Also at ]Rotman Institute of Philosophy, University of Western Ontario,
Canada.

# Curvature coupling, electromagnetic wave propagation, and the consistency of
the geometrical optics limit

Niels Linnemann<EMAIL_ADDRESS>University of Bremen
28359 Bremen, Germany  [  James Read<EMAIL_ADDRESS>Pembroke
College, University of Oxford
Oxford OX1 1DW, United Kingdom

###### Abstract

We study the propagation of Maxwellian electromagnetic waves in curved
spacetimes in terms of the appropriate geometrical optics limit, notions of
signal speed, and minimal coupling prescription from Maxwellian theory in flat
spacetime. In the course of this, we counter a recent major claim by Asenjo
and Hojman (2017) to the effect that the geometrical optics limit is partly
ill-defined in Gödel spacetime; we thereby dissolve the present tension
concerning established results on wave propagation and the optical limit.

##  I Introduction

The conventional wisdom that light in relativistic spacetimes propagates on
null geodesics can be substantiated in several ways. First: via the
observation that light, in the geometrical optics limit, propagates on null
geodesics (MTW, , p. 571). Second: via the recent technical result that small
bodies constructed from Maxwell fields in curved spacetimes ‘track’ null
geodesics GW . Third: via the core finding that solutions to the Maxwellian
wave equation in curved spacetimes have an idealised signal speed of exactly
$c$ Friedlander . A less well-known aspect of our understanding of the
propagation of electromagnetic waves in relativistic spacetimes is that this
propagation is associated with a ‘tail’, which moves behind the wavefront at a
speed less than $c$ DB ; Faraoni . Insofar as this tail becomes more
pronounced with an increase in gravitational field strength EllisSciama , tail
production can effectively retard the wavefront to a subluminal speed from a
metrological point of view: a detector only registers an incoming wave above a
certain threshold; if the power transmission linked to the tail becomes
significantly more relevant than that of the actual wavefront, the detector
will fire later than otherwise expected.

The recent works AH ; AH2  seek to further develop our understanding of the
propagation of light in curved spacetimes. In AH2 , it is shown that, before
taking the geometrical optics limit, light rays in generic spacetimes do not
propagate on null geodesics, in the sense that the dispersion relation
$K^{\mu}K_{\mu}\neq 0$, where $K^{\mu}$ is the wave vector. In AH , it is
argued in addition that at least in the Gödel solution, one cannot
consistently take the geometrical optics limit (in which, as already mentioned
above, the null propagation of light obtains).

These papers certainly constitute advances in our understanding of the
propagation of light in curved spacetimes. However, there remains much to be
done in analysing their results. This article seeks to clarify comprehensively
both the sense in which the claims made in AH ; AH2  are correct, and can be
bolstered, as well as the sense in which the claims made in those articles are
incorrect, and should be rejected. More specifically, in this paper we achieve
the following tasks: (A) Numerically solving the differential equations
describing the propagation of light in curved spacetimes presented in AH ; AH2
, in order to secure a better quantitative grasp of how the dispersion
relation (and thereby also the phase and group velocity) of such waves can
differ from that associated with null propagation (in this sense, we concur
with AH ; AH2 , and take our results to corroborate their claims). (B)
Understanding how different curvature couplings in matter dynamics can affect
the propagation of light, and thereby lead to distinct physics (again, in this
sense our results are consistent with, and develop further, AH ; AH2 ). (C)
Clarifying exactly when one can and cannot take the geometrical optics limit:
in this regard, we find the results presented in AH  to rest on a questionable
limit procedure; we instead side with the orthodoxy in concluding that the
geometrical optics limit can invariably be taken consistently, and moreover
that both group velocity and the causally relevant signalling speed are $c$ in
this limit. (D) Assessing the implications of the above work for the
possibility of super- or subluminal wavefront velocities, bearing in mind
classic work such as Brillouin , and associated more recent discussions (this
discussion is important, because explicit consideration of such different
notions of wave velocities is absent from AH ; AH2 ).

Overall, then, our purpose is to provide an exhaustive study of the claims
made in AH ; AH2  regarding the propagation of light in curved spacetimes. The
structure of the paper is as follows. In §III, we present our numerical
results which bolster the aforesaid claims regarding the propagation of light
outside of the geometrical optics regime. In §IV, we discuss the correct way
to take the geometrical optics limit in curved spacetimes. In §V, we consider
how different notions of wave velocities temper the claims made in AH ; AH2
regarding the possibility of superluminal propagation. Before turning to this
work in earnest, however, we must first recall some basic details regarding
Maxwell’s equations in curved spacetimes.

##  II Curvature-coupled Maxwell equations

The inhomogeneous Maxwell equation for the vector potential in Minkowski
spacetime reads

$-A\indices{{}^{\alpha,\mu}_{\mu}}+A\indices{{}^{\mu}{}^{,\alpha}_{\mu}}=4\pi
J^{\alpha}.$ (1)

On minimal coupling (i.e., applying the ‘comma-to-semicolon’ rule), one
obtains

$-A\indices{{}^{\alpha;\mu}_{\mu}}+A\indices{{}^{\mu}{}^{;\alpha}_{\mu}}=4\pi
J^{\alpha}.$ (2)

As has been well-appreciated since (at least) the elegant discussion of Misner
_et al._ (MTW, , §16.2), there arises in the implementation of the minimal
coupling procedure the possibility of factor-ordering ambiguities. Note, in
particular, that if (1) is written with its partial derivatives exchanged,
then one obtains the following curved-spacetime equation, on minimal coupling:

$-A\indices{{}^{\alpha;\mu}_{\mu}}+A\indices{{}^{\mu}{}^{;\alpha}_{\mu}}+R\indices{{}^{\alpha}_{\mu}}A^{\mu}=4\pi
J^{\alpha}.$ (3)

Typically, (2) is treated as the correct wave equation for $A^{\mu}$ in curved
spacetime; (3) is rejected. See, for example, (MTW, , p. 390), where it is
stated that “Coupling to curvature surely cannot occur without some physical
reason.” In this way, minimal coupling is regarded as a (fallible) heuristic
for generating equations of motion for matter fields in a curved spacetime
which at a point have the same form as their special relativistic counterparts
(and in particular do not feature explicit curvature terms): that is, a
fallible heuristic for implementing a form of the ‘strong equivalence
principle’ (SEP). As minimal coupling fails to implement unambiguously this
SEP with respect to (1) (it yields both (2) and (3)), the desired form of the
general relativistic equations—namely, (2)—has to be selected by fiat. Another
(again only) heuristic reason to reject (3) is its lack of gauge invariance
(there is no _a priori_ reason why Maxwell equations in curved spacetime have
to be gauge invariant). (Our thanks to Jacob Barandes for discussion on this
point.)

(An aside: Consider ‘standard’ minimal coupling, in which one obtains (2).
Even here, one must take care with claims that these equations are ‘locally
special relativistic’, as per the SEP. After all, at least in the gauge
$A\indices{{}^{\mu}_{;\mu}}=0$, (2) is equivalent, via

$R\indices{{}^{\alpha}_{\mu}}A^{\mu}=A\indices{{}^{\mu}_{;\mu}{}^{\alpha}}-A\indices{{}^{\mu}{}^{;\alpha}_{\mu}},$
(4)

to

$-A\indices{{}^{\alpha;\mu}_{\mu}}-R\indices{{}^{\alpha}_{\mu}}A^{\mu}=4\pi
J^{\alpha}.$ (5)

The curvature term in this latter presentation of (2) will, again, not vanish
at a point.)

In any case, (3) is a distinct equation from (2), and there is no reason to
expect _ab initio_ that the equations will make the same empirical
predictions; moreover, there is no logical bar to the correct general
relativistic versions of Maxwell’s equations being (3). Ultimately, only
experiment can adjudicate between these different possibilities. Thus, it is
incumbent on us to appreciate the empirical consequences of each. In this
article, we show that the differences between (2) and (3) manifest themselves
in the velocity of the propagation of light, at least outside of the
geometrical optics regime; in this sense, we develop further the claims made
in AH  in this regard (see §III).

There is one other point which is worth making here. In addition to the
factor-ordering ambiguities associated with minimal coupling, other authors
have motivated in other ways consideration of the relations between curvature
couplings and wave propagation. For example, Drummond and Hathrell consider
one-loop vacuum polarisation contribution to the QED effective action and the
additional terms in the resulting equations of motion derived therefrom DH .
The results which we present in this article are consistent with the
motivations of these investigations, insofar as one of our central interests
is also the effects that such additional couplings can have upon wave
propagation. For recent discussion of such work, see Shore ; Butterfield .
(Note that the putative superluminal signalling in the context of the Drummon-
Hathrell action has been called into question for making use of questionable
approximations (Shore, ). Interestingly, we will express a very similar
concern towards the claimed failure of the geometrical optics limit in Gödel
spacetime in AH .)

##  III Solving the Curvature-Coupled Equations

With this background in hand, we turn now to a quantitative discussion of the
propagation of Maxwellian waves in curved spacetimes; our intention is to
build upon the discussion of AH ; AH2  that this propagation need not
invariably be null.

The source-free version of (2) can be written in terms of partial (rather than
covariant) derivatives as

$\partial_{\alpha}\left[\sqrt{-g}g^{\alpha\mu}g^{\beta\nu}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\right]=0.$
(6)

By making a plane wave ansatz in the $\nabla_{\mu}A^{\mu}=0$ gauge, i.e.
$A_{\mu}=\xi_{\mu}(x^{\lambda})e^{iS(x^{\lambda})}$ where $\xi_{\mu}$ and $S$
are not necessarily constant and represent, respectively, the amplitude and
phase of the wave, it is shown in AH2  that the wave ‘vector’
$K_{\mu}:=\partial_{\mu}S$ obeys $K^{\mu}K_{\mu}=\chi$, where
($\tilde{\xi}:=\sqrt{\xi_{\mu}\xi^{\mu}}$)

$\chi=\frac{\xi_{\beta}}{\sqrt{-g}\tilde{\xi}^{2}}\partial_{\alpha}\left[\sqrt{-g}g^{\alpha\mu}g^{\beta\nu}\left(\partial_{\mu}\xi_{\nu}-\partial_{\nu}\xi_{\mu}\right)\right].$
(7)

(A general solution to the homogeneous, standard minimally coupled Maxwell
wave equation in curved spacetime was given in DB  in the context of the
bitensor formalism.) From this, one might already infer that in generic
spacetimes and before taking the geometrical optics limit, light does not
propagate on null geodesics: as pointed out in AH2 , (7) cannot in general be
solved consistently using a constant amplitude ansatz; this in turn, however,
implies that the dispersion relation for the wave is non-trivial. (That a non-
constant amplitude leads to a non-trivial dispersion relation holds even in
flat spacetime.)

One can generalise these results by writing the source-free version of (3) as

$\partial_{\alpha}\left[\sqrt{-g}g^{\alpha\mu}g^{\beta\nu}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\right]+zR\indices{{}^{\beta}_{\mu}}A^{\mu}=0;$
(8)

here, we have included a scalar parameter $z$, allowing us to ‘tune’ the
curvature-coupled term, the possibility of which arises due, for example, to
the factor-ordering ambiguities in the minimal coupling scheme, as discussed
in §II.

As we will now argue, including such a curvature-coupled term leads to a
further change in the dispersion relation. For this, we seek to solve (8) in
particular spacetimes. Selecting the FLRW spacetime (as also considered by AH2
in the case of $z=0$) and using spherical coordinates $(t,r,\phi,\theta)$, the
non-trivial components of the metric are

$\displaystyle g_{tt}$ $\displaystyle=-1,$ (9) $\displaystyle g_{rr}$
$\displaystyle=\frac{R^{2}(t)}{1-kr^{2}},$ (10) $\displaystyle g_{\phi\phi}$
$\displaystyle=R^{2}(t)r^{2}\sin^{2}(\theta),$ (11) $\displaystyle
g_{\theta\theta}$ $\displaystyle=R^{2}(t)r^{2}.$ (12)

Then, making the ansatz $A_{\mu}=A_{\phi}(r,t)\delta_{\mu\phi}$ (which is
slightly more restrictive than that used by AH2  in their treatment of the
$z=0$ case), (8) becomes

$\partial_{0}\left(\sqrt{-g}g^{tt}g^{\phi\phi}\partial_{0}A_{\phi}\right)\\\
+\partial_{r}\left(\sqrt{-g}g^{rr}g^{\phi\phi}\partial_{r}A_{\phi}\right)+zR^{\phi\phi}A_{\phi}=0.$
(13)

Recall from (Catalogue, , p. 38) that the $R_{\phi\phi}$ component of the
Ricci tensor is given by

$R_{\phi\phi}=r^{2}\sin^{2}(\theta)\left(R\ddot{R}+2(\dot{R}^{2}+k)\right).$
(14)

The square root of the metric determinant is given by

$\sqrt{-g}=R^{3}(t)r^{2}\frac{\sin{\theta}}{\sqrt{1-kr^{2}}}.$ (15)

For simplicity, we set $R=1$ from this point on (for instance, the constancy
of $R$ is typically assumed for the current cosmological era). Thereby,
raising indices using the metric,

$R^{\phi\phi}=\frac{R\ddot{R}+2(\dot{R}^{2}+k)}{R^{4}r^{2}\sin^{2}(\theta)}=\frac{2k}{r^{2}\sin^{2}(\theta)}.$
(16)

Upon further specifying the ansatz to be

$A_{\phi}(r,t)=\xi(r)\exp\left(iS(t)+iS(r)\right)$ (17)

with wave vector $K_{\mu}$ such that $K_{t}:=\partial_{t}S(t)$ and
$K_{i}:=\partial_{i}S(r)$, one finds that $K_{r}$, i.e. the (non-trivial
spatial) $r$-component of the wave vector, and $\xi(r)$ are governed by

$\partial_{t}\left(\sqrt{-g}g^{tt}g^{\phi\phi}\right)\left(\partial_{t}\xi+\xi
K_{t}i\right)+\left(\sqrt{-g}g^{tt}g^{\phi\phi}\right)\left(\partial_{t}^{2}\xi+2\left(\partial_{t}\xi\right)K_{t}i-\xi
K_{t}^{2}+\xi\left(\partial_{t}K_{t}\right)i\right)+\\\
\partial_{r}\left(\sqrt{-g}g^{rr}g^{\phi\phi}\right)\left(\partial_{r}\xi+\xi
K_{r}i\right)+\left(\sqrt{-g}g^{rr}g^{\phi\phi}\right)\left(\partial_{r}^{2}\xi+2\left(\partial_{r}\xi\right)K_{r}i-\xi
K_{r}^{2}+\xi\left(\partial_{r}K_{r}\right)i\right)+zR^{\phi\phi}\xi=0.$ (18)

Exploiting that $\partial_{t}(\sqrt{-g}g^{tt}g^{\phi\phi})=0$ for $R=1$, that
$K_{t}$ is constant, and that $\partial_{t}\xi(r)=0$, as well as splitting up
the real and imaginary parts of (18), yields two equations. First, the
imaginary part:

$\partial_{r}\left(\sqrt{-g}g^{rr}g^{\phi\phi}\right)K_{r}\xi+\\\
\sqrt{-g}g^{rr}g^{\phi\phi}\left(2\left(\partial_{r}\xi\right)K_{r}+\xi\left(\partial_{r}K_{r}\right)\right)=0$
(19)

Upon multiplication by $\xi$, one obtains
$\partial_{r}(\sqrt{-g}g^{rr}K_{r}\xi^{2})=0$; this is solved by

$\xi=\frac{a}{{\left(\sqrt{-g}g^{rr}g^{\phi\phi}K_{r}\right)}^{1/2}},$ (20)

where $a$ is a constant.

For the real part of (18), we have

$\sqrt{-g}g^{tt}g^{\phi\phi}\left(\partial_{t}^{2}\xi-\xi
K_{t}^{2}\right)+\partial_{r}\left(\sqrt{-g}g^{rr}g^{\phi\phi}\right)\partial_{r}\xi\\\
+\sqrt{-g}g^{rr}g^{\phi\phi}\partial_{r}^{2}\xi-\sqrt{-g}g^{rr}g^{\phi\phi}\xi
K_{r}^{2}+zR^{\phi\phi}\xi=0.$ (21)

Dividing by $\sqrt{-g}g^{\phi\phi}\xi$, one obtains

$\displaystyle K^{\mu}K_{\mu}$ $\displaystyle=g^{tt}K_{t}^{2}+g^{rr}K_{r}^{2}$
$\displaystyle=\frac{zR^{\phi\phi}}{\sqrt{-g}g^{\phi\phi}}+g^{rr}\frac{\partial_{r}^{2}\xi}{\xi}+\frac{\partial_{r}(\sqrt{-g}g^{rr}g^{\phi\phi})}{\sqrt{-g}g^{\phi\phi}}\frac{\partial_{r}\xi}{\xi},$
(22)

i.e. the dispersion relation for the wave. At this stage, note that for the
specific constant amplitude ansatz, i.e. $\xi(r)=\text{const}$, the wave
follows null geodesics for $z=0$ in the sense that the dispersion relation
vanishes (in particular, the group velocity becomes unity—this is shown in AH
), but also that in the case of FLRW spacetimes there is no consistent
solution for $K_{r}$ for $z\neq 0$. (In generic spacetimes, the specific
constant amplitude ansatz does not even lead to solutions for $z=0$, as
already mentioned above, and as pointed out in AH2 .)

We will, however, use the solution for $\xi=\xi(r)\neq\text{const}$ obtained
in terms of $K_{r}$ as given by (20); plugging into the dispersion relation
(III) gives

$-K_{t}^{2}+\left(1-kr^{2}\right)K_{r}^{2}=\frac{\left(kr^{2}-1\right)\partial^{2}_{r}K_{r}}{2K_{r}}-\frac{3\left(kr^{2}-1\right)(\partial_{r}K_{r})^{2}}{4K_{r}^{2}}-k\frac{2+r^{2}}{4\left(kr^{2}-1\right)}+zk\frac{8k^{2}r^{4}-16kr^{2}+8}{4r^{2}\left(1-kr^{2}\right)^{3/2}},$
(23)

where $z=0$ if the additional curvature coupling term is neglected. Note that
even for $z=0$, the wave exhibits non-trivial dispersion in these coordinates
(we discuss further the possibility of coordinate effects below), as long as
the amplitude is non-constant. (This is compatible with the finding by AH2
that in the $z=0$ but constant amplitude case the dispersion relation is
trivial again.)

Restricting to $S(t)=\omega t$ where $\omega$ is a constant, we now solve
numerically (using Mathematica) (23) in the domain $r\in\left[0.1,0.9\right]$
(in order to avoid issues of coordinate singularities) with $k=1$, subject to
the initial conditions $K_{r}\left(0.1\right)=\omega$ and
$K^{\prime}_{r}\left(0.1\right)=1$. One could equally well choose
$K^{\prime}_{r}\left(0.1\right)=0$, even though this seems to suggest that the
wave-vector is position-independent and that a trivial dispersion-relation
imposed at one point (as secured via $K_{r}(0.1)=\omega$) will thereby remain
trivial everywhere. This impression, however, is misleading insofar as we are
still considering at this point the wave in $\left(t,r\right)$ coordinates,
rather than in the $\left(\tau,\rho\right)$ coordinates to be discussed below.
(Only setting the analogous initial condition in terms of $\rho$ to $0$, i.e.
requiring $\partial K_{\rho}/\partial\rho=0$ would impose position-
independence of the dispersion relation in a physical sense.) Then, inputting
a fixed $r$ in the above domain, one can extract functional relationships
between $K_{r}$ and $\omega$ at different values of $z$. Choosing, for
instance, $r=0.2$, one finds the dependencies presented in figure 1.

__

Figure 1: $\omega=\frac{\partial S(t)}{\partial t}=K_{t}$ as a function of
$K_{r}$ for a transverse electromagnetic wave in FLRW spacetime ($k=1$).
($K_{r}$ is obtained numerically from (23) with $K_{r}(0.1)=\omega$,
$K_{r}^{\prime}(0.1)=1$, giving rise to the functional relationship between
$K_{r}$ and $\omega$.) Different values of $z$ correspond to the strength of
the additional curvature coupling.

From these results, one finds that although even for $z=0$ it is not the case
that the group velocity $\partial\omega/\partial K_{r}$ is exactly unity
(where we normalise $c=1$), dispersion effects increase with $z$. Choosing
greater $r$ exacerbates the dispersion effects, as one would expect, given the
greater radial distance from the point at which the above initial conditions
are imposed. From the above results, one also sees clearly that the phase
velocity $\omega/K_{r}$ of the wave is not constant. Finally: if one repeats
the calculations of this sections in other spacetimes, such as Gödel, one
finds analogous results.

Now, strictly speaking, in order to avoid being misled by coordinate effects
when making these judgements on the dispersion relation, one should switch to
coordinates that bring the wave equation into a form locally adapted to
Minkowski spacetime. Generally, for a spherically symmetric metric given as
(AH, , p. 4)

$\displaystyle g_{tt}$ $\displaystyle=f(t)q(r),$ (24) $\displaystyle g_{rr}$
$\displaystyle=h(t)b(r),$ (25) $\displaystyle g_{\theta\theta}$
$\displaystyle=h(t)r^{2},$ (26) $\displaystyle g_{\phi\phi}$
$\displaystyle=h(t)r^{2}\sin^{2}(\theta),$ (27)

the time coordinate $t$ and spatial coordinate $r$ need to be changed,
respectively, to (AH, , p. 5)

$\displaystyle\tau$ $\displaystyle:=\int dt\sqrt{-\frac{f}{h}},$ (28)
$\displaystyle\rho$ $\displaystyle:=\int dr\sqrt{\frac{b}{q}}.$ (29)

Thus, for FLRW spacetime with $R(t)=1$ and $k>0$,

$\displaystyle\tau$ $\displaystyle=t,$ (30) $\displaystyle\rho$
$\displaystyle=\int
dr\sqrt{\frac{1}{1-kr^{2}}}=\frac{1}{\sqrt{k}\sin{(\sqrt{k}r)}},$ (31)

setting all integration constants to zero. Inverting (31), one obtains

$r=\frac{1}{\sqrt{k}}\arcsin{\left(\frac{1}{\sqrt{k}\rho}\right)}.$ (32)

This being said, in the above we work in $\left(t,r\right)$ coordinates, for
two reasons. First, the form of the coordinate transformation between $r$ and
$\rho$, i.e. (32), leads to oscillatory functions in the transformed version
of (23); and unlike the $\left(t,r\right)$ coordinates, the coordinates
locally adapted to Minkowski spacetime can only be used to cover FLRW
spacetime for relatively small regions as the transformation rule from $r$ to
$\rho$ is periodic and thus non-monotonous. Second, given the particular form
of (32), one can be assured that tuning $z$ to different values will lead to
different physical effects even in the $\left(\tau,\rho\right)$ coordinates,
as the $z$-dependent term of (23) is non-constant in $\rho$ and not merely in
$r$ (this follows from plugging (32) into the $z$-dependent term). Note
finally that transforming to $\left(\tau,\rho\right)$ coordinates implies that
propagating electromagnetic waves can have dispersion even in familiar cases
of e.g. Schwarzschild spacetimes (and even having set $z=0$; see (AH, , §2)).

Thus, the above results corroborate the claims made in AH ; AH2  that, outside
of the geometrical optics regime, light does not invariably propagate on null
geodesics in general relativistic universes; they also make clear the tangible
physical effects of including extra curvature couplings in Maxwell’s equations
that potentially arise due to the ambiguities of the minimal coupling scheme
(and thus prior to any semi-classical corrections from QED, as for instance
accounted for by the Drummond-Hathrell action, or other sorts of matter
couplings Hertzberg ). There are, however, certain other claims made in AH
which are questionable: these regard the consistency of the geometrical optics
limit. It is to these issues that we now turn; ultimately, we will reject
these claims—this constitutes the negative part of our discussion of AH ; AH2
.

##  IV Spacetimes with inconsistent geometrical optics limits?

In AH , the authors suggest that certain wave solutions in Gödel spacetime do
not have a well-defined geometrical optics limit; that is, no exact wave
solutions are consistent with $K^{\mu}K_{\mu}=0$. They claim that there is
therefore no sense in which these electromagnetic waves can be said to move on
null geodesics at high frequencies.

These claims deserve greater scrutiny. To this end, recall the reasoning of AH
in more detail. For Gödel spacetime, the dispersion relation following from
(6) is

$-\omega^{2}+K_{x}^{2}=-\frac{\Omega^{2}}{2}+\frac{3K^{\prime
2}_{x}}{4K_{x}^{2}}-\frac{K^{\prime\prime}_{x}}{2K_{x}},$ (33)

where ${}^{\prime}\equiv\partial_{x}$. In AH , the geometrical optics limit is
referred to as the ‘high-frequency limit’, which one can take to be the limit
$\omega\rightarrow\infty$; it is then claimed that the limit is successfully
taken only if $K^{\mu}K_{\mu}\rightarrow 0$. Note, though, that the R.H.S. of
(33) is unequal to zero under $\omega\rightarrow\infty$ and is therefore
inconsistent with this requirement. The reason for this is not only due to the
presence of the $\Omega\neq 0$ term—as indicated in AH —but also due to the
presence of the terms in $K^{\prime}_{x}$ and $K^{\prime\prime}_{x}$, neither
of which need vanish when $\omega\rightarrow\infty$. To be fair to AH , these
latter two terms do vanish when amplitude variations are taken to be
negligible (a feature of the geometrical optics limit as discussed at (MTW, ,
§22.5) and cited in this context by AH ), but it is worth recognising that
this does not follow from the condition $\omega\rightarrow\infty$ alone.
Regardless of whether one takes this into account, however, the point made in
AH —that terms on the R.H.S. of (33) spoil the consistency of this limit—still
stands. This, indeed, is the origin of the claim in AH  that the geometrical
optics limit is inconsistent in such spacetimes.

Is it, however, correct to understand the geometrical optics regime in terms
of a high frequency (and potentially also constant amplitude) limit? Recall
from (MTW, , p. 571) that in the geometrical optics limit (traditionally
construed) one also requires that wavelength be much less than the
characteristic scale of curvature variations (in addition to taking amplitude
variations to be negligible—an assumption which we will continue to make in
this paragraph). Now, in Gödel spacetime, one has that
$G_{\mu\nu}u^{\mu}u^{\nu}=\Omega^{2}$, where $G_{\mu\nu}$ is the Einstein
tensor and $u^{\mu}$ is some normalised four-velocity vector (HE, , p. 168);
thus, we see that $\Omega$ is directly related to curvature effects, and so,
in the geometrical optics limit, it makes sense on physical grounds to impose
two conditions: $\omega\rightarrow\infty$ and $\Omega\rightarrow 0$;
alternatively—but not equivalently—one could take the limit
$\Omega/\omega\rightarrow 0$. (Whether these two approaches to the geometrical
optics limit are indeed equivalent depends upon the convergent series which
one selects in the latter: our thanks to Sam Fletcher for pointing this out to
us.) In the remainder of this section, we will for concreteness focus on the
limit $\Omega/\omega\rightarrow 0$. Taking this limit, one finds from (33)
that the geometrical optics limit can be taken consistently, and that light
does propagate on null geodesics in this limit. (An aside: taking a limit of
this kind, involving a comparison of scales, seems to us a promising way of
understanding the content of the equivalence principle as it is used in
physical practice—see e.g. Will .)

One might take the general point here to be this: a limiting procedure on a
physical quantity can be regarded as being physical only if formulated in
relation to some other physical quantity. (Compare, for example, the by-now
suspect $c\rightarrow\infty$ Newtonian limit in special relativity with the
better-accepted $v/c\rightarrow 0$ limit in this theory.) On these grounds,
one might claim that the $\omega\rightarrow\infty$ limit deployed in AH  is
not physically justified. While roughly correct, this way of putting things
remains a little too quick, for one can now ask: if, when considering some
particular limit, there is a sequence of models shown to converge in some
well-defined sense, and all of the models are interpreted (i.e., represent
some physical possibility), what more could be needed in declaring the limit
to be ‘physical’? Such certainly seems to be the case for the
$\omega\rightarrow\infty$ limit taken in AH , in which case it is not clear
that ‘physicality’ is the most appropriate way in which to couch the issue.
Indeed, we do not deny that studying the $\omega\rightarrow\infty$ limit can
advance our understanding of the behaviour of light in curved spacetimes:
there is nothing _per se_ wrong with the study of high frequencies in such
spacetimes. Rather, the issue with the limit taken in AH , as we see it, is
that it does not represent what the authors purport it to represent: in
particular, whether ‘physical’ or not, and whether physically illuminating in
certain respects or not, it nevertheless does not represent the geometrical
optics limit as traditionally understood. (Again, our thanks to Sam Fletcher
for discussions on these matters: see Fletcher2  for related discussions of
limiting relations in spacetime theories.)

For maximal clarity, one might introduce a parameter tracking the frequency
orders, as done in (MTW, , §22.5). It will then be clear that only terms in
the equations of motion involving at least two derivatives in the vector
potential term $A^{\mu}$ can, upon a wave ansatz, contribute at the relevant
order to the dispersion relation; at the frequency order two, we have
$K^{\mu}K_{\mu}=0$.

The following observation is also worthy of mention. Consider (23) in FLRW
spacetimes. As long as $k\neq 0$ (as discussed before, non-constant amplitude
waves are solutions in flat FLRW—they do not arise only in the geometrical-
optical limit approximation), the R.H.S. is unequal to zero under
$\omega\rightarrow\infty$ independently of the chosen curvature coupling
(i.e., value of $z$) for Maxwell’s equations. Therefore, non-flat FLRW
spacetime—which, of course, builds the modelling foundation for the standard
model of cosmology—would represent another physical example of there being
exact wave solutions in spacetimes which (i) do not propagate on null
geodesics, and which (ii) do not have a well-defined geometrical optics limit.
But again, our same reservations as above apply: if one takes what is arguably
the correct geometrical optics limit in this case, which here would
effectively involve taking $k\rightarrow 0$ (or $k/\omega\rightarrow 0$), one
recovers both the consistency of the limit and the null propagation of light.

In AH , approximations are invoked in order to derive from (33) the following
results for the phase and group velocities of light in Gödel spacetime in the
case $\Omega x\ll 1$:

$\displaystyle v_{\text{ph}}$ $\displaystyle=\frac{\omega}{K_{x}}\approx
1+\frac{1}{2}\Omega^{2}x^{2},$ (34) $\displaystyle v_{\text{gp}}$
$\displaystyle=\frac{\partial\omega}{\partial K_{x}}\approx
1+\frac{1}{2}\Omega^{2}x^{2}.$ (35)

This seems to corroborate the claim that, in the $\omega\rightarrow\infty$
limit, light still does not propagate on null geodesics. However, there are
two points to note here. First: if one also takes $\Omega\rightarrow 0$, then
null propagation is retained (the authors of AH  note this in the context of
flat spacetimes, but not in the context of the understanding of the
geometrical optics limit discussed above). Second: one might wonder whether
(34) and (35) are artefacts of truncating a series too early. Continuing the
expansion in (34) to higher order, one finds that

$K_{x}=\omega\left(1-\frac{x^{2}\Omega^{2}}{2}+\frac{x^{4}\Omega^{4}}{6}-\frac{17x^{6}\Omega^{6}}{360}+\cdots\right)+\omega^{3}\left(\frac{x^{4}\Omega^{2}}{6}-\frac{13x^{6}\Omega^{4}}{90}+\frac{8x^{8}\Omega^{6}}{105}-\frac{899x^{10}\Omega^{8}}{28350}+\cdots\right)\\\
+\omega^{5}\left(-\frac{x^{6}\Omega^{2}}{45}+\frac{x^{8}\Omega^{4}}{21}-\frac{1699x^{10}\Omega^{6}}{37800}+\frac{108593x^{12}\Omega^{8}}{3742200}-\cdots\right)+\cdots$
(36)

Noting the contribution of the additional terms in (36), it is not at all
clear that (34) and (35) are the final story on the phase and group velocities
of light in Gödel spacetime in the case $\Omega x\ll 1$.

To summarise: in §III we fortified the claims made in AH ; AH2  that light,
outside of the geometrical optics limit, need not propagate on null geodesics.
However, in this section we have called into question the claims made in those
articles that this limit is inconsistent in spacetimes such as Gödel.

##  V Propagation speed in curved spacetime

If one accepts that light need not propagate along null geodesics when
properly modelled by solving Maxwell’s equations in curved spacetimes, then
one decouples light propagation from null cone structure. In Hojman , Hojman
accordingly takes the above-discussed results regarding the varying group
velocity of light in curved spacetimes as motivation for redefining the proper
time interval: the proper time interval is to reflect the actual interaction
of light with the gravitational field (as represented by the background
metric) instead of just the idealised interaction of light with the
gravitational field in the geometrical optics limit (in which light indeed
moves on null geodesics). In response to this, however, one might argue that
it is not clear that null cone structure is most sensibly tied to light
structure to begin with: rather, the proper time interval might better be
understood through the major conceptual roles of encoding causal structure,
and quantifying distance between path-connectible points alone.

More significantly: the causal significance of the proper time interval is not
necessarily endangered by the mere finding that there are superluminal group
velocities—as for instance observed from the dispersion relations above for
Gödel and FLRW spacetimes—since superluminal propagation is not linked
straightforwardly to superluminal _signalling_. After all, signal propagation
speed (the propagation relevant for causal considerations) is bounded by the
front velocity which, for a general set of PDEs (Shore, , p. 15), is given by
(LIM, ; Shore, )

$v_{\text{wf}}=\lim_{\omega\rightarrow\infty}\frac{\omega}{|\vec{k}|}=\lim_{\omega\rightarrow\infty}v_{\text{ph}}\left(\omega\right).$
(37)

In fact, the front velocity can be shown to be unity in curved spacetime for a
wide class of second-order tensorial PDEs, including (2) and (3) (Friedlander,
, §3.2), in which case the upper bound to signalling speed indeed seems to be
unity, as argued in the classic work of Brillouin  (see Milonni  for further
discussion). That, of course, does not mean that this bound is always
saturated. In fact, it is generally accepted that in normal dispersive media,
such as water, causal influences propagate at velocities less than the front
velocity (cf. CausalLoop ); furthermore, it has been shown in the seminal work
Nature  that in anomalous dispersive media causal influences indeed propagate
at speeds slower than unity (despite a faster-than-unity group velocity).

Similarly, then, the instantaneous signal speed of an electromagnetic wave in
curved spacetime—the curvature of spacetime mimics across different
frequencies and curvature scales the effects of both normal and anomalous
dispersive media—might be seen generally to deviate downwards from unity.
(Such an effect can also be argued for on the basis of the tail problem, as
discussed above.) The precise amount of downwards deviation will be a function
of the curvature length scale relative to frequency at which the wave is
considered.

Now, upon taking the idealising geometrical optics limit, the causal structure
will still be given by the light cone structure as long as at most corrections
to the usual Maxwell equations (2) (or, for what it is worth, (3)) are added
that do not involve terms of equal to or higher order in $(\partial A)^{2}$ or
$\partial\partial A$ (Hertzberg, , §3.1). When terms of this form are added to
the Maxwellian equations of motion (as for instance the case when taking into
account semi-classical corrections from QED), the geometrical optical limit
generally results in a non-zero dispersion relation (in the case of the QED
corrections this is the already-discussed Dummond-Hathrell effect). The impact
of the aforementioned terms on the dispersion relation in the geometrical
optics limit is easy to understand from the fact that any derivative on
$A^{\mu}$ amounts, upon plugging in the wave ansatz, to pulling down a
frequency factor $\omega$. Thus, terms of the described form are of order
$\omega^{2}$ and cannot be neglected upon the geometrical-optical limit: see
Hertzberg  for a discussion.

This being said, even in the more general case in which these terms are
included, Hertzberg and Sandora (Hertzberg, , §4.2) have identified a back-
reaction mechanism such that in the context of general relativity (albeit not
for curved spacetimes in general) superluminal signalling can occur only at
scales smaller than a photon’s wavelength but is otherwise averaged out; given
that superluminal signalling events for a photon only endure at lengths scale
smaller than its own wavelength, such signalling might well be regarded as
being operationally meaningless.

Outside of the regime of the geometrical optics limit, a light clock realised
through periodic bouncing of a light signal (so-called ‘Langevin clocks’) will
in practice not read out reliably the proper time along its worldline
interval. But, to repeat: on taking the idealising geometrical optics limit
and setting aside the above-discussed additional coupling terms, contra clocks
, an idealised light clock construction modelled by Maxwellian theory in
curved spacetime will still read out the proper time interval along its
worldline (for, upon idealisation, the signal speed is unity, and the
assumptions of Fletcher  obtain). Moreover, even when corrections from QED are
taken into account, the aforementioned mechanism _à la_ Hertzberg and Sandora
guarantees that deviations from the light cone structure are minuscule, or
even operationally meaningless, in the context of general relativity. In the
context of AH ; AH2 , our point in this section is, ultimately, a simple one:
even granting the claims made in those papers, it is not at all clear that
superluminal _signalling_ is implicated.

##  VI Discussion

The finding of AH —that there are electromagnetic wave solutions in Gödel
spacetimes not amenable to the geometrical optics limit—seems to stand in
tension with other major results in the literature on electromagnetic wave
propagation in curved spacetimes, as were mentioned in the introduction: (i)
Friedlander shows that the front velocity of waves in curved spacetime is
always 1 (see (Friedlander, , Thm. 3.2.1)); (ii) in GW , the authors
demonstrate the small bodies constructed from Maxwell fields in curved
spacetimes ‘track’ (in a technical sense) null geodesics, and claim that this
is a special case of the geometrical optics limit. The present article should
ameliorate any such tension, for we have demonstrated that, given a physical
understanding of the geometrical optics limit, this limit can invariably be
taken consistently, that light invariably propagates along null geodesics in
this limit, that in particular the front velocity is unity, and that,
nevertheless, the signal velocity may be less than unity outside of the
geometrical optics limit.

## Acknowledgements

We are grateful to Jacob Barandes, Juliusz Doboszewski, John Dougherty, Sean
Gryb, Dennis Lehmkuhl, Tushar Menon, Erhard Scholz, Nic Teh, and Jim
Weatherall for valuable discussions, and to Sam Fletcher, Chris Smeenk, and
the three anonymous referees for very helpful feedback on previous drafts.
N.L. acknowledges financial support from the John Templeton Foundation (grant
61048).

## References

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# Observing dark matter clumps and asteroid-mass primordial black holes
in the solar system with gravimeters and GNSS networks

Bruno Bertrand1 Michal Cuadrat-Grzybowski2 Pascale Defraigne1 Michel Van Camp1
and Sébastien Clesse3 1Royal Observatory of Belgium, 1180 Brussels, Belgium
2Space Engineering Department, Delft University of Technology, 2629 HS Delft,
(the) Netherlands
3Service de Physique Théorique, Université Libre de Bruxelles (ULB), 1050
Brussels, Belgium.

###### Abstract

In this proceedings, we study the possible gravitational impact of primordial
black holes (PBHs) or dark matter (DM) clumps on GNSS satellite orbits and
gravimeter measurements. It provides a preliminary step to the future
exhaustive statistical analysis over 28 years of gravimeter and GNSS data to
get constraints over the density of asteroid-mass PBH and DM clumps inside the
solar system. Such constraints would be the first to be obtained by direct
observation on a terrestrial scale.

##  1 Introduction

Whereas there are multiple indirect probes of the existence of Dark Matter
(DM) from galactic to cosmological scales, there is no observational evidence
coming from smaller scales. It is plausible that DM sub-galactic clusters
fragments into smaller parts, referred here as DM clumps. There are various
theoretical scenarios in which a significant fraction of DM is made of dark
objects: primordial black holes (PBHs)[1, 2], dark quark nuggets or
strangelets [3], dark blobs or other composite states [4, 5], axion or scalar
miniclusters [6, 7], axion [8, 9] or boson stars [10, 11]… This work focuses
on DM clumps with a mass between 1010 to 1020 kg. This mass range is relevant
because microlensing of stars becomes ineffective in detecting compact DM
objects below 10-19 kg [12]. For what concern the PBHs, a suspected mass
threshold of $10^{11}$ kg comes from their evaporation through Hawking
radiation [13]. As those DM clumps travel in our galaxy, they eventually pass
through the Solar System (SS), itself in motion around the Galactic center. In
this work, we propose for the first time to jointly exploit gravimetry and
global navigation satellite systems (GNSS) data in order to track anomalies in
the gravitational potential induced by DM clumps passing sufficiently near the
Earth. For GNSS, this is the first study addressing their direct gravitational
influence on the satellite orbits. For gravimeters, we investigate the case of
transient signatures whereas previous work only focused on the periodic
signatures [14].

##  2 DM clump and PBH in the solar system

Our modeling of events rate is based on the usual value of the DM density in
the SS neighbourhood, $\rho_{\textrm{DM}}\simeq 0.4\,\textrm{GeV/cm}^{3}$.
However, our analysis goes beyond simple approximations of a constant DM flux,
by performing Monte-Carlo simulations of a realistic population of DM clumps
with asteroid masses and by numerically integrating their trajectories in the
SS. Their orbital motion is modeled as Keplerian, using a semi-analytical
approach. Only two input parameters are sufficient to fully define the 2D
orbit: the impact parameter $B$ and the hyperbolic excess velocity
$V_{\infty}$. Finally, the event rate and the phase space density close to the
Earth are determined from the output parameters which are: the fly-by distance
$d$, the relative velocity $V_{DM/{\text{\char 67\relax}}}$ at the distance
$d$, and the clump mass ${m}_{\textrm{DM}}$. Fig. 1 shows the output of our MC
simulation in terms of $d$ and the distribution of $V_{DM/{\text{\char
67\relax}}}$. The associated mass flux $\dot{m}_{\textrm{DM}}$ is presented as
a heat-map. Given that a model of point mass has been introduced, the notion
of mass flux is independent of an underlying model for DM halo fragmentation.
Indeed for example, a mass flux of $10^{10}$ kg/year cannot be distinguished
from a single clump with a mass of $10^{12}$ kg every century at the same
flyby distance $d$.

Figure 1: Monte-Carlo simulation of minimum Earth-Clump distance $d$ and
velocity $V_{DM/{\text{\char 67\relax}}}$, with mass flow
$\dot{m}_{\textrm{DM}}$ heat-map, for the range of impact parameters
($B\sim\mathcal{U}[0.98,1.025]$ AU).

##  3 Signature on GNSS and superconducting Gravimeters

###  3.1 GNSS

Our modeling shows that any change in the gravitational potential induced by a
DM clump should affect satellite orbits. We carried out a numerical
computation of the gravitational signal caused by the DM clump on GNSS
satellites, using the acceleration $\delta\vec{a}_{\textrm{c}}(t)$ in a third-
body perturbation. This simulation is performed by propagating in time a 3D
Keplerian osculating orbit for the GNSS satellite and one 3D unperturbed
hyperbolic orbit for the DM object. Given these orbital paths, the orbit
perturbation on GNSS satellite reads:

$\frac{\delta\vec{a}_{\textrm{c}}(t)}{g_{\oplus}(t)}=-\frac{\mu_{\textrm{DM}}}{g_{\oplus}(t)}\left(\frac{\vec{r}-\vec{r}_{\textrm{DM}}}{||\vec{r}-\vec{r}_{\textrm{DM}}||^{3}}+\frac{\vec{r}_{\textrm{DM}}}{||\vec{r}_{\textrm{DM}}||^{3}}\right)\,,$
(1)

where $\vec{r}$ and $\vec{r}_{\textrm{DM}}$ are the positions of the GNSS
satellite and DM Clump respectively, $\mu_{\textrm{DM}}$ is the DM celestial
parameter and $g_{\oplus}(t)$ is the reference gravity field. The subsequent
deviation of the GNSS reference orbit is performed by updating the Keplerian
elements using Gauss’ variational equations[15]. The semi-major axis $a$ is
chosen as observable since it is the most sensitive parameter:

$\frac{\textrm{d}a}{\textrm{d}t}=\frac{2\,a^{2}}{h}\cdot\left[e\,\text{sin}(\theta)\,\tilde{a}_{r}+\frac{h^{2}\,\tilde{a}_{\theta}}{\mu_{\oplus}\,r_{0}}\right]\,,$
(2)

where $\mu_{\oplus}$ refers to the Earth’s celestial parameter,
$\vec{r}_{0}(t)$ is the initial reference orbit, $h$ is the orbital angular
momentum, $e$ is the eccentricity, $\theta$ is the true anomaly and
($\tilde{a}_{r}$,$\tilde{a}_{\theta}$) are the radial and tangential
components within the satellite frame of the satellite acceleration caused by
the perturbation.

Figure 2: Signature of a DM clump of $10^{15}$ kg on the Galileo
constellation. Left: Impulse response to the semi-major axis for a flyby at
$B_{\text{\char 67\relax}}=15500$ km, $V_{\infty}=300$ km/s and
$i_{\textrm{DM}}=90$ deg. Right: Physical envelope of the max. $|\Delta a|$,
realised for 9 excess velocities ranging from 2.5 km/s to 600 km/s) with each
velocity associated to 20 values for $B_{\text{\char 67\relax}}$ in the range
[$10^{-4}$\- $2\cdot 10^{-2}$] AU. Added to the latter is the constellation
$\sigma_{PSD}$.

The signature of a DM clump flyby on the Galileo Constellation is modeled in
Fig. 2 (left), using $a$ as observable. The signal is a characteristic
impulse, slightly dephased for each satellite, with a maximum near the Earth
closest approach. Thanks to networks of fixed permanent GNSS stations, it is
possible to determine the GNSS satellites orbit at the cm level. Such products
are made available by e.g. the analysis center CODE[16] for the International
GNSS service (IGS). Fig. 2 (right) shows power spectral densities (PSD) of the
semi-major axis of each GNSS satellite (solid lines), in combination with
simulated events of duration $\Delta t_{\textrm{DM}}$ using DM clumps model
orbits, with different values for $B_{\text{\char 67\relax}}$ and
$V_{\infty}$.

###  3.2 Superconducting gravimeters (SG)

For SG, the perturbation induced by a transient DM clump is measured as the
radial component of the third-body perturbation. For flybys outside of the
Earth, the relation for the third-body acceleration is similar to the one
found for GNSS satellites (1). Fly-bys inside of Earth occur when the Earth
closest approach $d$ of the clump orbit is smaller than the Earth’s radius. In
that case, the time-dependent normalised gravimeter reading is computed as:

$\frac{\delta
g_{r}(t)}{g}=\frac{1}{g}\,\left[\mu_{\textrm{DM}}\,\frac{\vec{r}_{\textrm{g}}-\vec{r}_{\textrm{DM}}}{||\vec{r}_{\textrm{g}}-\vec{r}_{\textrm{DM}}||^{3}}+\frac{\textrm{d}^{2}X_{{\text{\char
67\relax}}}}{\textrm{d}t^{2}}\frac{\vec{r}_{DM}}{||\vec{r}_{\textrm{DM}}||}\right]\cdot\frac{\vec{r}_{\textrm{g}}}{||\vec{r}_{\textrm{g}}||}\,,$
(3)

where $\vec{r}_{g}$ is the position of the gravimeter and
$\vec{r}_{\textrm{DM}}(t)$ is the modeled hyperbolic orbit. It is also assumed
that $m_{\textrm{DM}}<<M_{enc}$ where $M_{enc}(r)$ is the Earth’s enclosed
mass at the radial distance $r_{\textrm{DM}}$. Two effects may be
distinguished in (3). The first one is a recoil of the Earth’s center of mass,
$d^{2}X_{{\text{\char 67\relax}}}/dt^{2}$. The second effect is the actual
acceleration caused by the DM clump. The signature of a DM clump flyby on the
gravimeter residuals is modeled in Fig. 3 using a worldwide network of 9
gravimeters. The nature of the perturbation is a transient peak reached near
the Earth closest approach. The level of precision is of the order of
$10^{-11}g$ within 1 minute, where $g$ is the Earth gravitational
acceleration. Fig. 3 (right) shows the PSD of residuals (observation - tidal
variations) time series in the Membach station [17] in Belgium, in combination
with simulated events of duration $\Delta t_{\textrm{DM}}$ using DM clumps
orbit models, with different values for $B_{\text{\char 67\relax}}$ and
$V_{\infty}$.

Figure 3: Signature of a DM clump of $10^{15}$ kg Left: Gravitational
perturbation induced by a flyby at $B_{\text{\char 67\relax}}=15500$ km,
$V_{\infty}=300$ km/s and $i_{\textrm{DM}}=0$ deg. Right: Physical envelope of
the max. $|\delta g_{r}/g|$, realised for 9 excess velocities ranging from 2.5
km/s to 600 km/s associated to 20 values for $B_{\text{\char 67\relax}}$ in
the range [$10^{-4}$\- $10^{-2}$] AU. Added to the latter is the Membach
station $\sigma_{PSD}/g$ (blue) with an additional $1/\sqrt{N}$ correction
factor (Orange), where $N$ is the number of gravimeters.

##  4 First results

A preliminary analysis based on the PSD of the gravimeter residuals and
Galileo orbital solutions provides a first assessment of the ‘one-probe’, i.e.
one satellite or one gravimeter, sensitivity. This sensitivity is determined
by the overlapping area between the PSD and the simulated events in Fig. 2 and
3. Focusing on GNSS, Fig. 4 relates the single-satellite sensitivity (blue
dots) to a minimum clump mass and a corresponding flyby distance. The ratio
with our simulated event rates (orange dots, Fig. 4) is around 6 orders of
magnitude, so that a single probe is not enough to hope a detection. However,
this single probe analysis paves the way for a future statistical analysis
based on correlations between a constellation of probes using the 28-years of
publicly available GNSS and gravimeter data. Such a statistical analysis would
enable to gain between 2 and 3 orders of magnitude in sensitivity (dashed blue
line, Fig. 4).

Finally, our one-probe sensitivity implies a state of equilibrium with a total
DM clump mass of less than 0.1 Ceres mass in a 1.5 AU sphere around the Sun.
This one-probe results would not compete with the bounds based from space
probes and planet ephemeris in the SS [18]. However, these latter limits apply
only to a permanent cloud of particles in the SS and are not transferable to
transient DM clumps. Anyway, a full statistical analysis of GNSS and SG data
would surpass the limits based on SS ephemeris [18] by at least one order of
magnitude. Hence, such constraints would be the first and the best to be
obtained by direct observation on a terrestrial scale, before LISA is
operational [19].

Figure 4: Orange dots: Simulation result of DM clump distance and integrated
mass computed over a $20$ year-period obtained from the Monte-Carlo
simulation. Blue dot: One-probe sensitivity distance as a function of the
accumulated clump mass. Blue dashed line: Expected sensitivity after a full
correlation analysis.

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