# Exact solutions for differentially rotating galaxies in general relativity

Marco Galoppo<EMAIL_ADDRESS>School of Physical & Chemical
Sciences, University of Canterbury,
Private Bag 4800, Christchurch 8140, New Zealand  David L. Wiltshire
<EMAIL_ADDRESS>School of Physical & Chemical Sciences,
University of Canterbury,
Private Bag 4800, Christchurch 8140, New Zealand

###### Abstract

A class of stationary axisymmetric solutions of Einstein’s equations for
isolated differentially rotating dust sources is presented. The low-energy
asymptotic regime is extracted, requiring a self-consistent coupling of
quasilocal energy and angular momentum. The Raychaudhuri equation reduces to a
balance equation, with two important limits. These limits can be interpreted
empirically for rotationally supported configurations such as galaxies. The
net energy including quasilocal kinetic contributions vanishes on the inner
vortex surface, and the outer rotosurface. These new geometrical objects
potentially shed light on virialization. Whether or not abundant collisionless
dark matter exists, the new solutions suggest that the phenomenology of
galactic rotation curves be fundamentally reconsidered, for consistency with
general relativity.

An exact solution of Einstein’s equations for an isolated galaxy sourced by a
realistic distribution of stars, treated as a pressureless fluid—dust—is a
decades-old open problem that has confounded mathematical relativists for
decades. In this Letter we present a new solution to this problem, which
unlike previous unsuccessful attempts incorporates the essential physical
feature that radial distributions of stars in galaxies rotate by varying
amounts. This statement will be precisely defined – but, roughly, each ring of
stars in an axially symmetric distribution rotates at slightly different
speeds to its neighbours. A consequence of our result is that the amount of
collisionless dark matter particles that is conventionally inferred needs to
be fundamentally revisited, with profound implications for astrophysics and
cosmology.

This problem has been the subject of controversy on account of researchers
artificially treating galaxies as rigidly rotating objects [1, 2, 3], which is
not only physically unrealistic but is also compounded by mathematical
problems [4, 5]. Remarkably, we will find that these issues can be sidestepped
by a refinement in understanding the consistent treatment of taking Newtonian
limits. In particular, conventionally researchers have naïvely applied
nonrelativistic limiting procedures to the gravitational metric before
attempting to include nonlinear terms. By contrast, we will include all
essential nonlinearities for stationary axisymmetric dust spacetimes before
considering the low–energy limit.

The nonlinearity of general relativity arises from the self–interaction of
matter and geometry via Einstein’s equations. In place of additive
gravitational potentials on a fixed background, the time–averaged motion of
matter sources defines regional backgrounds with their own quasilocal energy
and angular momentum content [6, 7]. Understanding the hierarchy of regional
scales, or the fitting of one regional geometry into another, is an important
foundational question in cosmology [8]. Conventionally one assumes the
existence of a global asymptotic Minkowski background before applying
nonrelativistic limits. Mass and angular momentum are then defined by ideal
asymptotic Killing vectors, which obscures their essential quasilocal origin
in general relativity. The manner in which quasilocal energy and angular
momentum approach asymptotic values in a nonempty universe is little
appreciated. It may nonetheless be key to understanding the biggest open
problems in physical cosmology.

In this Letter we derive new solutions with self-consistent coupling between
quasilocal energy and angular momentum, which can be applied in the low-energy
limit to systems at different scales in the fitting problem. Here our primary
interest is rotating galaxies. However, our approach will likely further
inform other attempts to model the dynamical properties of coarse-grained,
cosmological structures in general relativity [9, 10, 11, 12, 13].

Let us consider stationary axisymmetric solutions of the Einstein equations
with a dust source. Our solutions will thus apply to numerous astrophysical
systems including stellar systems, galaxies and putative dark matter halos on
larger scales. As applied to galaxies assuming stationarity—i.e., the presence
of a timelike Killing vector, $\partial/\partial t$—means the solutions apply
to short time scales relative to those involving galaxy formation and
evolution, denoted $t_{\text{evo}}$. Likewise, imposing axisymmetry—i.e., an
axial Killing vector, $\partial/\partial\phi$—means that our solutions then
apply to an effective dust fluid for time-averaged oscillatory motion,
$t_{\text{osc}}$, of individual stars and gas above and below the galactic
plane. At the present epoch $t_{\text{osc}}\mathop{\sim}\limits 10\,$Myr and
$t_{\text{evo}}\mathop{\sim}\limits 1\,$Gyr. Thus the effective dust fluid
applies on time scales, $10^{7}\mathop{\hbox{${\lower
3.8pt\hbox{$<$}}\atop{\raise 0.2pt\hbox{$\sim$}}$}}t\mathop{\hbox{${\lower
3.8pt\hbox{$<$}}\atop{\raise 0.2pt\hbox{$\sim$}}$}}10^{9}\,$yr for nearby
galaxies.

The spacetime metrics we consider can thus be written in the
Lewis–Papapetrou–Weyl form, viz.

$\mathop{\text{d}\\!}s^{2}=-c^{2}e^{2\Phi(r,z)/c^{2}}(\mathop{\text{d}\\!}t+A(r,z)\mathop{\text{d}\\!}\phi)^{2}+e^{-2\Phi(r,z)/c^{2}}\left[r^{2}\mathop{\text{d}\\!}\phi^{2}+e^{2k(r,z)/c^{2}}(\mathop{\text{d}\\!}r^{2}+\mathop{\text{d}\\!}z^{2})\right]\,,$
(1)

where $r$ is a radial coordinate, $z$ is an axial coordinate, $\Phi(r,z)$ is
related to the conventional Newtonian potential, $A(r,z)$ is related to frame-
dragging, and $k(r,z)$ is a conformal factor on the 2–dimensional space of
orbits of the isometry group generated by the Killing vectors.

The energy-momentum tensor takes the form

$T^{\mu\nu}=c^{2}\rho(r,z)\,U^{\mu}U^{\nu}\,,$ (2)

where $\rho(r,z)$ is the density of particles, each with a 4–velocity
$U^{\mu}$, given by

$U^{\mu}\partial_{\mu}=(-H)^{-1/2}\left(\partial_{t}+\Omega\,\partial_{\phi}\right)\,,$
(3)

where $d\phi/dt=\Omega(r,z)$ uniquely defines the angular speed of rotation at
any point, and $H(r,z)$ is a normalization factor. Since
$U^{\mu}U_{\mu}=-c^{2}$, it follows that

$H=-e^{2\Phi/c^{2}}(1+A\,\Omega)^{2}+e^{-2\Phi/c^{2}}r^{2}\,\Omega^{2}/c^{2}\,.$
(4)

Einstein’s equations are solved by a hierarchy of quadratures [14, 15, 16].
One finds that $H=H(\eta)$ and $\Omega=\Omega(\eta)$, where $\eta(r,z)$ is the
angular momentum of the dust particles measured by Zero Angular Momentum
Observers (ZAMOs), and

$\mathop{\text{d}\\!}\Omega=c^{2}\mathop{\text{d}\\!}H/({2\eta})\,.$ (5)

In view of (5) we call $H$ the differential rotation state function, since a
choice of $H$ fixes $\Omega$. The norms of the two Killing vectors,
$\xi^{\mu}\partial_{\mu}=\partial_{t}$ and
$\zeta^{\mu}\partial_{\mu}=\partial_{\phi}$, and their inner product are then
respectively given by

$\displaystyle\xi^{\mu}\xi_{\mu}=-c^{2}e^{2\Phi/c^{2}}=c^{2}\frac{\left(H-\eta\,\Omega/c^{2}\right)^{2}-\left(r\,\Omega/c\right)^{2}}{H}\,,$
(6)
$\displaystyle\zeta^{\mu}\zeta_{\mu}=r^{2}e^{-2\Phi/c^{2}}-c^{2}A^{2}e^{2\Phi/c^{2}}=\frac{r^{2}-(\eta/c)^{2}}{(-H)}\,,$
(7)
$\displaystyle\xi^{\mu}\zeta_{\mu}=-c^{2}Ae^{2\Phi/c^{2}}=\eta-\frac{(\eta/c)^{2}-r^{2}}{H}\,\Omega\,.$
(8)

The remaining Einstein equations yield

$\displaystyle\Xi_{,r}=\frac{1}{2r}\left[g_{tt,z}g_{\phi\phi,z}-g_{tt,r}g_{\phi\phi,r}-(g_{t\phi,z})^{2}+(g_{t\phi,r})^{2}\right]\,$
$\displaystyle\Xi_{,z}=\frac{1}{2r}\left[2g_{t\phi,r}g_{t\phi,z}+g_{tt,r}g_{\phi\phi,z}-g_{tt,z}g_{\phi\phi,r}\right],$
(9)

where $\Xi(r,z)=[\Phi(r,z)-k(r,z)]/c^{2}$. Finally, further requiring that
$\Xi_{,rz}=\Xi_{,zr}$ we find that the entire class of solutions is fully
determined by a choice of $H$ and a solution of the homogeneous Grad–Shafranov
equation [17, 18]

$\Psi_{,rr}-\frac{1}{r}\Psi_{,r}+\Psi_{,zz}=0,$ (10)

where

$\Psi=\eta+c^{2}\frac{r^{2}}{2}\int\frac{H^{\prime}}{H}\frac{d\eta}{\eta}-\frac{1}{2}\int\frac{H^{\prime}}{H}\eta\,d\eta\,.$
(11)

Eq. (11) is an integrability condition for the dust geodesic equations in the
absence of pressure in the effective fluid. Furthermore, to model galaxies, we
are interested in solutions with reflection symmetry with respect to the $z=0$
plane. The solution to (10) is then given by

$\displaystyle\Psi(r,z)=\,{\mathpzc C}_{0}\;+\;$ $\displaystyle
r\sum_{n=0}^{\infty}{\mathpzc A}_{n}I_{1}({\mathpzc a}_{n}r)\cos({\mathpzc
a}_{n}z)$ $\displaystyle+r\sum_{m=0}^{\infty}{\mathpzc B}_{m}K_{1}({\mathpzc
b}_{m}r)\cos({\mathpzc b}_{m}z),$ (12)

where $I_{1}$ and $K_{1}$ are the modified Bessel functions of the first and
second kind, the constants ${\mathpzc a}_{n},{\mathpzc b}_{m}$ are positive,
whilst the constants ${\mathpzc C}_{0}$, ${\mathpzc A}_{n}$ and ${\mathpzc
B}_{m}$ are arbitrary.

The trace of the Einstein equations gives

$8\pi
G\rho=\frac{\eta_{,r}^{2}+\eta_{,z}^{2}}{4\eta^{2}r^{2}}\left[\eta^{2}\Delta^{2}(\eta)-c^{4}r^{4}\ell(\eta)^{2}\right]e^{2\Xi}\\!,$
(13)

where $\ell(\eta)=H^{\prime}/H$ and $\Delta(\eta)=2-\eta\,\ell(\eta)$, with
$H^{\prime}=dH/d\eta$, while the Raychaudhuri equation reduces to

$\dot{\Theta}\equiv
U^{\nu}\nabla_{\nu}\Theta=-\frac{1}{3}\Theta^{2}+\omega^{2}-\sigma^{2}-R_{\mu\nu}U^{\mu}U^{\nu},$
(14)

where $\Theta=\nabla_{\mu}U^{\mu}$,
$\omega^{2}=\omega^{\mu\nu}\omega_{\mu\nu}$ and
$\sigma^{2}=\sigma^{\mu\nu}\sigma_{\mu\nu}$ are the expansion, vorticity and
shear scalars respectively. Here $\omega_{\mu\nu}=U_{[\mu;\nu]}$,
$\sigma_{\mu\nu}=U_{(\mu;\nu)}-\frac{1}{3}h_{\mu\nu}\Theta$,
$h_{\mu\nu}=g_{\mu\nu}+c^{-2}U_{\mu}U_{\nu}$. The dust is non-expanding,
$\Theta=0$, $\dot{\Theta}=0$, and $R_{\mu\nu}U^{\mu}U^{\nu}=4\pi\,G\rho$, so
that (14) reduces to a balance condition

$\omega^{2}-\sigma^{2}=4\pi G\rho\,.$ (15)

Significantly, since the effective dust energy density contains quasilocal
kinetic energy, $\rho$ may be negative when shear dominates over vorticity.
Indeed,

$\sigma^{2}=\frac{e^{2\Xi}c^{4}r^{2}(\eta_{,r}^{2}+\eta_{,z}^{2})\,\ell^{2}(\eta)}{8\eta^{2}},$
(16)

which combined with (13) yields

$\omega^{2}=\frac{e^{2\Xi}(\eta_{,r}^{2}+\eta_{,z}^{2})\,\Delta^{2}(\eta)}{8r^{2}}\,.$
(17)

Physical measurements are defined by relevant classes of observers. We first
introduce the ZAMO coframe [19]

$\displaystyle{\mathbf{\omega}}_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}=\Bigl{(}\frac{r\mathop{\text{d}\\!}t}{\sqrt{g_{\phi\phi}}}\,,\sqrt{g_{\phi\phi}}$
$\displaystyle\left(\mathop{\text{d}\\!}\phi\,-\,\chi\mathop{\text{d}\\!}t\right)\,,$
$\displaystyle
e^{(k-\Phi)/c^{2}}\mathop{\text{d}\\!}r\,,e^{(k-\Phi)/c^{2}}\mathop{\text{d}\\!}z\Bigr{)}\,,$
(18)

$g_{\phi\phi}=e^{-2\Phi/c^{2}}r^{2}-c^{2}A^{2}e^{2\Phi/c^{2}}$, and
$\chi=-g_{t\phi}/g_{\phi\phi}$ with $g_{t\phi}=-c^{2}\,A\,e^{2\Phi/c^{2}}$.
The dual ZAMO tetrad frame is then

$\displaystyle{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}=\Bigl{(}\frac{1}{r}{\sqrt{g_{\phi\phi}}}$
$\displaystyle\left(\partial_{t}\,+\,\chi\partial_{\phi}\right)\,,\frac{1}{\sqrt{g_{\phi\phi}}}\partial_{\phi}\,,$
$\displaystyle\qquad
e^{-(k-\Phi)/c^{2}}\partial_{r}\,,e^{-(k-\Phi)/c^{2}}\partial_{z}\Bigr{)}\,.$
(19)

Starting from the formula for the dust particles’ velocity measured by ZAMOs,
$v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}=({{{\mathbf{e}}_{\lower
2.0pt\hbox{$\scriptstyle Z$}}}^{1}}\cdot{\mathbf{U}})/({{{\mathbf{e}}_{\lower
2.0pt\hbox{$\scriptstyle Z$}}}^{0}}\cdot{\mathbf{U}})$, we find

$\eta(r,z)=r\,v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}(r,z)\,.$ (20)

Furthermore, let us define an effective Lorentz factor $\gamma_{\lower
2.0pt\hbox{$\scriptstyle Z$}}:={{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}}^{0}\cdot{\mathbf{U}}$, so that

$-H\,\gamma_{\mathrm{Z}}^{2}\,v_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}=r\,(\Omega-\chi)\,.$ (21)

The second congruence of relevance are ideal Stationary Observers (SOs), with
coframe [20]

$\displaystyle{\mathbf{\omega}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}=\Bigl{(}e^{\Phi/c^{2}}$
$\displaystyle\left(\mathop{\text{d}\\!}t\,+\,A\mathop{\text{d}\\!}\phi\right)\,,r\,e^{\Phi/c^{2}}\mathop{\text{d}\\!}\phi\,,$
$\displaystyle\qquad
e^{(k-\Phi)/c^{2}}\mathop{\text{d}\\!}r\,,e^{(k-\Phi)/c^{2}}\mathop{\text{d}\\!}z\Bigr{)}\,,$
(22)

The dual SO tetrad frame is then

$\displaystyle{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}=\Bigl{(}e^{-\Phi/c^{2}}$
$\displaystyle\partial_{t}\,,\frac{e^{\Phi/c^{2}}}{r}\left(\partial_{\phi}-A\,\partial_{t}\right)\,,$
$\displaystyle\qquad
e^{-(k-\Phi)/c^{2}}\partial_{r}\,,e^{-(k-\Phi)/c^{2}}\partial_{z}\Bigr{)}\,.$
(23)

By analogy to the ZAMO case for SOs we define,$v_{\lower
2.0pt\hbox{$\scriptstyle S$}}=({{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}^{1}}\cdot{\mathbf{U}})/({{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}^{0}}\cdot{\mathbf{U}})$ and $\gamma_{\lower 2.0pt\hbox{$\scriptstyle
S$}}:={{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}}^{0}\cdot{\mathbf{U}}$, so that

$r\,v_{\lower 2.0pt\hbox{$\scriptstyle
S$}}(r,z)=\gamma_{\mathrm{S}}^{-1}e^{-\Phi/c^{2}}r^{2}\Omega\,.$ (24)

We can define $\eta_{\lower 2.0pt\hbox{$\scriptstyle S$}}=v_{\lower
2.0pt\hbox{$\scriptstyle S$}}r$ by analogy to (20). However, it plays no
particular role here. Equivalently, we have

$v_{\lower 2.0pt\hbox{$\scriptstyle S$}}-v_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}=\frac{{{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}^{1}}\cdot{{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}^{0}}}{{{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
S$}}^{0}}\cdot{{\mathbf{e}}_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}^{0}}}=\frac{g_{\phi\phi}}{r^{2}}r\chi=\frac{-g_{t\phi}}{r},$ (25)

so that

$e^{2\Phi/c^{2}}\gamma_{\lower 2.0pt\hbox{$\scriptstyle S$}}v_{\lower
2.0pt\hbox{$\scriptstyle S$}}=-H\,\gamma_{\mathrm{Z}}^{2}\,(v_{\lower
2.0pt\hbox{$\scriptstyle Z$}}+r\,\chi)\,.$ (26)

Eq. (25) shows that ZAMOs and SOs measure differentdust velocities at any
spacetime event, with a difference directly proportional to the frame-dragging
contribution.

In the present framework it is clear how observations have been misapplied to
theoretical observables in the past. In particular, several analyses [1, 2, 3]
misapply $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}$ to the rotation curves of
distant galaxies, when $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$ given by (24)
is the relevant velocity, since

$\frac{v_{\lower 2.0pt\hbox{$\scriptstyle
S$}}}{c}=\frac{\sqrt{1+\left(2e^{2\Phi/c^{2}}r\Omega/c\right)^{2}}-1}{2e^{2\Phi/c^{2}}r\Omega/c}\simeq\frac{r\Omega}{c}e^{2\Phi/c^{2}}+\dots$
(27)

coinciding, at leading order, with the widely used special relativistic
interpretation of the redshift.

To complete our identification of physically relevant velocities in the
general case we supplement the velocities $v_{\lower 2.0pt\hbox{$\scriptstyle
Z$}}$ and $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$ by the kinetic and
dragging velocities [21, 18]

$\displaystyle v_{\lower 2.0pt\hbox{$\scriptstyle K$}}:=r\,\Omega\,,$
$\displaystyle v_{\lower 2.0pt\hbox{$\scriptstyle D$}}:=r\,\chi\,,$ (28)

respectively. While $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$ and $v_{\lower
2.0pt\hbox{$\scriptstyle K$}}$ coincide at low velocities, in general their
differences can be physically important.

Let us now derive the functional form applicable to systems such as galaxies,
with subrelativistic local relative speeds, $v\ll c$, weak pseudo-Newtonian
potentials, $\Phi\mathop{\sim}\limits v^{2}/c^{2}$, and nonrelativistic frame-
dragging.

From these conditions, we find $A=r\,v_{\lower 2.0pt\hbox{$\scriptstyle
D$}}/c^{2}+\mathcal{O}({v^{3}/c^{3}})$. Thus by (4),
$H=-1+\mathcal{O}({v^{2}/c^{2}})$. Since $H=H(\eta)$, using (20) and noting
that $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}=v_{\lower
2.0pt\hbox{$\scriptstyle K$}}-v_{\lower 2.0pt\hbox{$\scriptstyle
D$}}+\mathcal{O}(v^{3}/c^{3})$, we find

$H(\eta)=-1+\epsilon\,\eta^{2}/(b_{\epsilon}c)^{2}+\mathcal{O}(v^{3}/c^{3})\,,$
(29)

where $\epsilon\in\\{-1,0,1\\}$ and $b_{\epsilon}$ are constant lengths which
can be identified as impact parameters. We stress that the functional form of
$H(\eta)$ in (29) is the only possible choice that allows a dust system to be
consistently considered in a low-energy regime. Substituting just the first
two terms of (29) into (11) we obtain

$\eta(r,z)={b_{\epsilon}c}{\mathop{\text{\,tann}\\!}}\left({\frac{b_{\epsilon}\,\Psi(r,z)}{b_{\epsilon}^{2}c-\epsilon\,r^{2}c}}\right)\,,$
(30)

where

${\mathop{\text{\,tann}\\!}}\,(x)=\begin{cases}\tanh(x),&\epsilon=+1\\\
x,&\epsilon=0\\\ \tan(x),&\epsilon=-1\\\ \end{cases}.$ (31)

Moreover, by (5) it follows that

$\Omega=\Omega_{0}+\epsilon\frac{\eta}{b_{\epsilon}^{2}}.$ (32)

Furthermore, to ensure zero rotation on the symmetry axis we choose
$\Omega_{0}=0$. We thus find

$v_{\lower 2.0pt\hbox{$\scriptstyle
K$}}=\epsilon\frac{rc}{b_{\epsilon}}\,{\mathop{\text{\,tann}\\!}}\left({\frac{b_{\epsilon}\,\Psi(r,z)}{b_{\epsilon}^{2}c-\epsilon\,r^{2}c}}\right)\,,$
(33)

and

$v_{\lower 2.0pt\hbox{$\scriptstyle
D$}}=c\left(\frac{r}{b_{\epsilon}}\epsilon-\frac{b_{\epsilon}}{r}\right)\,{\mathop{\text{\,tann}\\!}}\left({\frac{b_{\epsilon}\,\Psi(r,z)}{b_{\epsilon}^{2}c-\epsilon\,r^{2}c}}\right).$
(34)

The presence of differential rotation is crucial mathematically since the term
$\epsilon r^{2}c$ in (30) regularizes potential divergences at infinity in
(Exact solutions for differentially rotating galaxies in general relativity).
In particular, we can choose solutions to (10) which are regular about the
$z$-axis and diverge as $\Psi\mathop{\sim}\limits r^{\alpha}$, $\alpha<2$ as
$r\to\infty$. All these cases are phenomenologically interesting as the sub-
quadratic divergences in the modified Bessel function series $I_{1}({\mathpzc
a}_{n}r)$ in (Exact solutions for differentially rotating galaxies in general
relativity), are thereby regularized. This option was unavailable in the
rigidly rotating Balasin-Grumiller (BG) model [3] where only the modified
Bessel function series $K_{1}({\mathpzc b}_{n}r)$ in (Exact solutions for
differentially rotating galaxies in general relativity) was chosen, being
regular as $r\to\infty$ but singular on the $z$-axis. Thus, we find solutions
that display self-consistency in the low-velocity regime.

By (13) and (29) in each limit the effective density is

$8\pi
G\rho_{\epsilon}=\frac{\eta_{,r}^{2}+\eta_{,z}^{2}}{2r^{2}}\left[\frac{b_{\epsilon}^{4}-\epsilon^{2}r^{4}}{\left(b_{\epsilon}^{2}-\eta^{2}/c^{2}\right)^{2}}\right]e^{2\Xi}\,,$
(35)

which vanishes on the cylinders $r=b_{\epsilon}$, independent of the choice of
solution to (10). The global structure of the spacetime is now determined by:
(i) choosing $\rho=\rho_{+}-\rho_{-}$, $b_{+}>b_{-}>0$ with an
$H(b_{+},b_{-},\eta)=-1+f(b_{+},b_{-})\eta^{2}$ such that $\lim_{b_{-}\to
0}f=1/(b_{+}c)^{2}$ and $\lim_{b_{-}\to\infty}f=-1/(b_{-}c)^{2}$; (ii)
requiring consistent coupling of quasilocal energy and angular momentum in the
ultraviolet and infrared limits that bound the interior and exterior
respectively. The ultraviolet limit involves the relevant physics of the
coarse-grained matter source, whilst the infrared limit involves the approach
to asymptotic infinity far from the compact source. While $r=b_{-}$ and
$r=b_{+}$ are precisely defined boundaries, a smooth transition from the
virial interior to exterior involves some phenomenological freedom. See Table
1.

Spacetime region | $\vphantom{\Big{|}}v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}$ | $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$ | $\rho$
---|---|---|---
UV limit: Vortex | $\vphantom{\Big{|}}\epsilon=-1,r<b_{-}$ | $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}<0$ | $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}>0$ | $\rho<0$
Virial interior | $\vphantom{\Big{|}}\epsilon=-1,r>b_{-}$ | $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}<0$ | $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}>0$ | $\rho>0$
Virial exterior | $\vphantom{\Big{|}}\epsilon=+1,r<b_{+}$ | $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}>0$ | $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}>0$ | $\rho>0$
IR limit: Virialized | $\vphantom{\Big{|}}\epsilon=+1,r>b_{+}$ | $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}<0$ | $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}<0$ | $\rho<0$
Table 1: ZAMO and SO velocities, $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}$ and
$v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$, and effective compact quasilocal
energy density, $\rho$.

There are only two cases in which $\Theta=0$, $\omega=\sigma$ and $\rho=0$;
i.e., the dust congruence has zero expansion, vorticity and shear exactly
cancel, and the net quasilocal energy density, $4\pi
G\rho=R_{\mu\nu}U^{\mu}U^{\nu}$, vanishes.

In the interior ($\epsilon=-1$), $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}$ and
$v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$ are antiparallel, since to maintain
zero angular momentum a ZAMO must counter the dragging of the central
rotation. The inner vortex surface arises when maintain this equilibrium
requires the net quasilocal energy density to change sign, becoming shear-
dominated. On the vortex boundary

$g_{\phi\phi}=b_{-}^{2}\left(1-\frac{1}{2}\frac{\Psi(r,z)^{2}}{c^{2}b_{-}^{2}}\right)+\mathcal{O}(v^{4}/c^{4})\,,$
(36)

by (7), (30), and other metric components are also regular, all relative
velocities, $v$, being subrelativistic. The inner vortex surface is found at
$r=b_{-}\gg L$, where $L$ is a typical scale length of the coarse-grained
dust. E.g., for the Milky Way, $L\mathop{\sim}\limits 50\,$kpc. This scale is
not unique, and depends crucially on phenomenological interpretation. Crosta
et al. [22, 23] used the BG model to fit a rotation curve to the Milky Way, as
inferred from the proper motion and redshifts of a sample of $7.2\times
10^{6}$ young stars from the GAIA–DR3 survey, lying within $r<19\,$kpc of the
galactic centre and $-1<z<1\,$kpc about the galactic plane. By (30) for large
values of the impact parameter, $b$, almost rigid rotation is obtained. This
explains why the $K_{1}({\mathpzc b}_{n}r)$ series alone can be fit to
observations close to the galactic plane and far from its centre. However, the
BG model will inevitably fail globally on account of its axial singularity,
which we sidestep via the differential rotation regularization term.

In the exterior ($\epsilon=+1$), $v_{\lower 2.0pt\hbox{$\scriptstyle Z$}}$ and
$v_{\lower 2.0pt\hbox{$\scriptstyle S$}}$ are parallel. At large spatial
distances $v_{\lower 2.0pt\hbox{$\scriptstyle S$}}-v_{\lower
2.0pt\hbox{$\scriptstyle Z$}}$ does not vanish in general, even for
asymptotically flat spacetimes. Their difference embodies the quasilocal
angular momentum that integrates to a global asymptotic charge in the case of
the Kerr geometry. However, the observed universe has zero global angular
momentum. To enable direct embedding into a nonrotating cosmological
background, differentially rotating distributed sources should consistently
couple quasilocal energy and angular momentum.

Our class of exact solutions does indeed admit such geometries, with a
bounding surface which we hereby name the rotosurface. It has the following
novel geometric properties. In the limit $r\to b_{+}$ at finite $z$, even
though the Killing vector norms diverge, the vorticity, shear, Ricci and
Kretschmann scalars converge to zero:
$\omega^{2}\mathop{\sim}\limits\sigma^{2}\mathop{\sim}\limits
R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\mathop{\sim}\limits
e^{2\Xi}\Psi^{2}/(8|b_{+}-r|^{4}])\to 0$, as $\Psi$ is finite and
$e^{2\Xi}\mathop{\sim}\limits e^{-\Psi^{2}/(bc^{2}|b_{+}-r|)}$ as $r\to
b_{+}$.

The rotosurface is a new mathematical object, whose topology, geometry,
fundamental physics and cosmological implications remain to be fully explored.
Analogously to an acceleration horizon for uniform linear acceleration in
Minkowski space, it is a limiting rotating surface where a fictitious observer
would need to rotate with $v\to c$ in order to have zero angular momentum with
respect to the vanishing local dust. However, it is a timelike surface.
Indeed, it provides a precise mathematical definition of one type of finite
infinity surface [6, 24, 25]. Furthermore, since relevant curvature scalars
vanish it provides a new mathematical setting for the concept of
virialization, and the nomenclature adopted in Table 1.

Beyond the rotosurface other quasilocal energy sources will dominate: the
effective thermal pressure of galaxies in clusters and the kinetic energy of
expansion in a void dominated universe [26]. The fact that the quasilocal
energy of the isolated system is negative beyond the rotosurface is consistent
with: (i) positive spatial curvature energy and binding energy being negative
relative to the background; (ii) a small kinematical backreaction,
$\Omega_{\cal Q}$, of virialized structures that is negative in the Buchert
averaging scheme [27, 28].

Since the new solutions successfully exhibit the essential physics of
differential rotation they naturally apply to rotationally supported systems,
e.g., disc galaxies. A sample exact solution for a Milky Way-like galaxy with
a disc mass of $10^{11}{\rm M}_{\odot}$ and rotosurface at $700\,$kpc (Fig. 1)
produces a realistic rotation curve (Fig. 2) consistent with our quasilocal
Newtonian limit. The interplay of central bulges and/or dark matter halos is
more complex. Nonetheless, having uncovered a consistent incorporation of the
low-energy regime in macroscopic general relativity, extensions to pressure-
supported systems follow naturally [29]. While the scales of globular
clusters, dwarf galaxies and virialized galaxy clusters are different—each
involving different particles—the underlying equations are universal.

Finally, we note that since the rotosurface is regular but timelike it may
provide a novel mathematical arena for understanding the asymptotic charges of
the Bondi–Metzner–Sachs group.

Figure 1: Angular momentum density for a Milky Way-like galaxy. The
rotosurface is found at 700 kpc. Figure 2: Rotation velocity curve for the
Milky Way-like galaxy of Fig. 1. The virial interior and exterior of Table 1
are found beyond the scales shown.

Acknowledgments DLW is supported by Marsden Fund grant M1271 administered by
the Royal Society of New Zealand, Te Apārangi. We thank Roy Kerr,Federico Re,
and Chris Stevens for important physical and mathematical insights that helped
us refine our understanding. We thank John Forbes, Christopher Harvey-Hawes,
Morag Hills, Emma Johnson, Zachary Lane, Antonia Seifert, Shreyas Tiruvaskar
and Michael Williams for useful discussions, and Frederic Hessman for
correspondence.

## References

  * Cooperstock and Tieu [2007] F. I. Cooperstock and S. Tieu, Int. J. Mod. Phys. A 22, 2293 (2007).
  * Cooperstock and Tieu [2008] F. I. Cooperstock and S. Tieu, Mod. Phys. Lett. A 23, 1745 (2008).
  * Balasin and Grumiller [2008] H. Balasin and D. Grumiller, Int. J. Mod. Phys. D 17, 475 (2008).
  * Vogt and Letelier [2005] D. Vogt and P. S. Letelier, arXiv:0510750 (2005).
  * Neill [2011] C. Neill, BSc Honours report, doi:10.26021/11331, U. Canterbury (2011).
  * Wiltshire [2007] D. L. Wiltshire, New J. Phys. 9, 377 (2007).
  * Wiltshire [2011] D. L. Wiltshire, Class. Quantum Grav. 28, 164006 (2011).
  * Ellis [1984] G. F. R. Ellis, in  _General Relativity and Gravitation Conference_ , edited by B. Bertotti, F. de Felice, and A. Pascolini (Reidel, Dordrecht, 1984) pp. 215–288.
  * Giani  _et al._ [2022] L. Giani, O. F. Piattella, and A. Y. Kamenshchik, J. Cosmol. Astropart. Phys. JCAP 03, 028 (2022).
  * Hills [2022] M. A. Hills, MSc thesis, doi:10.26021/12861, U. Canterbury (2022).
  * O’Keeffe [2023] F. C. O’Keeffe, MSc thesis, doi:10.26021/14659, U. Canterbury (2023).
  * Giani  _et al._ [2024] L. Giani, C. Howlett, K. Said, T. Davis, and S. Vagnozzi, J. Cosmol. Astropart. Phys. JCAP 01, 071 (2024).
  * Seifert [2024] A. Seifert, arXiv:2405.04933 (2024).
  * Geroch [1971] R. Geroch, J. Math. Phys. 12, 918 (1971).
  * Geroch [1972] R. Geroch, J. Math. Phys. 13, 394 (1972).
  * Winicour [1975] J. Winicour, J. Math. Phys. 16, 1806 (1975).
  * Ruggiero [2024] M. L. Ruggiero, J. Cosmol. Astropart. Phys. JCAP 02, 025 (2024).
  * Re and Galoppo [2024] F. Re and M. Galoppo, arXiv:2403.03227 (2024).
  * Bardeen  _et al._ [1972] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972).
  * Costa  _et al._ [2023] L. F. O. Costa, J. Natário, F. Frutos-Alfaro, and M. Soffel, Phys. Rev. D 108, 044056 (2023).
  * Astesiano  _et al._ [2022] D. Astesiano, S. Cacciatori, M. Dotti, F. Haardt, and F. Re, arXiv:2204.05143 (2022).
  * Crosta  _et al._ [2020] M. Crosta, M. Giammaria, M. Lattanzi, and E. Poggio, Mon. Not. R. Astr. Soc. 496, 2107 (2020).
  * Beordo  _et al._ [2024] W. Beordo, M. Crosta, M. Lattanzi, P. Re Fiorentin, and A. Spagna, Mon. Not. R. Astr. Soc. 529, 4681 (2024).
  * Wiltshire [2008] D. L. Wiltshire, Phys. Rev. D 78, 084032 (2008).
  * Wiltshire [2014] D. L. Wiltshire, in  _Proceedings of the XVth Brazilian School of Cosmology and Gravitation_ , edited by S. Perez Bergliaffa and M. Novello (Cambridge Scientific Publishers, 2014) pp. 203–244, arXiv:1311.3787 .
  * Williams  _et al._ [2024] M. J. Williams, H. J. Macpherson, D. L. Wiltshire, and C. Stevens, arXiv:2403.15134 (2024).
  * Buchert [2000] T. Buchert, Gen. Relativ. Gravitation 32, 105–125 (2000).
  * Buchert  _et al._ [2020] T. Buchert, P. Mourier, and X. Roy, Gen. Relativ. Gravitation 52 (2020), 10.1007/s10714-020-02670-6.
  * Galoppo  _et al._ [2024] M. Galoppo _et al._ , In preparation (2024).