path connecting $A_{i}$ to $x$. The level set of $\psi_{i}$ for any value
between $0$ and $1$ is homologous to $A_{i}$ (relative to
$\gamma^{0}:=\partial M\setminus\cup_{i}A_{i}$), so its area is bounded below
by $S(A_{i})$. For any $x\in M$ and $i\neq j$, $\psi_{i}(x)+\psi_{j}(x)\geq
1$, so the level sets of $\psi_{i}$ and $\psi_{j}$ for values between 0 and
$1/2$ do not intersect. This bounds the dual objective below by
$\sum_{i}S(A_{i})/2$.

####  5.1.1 Mapping between flows & thread distributions

Of course, it’s no coincidence that thread distributions obey the same
theorems as flows and multiflows. As we will now explain, a flow (or
multiflow) can be mapped to a thread distribution and vice versa.606060 A
similar mapping in the graph setting, between flows or multiflows and set of
threads (or paths), was discussed in Headrick:2020gyq  (see appendix A).
Roughly speaking, the field lines (or streamlines, or integral curves) of a
flow correspond to the curves in a thread distribution. However, there are a
few complications in this mapping. First, while a field line of a
divergenceless vector field cannot end in the interior of $M$, it need not end
on the boundary — it can be a loop, or keep going forever in both directions
without endpoints.616161 An example of a flow whose field lines go forever in
both directions is $dx^{1}+\alpha dx^{2}$ on the square torus, where $\alpha$
is irrational. Similar examples can be constructed on manifolds with boundary.
Note that the divergenceless condition prevents a field line from having just
one endpoint on the boundary. Second, the field lines are naturally oriented,
whereas threads are unoriented. Third, the field lines of a flow don’t
intersect, whereas the threads in a distribution may do so. These
complications will be dealt with as we go along.

We first define the map from a flow $v$ to a thread distribution $\upmu_{v}$.
As above, we fix boundary regions $A$, $B$ and define $\gamma^{0}:=\partial
M\setminus AB$ and $\mathscr{P}$ as the set of curves with one endpoint in $A$
and the other in $B$. We also assume that the flow $v$ obeys the no-flux
boundary condition on $\gamma^{0}$, $*v|_{\gamma^{0}}=0$. Let
$\mathscr{P}_{v}\subset\mathscr{P}$ be the set of field lines of $v$ running
from $A$ to $B$ (with their orientations dropped), $r$ be the set of points in
$M$ through which they pass, and $\tilde{v}$ the 1-form equal to $v$ on $r$
and $0$ elsewhere. We can then define the measure $\upmu_{v}$ on $\mathscr{P}$
by $\upmu_{v}(\mathscr{P}\setminus\mathscr{P}_{v})=0$ and
$\upmu_{v}(A^{\prime})=\int_{A^{\prime}}*\tilde{v}$ for any subregion
$A^{\prime}\subseteq A$. At any point $x\in M$, the density of threads equals
$|\tilde{v}(x)|$, so we have

$\int_{\mathscr{P}}d\upmu_{v}(\mathpzc{p})\Delta(x,\mathpzc{p})=|\tilde{v}(x)|\leq|v(x)|\leq
1\,.$ (183)

We conclude that $\upmu_{v}$ obeys the density bound and is therefore a thread
distribution. The total measure of $\mathscr{P}$ is at least the flux of $v$
on $A$:

$\upmu_{v}(\mathscr{P})\geq\int_{A}*v\,.$ (184)

The reason it’s not necessarily an equality is that any field lines running
from $B$ to $A$ contribute zero to the left-hand side but negatively to the
right-hand side. While both the set of flows and the set of thread
distributions are convex, the convexity is implemented differently in the two
cases; in general, given flows $v_{1},v_{2}$ and $\alpha\in(0,1)$,

$\upmu_{\alpha
v_{1}+(1-\alpha)v_{2}}\neq\alpha\upmu_{v_{1}}+(1-\alpha)\upmu_{v_{2}}\,.$
(185)

Equality holds if and only if $\tilde{v}_{1},\tilde{v}_{2}$ are everywhere
parallel (i.e.
$\tilde{v}_{1}\cdot\tilde{v}_{2}=|\tilde{v}_{1}||\tilde{v}_{2}|$). Otherwise
the distribution on the right-hand side includes intersecting threads, whereas
the left-hand side, like any distribution derived from a flow, does not.

To convert a thread distribution $\upmu$ into a flow $v_{\upmu}$,626262 The
subscript $\upmu$ in $v_{\upmu}$ is not an index but rather indicates the
dependence on the thread distribution $\upmu$. we can turn each thread into a
delta-function-localized “flux tube”, and then integrate over $\mathscr{P}$.
We then have

$v_{\upmu}(x):=\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(x,\mathpzc{p})\,\dot{x}\,,$
(186)

where $\dot{x}$ is the unit tangent covector to $\mathpzc{p}$ at $x$, with the
orientation along $\mathpzc{p}$ from $A$ to $B$. The integrand
$\Delta(x,\mathpzc{p})\dot{x}$, as a function of $x$ for fixed $\mathpzc{p}$,
is a flux tube supported on $\mathpzc{p}$, and is divergenceless by virtue of
the fact that $\mathpzc{p}$ has no endpoints in the interior of $M$. Hence
$v_{\upmu}$ is divergenceless. Using the Cauchy-Schwarz (or triangle)
inequality, we have

$\left|v_{\upmu}(x)\right|\leq\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(x,\mathpzc{p})\leq
1\,,$ (187)

so $v_{\upmu}$ is indeed a flow. With this construction, the flux of
$v_{\upmu}$ on $A$ equals the number of threads connecting $A$ and $B$:

$\int_{A}*v_{\upmu}=\upmu(\mathscr{P})\,.$ (188)

Unlike the above mapping from flows to thread distributions, this one does
preserve convex combinations:

$v_{\alpha\upmu_{1}+(1-\alpha)\upmu_{2}}=\alpha
v_{\upmu_{1}}+(1-\alpha)v_{\upmu_{2}}\,.$ (189)

The relations (184) and (188) together imply that the “max flow” is the same
whether computed using flows or thread distributions:

$\sup_{\upmu}\upmu(\mathscr{P})=\sup_{v}\int_{A}*v\,.$ (190)

Note also that these two mappings are not inverses of each other; given a flow
$\hat{v}$ and thread distribution $\hat{\upmu}$ we have in general neither
$v_{\upmu_{\hat{v}}}=\hat{v}$ (since flow lines running from $B$ to $A$ are
discarded in transforming $\hat{v}$ into a thread distribution) nor
$\upmu_{v_{\hat{\upmu}}}=\hat{\upmu}$ (since any intersecting threads in
$\hat{\upmu}$ are recombined in transforming $\hat{\upmu}$ into a flow).

Similar constructions convert a multiflow into a thread distribution and vice
versa. We remind the reader that, given a decomposition $\\{A_{i}\\}$ of
$\partial M$, a multiflow is a set of 1-forms $v_{ij}$ ($i<j$) obeying

$d{*v_{ij}}=0\,,\qquad\sum_{i,j}|v_{ij}|\leq 1\,,\qquad*v_{ij}|_{\partial
M\setminus(A_{i}\cup A_{j})}=0\,.$ (191)

The corresponding thread distribution is the sum of the distributions
$\upmu_{v_{ij}}$ for the component flows. Via the analogue of (186) for each
pair of distinct boundary regions, a thread distribution can also be converted
into a multiflow.

###  5.2 V-thread & U-thread distributions

In this subsection we return to the Lorentzian setting of the rest of the
paper, as laid out in subsection 2.2. We also continue to fix a decomposition
of the boundary $\mathcal{N}$ into $D(A)$ and $D(B)$, and to suppress the
$A$\- and $B$-dependence of various quantities, for example writing $S_{\rm
c}$ for $S_{\rm c}(A:B)$. In section 4, we defined V-flows and U-flows, and
will correspondingly define two types of threads, V-threads and U-threads.
Like the Riemannian threads of the previous subsection, both types of threads
will be subject to density bounds; unlike in that case, however, these density
bounds will be non-local. It will turn out that the density bound for
V-threads is enforced by U-threads and vice versa, and the problem of
maximizing the number of V-threads is dual to the problem of minimizing the
number of U-threads (as was the case for the max V-flow and min U-flow
programs (54), (78)). The solution to both of these problems will also equal
the quantity $S_{\rm c}$ defined in (49), as we will show by explaining how a
V/U-thread distribution can be converted into a V/U-flow.

We define a _V-thread_ $\mathpzc{p}$ as an unoriented open curve in
$\mathcal{M}$ with one end in $D(A)$ and the other in $D(B)$, and a _U-thread_
$\mathpzc{q}$ as an inextendible causal curve in $\mathcal{M}$ (which
necessarily has one end on $\mathcal{I}^{+}$ and the other on
$\mathcal{I}^{-}$). V-threads are not required to have any particular causal
character. $\mathscr{P}$ and $\mathscr{Q}$ are the set of all V- and U-threads
respectively. We define the following function on
$\mathscr{Q}\times\mathscr{P}$:

$\Delta(\mathpzc{q},\mathpzc{p}):=\int{q}dt\int{p}ds\,\delta(x(t),y(s))\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}\dot{y}|\,,$ (192)

where $t$ is a parameter along $\mathpzc{q}$, $x(t)$ is the corresponding
point in $\mathcal{M}$, and $\dot{x}$ is dual covector to the tangent vector
$dx(t)/dt$; similarly for $\mathpzc{p}$, $y$, and $s$. (For simplicity we use
the overdot for both $d/dt$ and $d/ds$.) Since the wedgedot function $|\
\mathclap{\hskip 6.75pt\cdot}{\wedge}\ |$ is bihomogeneous,
$\Delta(\mathpzc{q},\mathpzc{p})$ is independent of the parametrization of
$\mathpzc{q}$ and $\mathpzc{p}$. $\Delta$ is a delta-function supported on the
codimension-$(D-2)$ locus in $\mathscr{Q}\times\mathscr{P}$ on which
$\mathpzc{q}$ and $\mathpzc{p}$ intersect. Given a measure $\upmu$ on
$\mathscr{P}$, we define the density of V-threads along a given U-thread
$\mathpzc{q}$ as
$\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(\mathpzc{q},\mathpzc{p})$.
(Note that, given the definition of $\Delta$, this definition automatically
includes an integration along $\mathpzc{q}$, so this is an integrated, not
pointwise, density.) We define a _V-thread distribution_ as a measure $\upmu$
on $\mathscr{P}$ such that this density never exceeds 1:

$\forall\,\mathpzc{q}\in\mathscr{Q}\,,\quad\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(\mathpzc{q},\mathpzc{p})\leq
1\,.$ (193)

Similarly, given a measure $\upnu$ on $\mathscr{Q}$, we define the density of
U-threads along a given V-thread $\mathpzc{p}$ as
$\int_{\mathscr{Q}}d\upnu(\mathpzc{q})\,\Delta(\mathpzc{q},\mathpzc{p})$, and
a _U-thread distribution_ (for $A$) as a measure $\upnu$ on $\mathscr{Q}$
satisfying

$\forall\,\mathpzc{p}\in\mathscr{P}\,,\quad\int_{\mathscr{Q}}d\upnu(\mathpzc{q})\,\Delta(\mathpzc{q},\mathpzc{p})\geq
1\,.$ (194)

Essentially, the U-threads have to form a sufficient “barrier” separating
$D(A)$ from $D(B)$.

The problem of maximizing $\upmu(\mathscr{P})$ over V-thread distributions
defines a linear program:

$\text{Maximize}\quad\upmu(\mathscr{P})\quad\text{over measure $\upmu$ on
$\mathscr{P}$ subject
to:}\quad\forall\,\mathpzc{q}\in\mathscr{Q}\,,\quad\int_{\mathscr{P}}d\upmu(\mathpzc{p})\,\Delta(\mathpzc{q},\mathpzc{p})\leq
1\,.$ (195)

The problem of minimizing $\upnu(\mathscr{Q})$ over U-thread distributions
also defines a linear program:

$\text{Minimize}\quad\upnu(\mathscr{Q})\quad\text{over measure $\upnu$ on
$\mathscr{Q}$ subject
to:}\quad\forall\,\mathpzc{p}\in\mathscr{P}\,,\quad\int_{\mathscr{Q}}d\upnu(\mathpzc{q})\,\Delta(\mathpzc{q},\mathpzc{p})\geq
1\,.$ (196)

The programs (195), (196) are dual to each other. Strong duality is again
guaranteed by the existence of a feasible point for one of the programs,
namely $\upmu=0$ for (195), so we have

$\sup_{\upmu}\upmu(\mathscr{P})=\operatorname*{inf\vphantom{p}}_{\upnu}\upnu(\mathscr{Q})\,.$
(197)

Next we will show that these quantities both equal $S_{\rm c}$.636363 In the
subsection on Riemannian thread distributions, below (182), it was explained
how to define a thread distribution for multiple boundary regions, the
analogue of a multiflow. One may be tempted to do the same in the Lorentzian
setting. However, for the reasons given in footnote 46, this is not
straightforward.

####  5.2.1 Mapping between flows & thread distributions

Just as, in subsection 5.1.1, we converted Riemannian flows to thread
distributions and vice versa, we can convert V/U-flows to V/U-thread
distributions and vice versa. We start with the V-flows, which are more
parallel with the Riemannian case. To convert a V-flow $V$ into a V-thread
distribution, we first strip away any field lines that do not run from $D(A)$
to $D(B)$. Call the resulting 1-form $\tilde{V}$, and set of field lines
$\mathscr{P}_{V}\subset\mathscr{P}$. We then define the measure $\upmu_{V}$ on
$\mathscr{P}$ so that $\upmu_{V}(\mathscr{P}\setminus\mathscr{P}_{V})=0$ and
$\upmu_{V}(D^{\prime})=\int_{D^{\prime}}*\tilde{V}$ for any subregion
$D^{\prime}\subseteq D(A)$. For any U-thread $\mathpzc{q}$, we then have,
similarly to (183),

$\int_{\mathscr{P}}d\upmu_{V}(\mathpzc{p})\,\Delta(\mathpzc{q},\mathpzc{p})=\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}\tilde{V}|\leq\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V|\leq 1\,.$ (198)

The first equality follows from the definition (192) of the function $\Delta$,
the first inequality from the fact that $\tilde{V}$ equals either $V$ or 0 at
every point in $\mathcal{M}$, and the last inequality from the bound (68)
obeyed by any V-flow. Thus $\upmu_{V}$ is indeed a thread distribution.
Furthermore, similarly to (184),

$\upmu_{V}(\mathscr{P})\geq\int_{D(A)}*V\,.$ (199)

To go from a V-thread distribution $\upmu$ to a V-flow $V_{\upmu}$,646464 The
subscript on $V_{\upmu}$ is not an index but rather denotes the dependence on
the measure $\upmu$. we orient the threads from $D(A)$ to $D(B)$ and put a
delta-function flux tube on each:

$V_{\upmu}(x):=\int_{\mathscr{P}}d\upmu(\mathpzc{p})\int{p}ds\,\delta(x,y(s))\,\dot{y}\,.$
(200)

For any U-thread $\mathpzc{q}$, we have

$\int{q}dt\,|(-\dot{x})\mathclap{\hskip
6.75pt\cdot}{\wedge}V_{\upmu}|\leq\int_{\mathscr{P}}d\upmu(\mathpzc{p})\Delta(\mathpzc{q},\mathpzc{p})\leq
1\,,$ (201)

where in the first inequality we used the convexity of $|\ \mathclap{\hskip
6.75pt\cdot}{\wedge}\ |$ in its second argument; hence $V_{\upmu}$ is indeed a
V-flow. We also have

$\int_{D(A)}*V_{\upmu}=\upmu(\mathscr{P})\,.$ (202)

The mappings between V-flows and V-thread distributions, and the relations
(202), (199) imply that the maximum number of threads in $\mathscr{P}$, for
any V-thread distribution, equals $S_{\rm c}$:

$\sup_{\upmu}\upmu(\mathscr{P})=\sup_{V\in\mathscr{F}}\int_{D(A)}*V=S_{\rm
c}\,.$ (203)

Similarly, we can convert a U-flow into a U-thread distribution and back. One
difference with the V case is that every U-thread and every field line of a
U-flow necessarily goes from $\mathcal{I}^{-}$ to $\mathcal{I}^{+}$, so there
is no need to select a subset of the threads or field lines, or to fuss with
orientations; the analogue of (199) is therefore an equality. We leave the
details as an exercise to the reader. In the end, similarly to (203), we have

$\inf_{\upnu}\upnu(\mathscr{Q})=\inf_{U\in\mathscr{G}}\int_{\mathcal{I}^{+}}*U=S_{\rm
c}\,.$ (204)

With (203) and (204) in hand, we can add another row to our diagram (81) of
formulas for the quantity $S_{\rm c}$:

$\minCDarrowwidth 50pt\CD\boxed{\sup_{\phi\in\mathscr{S}_{\rm
c}}\operatorname*{inf\vphantom{p}}_{\psi\in\mathscr{T}_{\rm
c}}\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|}@Z\text{minimax}Z\text{theorem}Z\boxed{\operatorname*{inf\vphantom{p}}_{\psi\in\mathscr{T}_{\rm
c}}\sup_{\phi\in\mathscr{S}_{\rm
c}}\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|}\\\
@X{\scriptsize\begin{array}[]{r}\psi\leftrightarrow V\\\
\text{duality}\end{array}}XX@XX{\scriptsize\begin{array}[]{r}\phi\leftrightarrow
U\\\ \text{duality}\end{array}}X\\\
\boxed{\sup_{V\in\mathscr{F}}\int_{D(A)}*V}@Z{(V,\phi)\leftrightarrow(U,\psi)}Z\text{duality}Z\boxed{\inf_{U\in\mathscr{G}}\int_{\mathcal{I}^{+}}*U}\\\
@X{\scriptsize\begin{array}[]{r}\text{V-flow$\leftrightarrow$V-thread}\\\
\text{conversion}\end{array}}XX@XX{\scriptsize\begin{array}[]{l}\text{U-flow$\leftrightarrow$U-thread}\\\
\text{conversion}\end{array}}X\\\
\boxed{\sup_{\upmu}\upmu(\mathscr{P})}@Z{\upmu\leftrightarrow\upnu}Z\text{duality}Z\boxed{\inf_{\upnu}\upnu(\mathscr{Q})}$
(205)

##  6 Holographic spacetimes

In the previous sections, for given boundary regions $A$, $B$, we defined the
three quantities $S_{+}$, $S_{-}$, $S_{\rm c}$, gave several formulas for
each, and showed that they obey the relation

$S_{-}\leq S_{\rm c}\leq S_{+}\,.$ (206)

All of this was essentially assuming just global hyperbolicity for the bulk
spacetime — no field equations, energy conditions, or boundary conditions were
invoked.

Starting in this section, we specialize to “standard” classical holographic
spacetimes, in other words ones in which the metric obeys AdS boundary
conditions as well as the null energy (or curvature) condition (NEC). In such
spacetimes, the maximin quantity $S_{-}$ equals the area
$S_{{}_{\text{HRT}}}=S(A:B)$ of the HRT (or entanglement wedge cross-section,
EWCS) surface ${\gamma_{{}_{\text{HRT}}}}$ Wall:2012uf ; paper1 . In
subsection 6.1, we will show that both inequalities in (206) are saturated, so
$S_{\rm c}$ and $S_{+}$ also equal $S_{{}_{\text{HRT}}}$. Altogether, counting
the two new formulas for $S_{-}$ and two formulas for $S_{+}$ given in
subsection 3.1, as well as the six formulas for $S_{\rm c}$ shown in (205), we
have therefore provided ten new covariant formulas for the holographic EE. In
subsection 6.2, we will focus on the V-flow and U-flow formulas of section 4,
and see what the optimal flows look like. Then, in subsection 6.3, we will
generalize our results to multiple boundary regions.

###  6.1 Relation to the HRT formula

To orient the reader, we start by previewing the logic of the argument. There
are two crucial features of the holographic setup which allow us to saturate
both inequalities in (206) and thereby collapse the maximin, minimax, and
convex-relaxed values into one quantity. The first is the NEC which ensures
that each null normal congruence from any extremal surface has non-positive
and decreasing expansion. The second is less familiar, involving AdS boundary
conditions in the more general context of allowing end-of-the-world branes
comprising $\mathcal{I}^{0}$. Together, these ensure that within the time-
sheet formed by the bulk part of the entanglement wedge boundary, the HRT
surface is area-maximizing within its relative homology class. On the other
hand, there exists a maximin Cauchy slice containing the HRT surface, on which
the latter is area-minimizing within its relative homology class. Taken
together, these two observations will then suffice to establish the HRT
surface as a global saddle point.

To flesh this out more explicitly, it will be useful to set up a few
deﬁnitions. Recall that, given a slice $\sigma$, $\Gamma_{\sigma}$ (defined
below (26)) is the set of surfaces $\gamma$ in $\sigma$ homologous to
$A_{\sigma}:=D(A)\cap\sigma$ (relative to
$\gamma^{0}_{\sigma}:=\mathcal{I}^{0}\cap\sigma$); and, given a time-sheet
$\tau$ homologous to $D(A)$ (relative to $\mathcal{I}$), $\Gamma_{\tau}$
(defined at the end of section 2.2) is the set of surfaces of the form
$\tau\cap\sigma$ for some slice $\sigma$. The union of the former over all
slices, or equivalently the union of the latter over all time-sheets, defines
the full set of surfaces that are spacelike-homologous to $A$, which we call
$\Gamma$:

$\Gamma:=\bigcup_{\sigma\in\mathscr{S}}\Gamma_{\sigma}=\bigcup_{\tau\in\mathscr{T}}\Gamma_{\tau}=\left\\{\sigma\cap\tau|\,\sigma\in\mathscr{S},\tau\in\mathscr{T}\right\\}.$
(207)

We denote by $\Gamma_{\rm ext}$ the set of surfaces in $\Gamma$ that are
extremal.656565 By an extremal surface we mean one that extremizes the area
with respect to all variations in position, including those that move the
intersection with $\mathcal{I}^{0}$ (if any). Such a surface has vanishing
mean curvature vector and intersects $\mathcal{I}^{0}$ orthogonally. We then
have

$S_{{}_{\text{HRT}}}:=\inf_{\gamma\in\Gamma_{\rm
ext}}\operatorname{area}(\gamma)\,.$ (208)

The minimizer (or any minimizer, if there is more than one) is the (or an)
HRT/EWCS surface ${\gamma_{{}_{\text{HRT}}}}$.

It will also be useful to slightly expand the definition of a time-sheet to
allow null pieces. Specifically, we write $\bar{\mathscr{T}}$ for the set of
hypersurfaces that are piecewise timelike or null and homologous to $D(A)$
(whereas $\mathscr{T}$ only includes piecewise timelike hypersurfaces). Since
any element of $\bar{\mathscr{T}}$ is a limit of (piecewise) timelike
hypersurfaces, this does not change the infimum in the minimax formula:

$S_{+}=\operatorname*{inf\vphantom{p}}_{\tau\in\bar{\mathscr{T}}}\
\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,.$ (209)

Figure 9:  Illustration of geometric constructs associated to an element
$\gamma\in\Gamma_{\rm ext}$, an extremal surface spacelike-homologous to $A$,
described in item (1): $\sigma$ is a slice containing $\gamma$;
$A_{\sigma}:=\sigma\cap D(A)$; $r_{\gamma}$ is the homology region on $\sigma$
interpolating between $\gamma$ and $A_{\sigma}$; $D(r_{\gamma})$ is the bulk
domain of dependence of $r_{\gamma}$; and the null surface $H_{\gamma}$ is the
bulk part of the boundary of $D(r_{\gamma})$. The following general facts
about these constructs are important: $D(r_{\gamma})$ and $H_{\gamma}$ are
independent of the choice of $\sigma$; $H_{\gamma}$ is homologous to $D(A)$
(relative to $\mathcal{I}$); and $\gamma$ is maximal on $H_{\gamma}$.

Two important sets of facts about surfaces in $\Gamma_{\rm ext}$, hinging on
AdS boundary conditions and the NEC are the basis for what follows (proofs and
discussion can be found in Wall:2012uf ; Marolf:2019bgj ; Grado-White:2020wlb
; paper1 ):

  1. (1)

Given any surface $\gamma\in\Gamma_{\rm ext}$, let $\sigma$ be a slice
containing $\gamma$, $r_{\gamma}\subset\sigma$ the homology region on $\sigma$
between $\gamma$ and $A_{\sigma}$, $D(r_{\gamma})$ its causal domain, and
$H_{\gamma}$ the bulk part of the boundary of $D(r_{\gamma})$, i.e.
$H_{\gamma}:=\partial D(r_{\gamma})\setminus(\mathcal{I}\cup\mathcal{N})$.
$D(r_{\gamma})$ and $H_{\gamma}$ are independent of the choice of $\sigma$,
and furthermore:

    1. (a)

$H_{\gamma}$ is made up of congruences of null geodesics shot orthogonally
from $\gamma$ toward the past and future in the direction of $D(A)$.666666
This requires, in addition to AdS boundary conditions and NEC, the assumption
that $\mathcal{I}^{0}$ (if non-empty) is “well-behaved”, in the sense that
while generators of $H_{\gamma}$ may fall into $\mathcal{I}^{0}$, they never
emerge from $\mathcal{I}^{0}$. (On the other hand, in the case
$\mathcal{I}^{0}=\gamma^{0}$ where the above-mentioned energy conditions don’t
enter, any generators of $\partial D(r_{\gamma})$ that emanate from this
$\gamma^{0}$ are part of $\mathcal{I}$ and hence not included in
$H_{\gamma}$.)  Since $\gamma$ is extremal, the expansion $\theta$ on these
congruences is initially zero, and by virtue of the NEC and Raychaudhuri and
Einstein equations, $\dot{\theta}\leq 0$. Therefore, $\gamma$ has maximal area
among surfaces in $\Gamma$ contained in $H_{\gamma}$.676767 A-priori, if one
just arbitrarily extends $\gamma^{0}$ in a time direction and declares it to
be $\mathcal{I}^{0}$, one would have to worry that $H_{\gamma}$ contains
future light cone from $\gamma\cap\mathcal{I}^{0}$, i.e. that $\partial
I^{+}(\sigma\setminus r_{\gamma})\setminus(\mathcal{I}\cup\mathcal{N})$ does
in fact contain some generators emanating from $\mathcal{I}^{0}$, in which
case $H_{\gamma}$ would have an expanding part, and it would no longer follow
that $\gamma$ is maximal on $H_{\gamma}$. However, this contingency is
excluded by the previous footnote. The underlying assumption that generators
of $H_{\gamma}$ can’t emanate from $\mathcal{I}^{0}$ is therefore a strong
(and crucial) one.

    2. (b)

$D(r_{\gamma})\cap\mathcal{N}=D(A)$, implying that $H_{\gamma}$ is homologous
to $D(A)$ (relative to $\mathcal{I}$). Since $H_{\gamma}$ is null,
$H_{\gamma}\in\bar{\mathscr{T}}$.

An important special case is the HRT surface ${\gamma_{{}_{\text{HRT}}}}$, for
which $D(r_{\gamma_{{}_{\text{HRT}}}})=:\mathcal{W}(A)$ is the entanglement
wedge and $H_{\gamma_{{}_{\text{HRT}}}}=:\mathcal{H}(A)$ is the entanglement
horizon. However, it is important that properties (a) and (b) hold for any
surface $\gamma\in\Gamma_{\rm ext}$, not just the HRT surface. See figure 9
for a sketch illustrating these constructs.

  2. (2)

The sup and inf in the maximin formula (26) are attained, and the maximin
surface $\gamma_{-}$ is the HRT surface,
$\gamma_{-}={\gamma_{{}_{\text{HRT}}}}$, so $S_{-}=S_{{}_{\text{HRT}}}$. The
hard parts here are showing that the sup is attained and that the maximin
surface is extremal (which naively follows from the fact that it is extremal
with respect to variations in both the space and time normal directions;
however there are subtleties to this argument). The fact that $\gamma_{-}$ is
minimal among extremal surfaces is argued as follows. For
$\gamma\in\Gamma_{\rm ext}$, let $\gamma^{\prime}:=H_{\gamma}\cap\sigma_{-}$
(where $\sigma_{-}$ is a maximin slice). Then by the fact that $\gamma$ is
maximal on $H_{\gamma}$,
$\operatorname{area}(\gamma)\geq\operatorname{area}(\gamma^{\prime})$; and by
the fact that $\gamma_{-}$ is minimal on $\sigma_{-}$,
$\operatorname{area}(\gamma^{\prime})\geq\operatorname{area}(\gamma_{-})$.

The fact that ${\gamma_{{}_{\text{HRT}}}}$ is, by (1), maximal on the
hypersurface $\mathcal{H}(A)$ and, by (2), minimal on the slice $\sigma_{-}$,
makes it a global saddle point, establishing that it is not only the maximin
but also the minimax surface:
$\gamma_{-}=\gamma_{+}={\gamma_{{}_{\text{HRT}}}}$ and
$S_{-}=S_{+}=S_{{}_{\text{HRT}}}$ (see (37)). We can also make the argument a
slightly different way: by property (1), any surface $\gamma\in\Gamma_{\rm
ext}$ is maximal on the hypersurface $H_{\gamma}$, so by (208) $S_{+}\leq
S_{{}_{\text{HRT}}}$; the min-max inequality $S_{-}\leq S_{+}$ and the fact
$S_{-}=S_{{}_{\text{HRT}}}$ then imply the equality of all three. Finally,
since $S_{\rm c}$ is caught between $S_{-}$ and $S_{+}$, we have

$S_{-}=S_{\rm c}=S_{+}=S_{{}_{\text{HRT}}}\,,$ (210)

Eq. (210) opens the door to using the flow- and thread-based convex programs
to calculate the HRT area.

One way to think about the collapse of the three quantities $S_{-}$, $S_{\rm
c}$, and $S_{+}$ into a single quantity is that the contents of sections 3–5
are essentially “kinematics” — things that follow just from having a (globally
hyperbolic) Lorentzian spacetime. The “dynamics” that collapses those three
quantities into a single one stems from the properties of the entanglement
horizon, which require field equations, energy conditions, and boundary
conditions.

###  6.2 Optimal flows

In this subsection we will describe solutions, or optimal configurations, for
the V-flow and U-flow formulas (54), (78). The solutions for the other
formulas can be deduced from these. As we will see, the optimal configurations
are strongly constrained by the properties of holographic spacetimes. These
constraints are simplest to describe in the generic case, by which we mean
that (1) ${\gamma_{{}_{\text{HRT}}}}$ is the unique minimal surface on some
slice $\sigma_{-}$, and (2) $\dot{\theta}<0$ on both the future and past
branches of the entanglement horizon $\mathcal{H}(A)$, implying that
${\gamma_{{}_{\text{HRT}}}}$ is the unique maximal-area acausal surface in
$\mathcal{H}(A)$.686868 A necessary and sufficient condition for
$\dot{\theta}<0$ everywhere on $\mathcal{H}(A)$ is for the following
inequality to hold at every point on ${\gamma_{{}_{\text{HRT}}}}$ and for both
null normals $k^{\mu}$:
$k^{\mu}k^{\nu}(R_{\mu\nu}+K_{\mu\lambda\rho}{K_{\nu}}^{\lambda\rho})>0\,,$
(211) where $K_{\mu\lambda\rho}$ is the extrinsic curvature tensor of
${\gamma_{{}_{\text{HRT}}}}$. Here we used the fact that the initial shear of
the congruence is $\Sigma_{\lambda\rho}=k^{\mu}K_{\mu\lambda\rho}$.

We start with the V-flow case. In order for the flux of $V$ through $D(A)$,
and therefore through $\mathcal{H}(A)$, to equal the area of
${\gamma_{{}_{\text{HRT}}}}$, all of that flux must pass through
${\gamma_{{}_{\text{HRT}}}}$, and there must be no flux elsewhere on
$\mathcal{H}(A)$; in other words, the flux $*V|_{\mathcal{H}(A)}$ must be
proportional to a delta-function supported on ${\gamma_{{}_{\text{HRT}}}}$.
The same reasoning applies, of course, to $\mathcal{H}(B)$; thus, the flux
squeezes from $\mathcal{W}(A)$ into $\mathcal{W}(B)$ through
${\gamma_{{}_{\text{HRT}}}}$, without passing through its past or future
$I^{\pm}({\gamma_{{}_{\text{HRT}}}})$.

Figure 10:  Cross section of generic max V-flow [Left] and min U-flow [Right].
The max V-flow squeezes through the HRT surface, being excluded from its past
and future. The min U-flow also squeezes through the HRT surface, being
excluded from the entanglement wedges. Notice that while the V-flow can have
somewhere-timelike flow lines, the U-flow lines cannot be spacelike anywhere.

The function $\phi$ that witnesses the norm bound on $V$ is also strongly
constrained. For a thin ribbon $\tau$ that follows a timelike curve starting
at $x$ and ending at $y$, with constant spatial area $a$, the same reasoning
that gave (75) implies

$\int_{\mathcal{T}}*V\leq a\,(\phi(y)-\phi(x))\,.$ (212)

If the ribbon passes through ${\gamma_{{}_{\text{HRT}}}}$, spatially parallel
to it, then the flux through it equals $a$, so $\phi(y)-\phi(x)\geq 1$. But
given that $|\phi|\leq 1/2$ everywhere in $\bar{\mathcal{M}}$, this implies

$\phi|_{I^{\pm}({\gamma_{{}_{\text{HRT}}}})}=\pm\frac{1}{2}\,.$ (213)

In other words, all of the level sets of $\phi$ pass through
${\gamma_{{}_{\text{HRT}}}}$. Given the condition $d\phi\pm
V\in\mathfrak{j}^{+}$, $V$ must vanish in any region where $\phi$ is constant,
so (213) implies

$V|_{I^{\pm}({\gamma_{{}_{\text{HRT}}}})}=0\,.$ (214)

Although we already showed that none of the flux from $D(A)$ to $D(B)$ passes
through $I^{\pm}({\gamma_{{}_{\text{HRT}}}})$ (since it all passes through
${\gamma_{{}_{\text{HRT}}}}$), (213) implies that there can be no stray flux
circulating in closed loops in those regions. (213), (214) only constrain $V$
and $\phi$ in the regions $I^{\pm}({\gamma_{{}_{\text{HRT}}}})$. In the
complementary regions $\mathcal{W}(A)$, $\mathcal{W}(B)$, there remains
enormous freedom in the solution, including not only how the flux is spread
out in space — as for Riemannian flows (or bit threads) — but also how it is
spread out in time. See figure 10 for a picture of a “typical” max V-flow.

We want to emphasize that (213), (214) are properties of _solutions_ to the
V-flow program. There are many perfectly acceptable V-flows that do not obey
those equations — but they will not have maximal flux. The V-flow program,
which in its definition knows nothing about ${\gamma_{{}_{\text{HRT}}}}$, in
maximizing the flux of $V$ naturally discovers not only the HRT surface
${\gamma_{{}_{\text{HRT}}}}$ but also the entanglement wedges
$\mathcal{W}(A)$, $\mathcal{W}(B)$.696969 In fact, in this regard, the full
collection of maximizing flows does better than the collection of HRT surfaces
anchored on all possible subregions: The latter, being smooth, generically
can’t probe arbitrarily close to the crossover seams typically present on the
entanglement horizon. Hence only a subset of the entanglement wedge would be
reachable by HRT surfaces, whereas the entire entanglement wedge is reachable
by maximizing flows. An avatar of this was seen already in the Riemannian
context Freedman:2016zud : the bit threads can penetrate ‘entanglement
shadows’, namely bulk regions inaccessible by RT surfaces.

Optimal U-flows, which minimize the flux of $U$ through $\mathcal{I}^{+}$,
have a similar structure, but “turned on the side”.707070 One additional
difference being that while the V-flows can have timelike parts, the U-flows
cannot have spacelike parts, so in this sense the floppiness of U-flows is
slightly more rigidified. Given a slice $\sigma_{-}$ on which
${\gamma_{{}_{\text{HRT}}}}$ is the unique minimal-area surface, a minimal
U-flow $U$ saturates (99), implying that the flux $*U|_{\sigma_{-}}$ is a
delta-function supported on ${\gamma_{{}_{\text{HRT}}}}$. The conditions (79)
and $U\in\mathfrak{j}^{+}$ then imply that $U$ vanishes everywhere in the
causal domain of $\sigma_{-}\setminus{\gamma_{{}_{\text{HRT}}}}$, which is to
say the union of the entanglement wedges $\mathcal{W}(A)$, $\mathcal{W}(B)$,
or more precisely their interiors:

$U|_{\text{int}\mathcal{W}(A)\cup\text{int}\mathcal{W}(B)}=0\,.$ (215)

At any point where $U=0$, $d\psi=0$ as well, so $\psi$ is constant on
$\mathcal{W}(A)$ and $\mathcal{W}(B)$. Given the boundary conditions on
$\psi$, we have

$\psi|_{\mathcal{W}(A)}=-\frac{1}{2}\,,\qquad\psi|_{\mathcal{W}(B)}=\frac{1}{2}\,.$
(216)

See figure 10 for a typical min U-flow. As for V-flows, the U-flow program
does not know, in its definition, where the entanglement wedges are, and a
non-minimizing U-flow need not obey (215), (216). Rather, in minimizing the
flux of $U$ the U-flow program naturally discovers the HRT surface and
entanglement wedges.

Among the optimal V-flows are slice-flows, where the slice is any maximin
slice. Among the optimal U-flows are time-sheet-flows, where the time-sheet
for example is the entanglement horizon $\mathcal{H}(A)$.

###  6.3 Multiple regions

So far, in most of the paper (except for section 4.3), we have for clarity
considered a single boundary region $A$ and its complement $B:=A^{c}$.
However, as explained in section 4.3, all our constructions can be generalized
to the case of multiple regions of interest $A,B,C,\ldots$ sharing a boundary
Cauchy slice. We assume these regions are separated by buffers, and take our
regulated spacetime $\mathcal{M}$ to be the joint entanglement wedge of all
these regions $\mathcal{W}(ABC\cdots)$. We also define the complement of any
given region to consist of the union of all the other ones, $A^{c}=BC\cdots$
etc. In this context it is interesting to ask when can we utilize a single
optimal flow configuration — either U- or V-flow — for multiple regions, and
what would such a flow look like.

Let us first recall the Riemannian case. In the RT (min cut) formulation, the
inclusion-exclusion argument of Headrick:2013zda  shows that homology regions
are monotonic under inclusion, hence that RT surfaces for nested regions (say
$A$ and $AB$) do not intersect transversely (though they can be coincident on
connected components). As shown in Headrick:2017ucz , strong duality relates
the nesting property for homology regions to nesting property for flows. In
the latter case, this states that there exists a flow $v$ that simultaneously
maximizes on any collection of nested regions. On the other hand, when RT
surfaces do intersect transversely (say for regions $AB$ and $BC$), then it is
clear that there cannot exist any flow simultaneously maximizing on both
crossing regions. This is because at the intersection of the RT surfaces, the
normals point in different directions, while a single maximizing flow has to
be of unit norm and normal to the RT surfaces everywhere.

Figure 11:  [Left]: Illustration of a max V-flow for nested regions $A$ and
$AB$ on the Poincare disk. The dashed curves indicate the HRT surfaces for the
respective regions, and the solid thin curves indicate the V-flow lines,
color-coded by which regions they connect. [Right]: To illustrate the temporal
behavior of the flow lines for a generic max V-flow (green) as well as a
generic min U-flow (blue), we take a cross-section of the left diagram between
$A$ and $C$, so that only the $A-C$ V-flow lines (green in the left figure)
are visible.

Let us now return to the Lorentzian case. In the holographic context,
entanglement wedge nesting applies whenever the NEC is satisfied Wall:2012uf .
In other words, for nested boundary regions $A$ and $AB$, the entanglement
wedges are nested, which automatically implies that the respective HRT
surfaces ${\gamma_{{}_{\text{HRT}}}}(A)$ and ${\gamma_{{}_{\text{HRT}}}}(AB)$
cannot be timelike-separated, and in fact lie on a common maximin slice. Hence
there exists a V-flow configuration (namely a slice flow) which passes through
both HRT surfaces. A generic maximizing V-flow would be dispersed everywhere
outside the future and past of the two HRT surfaces
${\gamma_{{}_{\text{HRT}}}}(A)$, ${\gamma_{{}_{\text{HRT}}}}(AB)$ (and
collimated through both of them), in other words within the following
spacetime region:

$\mathcal{W}(A)\cup\mathcal{W}((AB)^{c})\cup(W(AB)\cap W(A^{c}))\,.$ (217)

Similarly, one can combine the minimizing U-flows in the complementary
regions, likewise collimated through both HRT surfaces. However, whereas a
single V-flow line can pass through both HRT surfaces, a single U-flow line
must pass through exactly one of them. This is illustrated in figure 11.

Now what about crossing regions, e.g. $AB$ and $BC$? Since HRT surfaces are
bulk codimension-2, two HRT surfaces anchored on crossing regions will
generically not intersect. If they do, the above argument for the Riemannian
case immediately generalizes to the present context. However, when the HRT
surfaces ${\gamma_{{}_{\text{HRT}}}}(AB)$ and ${\gamma_{{}_{\text{HRT}}}}(BC)$
do not intersect transversely, they must be timelike-separated somewhere,
which means they would violate the V-norm bound.

##  7 Embedding in the full spacetime & removing the regulator

In subsection 2.1, we explained that, by focusing on the EWCS (either as a
regulator for the EE or for its own sake), we could work entirely in the joint
$AB$ entanglement wedge, discarding the rest of the original spacetime. But,
in the other direction, we can also easily translate the various new formulas
for the EWCS that we’ve developed in this paper back to the original
spacetime. Here we will focus on three formulas in particular: minimax, max
V-flow, and min U-flow; the others follow the same pattern, and we leave it to
the reader to work out the details. We can even remove the regulator entirely,
and define max V-flows and min U-flows when the EE is infinite.

###  7.1 Embedding in the full spacetime

Figure 12:  Schematic illustration of the full spacetime $\hat{\mathcal{M}}$,
partitioned into 4 causal domains specified by
${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$, so that the regulated spacetime
(shaded) is $\mathcal{M}=\hat{\mathcal{W}}(AB)$. This is to be visualized as a
cross-section of AdS (cf. right panel of figure 3, but rotated by 90∘, with
$A$ in front and $B$ in back).

Let us first set up some notation for this purpose. Let $\hat{\mathcal{M}}$ be
the original holographic spacetime containing $\mathcal{M}$, with future/past
boundaries $\hat{\mathcal{I}}^{\pm}$ (which may be singularities or at
infinite time) and timelike spatial boundary (end-of-the-world brane)
$\hat{\mathcal{I}}^{0}$ (which may be empty); define
$\hat{\mathcal{I}}:=\hat{\mathcal{I}}^{+}\cup\hat{\mathcal{I}}^{-}\cup\hat{\mathcal{I}}^{0}$.
(In general we use hatted symbols to denote geometric constructs in the
original, unregulated spacetime.) Let ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$
be the HRT surface in $\hat{\mathcal{M}}$ for $AB$. We assume that
$\hat{\mathcal{M}}$ obeys the null energy condition and asymptotically AdS
boundary conditions, so it is decomposed along the $AB$ entanglement horizons
$\hat{\mathcal{H}}(AB)$, $\hat{\mathcal{H}}((AB)^{c})$ into four spacetime
regions: the entanglement wedge of $AB$, which is $\mathcal{M}$; the
complementary entanglement wedge $\hat{\mathcal{W}}((AB)^{c})$; and
$J^{\pm}({\hat{\gamma}_{{}_{\text{HRT}}}}(AB))$. (Cf. figure 12.) The
future/past boundary $\mathcal{I}^{\pm}$ of $\bar{\mathcal{M}}$ is made up of
the future/past entanglement horizons $\hat{\mathcal{H}}^{\pm}(AB)$ possibly
along with part or all of $\hat{\mathcal{I}}^{\pm}$.

Everything we discuss below can be done for multiple boundary regions
$A,B,\ldots$, as discussed in subsection 6.3, provided of course that they lie
on a common boundary Cauchy slice. Then the joint entanglement wedge
$\hat{\mathcal{W}}(AB)$ is replaced by the entanglement wedge of their union
$\hat{\mathcal{W}}(AB\cdots)$, and similarly for their joint HRT surface.

##### Minimax

Here we start from the minimax formula in the regulated spacetime (33), which
we repeat here for convenience:

$S_{+}=\operatorname*{inf\vphantom{p}}_{\tau\in\mathscr{T}}\
\sup_{\begin{subarray}{c}\gamma\subset\tau\\\
\text{achronal}\end{subarray}}\operatorname{area}(\gamma)\,.$ (218)

Any time-sheet $\tau\in\mathscr{T}$ may be extended to a piecewise timelike
hypersurface $\hat{\tau}$ in $\hat{\mathcal{M}}$ that is homologous to $D(A)$
relative to $\partial\hat{\mathcal{M}}\setminus(D(A)\cup D(B))$. Conversely,
any such hypersurface, when restricted to $\mathcal{M}$, is in $\mathscr{T}$.
We therefore define $\hat{\mathscr{T}}$ as the set of such hypersurfaces.
Within such a hypersurface, we want to maximize the area of an achronal
surface contained in $\mathcal{W}(AB)$:

$S_{\rm
HRT}=\inf_{\hat{\tau}\in\hat{\mathscr{T}}}\sup_{\begin{subarray}{c}\gamma\subset\tau\cap\mathcal{W}(AB)\\\
\text{achronal}\end{subarray}}\operatorname{area}(\gamma)\,.$ (219)

(The condition that $\hat{\tau}$ is piecewise timelike outside of
$\hat{\mathcal{W}}(AB)$ is actually unnecessary, since we are only measuring
the area of a surface within $\hat{\mathcal{W}}(AB)$.)

##### V-flows

Starting with the definition (55), (56) of a V-flow in the regulated
spacetime, we can define a V-flow in $\hat{\mathcal{M}}$. Any slice for
$\hat{\mathcal{M}}$ that contains ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$
becomes a slice for $\bar{\mathcal{M}}$ when restricted to it. We therefore
define $\hat{\mathscr{S}}$ as the set of slices for $\hat{\mathcal{M}}$
containing ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$. Its relaxation
$\hat{\mathscr{S}}_{\rm c}$ is then given by the following set of functions:

$\hat{\mathscr{S}}_{\rm
c}:=\bigg{\\{}\phi:\hat{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\phi|_{J^{\pm}({\hat{\gamma}_{{}_{\text{HRT}}}}(AB))}=\phi|_{\hat{\mathcal{I}}^{\pm}}=\pm\frac{1}{2}\,,\,d\phi\in\mathfrak{j}^{+}\right\\}\
.$ (220)

We can in fact be far more parsimonious in restricting $\phi$. Note that,
given the conditions $d\phi\in\mathfrak{j}^{+}$ and
$\phi|_{\hat{\mathcal{I}}^{\pm}}=\pm 1/2$, the condition
$\phi|_{J^{\pm}({\hat{\gamma}_{{}_{\text{HRT}}}}(AB))}=\pm 1/2$ follows just
from requiring that $\phi=-1/2$ infinitesimally to the past of
${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$ and $\phi=+1/2$ infinitesimally to the
future of ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$. We can then rewrite (220) as

$\hat{\mathscr{S}}_{\rm
c}=\bigg{\\{}\phi:\hat{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\phi|_{{\gamma_{{}_{\text{HRT}}}}(AB)^{\pm}}=\phi|_{\hat{\mathcal{I}}^{\pm}}=\pm\frac{1}{2}\,,\,d\phi\in\mathfrak{j}^{+}\right\\},$
(221)

where ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)^{\pm}$ are infinitesimally to the
future and past of ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$.

We then define a V-flow on $\hat{\mathcal{M}}$ as a 1-form $V$ obeying

$d{*V}=0\,,\qquad*V|_{\hat{\mathcal{I}}}=0$ (222)
$\exists\,\phi\in\hat{\mathscr{S}}_{\rm c}\text{ s.t. }d\phi\pm
V\in\mathfrak{j}^{+}\,.$ (223)

Let $\hat{\mathscr{F}}$ be the set of such 1-forms.

A V-flow $V\in\mathscr{F}$ can be extended to one in $\hat{\mathscr{F}}$
simply by setting $V=0$ on $\hat{\mathcal{M}}\setminus\mathcal{M}$. The
resulting 1-form on $\hat{\mathcal{M}}$ is divergenceless and obeys
$*V|_{\hat{\mathcal{I}}^{\pm}}=0$ by virtue of the no-flux boundary condition
on $\mathcal{I}^{\pm}$. The witness function $\phi\in\mathscr{S}_{\rm c}$ can
similarly be extended to one in $\hat{\mathscr{S}}_{\rm c}$ by setting it
equal to $\pm 1/2$ on $J^{\pm}({\hat{\gamma}_{{}_{\text{HRT}}}}(AB))$ and to
any function obeying $d\phi\in\mathfrak{j}^{+}$ on the complementary
entanglement wedge $\hat{\mathcal{W}}((AB)^{c})$.

Conversely, any $V\in\hat{\mathscr{F}}$, when restricted to $\mathcal{M}$,
gives a $V\in\mathscr{F}$. To show this, we restrict
$\phi\in\hat{\mathscr{S}}_{\rm c}$ to $\mathcal{M}$ to obtain
$\phi\in\mathscr{S}_{\rm c}$, and similarly restrict $V$ to
$\bar{\mathcal{M}}$. The only thing that needs to be checked is the boundary
condition $*V|_{\mathcal{I}^{\pm}}=0$. As mentioned above, $\mathcal{I}^{\pm}$
is made up of $\hat{\mathcal{H}}(AB)$ together with (part or all of)
$\hat{\mathcal{I}}^{\pm}$. We already have the boundary condition on
$\hat{\mathcal{I}}^{\pm}$ from (222), and the boundary condition on
$\hat{\mathcal{H}}(AB)$ is enforced by the condition $d\phi\pm
V\in\mathfrak{j}^{+}$, given that $d\phi=0$ in
$J^{\pm}({\hat{\gamma}_{{}_{\text{HRT}}}}(AB))$.

These mappings between $\mathscr{F}$ and $\hat{\mathscr{F}}$ do not change the
flux on $D(A)$, so we have

$S_{{}_{\text{HRT}}}=\sup_{V\in\mathscr{F}}\
\int_{D(A)}*V=\sup_{V\in\hat{\mathscr{F}}}\ \int_{D(A)}*V\,.$ (224)

##### U-flows

Starting from the definition (79), (80) of a U-flow in the regulated
spacetime, we can also easily define a U-flow on $\hat{\mathcal{M}}$. First we
define the relaxed time-sheet set $\hat{\mathscr{T}}_{\rm c}$ as any function
$\psi:\hat{\mathcal{M}}\to[-1/2,1/2]$ that, when restricted to
$\bar{\mathcal{M}}$, is an element of $\mathscr{T}_{\rm c}$ (as defined in
(45)):

$\hat{\mathscr{T}}_{\rm
c}:=\bigg{\\{}\psi:\hat{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\psi|_{D(A)}=-\frac{1}{2}\,,\,\psi|_{D(B)}=\frac{1}{2}\right\\}.$
(225)

Imposing the condition $U\pm d\psi\in\mathfrak{j}^{+}$ along with the no-flux
condition on $D(A)\cup D(B)\cup\hat{\mathcal{I}}^{0}$ gives us the definition
of a U-flow:

$d{*U}=0\,,\qquad*U|_{D(A)}=*U|_{D(B)}=*U|_{\hat{\mathcal{I}}^{0}}=0$ (226)
$\exists\,\psi\in\hat{\mathscr{T}}_{\rm c}\text{ s.t. }U\pm
d\psi\in\mathfrak{j}^{+}\,.$ (227)

Let $\hat{\mathscr{G}}$ be the set of U-flows.

Notice that there is no boundary condition on $\psi$ or $U$ except on $D(A)$,
$D(B)$, and $\hat{\mathcal{I}}^{0}$. In particular, the flux of $U$ is free to
enter or leave the spacetime on the rest of the boundary. This means that the
total flux must be measured on a slice for $\mathcal{M}$, or on
$\mathcal{I}^{\pm}$, not on a slice for $\hat{\mathcal{M}}$.

Any U-flow on $\hat{\mathcal{M}}$, $U\in\hat{\mathscr{G}}$, can be restricted
to a U-flow on $\bar{\mathcal{M}}$. Conversely, a U-flow on
$\bar{\mathcal{M}}$, $U\in\mathscr{G}$, can be extended to one on
$\hat{\mathcal{M}}$. There are many ways to carry out this extension
consistent with the conditions $d*U=0$, $U\pm d\psi\in\mathfrak{j}^{+}$. Here
we will present one way that is relatively simple, although the resulting
1-form is singular on the boundary $\hat{\mathcal{H}}(AB)$ between
$\mathcal{M}$ and $\hat{\mathcal{M}}\setminus\bar{\mathcal{M}}$. First, we set
$\psi=0$ on $\hat{\mathcal{M}}\setminus\bar{\mathcal{M}}$. As a result, the
only condition on $U$ in that region is $U\in\mathfrak{j}^{+}$, which is
trivial to satisfy simply by extending the field lines from
$\bar{\mathcal{M}}$ along any set of non-intersecting causal curves. Call the
resulting 1-form on $\hat{\mathcal{M}}$ $U_{0}$. By construction, $U_{0}$ is
divergenceless and obeys $U_{0}\pm d\psi\in\mathfrak{j}^{+}$ everywhere,
except possibly on $\hat{\mathcal{H}}(AB)$. By setting $\psi=0$ in
$\hat{\mathcal{M}}\setminus\bar{\mathcal{M}}$, we have potentially introduced
a discontinuity in $\psi$ on $\hat{\mathcal{H}}(AB)$. Let $\lambda$ be a local
coordinate in a neighborhood of a point on $\hat{\mathcal{H}}(AB)$, such that
$\lambda=0$ on $\hat{\mathcal{H}}(AB)$; the discontinuity implies a delta-
function in $d\psi$ proportional to $\delta(\lambda)d\lambda$. Since, on
$\mathcal{M}$, $\psi\in[-1/2,1/2]$, the largest the discontinuity can be (in
absolute value) is $1/2$. So by adding the 1-form
$U_{1}:=\pm(1/2)\delta(\lambda)d\lambda$ (where the sign is determined by the
condition $U_{1}\in\mathfrak{j}^{+}$) to $U_{0}$, we are guaranteed to satisfy
the condition $(U_{0}+U_{1})\pm d\psi\in\mathfrak{j}^{+}$. $U_{1}$, however,
is not divergenceless, since the transverse area element is not constant along
the null generators of $\hat{\mathcal{H}}(AB)$, and those null generators
begin (on ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$) and end (on caustics and
crossover seams). Again, this can be fixed by adding causal field lines in
$\hat{\mathcal{M}}\setminus\bar{\mathcal{M}}$ to $U_{1}$. Finally, we set
$U=U_{0}+U_{1}$.

The important thing about the above construction is that the 1-form $U$ is
unchanged in $\mathcal{M}$ itself. Therefore its flux across any slice
$\sigma$ for $\mathcal{M}$ is unchanged. So we have

$S_{{}_{\text{HRT}}}=\inf_{U\in\mathscr{G}}\
\int_{\sigma}*U=\inf_{U\in\hat{\mathscr{G}}}\ \int_{\sigma}*U\,.$ (228)

###  7.2 Removing the regulator

The quantity $S_{{}_{\text{HRT}}}$ computed in (219), (224), (228) is
obviously most meaningful when it is finite, which is the case when either the
EE is finite (e.g. in a multiboundary wormhole) or we are computing a (finite)
EWCS. Nonetheless, it is possible to define a max V-flow and a min U-flow in
the absence of a regulator, and these can be used to compute naturally finite
quantities like the mutual information of separated boundary regions.

In the limit where $AB$ covers an entire boundary slice, in other words we
remove the regulating buffer separating them, the joint HRT surface
${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$ retreats to the boundary and becomes
the mutual entangling surface $\partial A=\partial B$. This entangling surface
partitions the conformal boundary into four regions: $D(A)$, $D(B)$,
$J^{\pm}(\partial A)$.

##### Minimax:

In this limit, the time-sheet set $\hat{\mathscr{T}}$ becomes the set of
piecewise-timelike hypersurfaces in $\hat{\mathcal{M}}$ homologous to $D(A)$
relative to
$\hat{\mathcal{I}}^{+}\cup\hat{\mathcal{I}}^{-}\cup\hat{\mathcal{I}}^{0}\cup
J^{+}(\partial A)\cup J^{-}(\partial A)$. Note that this homology condition
forces the time-sheet $\hat{\tau}$ to contain $\partial A$. Furthermore, in
order to ensure that the surface $\gamma\subset\hat{\tau}$ is contained in the
$AB$ entanglement wedge — i.e. is excluded from the _bulk_ chronal future and
past of $\partial A$ — we require it to include $\partial A$:

$S_{\rm
HRT}=\inf_{\hat{\tau}\in\hat{\mathscr{T}}}\sup_{\begin{subarray}{c}\partial
A\subset\gamma\subset\hat{\tau}\\\
\text{achronal}\end{subarray}}\operatorname{area}(\gamma)\,.$ (229)

Of course, for any $\hat{\tau}$, the area of $\gamma$ is unbounded above, so
the minimax is infinite.

##### V-flows:

In this limit, the boundary condition on $\phi$ becomes
$\phi|_{\hat{I}^{\pm}}=\phi|_{\partial A^{\pm}}=\pm 1/2$:

$\hat{\mathscr{S}}_{\rm
c}:=\bigg{\\{}\phi:\hat{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\phi|_{\hat{I}^{\pm}}=\phi|_{\partial
A^{\pm}}=\pm\frac{1}{2}\,,\,d\phi\in\mathfrak{j}^{+}\right\\}.$ (230)

In view of the condition $d\phi\in\mathfrak{j}^{+}$, the condition
$\phi|_{\partial A^{\pm}}=\pm 1/2$ is equivalent to $\phi|_{J^{\pm}(\partial
A)}=\pm 1/2$. The rest of the V-flow definition (222), (223) remains
unchanged.

For multiple boundary regions, the entangling surface $\partial A$ in (230) is
replaced by the union of all the entangling surfaces. Thus, as in the
regulated spacetime, the definition of a V-flow is uniform for all the regions
and groupings thereof.

Despite the fact that $S_{{}_{\text{HRT}}}$, and therefore the maximal flux of
$V$, is infinite, it is nonetheless possible to define the notion of a _max
V-flow_. We say that a 1-form $\Delta V$ is an _augmentation_ of the V-flow
$V$ if it has positive flux, $\int_{D(A)}*\Delta V>0$, and $V+\Delta V$ is a
V-flow. $V$ is a max V-flow if it does not admit an augmentation. Clearly when
the flux is finite this definition agrees with the usual one — a V-flow with
maximal flux — but it also makes sense when the flux is infinite. We can use
it, for example to define the mutual information between separated regions,
without ever introducing a regulator at an intermediate step; see
Freedman:2016zud .

##### U-flows:

The U-flow definition (225), (226), (227) is unchanged in this limit. We can
measure the flux of $U$ through any slice $\sigma$ for $\hat{\mathcal{M}}$