2022

[1]David Rumler

[1]Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,
Max-Wien-Platz 1, 07743 Jena, Germany

# Geometry of charged rotating discs of dust in Einstein-Maxwell theory

<EMAIL_ADDRESS>Andreas Kleinwächter  Reinhard Meinel  *

###### Abstract

Within the framework of Einstein-Maxwell theory geometric properties of
charged rotating discs of dust, using a post-Newtonian expansion up to tenth
order, are discussed. Investigating the disc’s proper radius and the proper
circumference allows us to address questions related to the Ehrenfest paradox.
In the Newtonian limit there is an agreement with a rotating disc from special
relativity. The charged rotating disc of dust also possesses material-like
properties. A fundamental geometric property of the disc is its Gaussian
curvature. The result obtained for the charged rotating disc of dust is
checked by additionally calculating the Gaussian curvature of the analytic
limiting cases (charged rotating) Maclaurin disc, electrically counterpoised
dust-disc and uncharged rotating disc of dust. We find that by increasing the
disc’s specific charge there occurs a transition from negative to positive
curvature.

###### keywords:

charged rotating disc of dust, post-Newtonian expansion, Ehrenfest paradox,
Gaussian curvature, Maclaurin disc, electrically counterpoised dust

##  1 Introduction

In 1909 Ehrenfest formulated a famous paradox concerning a rigidly rotating
disc (or a cylinder in the original version) within special relativity
Ehrenfest . He pointed out that special relativistic rigidity, introduced by
Born Born , easily leads to two contradicting statements about radius and
circumference measured by a non-rotating observer before and after the disc is
set into rotation.

The paradox gave rise to a long lasting debate on how to resolve it, see,
e.g., Gron2004 . Numerous physicists came up with solution approaches,
however, they turned out to be wrong in most cases. Grøn presented in 1975 a
resolution solely based on kinematics Gron1975 . Initially, it was not clear
whether this would be sufficient or whether a dynamical treatment would be
necessary to solve the problem completely. Today, most physicists (including
the authors of this paper) accept Grøn’s kinematic considerations as the
correct way to resolve Ehrenfest’s paradox.

Nevertheless there are related questions to the paradox, as how the ratio of
circumference to radius is observed from the point of view of non-rotating
versus co-rotating observers (on the disc), how those measurements change when
the rotation speed is altered and how material properties affect more
realistic discs.

With the help of the available semi-analytic solution of the charged rotating
disc of dust, which is a concrete, physically relevant solution of the
Einstein-Maxwell equations in terms of a post-Newtonian expansion
Breithaupt_2015 , we can tackle the above questions in a very direct way.

Motivated by those geometric investigations it is also natural to ask about
the intrinsic curvature of the charged rotating disc of dust. For a two-
dimensional disc the intrinsic curvature is simply the Gaussian curvature. In
particular, the post-Newtonian expansion allows us not only to study special
relativistic effects, but also investigate the influence of higher order
corrections coming from general relativity. A key observation is that the
specific charge of the disc acts like a regulator to adjust the curvature of
the disc. More precisely, by increasing the specific charge from $0$ to $1$
there is a critical value at which a transitions from negative to positive
curvature occurs.

The present paper is structured as follows. In section 2 we briefly introduce
the mathematical formulation of the disc problem in terms of a boundary value
problem, followed by a short overview of the used post-Newtonian expansion in
section 3. After defining the proper radius and the proper circumference of
the rotating disc in section 4, we discuss the above mentioned questions
related to the Ehrenfest paradox based on the charged rotating disc of dust.
Section 5 is devoted to the curvature of the disc. The result will be compared
with the ones for a Maclaurin disc within Newtonian theory, a disc with a
concrete electrically counterpoised dust-configuration (having maximal charge)
and the uncharged disc of dust.

##  2 Boundary value problem for the charged disc of dust

We consider an equilibrium configuration of an infinitesimally thin disc of
dust (perfect fluid with vanishing pressure) with constant specific charge
$\epsilon=\frac{\rho_{\text{el}}}{\mu}\in[-1,1]$ and constant angular velocity
$\Omega$ Palenta_2013 ; Breithaupt_2015 ; Meinel_2015 . $\rho_{\text{el}}$ and
$\mu$ denote the charge density and the baryonic mass density of the disc,
respectively.

The corresponding disc problem is formulated in the framework of Einstein-
Maxwell equations111We use units in which $c=G=4\pi\epsilon_{0}=1$.:

$\displaystyle R_{ab}-\frac{1}{2}R\,g_{ab}=8\pi T_{ab}\,,$ (1)
$\displaystyle\quad{F^{ab}}_{\mathchar 24635\relax\;\negthickspace b}=4\pi
j^{a}\,,$ (2)

where

$\displaystyle
T_{ab}=T_{ab}^{(\text{em})}+T_{ab}^{(\text{dust})}=\frac{1}{4\pi}\left(F_{ac}{F_{b}}^{c}-\frac{1}{4}g_{ab}F_{cd}F^{cd}\right)+\mu
u_{a}u_{b}\,,$ (3) $\displaystyle j^{a}=\rho_{\text{el}}u^{a}=\epsilon\mu
u^{a}\,,$ (4) $\displaystyle F_{ab}=A_{b\mathchar 24635\relax\;\negthickspace
a}-A_{a\mathchar 24635\relax\;\negthickspace b}=A_{b,a}-A_{a,b}\,.$ (5)

To the energy momentum tensor, $T_{ab}$, both the electromagnetic field and
matter (dust) contribute. $j^{a}$ represents a purely convective four-current
density, where $u^{a}$ is the four-velocity. As is common practice, the field
strength tensor $F_{ab}$ is expressed in terms of the four-potential $A_{a}$.

For an equilibrium configuration, spacetime is stationary, implying the
existence of an associated Killing vector $\boldsymbol{\xi}$. We also demand
axisymmetry, characterized by another Killing vector $\boldsymbol{\eta}$ that
commutes with $\boldsymbol{\xi}$. Furthermore, the problem is reflection
symmetric with respect to the equatorial plane of the disc.

Exploiting stationarity and axisymmetry the line element can globally be
written in terms of Weyl-Lewis-Papapetrou coordinates:

$\mathrm{d}s^{2}=f^{-1}\left[h\left(\mathrm{d}\rho^{2}+\mathrm{d}\zeta^{2}\right)+\rho^{2}\mathrm{d}\varphi^{2}\right]-f\left(\mathrm{d}t+a\,\mathrm{d}\varphi\right)^{2}\,.$
(6)

Using Lorenz-gauge the four-potential $A_{a}$ takes the form:

$A_{a}=(0,0,A_{\varphi},A_{t})\,.$ (7)

The coordinates $t$ and $\varphi$ used in the line element are adapted to the
Killing vectors, such that

$\boldsymbol{\xi}=\frac{\partial}{\partial
t}\quad\text{and}\quad\boldsymbol{\eta}=\frac{\partial}{\partial\varphi}\,.$
(8)

As a consequence, the metric functions $f$, $h$, and $a$ as well as the
potentials $A_{\varphi}$ and $A_{t}$ depend on the coordinates $\rho$ and
$\zeta$ only. Instead of $f$ and $h$ we will also make use of the functions
$U$ and $k$, where $f=e^{2U}$ and $h=e^{2k}$.

In the subsequent discussion we heavily rely on the co-rotating frame of
reference defined by $\varphi^{\prime}=\varphi-\Omega t$. As usual, the
covariance of the metric and the four-potential imply the following
transformation laws:

$\displaystyle f^{\prime}=f\left[\left(1+\Omega
a\right)^{2}-\Omega^{2}\rho^{2}f^{-2}\right]\,,\quad\left(1-\Omega
a^{\prime}\right)f^{\prime}=\left(1+\Omega
a\right)f\,,\quad\frac{h^{\prime}}{f^{\prime}}=\frac{h}{f}\,,$ (9)
$\displaystyle A^{\prime}_{\varphi^{\prime}}=A_{\varphi}\,,\quad
A^{\prime}_{t}=\Omega A_{\varphi}+A_{t}\,.$ (10)

It should be noted that the mass density $\mu$ is related to the surface mass
density $\sigma$ via

$\mu=\frac{f}{h}\sigma(\rho)\delta(\zeta)\,.$ (11)

However, $\sigma$ is not coordinate independent in contrast to the proper
surface mass density

$\sigma_{\text{p}}=\sqrt{\frac{f}{h}}\sigma\,.$ (12)

In (electro-)vacuum, assuming axisymmetry and stationarity, the combined
Einstein-Maxwell equations in terms of $f$, $h$, $a$, $A_{\varphi}$ and
$A_{t}$, that can be found in Palenta_2013 , can be reduced to the Ernst
equations PhysRev.168.1415 :

$\displaystyle\left(\Re\mathcal{E}+\lvert\Phi\rvert^{2}\right)\Delta\mathcal{E}$
$\displaystyle=\left(\nabla\mathcal{E}+2\bar{\Phi}\nabla\Phi\right)\cdot\nabla\mathcal{E}\,,$
(13) $\displaystyle\left(\Re\mathcal{E}+\lvert\Phi\rvert^{2}\right)\Delta\Phi$
$\displaystyle=\left(\nabla\mathcal{E}+2\bar{\Phi}\nabla\Phi\right)\cdot\nabla\Phi\,,$
(14)

where

$\Phi=\alpha+\mathrm{i}\beta\quad\text{and}\quad\mathcal{E}=\left(f-\lvert\Phi\rvert^{2}\right)+\mathrm{i}b\,,$
(15)

with $\alpha\coloneqq\Re\Phi=-A_{t}$, are the Ernst potentials. The potentials
$\beta$ and $b$ are defined via

$\beta_{,\rho}=\frac{f}{\rho}\left(A_{\varphi,\zeta}-aA_{t,\zeta}\right)\,,\quad\beta_{,\zeta}=-\frac{f}{\rho}\left(A_{\varphi,\rho}-aA_{t,\rho}\right)$
(16)

and

$b_{,\rho}=-\frac{f^{2}}{\rho}a_{,\zeta}-2\left(\beta
A_{t,\rho}-A_{t}\beta_{,\rho}\right)\,,\quad
b_{,\zeta}=\frac{f^{2}}{\rho}a_{,\rho}-2\left(\beta
A_{t,\zeta}-A_{t}\beta_{,\zeta}\right)\,.$ (17)

$\nabla$ and $\Delta$ are meant as operators in 3-dimensional Euclidean space
with cylindrical coordinates $\rho,\zeta$ and $\varphi$. In the Ernst
equations the function $h$ is absent, since it can be determined by a path-
independent line integration from the other functions.

Using reflection symmetry the following boundary conditions on the disc, i.e.
for $0\leq\rho\leq\rho_{0}$ and $\zeta=0$, where $\rho_{0}$ is the coordinate
radius, can be derived:

$b^{\prime}=0\,,\quad\beta^{\prime}=0\,,\quad\left(e^{U^{\prime}}-\epsilon
A^{\prime}_{t}\right)_{,\rho}=0\,,\quad\left(A^{\prime}_{t}-\epsilon
e^{U^{\prime}}\right)_{,\zeta}=0\,.$ (18)

The Ernst equations (13) and (14) together with the boundary conditions (18),
as well as the asymptotic flatness condition ($\mathcal{E}\to 1$ and $\Phi\to
0$ for $\rho^{2}+\zeta^{2}\to\infty$) and the regularity condition at the rim
of the disc, form a well-defined boundary value problem for the charged disc
of dust.

##  3 Post-Newtonian expansion

In case of the rotating disc of dust without charge the corresponding boundary
value problem could be solved analytically PhysRevLett.75.3046 ; RFE . For the
charged rotating disc of dust a semi-analytic solution in terms of a post-
Newtonian expansion up to tenth order is available Palenta_2013 ;
Breithaupt_2015 .

This expansion uses a relativity parameter $\gamma$, defined by

$\gamma\coloneqq 1-\sqrt{f_{c}}\,,\quad\text{with}\quad f_{c}\coloneqq
f(\rho=0,\zeta=0)\,.$ (19)

Equivalently, $\gamma$ can be expressed in terms of the redshift, $Z_{c}$, of
a photon emitted at $\rho=0$, $\zeta=0$ and observed at
$\rho^{2}+\zeta^{2}\to\infty$:

$\gamma=\frac{Z_{c}}{1+Z_{c}}\,.$ (20)

From both expressions, (19) and (20), it is evident that $\gamma$ takes values
between $0$ and $1$, where $\gamma\approx 0$ represents Minkowski spacetime
and $\gamma\to 1$ corresponds to the ultra-relativistic limit. At the ultra-
relativistic limit we assume the formation of black holes Breithaupt_2015 .
Note that $U$, appearing in $f_{c}=e^{2U_{c}}$, can be interpreted as a
generalized Newtonian potential.

The parameter space of the charged disc solution is spanned by
$\gamma\in[0,1]$, $\epsilon\in[0,1]$ and the coordinate radius $\rho_{0}$ as a
scaling parameter. By restricting to positive charges we do not loose
generality.

A convenient choice of coordinates is given in terms of elliptic coordinates
$\eta$ and $\nu$, defined through

$\rho=\rho_{0}\sqrt{\left(1-\eta^{2}\right)(1+\nu^{2})}\,,\quad\zeta=\rho_{0}\eta\nu\,,$
(21)

with $\eta\in[-1,1]$ and $\nu\in[0,\infty]$. Additionally normalizing all
dimensioned quantities with suitable powers of $\rho_{0}$ leads to a
dimensionless formulation of the disc problem. The normalized quantities are
denoted by ∗, e.g.

$a^{*}=\frac{a}{\rho_{0}}\,,\quad
A^{*}_{\varphi}=\frac{A_{\varphi}}{\rho_{0}}\quad\text{or}\quad\Omega^{*}=\rho_{0}\Omega\,.$
(22)

As a starting point of the post-Newtonian expansion we utilize the relation

$\gamma=-U_{N}(\rho=0,\zeta=0)=\frac{{\Omega^{*}_{N}}^{2}}{1-\epsilon^{2}}\,,$
(23)

between the Newtonian potential $U_{N}$ and the angular frequency $\Omega_{N}$
that holds in Newtonian limit only. As a consequence, we have
${\Omega^{*}}^{2}=\left(1-\epsilon^{2}\right)\gamma+\mathcal{O}(\gamma^{2})$
in general (beyond the Newtonian limit). Expanding this relation implies that
$\Omega^{*}$ is an odd function in $g\coloneqq\sqrt{\gamma}$, i.e.

$\Omega^{*}=\sum_{n=0}^{\infty}\Omega^{*}_{2n+1}g^{2n+1}\,,$ (24)

where $\Omega^{*}_{2n+1}$ is a function of $\eta$ and $\nu$ and
$\Omega^{*}_{1}=\sqrt{1-\epsilon^{2}}$.

All the discussed potentials and metric functions can be expanded in
$\Omega^{*}$. Examining their symmetry properties reveals a purely symmetric
or antisymmetric behaviour under the transformation $\Omega^{*}\to-\Omega^{*}$
that changes the sense of rotation. Therefore they can also be written as a
series in $g$, containing only even or odd powers of $g$, respectively. For
the metric functions we get:

$f=1+\sum_{n=1}^{\infty}f_{2n}g^{2n}\,,\quad
h=1+\sum_{n=2}^{\infty}h_{2n}g^{2n}\,,\quad
a^{*}=\sum_{n=1}^{\infty}a^{*}_{2n+1}g^{2n+1}\,.$ (25)

All the coefficient functions $f_{2n}$, $h_{2n}$ and $a^{*}_{2n+1}$ depend on
the coordinates $\eta$ and $\nu$ only.

Palenta and Breithaupt were able to use this post-Newtonian expansion to
derive analytic solutions for those coefficient functions. First, Palenta
solved the boundary value problem in terms of this expansion up to eighth
order in $n$ and Breithaupt later on up to tenth Palenta_2013 ;
Breithaupt_2015 . The subsequent calculations and discussions of the present
paper build on Breithaupt’s semi-analytic solution of the charged rotating
disc of dust.

##  4 Related questions to Ehrenfest’s paradox

In 1909 Ehrenfest pointed out that rigid rotation of discs (or cylinders)
within special relativity involving a period of angular acceleration
inevitably leads to a paradox Ehrenfest .

The concept of rigidity in special relativity is defined by Born as follows:
Each infinitesimal neighbourhood of an arbitrarily moving body, as measured by
co-moving observers at each point of the body, should appear permanently
undeformed Born .

Relying on this definition, Ehrenfest formulated a paradox: A relativistic
cylinder with radius $R$ is given a rotating motion about its axis, which
finally becomes constant - while satisfying Born’s rigidity at all times. Let
$R^{\prime\prime}$ be the radius of the constantly rotating cylinder measured
by an observer at rest, then two contradicting requirements need to be
fulfilled:

  1. (i)

$2\pi R^{\prime\prime}<2\pi R$ ,

  2. (ii)

$R^{\prime\prime}=R$ .

While to this day there is still no common agreement on its solution, there is
a kinematic resolution of the paradox by Grøn most physicists do indeed accept
Gron1975 ; Gron1977 ; Gron2004 . He showed that it is not possible to
synchronize clocks of successive inertial rest frames momentarily at rest to
points around the periphery of the rotating disc. This means that setting a
disc in rotational motion, while satisfying Born’s rigidity condition, is
kinematically impossible.

Despite its resolution, there are still interesting questions related to
Ehrenfest’s paradox:

  1. 1)

How does the spatial geometry of an accelerated disc which is initially at
rest evolves as measured by non-rotating and co-rotating observers?

  2. 2)

What is the spatial geometry of a disc that is already set into rotation as
seen by non-rotating and co-rotating observers?

  3. 3)

A real disc has material properties, how do they influence the rotating disc?

The uniformly rotating disc of dust is relativistically rigid in the sense
that it rotates rigidly, meaning $\Omega=\text{const.}$ Rigidity .

In the present section we want to describe geometric quantities like radius
and circumference of the rotating disc and address the above posed questions.
Those quantities, however, only have meaning if the 3-dimensional space of the
disc is defined correctly.

The 3-dimensional space of the rotating disc is defined to be the surface
orthogonal to world lines of fixed particles on the disc. Or equivalently the
disc-space is defined by simultaneous events measured on usual Einstein
synchronized clocks on the disc Gron2004 .

Locally, it is possible to decompose the 4-dimensional line element into
$\mathrm{d}s^{2}=\mathrm{d}\sigma^{2}-\mathrm{d}\tau^{2}$, where
$\mathrm{d}\sigma^{2}$ is the coordinate-invariant proper spatial line element
and $\mathrm{d}\tau$ is the infinitesimal time interval measured in an
inertial frame instantaneously at rest relative to a particle on the rotating
disc Langevin ; Gron1979 .222I should be stressed, that $\mathrm{d}\tau$ is
not a perfect differential and thus cannot be integrated around a closed curve
on the rotating disc. As a consequence, the surface represented by
$\mathrm{d}\tau=0$, i.e. the surface characterized by $\mathrm{d}\sigma^{2}$,
is discontinuous. This also prevents the Einstein synchronization of clocks
around the periphery of the rotating disc. The proper spatial line element can
be calculated via the radar method of Landau and Lifschitz LL :

$\mathrm{d}\sigma^{2}=\left(g^{\prime}_{\alpha\beta}-\frac{g^{\prime}_{\alpha
4}g^{\prime}_{\beta
4}}{g^{\prime}_{44}}\right)\mathrm{d}x^{\prime\alpha}\mathrm{d}x^{\prime\beta}\,,\quad\text{where}\;\;\alpha,\beta=1,2,3\,.$
(26)

As a consequence, the surface characterized by $\mathrm{d}\tau^{2}=0$ is,
according to the above definition, the 3-dimensional space of the rotating
disc and its intrinsic geometry is described by the proper spatial line
element.

By using the radar method of Landau and Lifschitz, the 2-dimensional proper
spatial line element of the metric (6) in elliptic coordinates takes the form

$\mathrm{d}\sigma^{2}\Big{|}_{\nu=0}=f^{\prime-1}h^{\prime}\rho_{0}^{2}\frac{\eta^{2}}{1-\eta^{2}}\mathrm{d}\eta^{2}+f^{\prime-1}\rho_{0}^{2}(1-\eta^{2})\mathrm{d}\varphi^{\prime
2}\,.$ (27)

From now on we are only interested in the proper spatial geometry of the disc
itself, as a 2-dimensional object. Note that, since $\nu=0$ on the disc, the
elliptic coordinates decouple and
$\mathrm{d}\rho^{2}=\rho_{0}^{2}\frac{\eta^{2}}{1-\eta^{2}}\mathrm{d}\eta^{2}$
and $\rho^{2}=\rho_{0}^{2}(1-\eta^{2})$.

Integration of the line element (27) then gives the
proper circumference

$C^{\prime}(\eta)=2\pi\rho_{0}\sqrt{1-\eta^{2}}f^{\prime-1/2}\,,$ (28)

the proper area

$A^{\prime}(\eta)=2\pi\rho_{0}^{2}\int_{\eta}^{1}\\!\mathrm{d}\tilde{\eta}\,\tilde{\eta}\,f^{\prime-1}h^{\prime
1/2}$ (29)

and the proper radius

$R(\eta)=R^{\prime}(\eta)=\rho_{0}\int_{\eta}^{1}\\!\mathrm{d}\tilde{\eta}\,\frac{\tilde{\eta}}{\sqrt{1-\tilde{\eta}^{2}}}\left(f^{\prime-1}h^{\prime}\right)^{1/2}$
(30)

of the rotating disc of charged dust.444The proper spatial line element has to
be positive definite, which is fulfilled trivially here, since radius and
circumference are positive definite. Furthermore, integration of the line
element is only possible due to the time-independent metric. By inserting the
post-Newtonian expansions of the metric functions in the co-rotating frame,
using (25), (24) and (9), also (28), (29) and (30) can be written as series
expansions in $g$.555Accordingly, in the non-rotating frame of reference
circumference and radius are defined by
$C(\eta)=2\pi\rho_{0}\sqrt{1-\eta^{2}}f^{-1/2}$ and
$R(\eta)=\rho_{0}\int_{\eta}^{1}\\!\mathrm{d}\tilde{\eta}\,\frac{\tilde{\eta}}{\sqrt{1-\tilde{\eta}^{2}}}\left(f^{-1}h\right)^{1/2}$,
respectively.

In order to compare the results for $C^{\prime}$, $A^{\prime}$ and $R$ of the
charged disc of dust with those of a standard disc within the framework of
special relativity, described by Grøn, we have to take the Newtonian limit.
Thus, we require that both the metric deviates only slightly from the
Minkowski metric and the rotational velocities are small compared to the speed
of light.
Charged disc of dust (Newtonian limit): $\displaystyle
C^{\prime}=2\pi\rho_{0}\left[{\rho^{\star}}+\left(\frac{1}{2}{\Omega_{1}^{\star}}^{2}{\rho^{\star}}^{2}-\frac{1}{2}f_{2}\right){\rho^{\star}}g^{2}\right]\,,$
$\displaystyle
A^{\prime}=\pi\rho_{0}^{2}\left[{\rho^{\star}}^{2}+\left(\frac{1}{4}{\Omega_{1}^{\star}}^{2}{\rho^{\star}}^{2}+\frac{1}{2}\left(2-f_{2}\right)\right)\\!{\rho^{\star}}^{2}g^{2}\right]\,,$
$\displaystyle
R=\rho_{0}\left[{\rho^{\star}}+\frac{1}{6}\left(4-f_{2}\right)\rho^{\star}g^{2}\right]\,.$
Disc within SR (Newtonian limit): $\displaystyle C^{\prime}=2\pi
r\left[1+\frac{1}{2}\left(\omega r\right)^{2}\right]\,,$ $\displaystyle
A^{\prime}=\pi r^{2}\left[1+\frac{1}{4}\left(\omega r\right)^{2}\right]\,,$
$\displaystyle R=r\,.$
Crucial to the discussion of Ehrenfest’s paradox and the closely related
geometric questions is the ratio of circumference to radius as seen from co-
rotating and non-rotating observers.
Charged disc of dust (Newtonian limit):
$\displaystyle\frac{C^{\prime}}{R^{\prime}}=2\pi\left[1+\left(\frac{1}{2}{\Omega_{1}^{\star}}^{2}{\rho^{\star}}^{2}-\frac{1}{3}\left(2+f_{2}\right)\right)g^{2}\right]\,,$
$\displaystyle\frac{C}{R}=2\pi\left[1-\frac{1}{3}\left(2+f_{2}\right)g^{2}\right]\,.$
Disc within SR (Newtonian limit):
$\displaystyle\frac{C^{\prime}}{r}=2\pi\left[1+\frac{1}{2}\left(\omega
r\right)^{2}\right]\,,$ $\displaystyle\frac{C}{r}=2\pi\,.$
The first-order ($n=1$) coefficient function $f_{2}$ is given by
$f_{2}\\!\left(\eta,0\right)=-1-\eta^{2}$ and
$\rho^{\star}=\frac{\rho}{\rho_{0}}=\sqrt{1-\eta^{2}}$.

It should be noted that the above expansions utilize different expansion
parameters and therefore only in the Newtonian limit a comparison is
meaningful. For the disc of dust we demand the gravitational potential to be
small compared to the speed of light $c$ and the expansion parameter is
$g=\frac{\sqrt{-U_{c}}}{c}\ll 1$. On the other hand, for the SR-disc we have
expansions around small rotational velocities, $\frac{\omega r}{c}\ll 1$.

Grøn discusses only kinematic aspects of the disc and material properties are
absent in Gron1975 ; Gron1977 . However, a real disc is hold together by
attractive forces originating from the rigid material itself. In case of the
disc of dust attractive gravitational forces, mediated through the terms
involving $f_{2}$, play the role of those material forces.

Taking the different expansions and the appearance of the terms with $f_{2}$
into account, the disc of dust agrees with Grøn’s investigated disc in the
Newtonian limit.

Furthermore, an interesting observation can be made. The quantities
$C^{\prime},A^{\prime}$ and $R$ of the disc of dust are all larger than those
of Grøn’s analysed disc (for $\eta^{2}<1$ and $r>0$). Although dust is by
definition a pressure-less fluid without any elastic properties, the rotating
disc of dust nevertheless behaves to some extend as if it had material
properties, resulting in a kind of "elastic" expansion.

Explicitly writing out the ratios of circumference to radius leads to:

$\displaystyle\frac{C^{\prime}}{R^{\prime}}=2\pi\left[1+\frac{1}{6}\left(1-\eta^{2}\right)\\!\left(1-3\epsilon^{2}\right)g^{2}\right]\,,$
(31)
$\displaystyle\frac{C}{R}=2\pi\left[1-\frac{1}{3}\left(1-\eta^{2}\right)g^{2}\right]\,.$
(32)

Remarkably, for $\eta^{2}<1$, there is a geometrical transition induced by an
alteration of the specific charge: $\frac{C^{\prime}}{R^{\prime}}\gtrless
2\pi$ for $\epsilon^{2}\lessgtr\frac{1}{3}$. This transition is also present
beyond the Newtonian limit, as can be seen in figure 2. All figures in this
paper are created using all available orders, i.e. up to $n=10$. For vanishing
charge the ratio of proper circumference to proper radius becomes larger than
$2\pi$, just as in the case of Grøn’s result for this ratio.

In the non-rotating frame there is no such transition, instead the ratio of
circumference to radius is always smaller than $2\pi$, independent of the
specific charge (excluding $\eta=\pm 1$). This also stays true beyond the
Newtonian limit (apart from high values of $g$), see figure 2. Within special
relativity the ratio is exactly equal to $2\pi$, as Grøn showed. Those
results, however, do not contradict each other, since values smaller than
$2\pi$ are caused by the gravitational correction term involving $f_{2}$. As
discussed above, Grøn’s analysis of the rotating disc does not involve
corresponding terms.666In the framework of special relativity there is an
enlightening explanation for Grøn’s results: Observed in the non-rotating
frame, not the circumference of the disc is being contracted due to rotation,
but the infinitesimal measuring rods placed along the circumference. As a
consequence observers on the rotating disc measure a ratio larger than $2\pi$
(they have to use more measuring rods), while an observer in the non-rotating
frame still measures exactly $2\pi$.

$\frac{1}{2\pi}\frac{C^{\prime}}{R^{\prime}}$$g$$\epsilon$ Figure 1:
Normalized ratio of circumference to radius in the rotating frame plotted for
$\eta=0$.

$\frac{1}{2\pi}\frac{C}{R}$$g$$\epsilon$ Figure 2: Normalized ratio of
circumference to radius in the non-rotating frame plotted for $\eta=0$.

Now, let us address the questions raised in the beginning of this section,
starting with the first one. Formulated in the Newtonian language, a dust
particle in the rotating disc is in an equilibrium state of gravitational,
electric and centrifugal force. It is therefore evident that changing the
parameter $\epsilon$ directly affects the rotational velocity $\Omega$. In
particular, by decreasing $\epsilon$ we can transition to disc configurations
with higher rotational velocities $\Omega$. This, however, is achieved in a
quasi-stationary way and cannot be seen as setting the disc into rotation as
we alter the specific charge of the disc and thus the disc itself.
Nevertheless, we can compare disc configurations with increasing rotational
velocities with each other and examine the effect on the spatial geometry. As
can be seen in figure 2, for the co-rotating observer the geometric ratio
$\frac{C^{\prime}}{R^{\prime}}$ monotonically increases from values smaller to
greater than $2\pi$. In fact, as will be shown in section 6, a simultaneous
transition from positive to negative curvature also occurs. The non-rotating
observer also measures increasing values of $\frac{C}{R}$. They, however,
remain below $2\pi$, apart from high values of $g$ due to the emergence of an
ergosphere.

Concerning the second question, we have seen that the ratio
$\frac{C^{\prime}}{R^{\prime}}$, that characterizes the proper spatial
geometry and is measured by a co-rotating observer, can be less than, equal to
or greater than $2\pi$ depending on the specific charge $\epsilon$. For a
ratio equal to $2\pi$ the critical value of the specific charge is
$\epsilon_{\text{crit}}=\frac{1}{\sqrt{3}}$ in the Newtonian limit. With
growing $g$, $\epsilon_{\text{crit}}$ decreases slightly and becomes dependent
on the radial coordinate $\eta$ (see figures 16 and 16). A non-rotating
observer, on the other hand, always perceives smaller values than $2\pi$ for
$\frac{C}{R}$ (again, apart from high values of $g$).

$\frac{R(\eta=0)}{\rho_{0}}$$\epsilon$$g=0.8$$g=0.7$$g=0.6$...$g=0$ Figure 3:
Normalized proper radius, evaluated at the rim of the disc, as a function of
$\epsilon$.

Finally, the third question resolves around the influence of material
properties present in a real rotating disc. Indeed, as we have already seen in
the Newtonian limit, the disc of dust mimics "elastic" material properties and
shows an "elastic" expansion as response to the rotational motion. But also
beyond Newtonian physics, according to figure 3, the disc’s total proper
radius $R_{0}\coloneqq R(\eta=0)$ grows with decreasing specific charge
$\epsilon$ and in turn increasing rotational velocities $\Omega$. The more
relativistic the disc, the more pronounced is the effect.

##  5 Curvature

Motivated by the found geometric transition induced by a change of the
specific charge $\epsilon$, we now examine the intrinsic curvature of rotating
discs more closely.

As the domain of a rotating disc is simply a two-dimensional spatial surface,
denoted by $\Sigma_{2}$, the geometric quantity that describes its intrinsic
curvature is the Gaussian curvature $K$.

In the following subsections we will calculate the Gaussian curvature of the
charged rotating disc of dust (using the post-Newtonian expansion up to
$n=10$) and various analytic limiting cases. The findings will be compared and
reviewed.

###  5.1 Gaussian curvature of the charged disc of dust

By using the derived expressions for the proper radius $R$ and the proper
circumference $C^{\prime}$, (30) and (28), the proper spatial line element of
the charged rotating disc of dust can be written in the following instructive
and simple way:

$\mathrm{d}\sigma^{2}\Big{|}_{\Sigma_{2}}=\mathrm{d}R^{2}+\left(\frac{C^{\prime}(R)}{2\pi}\right)^{\\!2}\mathrm{d}\varphi^{\prime
2}\,,$ (33)

where at least formally $C^{\prime}(R)=C^{\prime}(\eta(R))$.

With this condensed version of the proper spatial line element also the
Gaussian curvature takes a very simply form, i.e.

$K=-\frac{C^{\prime}(R)_{,RR}}{C^{\prime}(R)}\,.$ (34)

It thus follows immediately that the Gaussian curvature of the charged
rotating disc of dust is determined primarily by the second derivative of the
proper circumference $C^{\prime}$ with respect to the proper radius $R$.

Even though the above formula for $K$, (34), is quite insightful, it is not
that practical, since we do not know the functional dependence of $C^{\prime}$
on $R$. By using the chain rule we can, however, transform formula (34) into
the slightly less appealing but all the more useful form

$K=\frac{1}{C^{\prime}(\eta)}\left(\frac{\mathrm{d}R}{\mathrm{d}\eta}\right)^{\\!-3}\left[\frac{\mathrm{d}^{2}R}{\mathrm{d}\eta^{2}}C^{\prime}(\eta)_{,\eta}-\frac{\mathrm{d}R}{\mathrm{d}\eta}C^{\prime}(\eta)_{,\eta\eta}\right]\,.$
(35)

Inserting the solutions for the post-Newtonian expansion leads to

$\displaystyle K=\frac{1}{\rho_{0}^{2}}\biggl{[}$
$\displaystyle\left(3\epsilon^{2}-1\right)g^{2}$
$\displaystyle+\left(\left(6\eta^{2}-5\right)\epsilon^{4}+\left(\frac{7}{6}-8\eta^{2}\right)\epsilon^{2}+3\eta^{2}+\frac{1}{2}+\frac{16\eta^{2}}{\pi^{2}}\left(\epsilon^{2}-1\right)\right)g^{4}$
$\displaystyle+\mathcal{O}\left(g^{6}\right)\biggr{]}\,.$ (36)

As can be seen in figure 5, there is a characteristic transition curve in
parameter space where the curvature $K$ changes its sign. The same transition
curve appears in figure 2 depicting $\frac{C^{\prime}}{R^{\prime}}$.
Explicitly written out expansions up to fifth order of
$\frac{C^{\prime}}{R^{\prime}}$ and $R_{0}^{2}K$ are listed in appendix A.

$R_{0}^{2}K$$\epsilon$$g$ Figure 4: Gaussian Curvature $K$ normalized by the
total proper radius, $R_{0}$, squared and evaluated at $\eta=0$.

$R_{0}^{2}K$$\eta$$\epsilon=1$$\epsilon=0$ Figure 5: Radial dependence of the
normalized Gaussian curvature for the special cases $\epsilon=0$ and
$\epsilon=1$, where $g=0.6$.

Figure 5 shows the radial dependence of $R_{0}^{2}K$ for the special cases
$\epsilon=1$ and $\epsilon=0$. In case of maximal charge, i.e. $\epsilon=1$,
the disc has no rotation at all and general relativistic effects cause
positive curvature. For $\epsilon=0$ the rotation is maximal and special
relativistic effects are dominant and lead, in accordance with the disc
described by Grøn, to negative curvature. In both cases the curvature $K$ gets
more positive towards the centre of the disc (where $\eta=1$). This is
expected, since in the non-rotating case general relativistic effects increase
towards the centre of a massive object, where the gravitational potential is
the deepest. At maximal rotation, on the other hand, the rotational speed
decreases towards the centre while general relativistic effects increase.

$\frac{C^{\prime}}{2\pi R_{0}}$$\frac{R}{R_{0}}$$g=0.8$$g=0$ Figure 6:
Functional dependence of $C^{\prime}$ (normalized by $2\pi R_{0}$) on $R$
(normalized by $R_{0}$) plotted for $\epsilon=0$ and
$g=\\{0,0.2,0.4,0.6,0.8\\}$.

$\frac{C^{\prime}}{2\pi R_{0}}$$\frac{R}{R_{0}}$$g=0.8$$g=0$ Figure 7:
Functional dependence of $C^{\prime}$ (normalized by $2\pi R_{0}$) on $R$
(normalized by $R_{0}$) plotted for $\epsilon=0.5$ and
$g=\\{0,0.2,0.4,0.6,0.8\\}$.

$\frac{C^{\prime}}{2\pi R_{0}}$$\frac{R}{R_{0}}$$g=0$$g=0.8$ Figure 8:
Functional dependence of $C^{\prime}$ (normalized by $2\pi R_{0}$) on $R$
(normalized by $R_{0}$) plotted for $\epsilon=1$ and
$g=\\{0,0.2,0.4,0.6,0.8\\}$.

Interesting to the current discussion is also the functional dependence of
$C^{\prime}$ on $R$ that is visualized in terms of parametric plots (including
appropriate normalizations) in figures 8 \- 8. As can be observed there,
$C^{\prime}_{,RR}>0$ for small $\epsilon$ and $C^{\prime}_{,RR}<0$ for large
$\epsilon$. Considering formula (34), this perfectly coincides with the plot
of the normalized curvature in figure 5. The plots in figures 8 \- 8 are also
in line with the $\frac{C^{\prime}}{R^{\prime}}$-plot in figure 2.

###  5.2 Newtonian theory: Maclaurin discs and their Gaussian curvature

In the Newtonian limit Einstein and Maxwell equations decouple and reduce to
the Poisson equations for the gravitational and the electric potential. Thus,
in Newtonian theory the disc of dust is fully described by

$\displaystyle\Delta
U=4\pi\mu^{\text{Mld}}\,,\quad\text{with}\quad\mu^{\text{Mld}}=\begin{cases}\sigma^{\text{Mld}}(\rho)\delta(\zeta)&\text{for}\,\,\,0\leq\rho\leq\rho_{0}\\\
0&\text{otherwise}\end{cases}\,,$ (37) $\displaystyle\Delta
U^{\text{E}}=-4\pi\epsilon\mu^{\text{Mld}}\,,$ (38)

and

$\epsilon{U^{\text{E}}}_{,\rho}=-\left(U-\frac{1}{2}\Omega^{2}\rho^{2}\right)_{\\!,\rho}\,,$
(39)

where $U$ is the gravitational and $U^{\text{E}}\coloneqq\alpha=-A_{t}$ the
electric potential. Equation (39) shows that the dust particles in the disc
are in an equilibrium of electric, gravitational and centrifugal force.

Rotating discs in the framework of Newtonian theory characterized by (37) -
(39) are also known as charged Maclaurin discs.

The solution for the exterior Newtonian gravitational potential of Maclaurin
spheroids, in terms of elliptic coordinates, can be found in RFE . In the
limit where the spheroid shrinks to a disc, this potential is given
everywhere. As before, we are interested in the solution on the disc itself.
On $\Sigma_{2}$, where $\nu=0$, the gravitational potential reduces to

$U|_{\nu=0}=\frac{1}{2}U_{c}\left(1+\eta^{2}\right)\,,$ (40)

with $U_{c}=-g^{2}$ in the Newtonian limit. According to equations (37) and
(38), $U^{\text{E}}$ trivially follows from equation (40):
$U^{\text{E}}=-\epsilon U$.

Furthermore, using the solution for $U$, (39) can be rewritten to the already
known equation,

$U_{c}=-\frac{\Omega^{2}\rho_{0}^{2}}{1-\epsilon^{2}}\,,$ (41)

that was used as a starting point for the post-Newtonian expansion.

Integrating (37) over an infinitesimal $\zeta$-interval and exploiting
reflection symmetry reveals the surface mass density

$\sigma^{\text{Mld}}=\frac{1}{2\pi}U_{,\zeta}\Big{|}_{\zeta=0^{+}}=\frac{3M}{2\pi\rho_{0}^{2}}\eta\,,$
(42)

where $M$ is the mass of the disc. It can be easily verified that

$U^{\prime}=U-\frac{1}{2}\Omega^{2}\rho^{2}$ (43)

is the corotating potential in the Newtonian limit.

By understanding the Newtonian theory as a limit of the full general
relativistic theory we can reuse the line element. The 4-dimensional line
element in the Newtonian limit reads

$\mathrm{d}s^{2}\Big{|}_{\nu=0}=\left(1-2U\right)\rho_{0}^{2}\frac{\eta^{2}}{1-\eta^{2}}\mathrm{d}\eta^{2}+\left(1-2U^{\prime}\right)\rho_{0}^{2}\left(1-\eta^{2}\right)\mathrm{d}\varphi^{\prime
2}-(1+2U^{\prime})\mathrm{d}t^{2}\,.$ (44)

Due to the absence of an off-diagonal term, the proper spatial line element of
the Maclaurin disc follows immediately777As can be easily verified, here the
proper spatial line element is simply the spatial part of (44).:

$\displaystyle d\sigma^{2}\Big{|}_{\nu=0}=$
$\displaystyle\left[1+\left(1+\eta^{2}\right)g^{2}\right]\rho_{0}^{2}\frac{\eta^{2}}{1-\eta^{2}}\mathrm{d}\eta^{2}$
$\displaystyle+\left[1+\left(2-\epsilon^{2}\left(1-\eta^{2}\right)\right)g^{2}\right]\rho_{0}^{2}\left(1-\eta^{2}\right)\mathrm{d}\varphi^{\prime
2}\,.$ (45)

As expected, the Gaussian curvature of the Maclaurin disc resulting from (5.2)
is given by:

$K^{\text{Mld}}=\frac{1}{\rho_{0}^{2}}\left(3\epsilon^{2}-1\right)g^{2}\,.$
(46)

In conclusion, the Gaussian curvature of the charged Maclaurin disc is in
perfect agreement with the Gaussian curvature of the disc of dust in the
Newtonian limit.

###  5.3 Gaussian curvature of a specific ECD-disc configuration

For general (not necessarily disc-) configurations with $\epsilon=\pm 1$ one
obtains electrically counterpoised dust (ECD), see, e.g., Meinel_2011 . Those
solutions are static. The Papapetrou-Majumdar class Papapetrou ; Majumdar
contains such static solutions to the Einstein-Maxwell equations and the
corresponding line element is of the form

$\mathrm{d}s^{2}=f^{-1}\left[\left(\mathrm{d}\rho^{2}+\mathrm{d}\zeta^{2}\right)+\rho^{2}\mathrm{d}\varphi^{2}\right]-f\mathrm{d}t^{2}\,,$
(47)

where

$f^{1/2}=e^{U}=1-\epsilon\alpha$ (48)

is the defining relation between the metric function $f$ and the electrostatic
potential $\alpha$. Due to the static spacetime the electromagnetic four-
potential has only one non-vanishing component:

$A_{a}=(0,0,0,-\alpha)\,.$ (49)

Starting from line element (47), the Einstein-Maxwell equations reduce to the
surprisingly simple equation

$\Delta e^{-U}=-4\pi\mu^{\text{ECD}}e^{-3U}\,.$ (50)

By introducing a new potential $V$ and a redefined mass density
$\mu^{\text{ECD}}_{\text{st}}$ as

$V\coloneqq
1-e^{-U}\quad\text{and}\quad\mu^{\text{ECD}}_{\text{st}}\coloneqq\mu^{\text{ECD}}e^{-3U}\,,$
(51)

equation (50) can be transformed into the Poisson equation

$\Delta V=4\pi\mu^{\text{ECD}}_{\text{st}}\,.$ (52)

The solution can consequently be represented as a Poisson integral and based
on its asymptotic behaviour we can identify the mass:

$M=\int\mu^{\text{ECD}}_{\text{st}}\\!\left(\rho,\zeta\right)\rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}\zeta\,.$
(53)

For ECD the mass density $\mu^{\text{ECD}}_{\text{st}}$ is not predetermined
by the theory, but can rather be freely chosen. As a physically motivated toy
model, we consider a specific ECD-disc configuration equipped with the mass
density of a Maclaurin disc:

$\mu^{\text{ECD}}_{\text{st}}=\mu^{\text{Mld}}\,.$ (54)

Note, that this means
$\mu^{\text{ECD}}_{\text{st}}=e^{2U}\sigma^{\text{ECD}}_{\text{st}}(\rho)\delta(\zeta)=\sigma^{\text{Mld}}(\rho)\delta(\zeta)$.
Choosing the mass density in this way is justified, since then the mass of the
ECD-disc also coincides with the one of the Maclaurin disc:
$M=\int\mu^{\text{ECD}}\\!\left(\rho,\zeta\right)\rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}\zeta=\int\mu^{\text{Mld}}_{\text{st}}\\!\left(\rho,\zeta\right)\rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}\zeta$.

In case of the Maclaurin disc, the equation of motion $\Delta
U=4\pi\mu^{\text{Mld}}$ leads to the solution
$U|_{\nu=0}=\frac{1}{2}U_{c}\left(1+\eta^{2}\right)$, see equations (37) and
(40). Analogously, for ECD the solution of (52) with the chosen mass density
(54) is given by

$V|_{\nu=0}=\frac{1}{2}V_{c}\left(1+\eta^{2}\right)\,.$ (55)

However, now we get $V_{c}=-\frac{g^{2}}{1-g^{2}}$.

The proper spatial line element, stemming from (47), for the chosen ECD-disc
configuration in terms of elliptic coordinates then reads

$\mathrm{d}\sigma^{2}\Big{|}_{\nu=0}=f^{-1}\Big{|}_{\nu=0}\left(\rho_{0}^{2}\frac{\eta^{2}}{1-\eta^{2}}\mathrm{d}\eta^{2}+\rho_{0}^{2}\left(1-\eta^{2}\right)\mathrm{d}\varphi^{2}\right)\,,$
(56)

where

$f^{-1/2}=1-V\,,\quad
V|_{\nu=0}=-\frac{1}{2}\frac{g^{2}}{1-g^{2}}\left(1+\eta^{2}\right)\,.$

It implies the following Gaussian curvature:

$K^{\text{ECD}}=\frac{1}{\rho_{0}^{2}}\frac{32\left(1-g^{2}\right)^{2}g^{2}}{\left[2-\left(1-\eta^{2}\right)g^{2}\right]^{4}}\,.$
(57)

$\left(R_{0}^{\text{ECD}}\right)^{2}K^{\text{ECD}}$$g$$\eta$ Figure 9:
Normalized Gaussian Curvature of the chosen ECD-configuration.

The corresponding plot of the normalized Gaussian curvature can be found in
figure 9. There we introduced $R_{0}^{\text{ECD}}\coloneqq
R^{\text{ECD}}(\eta=0)=\rho_{0}\frac{6-g^{2}}{6\left(1-g^{2}\right)}$.

Interestingly, in contrast to the charged rotating disc of dust (see figure
5), the chosen ECD-disc configuration possesses higher (positive) curvature at
the rim of the disc and lower at the centre. But nevertheless, it has a
positive curvature in agreement with the $\epsilon\\!=\\!1$-limit of the
charged rotating disc of dust.

To gain a better understanding of this radial curvature behaviour, we
investigate and compare the proper surface mass densities of the so far
discussed discs. The resulting proper surface mass densities are

$\displaystyle\sigma_{\text{p}}$
$\displaystyle=\sigma\sqrt{fh^{-1}}=\frac{1}{2\pi}U^{\prime}_{,\zeta}\Big{|}_{\zeta=0^{+}}\sqrt{fh^{-1}}$
$\displaystyle=\frac{2}{\pi^{2}\rho_{0}}\left[\eta
g^{2}+\frac{1}{6}\left(\left(\frac{23}{3}\eta^{2}-5\right)\epsilon^{2}-7\eta^{2}\right)\eta
g^{4}+\mathcal{O}\left(g^{6}\right)\right]\,,$ (58)
$\displaystyle\sigma_{\text{p}}^{\text{Mld}}$
$\displaystyle=\sigma^{\text{Mld}}$
$\displaystyle=\frac{2}{\pi^{2}\rho_{0}}g^{2}\eta\,,$ (59)
$\displaystyle\sigma_{\text{p}}^{\text{ECD}}$
$\displaystyle=\sigma^{\text{ECD}}e^{U}$
$\displaystyle=\frac{2}{\pi^{2}\rho_{0}}\left[1+\frac{1}{2}\frac{g^{2}}{1-g^{2}}\left(1+\eta^{2}\right)\right]^{-2}\frac{g^{2}}{1-g^{2}}\eta$
$\displaystyle=\frac{2}{\pi^{2}\rho_{0}}\left[\eta
g^{2}-\eta^{3}g^{4}+\mathcal{O}\left(g^{6}\right)\right]\,.$ (60)

As it should be, $\sigma_{\text{p}}^{\text{Mld}}$ represents the Newtonian
limit of $\sigma_{\text{p}}$ and $\sigma_{\text{p}}^{\text{ECD}}$.

$R_{0}^{\text{Mld}}\sigma_{\text{p}}^{\text{Mld}}$$R_{0}\sigma_{\text{p}}$$R_{0}^{\text{ECD}}\sigma_{\text{p}}^{\text{ECD}}$$R_{0}^{\text{Mld}}$$R_{0}$$R_{0}^{\text{ECD}}$
Figure 10: Comparison of the normalized proper surface mass densities
$R_{0}^{\text{Mld}}\sigma_{\text{p}}^{\text{Mld}}$, $R_{0}\sigma_{\text{p}}$
and $R_{0}^{\text{ECD}}\sigma_{\text{p}}^{\text{ECD}}$ for $g=0.7$ and
$\epsilon=1$.

$R_{0}\sigma_{\text{p}}$$R_{0}^{\text{ECD}}\sigma_{\text{p}}^{\text{ECD}}$$R_{0}$$R_{0}^{\text{ECD}}$$g=0.8$$g=0.6$$g=0.4$$g=0.2$
Figure 11: Comparison of the normalized proper surface mass densities
$R_{0}\sigma_{\text{p}}$ and $R_{0}^{\text{ECD}}\sigma_{\text{p}}^{ECD}$ for
different values of $g$ and $\epsilon=1$.

In figure 11 all three normalized proper surface mass densities are plotted
and in figure 11 $R_{0}\sigma_{\text{p}}|_{\epsilon=1}$ and
$R_{0}^{\text{ECD}}\sigma_{\text{p}}^{ECD}$ are depicted for different values
of the relativity parameter $g$.

From figure 11 it is evident that $\sigma_{\text{p}}^{\text{ECD}}$ is denser
at the rim and less dense at the centre compared to
$\sigma_{\text{p}}|_{\epsilon=1}$. The higher $g$, the more extreme is this
behaviour of the chosen ECD-configuration. This dominance of the proper
surface mass density $\sigma_{\text{p}}^{\text{ECD}}$ at the rim as opposed to
the centre leads to the observed higher curvature $K^{\text{ECD}}$ at the rim.
In contrast, the curvature $K|_{\epsilon=1}$ generated by the surface mass
density $\sigma_{\text{p}}|_{\epsilon=1}$ of the maximally charged disc of
dust increases towards the centre.

###  5.4 Gaussian curvature of the uncharged disc of dust

By means of the inverse scattering method that originates from soliton theory,
the global problem of the uniformly rotating disc of dust without charge was
solved rigorously by Neugebauer and Meinel Neugebauer:1993ct ;
PhysRevLett.75.3046 ; RFE .

An essential part of this method is to formulate and to solve a Riemann-
Hilbert problem. It turns out that this Riemann-Hilbert problem has a unique
solution in the parameter region $0<\mu<\mu_{0}\coloneqq 4.629...\,$, where
$\mu\coloneqq 2\left(\rho_{0}\Omega\right)^{2}e^{-2V_{0}}$ and $V_{0}\coloneqq
U^{\prime}(\rho=0,\zeta=0)$. $\mu\to 0$ corresponds to the Newtonian limit and
for $\mu\to\mu_{0}$ the formation of a Kerr-black hole was proven
Neugebauer:1993ct ; PhysRevLett.75.3046 , see also Meinel2002 .

Analogous to the charged disc, the line element can globally be written in
Weyl-Lewis-Papapetrou form:

$\mathrm{d}s^{2}=e^{-2U^{\prime}}\left[e^{2k^{\prime}}\left(\mathrm{d}\rho^{2}+\mathrm{d}\zeta^{2}\right)+\rho^{2}\mathrm{d}\varphi^{\prime
2}\right]-e^{2U^{\prime}}\left(\mathrm{d}t+a\,\mathrm{d}\varphi^{\prime}\right)^{2}\,.$
(61)

Denoting $x\coloneqq\frac{\rho}{\rho_{0}}$, as used in the literature above,
the resulting proper spatial line element reads

$\mathrm{d}\sigma^{2}\Big{|}_{\Sigma_{2}}=e^{-2\left(U-k\right)}\rho_{0}^{2}\,\mathrm{d}x^{2}+e^{-2V_{0}}\rho_{0}^{2}x^{2}\mathrm{d}\varphi^{\prime
2}\,,$ (62)

utilising the boundary condition $e^{2U^{\prime}}=e^{2V_{0}}$ and the
transformation law $k^{\prime}-U^{\prime}=k-U$.

For the associated Gaussian curvature we obtain the compact formula

$K^{\text{uncharged}}=-\frac{1}{2\rho_{0}^{2}x}\frac{\partial}{\partial
x}e^{2(U-k)}=-\frac{1}{2\rho_{0}^{2}x}\frac{\partial}{\partial
x}e^{2(V_{0}-k^{\prime})}\,.$ (63)

Hence the normalized Gaussian curvature plotted in figure 13 can be calculated
by

$\left(R_{0}^{\text{uncharged}}\right)^{2}K^{\text{uncharged}}(g,x)=-2\mu
k^{\prime}_{,\tilde{\mu}}(\tilde{\mu})\frac{\left[\int_{0}^{1}dx\,e^{k^{\prime}}(\tilde{\mu})\right]^{2}}{e^{2k^{\prime}}(\tilde{\mu})}\,,$
(64)

with $\tilde{\mu}:=\mu(1-x^{2})$ and the relation
$g(\mu)=\sqrt{1-e^{V_{0}}(\mu)}$ between the parameters $g$ and $\mu$. The
only occurring metric potential $k^{\prime}$ is related (within the disc) to
the Ernst potential $e^{2V_{0}}+ib_{0}$ in the origin ($x=0$) by

$k^{\prime}_{,\tilde{\mu}}(\tilde{\mu})=\frac{1}{4}\frac{\left[{e^{2V_{0}}}_{,\tilde{\mu}}(\tilde{\mu})\right]^{2}+\left[{b_{0}}_{,\tilde{\mu}}(\tilde{\mu})\right]^{2}}{e^{2V_{0}}(\tilde{\mu})}\,.$
(65)

These parameter functions $e^{2V_{0}}$ and $b_{0}$ itself are given in terms
of Jacobi’s elliptic functions.

$\left(R_{0}^{\text{uncharged}}\right)^{2}K^{\text{uncharged}}$$x$$g$ Figure
12: Normalized Gaussian curvature of the uncharged disc of dust.

$R_{0}^{2}K$$x$$g$ Figure 13: Normalized Gaussian curvature of the charged
disc of dust in the limit of vanishing charge.

Comparing the plot of the normalized Gaussian curvature of the uncharged disc
(exact solution), figure 13, with the one of the charged disc evaluated at
$\epsilon=0$ (post-Newtonian expansion up to tenth order), figure 13, shows a
good qualitative agreement.

Checking the more conclusive numerical values, certifies an excellent
coincidence between the (exact) analytic and the semi-analytic solution for
$\epsilon=0$. Averaged over the disc (using $x=\\{0,0.3,0.7,1\\}$), the
percent deviation of $R_{0}^{2}K|_{\epsilon=0}$ from
$\left(R_{0}^{\text{uncharged}}\right)^{2}K^{\text{uncharged}}$ is
$2.29\text{\times}{10}^{-6}$ for $g=0.6$, $3.50\text{\times}{10}^{-5}$ for
$g=0.7$ and still only $4.72\text{\times}{10}^{-4}$ for $g=0.8$. See appendix
B for the corresponding numerical values of the normalized Gaussian curvature.

###  5.5 Visualization

Similar to Flamm’s paraboloid in case of Schwarzschild spacetime, we want to
visualize the spatial curvature of the charged disc of dust by an isometric
embedding.

The isometric embedding of the proper 2-dimensional disc space, characterized
by (33), into 3-dimensional Euclidean space, furnished with the line element
$\mathrm{d}l^{2}=\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2}+\mathrm{d}z^{2}$ is
achieved by the identifications

$\displaystyle\phi=\varphi^{\prime}\,,$ (66) $\displaystyle
r=\frac{C^{\prime}(R)}{2\pi}\,,$ (67) $\displaystyle
z(R)=\int_{0}^{R}\left[1-\frac{C^{\prime}(\tilde{R})_{,\tilde{R}}}{2\pi}\right]^{1/2}\mathrm{d}\tilde{R}\,,$
(68)

if the condition $C^{\prime}(R)_{,R}\leq 2\pi$ is satisfied.

This means that this embedding only works for sufficiently high values of
$\epsilon$. Based on to the fact that the curvature becomes first negative at
the centre of the disc by lowering $\epsilon$ (if initially it is positive
everywhere), see section 5.6, it can be shown that $K\geq 0$ is not only a
sufficient but also a necessary condition for the embedding constraint,
$C^{\prime}(R)_{,R}\leq 2\pi$, to be fulfilled.

$z(r)$ Figure 14: Isometric embedding of the proper 2-dimensional disc space
into 3-dimensional Euclidean space for $g=0.6$ and $\epsilon=1$. The scaling
parameter $\rho_{0}$ is set to $1$.

In figure 14 the embedding of the charged disc of dust is depicted for the
values $g=0.6$ and $\epsilon=1$.

###  5.6 Transition curve

In three different cases we have seen the occurrence of a transition curve in
the parameter space $(\epsilon,g)$:

$\text{1)}\;\,\frac{C^{\prime}}{R^{\prime}}=2\pi\;,\qquad\text{2)}\;\,K=0\;,\qquad\text{3)}\;\,C^{\prime}_{,R}=2\pi\;.$

It turns out that all those three conditions are equivalent and indeed all the
curves are identical. This is also backed up by figure 16.

$\epsilon$$g$2)3)1) Figure 15: Conditions 1), 2), and 3) lead to the same
transition curve in the parameter space. W.l.o.g. $\eta=0$.

$\epsilon$$\eta$$g=0.7$$g=0.5$$g=0.3$$g=0.1$ Figure 16: Radial dependence of
the curve $K=0$ plotted for different values of the relativistic parameter
$g$.

The radial dependence of the transition curve can be read off from figure 16.
As can be seen there, changing $\epsilon$ does not cause a transition
throughout the whole disc at the same time. In fact, by starting with positive
curvature, i.e. high values of $\epsilon$, the transition to negative
curvature happens first at the centre and last at the rim.

##  6 Discussion

A central assumption in cosmology is that the universe is spatially
homogeneous and isotropic on large length scales. This is the cosmological
principle and indeed observations confirm that it is fulfilled on scales of
about 100 $\mathrm{M}\mathrm{p}\mathrm{c}$. Using the cosmological principle
one can straightforwardly derive the Friedmann-Robertson-Walker metric that
reduces the Einstein equations to the famous Friedmann equations. Due to
homogeneity and isotropy the solutions exhibit a globally constant geometry
with only three possible spatial curvatures: flat ($k=0$), positive ($k=+1$)
or negative ($k=-1$). Remarkably, observations of the CMB suggest that our
universe might indeed be flat. The first Friedmann equation reveals that there
is a critical energy density that leads to a flat universe. If the density is
higher than the critical density the universe is positively curved and if it
is lower it is negatively curved. For more details see, e.g., mukhanov2005 .

In some sense the proper spatial geometry of the charged rotating disc of dust
behaves analogous to the spatial curvature of the Friedmann universe. High
values of the specific charge $\epsilon$ (defined as the ratio of electric
charge to baryonic mass) cause positive and low values negative curvature.
Furthermore, for given $g$ and $\eta$ there is a critical value of $\epsilon$
that gives rise to Euclidean geometry. However, unlike in Friedmann cosmology
the geometry of the charged disc of dust is not globally constant. The
geometry remains unchanged only in angular direction, but not in radial one. A
beautiful exception of this represents the Newtonian limit with a radially
independent curvature. In fact, for the critical value
$\epsilon=\frac{1}{\sqrt{3}}$ the disc is globally flat.

Acknowledgments

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) under
Grant No. 406116891 within the Research Training Group RTG 2522/1. The authors
would like to thank Martin Breithaupt for the provided results of the post-
Newtonian expansion up to tenth order.

## Declarations

Conflict of interest: The authors have no relevant financial or non-financial
interests to disclose.
Data availability: The datasets generated during and/or analysed during the
current study are available from the corresponding author on reasonable
request.

##  Appendix A Expansions up to fifth order

$\displaystyle\frac{1}{2\pi}$
$\displaystyle\frac{C^{\prime}}{R^{\prime}}\left(\eta=0\right)$
$\displaystyle={}$ $\displaystyle
1-\frac{1}{2}\left(\epsilon^{2}-\frac{1}{3}\right)g^{2}+\frac{5}{24\pi^{2}}\left[\left(\epsilon^{4}-\frac{52}{75}\epsilon^{2}-\frac{53}{75}\right)\pi^{2}-\frac{224\epsilon^{2}}{25}+\frac{224}{25}\right]g^{4}$
$\displaystyle-\frac{223}{1920\pi^{2}}\left[\left(\epsilon^{6}-\frac{3457}{892}\epsilon^{4}+\frac{113453}{112392}\epsilon^{2}+\frac{31336}{14049}\right)\pi^{2}\right.$
$\displaystyle\left.-\,\frac{2560}{669}\left(\epsilon^{2}-1\right)\left(\epsilon^{4}-\frac{313}{70}\epsilon^{2}-\frac{257}{42}\right)\right]g^{6}$
$\displaystyle+\frac{3145}{48384\pi^{4}}\left[\left(\epsilon^{8}-\frac{313007839}{40256000}\epsilon^{6}+\frac{3055678201}{193228800}\epsilon^{4}-\frac{6005877459}{644096000}\epsilon^{2}-\frac{29078}{235875}\right)\pi^{4}\right.$
$\displaystyle\left.-\,\frac{111104}{15725}\left(\epsilon^{2}-1\right)\left(\epsilon^{6}-\frac{69012347}{9374400}\epsilon^{4}+\frac{113190529}{24998400}\epsilon^{2}+\frac{78691}{13020}\right)\pi^{2}\right.$
$\displaystyle\left.+\,\frac{802816}{15725}\left(\epsilon^{2}-1\right)^{2}\left(\epsilon^{2}-\frac{15227}{1960}\right)\right]g^{8}$
$\displaystyle-\frac{2815391}{66355200\pi^{4}}\left[\left(\epsilon^{10}-\frac{379034234759}{41622740544}\epsilon^{8}+\frac{95728571097211}{3874698756096}\epsilon^{6}\right.\right.$
$\displaystyle\left.\left.-\,\frac{103871864922476873}{2727787924291584}\epsilon^{4}+\frac{14955289324645259}{495961440780288}\epsilon^{2}-\frac{5355807104}{650355321}\right)\pi^{4}\right.$
$\displaystyle\left.-\,\frac{238424064}{19707737}\left(\epsilon^{2}-1\right)\left(\epsilon^{8}-\frac{10000383870041}{1084307938560}\epsilon^{6}+\frac{104207736848353}{5046960586752}\epsilon^{4}\right.\right.$
$\displaystyle\left.\left.-\,\frac{49539410937953809}{5921767088455680}\epsilon^{2}-\frac{91863454}{13446279}\right)\pi^{2}\right.$
$\displaystyle\left.+\,\frac{78643200}{2815391}\left(\epsilon^{2}-1\right)^{2}\left(\epsilon^{6}-\frac{4169}{350}\epsilon^{4}+\frac{6055401553}{271656000}\epsilon^{2}+\frac{2656693}{46200}\right)\right]g^{10}$
$\displaystyle R_{0}^{2}$ $\displaystyle K\left(\eta=0\right)$
$\displaystyle={}$
$\displaystyle\left(3\epsilon^{2}-1\right)g^{2}-5\left(\epsilon^{2}-\frac{7}{30}\right)\left(\epsilon^{2}-1\right)g^{4}$
$\displaystyle+\,\frac{1703}{320\pi^{2}}\left[\left(\epsilon^{6}-\frac{52261}{20436}\epsilon^{4}+\frac{219665}{122616}\epsilon^{2}-\frac{3968}{15327}\right)\pi^{2}-\frac{2560\epsilon^{6}}{5109}+\frac{34816\epsilon^{4}}{15327}\right.$
$\displaystyle\left.-\,\frac{11776\epsilon^{2}}{5109}+\frac{8192}{15327}\right]g^{6}$
$\displaystyle-\,\frac{9769}{2016\pi^{2}}\left[\left(\epsilon^{8}-\frac{274414949}{60020736}\epsilon^{6}+\frac{119546257}{18756480}\epsilon^{4}-\frac{249550409}{80027648}\epsilon^{2}+\frac{16824}{48845}\right)\pi^{2}\right.$
$\displaystyle\left.-\,\frac{94976}{48845}\left(\epsilon^{6}-\frac{964087}{142464}\epsilon^{4}+\frac{1525885}{284928}\epsilon^{2}-\frac{856}{1113}\right)\left(\epsilon^{2}-1\right)\right]g^{8}$
$\displaystyle+\,\frac{316977977}{77414400\pi^{4}}\left[\left(-\frac{158969856}{316977977}+\epsilon^{10}-\frac{425179610969}{60859771584}\epsilon^{8}+\frac{1000003956304379}{62320406102016}\epsilon^{6}\right.\right.$
$\displaystyle\left.\left.-\,\frac{61149890345347447}{3988505990529024}\epsilon^{4}+\frac{15274226506312301}{2659003993686016}\epsilon^{2}\right)\pi^{4}-\frac{1296420864}{316977977}\left(\epsilon^{2}-1\right)\left(\epsilon^{8}\vphantom{\frac{1129856}{949527}}\right.\right.$
$\displaystyle\left.\left.-\,\frac{32218659817}{3646183680}\epsilon^{6}+\frac{47760895613989}{2800269066240}\epsilon^{4}-\frac{566746281193567}{59739073413120}\epsilon^{2}+\frac{1129856}{949527}\right)\pi^{2}\right.$
$\displaystyle\left.+\,\frac{550502400}{316977977}\left(\epsilon^{4}-\frac{56}{5}\epsilon^{2}+\frac{5296}{175}\right)\left(\epsilon^{2}-1\right)^{2}\left(\epsilon^{2}-\frac{1}{3}\right)\right]g^{10}$

##  Appendix B Gaussian curvature

Table 1: Gaussian curvature: uncharged disc (exact solution) versus charged disc in the limit $\epsilon\to 0$ (series expansion up to tenth order). |  | $K^{\text{uncharged}}\left(R_{0}^{\text{uncharged}}\right)^{2}$ | $KR_{0}^{2}\Big{|}_{\epsilon\to 0}$
---|---|---|---
$g=0.6$ | $x=0$ | \- 0.333602 | \- 0.333602
| $x=0.3$ | \- 0.349614 | \- 0.349614
| $x=0.7$ | \- 0.434073 | \- 0.434072
| $x=1$ | \- 0.584301 | \- 0.584297
$g=0.7$ | $x=0$ | \- 0.441980 | \- 0.441979
| $x=0.3$ | \- 0.471784 | \- 0.471784
| $x=0.7$ | \- 0.640493 | \- 0.640487
| $x=1$ | \- 0.988126 | \- 0.987999
$g=0.8$ | $x=0$ | \- 0.560113 | \- 0.560108
| $x=0.3$ | \- 0.611598 | \- 0.611586
| $x=0.7$ | \- 0.930419 | \- 0.930281
| $x=1$ | \- 1.73047 | \- 1.72751
|  |  |

## References

  *   * (1) Ehrenfest, P.: Gleichförmige Rotation starrer Körper und Relativitätstheorie. Physikalische Zeitschrift 10, 918 (1909)
  * (2) Born, M.: Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. Annalen der Physik 335, 1–56 (1909). https://doi.org/10.1002/andp.19093351102
  * (3) Grøn, Ø.: In: Rizzi, G., Ruggiero, M.L. (eds.) Space Geometry in Rotating Reference Frames: A Historical Appraisal, pp. 285–333. Springer, Dordrecht (2004). https://doi.org/10.1007/978-94-017-0528-8_17
  * (4) Grøn, Ø.: Relativistic description of a rotating disk. American Journal of Physics 43(10), 869–876 (1975). https://doi.org/10.1119/1.9969
  * (5) Breithaupt, M., Liu, Y.-C., Meinel, R., Palenta, S.: On the black hole limit of rotating discs of charged dust. Classical and Quantum Gravity 32(13), 135022 (2015). https://doi.org/10.1088/0264-9381/32/13/135022
  * (6) Palenta, S., Meinel, R.: Post-Newtonian expansion of a rigidly rotating disc of dust with a constant specific charge. Classical and Quantum Gravity 30(8), 085010 (2013). https://doi.org/10.1088/0264-9381/30/8/085010
  * (7) Meinel, R., Breithaupt, M., Liu, Y.-C.: Black holes and quasiblack holes in Einstein-Maxwell theory. In: 13th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, pp. 1186–1188 (2015). https://doi.org/10.1142/9789814623995_0117
  * (8) Ernst, F.J.: New formulation of the axially symmetric gravitational field problem. ii. Phys. Rev. 168, 1415–1417 (1968). https://doi.org/10.1103/PhysRev.168.1415
  * (9) Neugebauer, G., Meinel, R.: General relativistic gravitational field of a rigidly rotating disk of dust: Solution in terms of ultraelliptic functions. Phys. Rev. Lett. 75, 3046–3047 (1995). https://doi.org/10.1103/PhysRevLett.75.3046
  * (10) Meinel, R., Ansorg, M., Kleinwächter, A., Neugebauer, G., Petroff, D.: Relativistic Figures of Equilibrium, (2008). https://doi.org/10.1017/CBO9780511535154
  * (11) Grøn, Ø.: Rotating frames in special relativity analyzed in light of a recent article by M. Strauss. International Journal of Theoretical Physics 16(8), 603–614 (1977). https://doi.org/%****␣Geometry_charged_disc_of_dust.bbl␣Line␣200␣****10.1007/BF01811093
  * (12) Pirani, F.A.E., Williams, G.: Rigid motion in a gravitational field. Séminaire Janet. Mécanique analytique et mécanique céleste 5 (1961-1962). talk:8-9
  * (13) Langevin, P.: Remarques au sujet de la note de M. Prunier. Comptes Rendus des Séances de l’Académie des Sciences 200(48) (1935)
  * (14) Grøn, Ø.: Relativistic description of a rotating disk with angular acceleration. Foundations of Physics 9(5), 353–369 (1979). https://doi.org/10.1007/BF00708527
  * (15) Landau, L.D., Lifschitz, E.M.: The Classical Theory of Fields, (1971)
  * (16) Meinel, R., Hütten, M.: On the black hole limit of electrically counterpoised dust configurations. Classical and Quantum Gravity 28(22), 225010 (2011). https://doi.org/10.1088/0264-9381/28/22/225010
  * (17) Papapetrou, A.: A static solution of the equations of the gravitational field for an arbitary charge-distribution. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 51, 191–204 (1945)
  * (18) Majumdar, S.D.: A class of exact solutions of Einstein’s field equations. Phys. Rev. 72, 390–398 (1947). https://doi.org/10.1103/PhysRev.72.390
  * (19) Neugebauer, G., Meinel, R.: The Einsteinian gravitational field of the rigidly rotating disk of dust. Astrophys. J. Lett. 414, 97–99 (1993). https://doi.org/10.1086/187005
  * (20) Meinel, R.: Black holes: a physical route to the Kerr metric. Annalen der Physik 514, 509–521 (2002). https://doi.org/10.1002/andp.20025140704
  * (21) Mukhanov, V.: Physical Foundations of Cosmology, (2005). https://doi.org/10.1017/CBO9780511790553