# Re-weighting dark matter in disc galaxies: a new general relativistic
observational test

D. Astesianoa,b, S.L. Cacciatori a,b, M. Dottic,d, F. Haardta,d and F. Rea,b,c
a Department of Science and High Technology, Università dell’Insubria, Via
Valleggio 11, 22100, Como, Italy
b INFN, sezione di Milano, Via Celoria 16, 20133, Milano, Italy
c Department of Physics Giuseppe Occhialini, Università di Milano-Bicocca,
Piazza della Scienza 3, 20126, Milano, Italy
d INFN, sezione di Milano-Bicocca, Piazza della Scienza 3, 20126, Milano,
Italy

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A recent analysis of data from the ESA Gaia mission demonstrated that the
kinematics of stars in the Milky Way can be modelled without invoking the
presence of dark matter whatsoever. Indeed, the higher-than-Keplerian
velocities observed in outer stars can be ascribed to the properties of the
general relativistic (GR) metric assumed to describe the Galaxy. Here we
generalize the concept, and derive the most general exact GR model for a
stationary axi-symmetric dynamically cold dust structure. We explicitly show
how deviations from the commonly adopted Newtonian dynamics are indeed
manifest even at low velocities and low densities. We provide for the first
time a detailed description of the frequency shift experienced by photons
travelling from any emission site within an external disc galaxy to the
observer, relating the outcome of the shift measurement to the gravitational
properties of the galaxy. Finally, we devise a novel, groundbreaking
observational test potentially able to fully characterize the GR metric under
the minimal set of assumptions mentioned above. The proposed experiment
exploits the effects non-diagonal GR terms have on the frequency shift of
photons, ultimately providing a test to evaluate whether dark matter is
actually required by disc galaxy kinematics.

###### keywords:

Galaxy:kinematics and dynamics– Galaxy:general – dark matter

††pubyear: 2022††pagerange: Re-weighting dark matter in disc galaxies: a new
general relativistic observational test–Re-weighting dark matter in disc
galaxies: a new general relativistic observational test

##  1 Introduction and presentation of the results

The dark matter (DM) hypothesis, i.e., the hypothesis of the existence of a
non-baryonic component of mass dominating the matter density budget in the
Universe, is one of the foundations of the widely accepted cosmological model
(e.g. Planck Collaboration et al., 2020) and, at the same time, one of the
greatest unknown in physics. The DM hypothesis has been incredibly successful
in interpreting different astrophysical observables, such as the velocity
distribution of galaxies in galaxy clusters (GCs,, since Zwicky, 1933), the
rotation curves (RCs) of disc galaxies (e.g. Corbelli & Salucci, 2000), the
thermodynamical properties of X-ray emitting gas in GCs (Allen et al., 2011)
and the gravitational lensing produced by their mass distributions (Taylor,
1999), as well as cosmological ones, e.g., the anisotropies observed in the
cosmic microwave background (Hinshaw et al., 2009; Planck Collaboration et
al., 2016) and the growth of cosmic structures from such anisotropies (Conroy
et al., 2006; Piattella, 2018). However, despite such remarkable successes,
results of the many enterprises aimed at the direct detection of DM particles
are, to date, inconclusive (e.g., Bertone & Hooper, 2018; Liu et al., 2017;
Undagoitia & Rauch, 2015).

Because of the DM relevance in our understanding of the laws of nature, it is
then of paramount importance to fully characterise such elusive component of
our Universe, making sure to maximise the outcome of the available indirect
astrophysical and cosmological tests. Indeed, the characterization of the
nature of DM through the study of the large scale distribution of galaxies is
one of the scientific drivers of the medium-class ESA mission Euclid (Laureijs
et al., 2011; Borlaff, 2022). With the present study we demonstrate that a
fully general relativistic (GR) description of individual disc galaxies can
also be used to improve considerably our estimates of their DM content.

As mentioned above, the current estimates of the DM content in disc galaxies
are based on the observed rotation curves, whose flatness at large radii
cannot be explained by the distribution of visible (baryonic) matter alone.
This assertion, however, relies on a purely Newtonian description of the
system. The simplifying assumption of Newtonian gravity is commonly justified
by the sub-relativistic velocity of the stars in galaxies, and by the weak
gravitational potential involved.

It has been noticed, however, that even in these limits GR corrections may be
not-negligible, and, as a consequence, relativistic corrections to Newtonian
dynamics (Ruggiero et al., 2021; Ludwig, 2021), as well as an exact GR model
(Balasin & Grumiller, 2008) have been proposed. All these models are static
and axi-symmetric. The exact model by Balasin & Grumiller (2008) has
additional simplifying assumptions, considering only dust with zero velocity
dispersion, zero mean velocity in the radial and vertical direction and null
deformation tensor ($H=-1$ in the following, see section 2 and Astesiano et
al. 2021). Despite these additional assumptions, the model has been shown to
reproduce the kinematic data obtained by Gaia for stars in the Milky Way
without invoking any additional component other than visible matter (Crosta et
al., 2020).

In the present work we continue the generalization of the Balasin & Grumiller
(2008) model started in Astesiano et al. (2021). While preserving the
assumption of axi-symmetry, we relax the assumption of null deformation
tensor, and derive, for the first time, the frequency shift of a generic
spectral line emitted in our GR galaxy model. Making use of such observable,
we devise an observational test that could, in principle, fully characterize
the metric of any axi-symmetric disc galaxy, therefore providing an accurate
estimate of the DM content needed to fit radial velocity profiles.

The outline of the work is as follows: in §2, we describe the GR model for
disc galaxy assuming a stationary and axisymmetric metric filled with dust
111With the term “dust” we refer here to the GR definition, i.e., the energy
momentum tensor of the matter is given by (3), with zero pressure and zero
stress.; in §3 we provide general formulae for the description of the motion
of the dust; in §4, we calculate the frequency shift that a photon undergoes
from the emission in the galaxy until its observation by an asymptotic
inertial observer, using the eikonal approximation; the main observational
test is presented in §5; finally, our conclusions are presented in §6. The
reader uninterested in the actual derivation of the model can go directly to
§5, as the mathematical details of the GR solutions in sections §2, §3 and §4
are not necessary to understand the logic behind the observational test we are
proposing.

##  2 Preliminary set up

In the derivation of the metric any internal symmetry-breaking structure, such
as spirals or stellar bars, as well as any component with significant velocity
dispersion such as, e.g., stellar bulges, are neglected in favor of the
analytic results. Moreover, we consider the velocity dispersion in the disc to
be negligible compared to the main global rotational motion. Unlike other
general-relativistic galaxy models in the literature (e.g., Balasin &
Grumiller, 2008; Cooperstock & Tieu, 2005) we do not assume a rigidly rotating
dust; conversely, we consider the most general differential rotation which is
allowed by stationarity and axisymmetry.

It is well known (Hansen & Winicour, 1977; Misner et al., 1973) that a
stationary, axisymmetric metric can be written in the form

$\displaystyle ds^{2}=$
$\displaystyle-e^{2U(r,z)}\left(dt+A(r,z)d\phi\right)^{2}+$ (1)
$\displaystyle+$ $\displaystyle
e^{-2U(r,z)}\left[r^{2}d\phi^{2}+e^{2k(r,z)}\left(dr^{2}+dz^{2}\right)\right].$
(2)

Indeed, stationarity and axial symmetry ensure that it is possible to choose
local coordinates such that the coefficients of the metric are independent
from the time coordinate $t$ and from the angle $\phi$ of rotation around the
symmetry axis. Non staticity implies that in such coordinates the metric is
non diagonal, with a $2g_{t\phi}(r,z)dtd\phi$ term. After a suitable choice of
the coordinates the elements of the metric can be then expressed in terms of
three arbitrary functions of $r$ and $z$. These are the functions $A(r,z)$
which describes the rotational properties (i.e., non staticity), $U(r,z)$
which essentially measures the time lapses, and $k(r,z)$ which is related to
spatial scales.

As in Stephani et al. (2009), we consider when gravity is coupled to dust
through the stress-energy tensor

$\displaystyle T^{\mu\nu}=\rho(r,z)u^{\mu}u^{\nu},$ (3)

where the mass density $\rho$ and four velocity $u$ are

$\displaystyle
u=\frac{1}{\sqrt{-H}}\left(\partial_{t}+\Omega(r,z)\partial_{\phi}\right).$
(4)

Here $\Omega$ represents the angular velocity of the dust with respect to
coordinates $t,\phi$. The function $H$ is defined by the condition222Note that
$H$ is defined as negative, consistent with the notation used in Stephani et
al. (2009)

$\displaystyle-1=g_{\mu\nu}u^{\mu}u^{\nu}=-\frac{1}{H}(g_{tt}+\Omega^{2}g_{\phi\phi}+2g_{t\phi}\Omega).$
(5)

Therefore,

$\displaystyle
H=-e^{2U(r,z)}\left(1+A(r,z)\Omega(r,z)\right)^{2}+e^{-2U(r,z)}r^{2}\Omega(r,z)^{2}$
(6)

is a well defined combination of the coefficients of the metric.

The equations of motion are completely integrable and depend on the choice of
arbitrary solutions of the equations of motion, as shown below. Therefore the
Einstein equations do not fix $H$ completely, rather they imply that $H$ must
be an a priory arbitrary function of another function $\eta(r,z)$. The
importance of passing through $\eta$ is that it has a direct physical
interpretation, as we will see in (16).

Once the negative function $H(\eta)$ is chosen, we introduce the auxiliary
function333For any function of one variable, like $H(\eta)$, we use a prime
sign for the derivative w.r.t. the variable.

$\mathcal{F}(r,z)=2\eta+r^{2}\int\frac{H^{\prime}}{H}\frac{d\eta}{\eta}-\int\frac{H^{\prime}}{H}\eta
d\eta,$ (7)

so then $\eta$ is set by the following equation:

$\mathcal{F}_{,rr}-\frac{1}{r}\mathcal{F}_{,r}+\mathcal{F}_{,zz}=0,$ (8)

where comma sign indicates partial derivative w.r.t. the specified coordinate.
In a practical procedure, it is convenient to solve first the latter equation
for $\mathcal{F}$ and then use (7) as a functional equation to determine
$\eta$ and then $H$.

Once $H(\eta)$ and the solution to eq.(8) are known, we are in the position to
determine the metric components:

$\displaystyle g_{tt}$
$\displaystyle=\frac{(H-\eta\Omega)^{2}-r^{2}\Omega^{2}}{H},$ (9)
$\displaystyle g_{t\phi}$ $\displaystyle=\frac{r^{2}-\eta^{2}}{H}\Omega+\eta,$
(10) $\displaystyle g_{\phi\phi}$ $\displaystyle=\frac{r^{2}-\eta^{2}}{(-H)},$
(11) $\displaystyle\Omega$ $\displaystyle=\frac{1}{2}\int
H^{\prime}\frac{d\eta}{\eta}.$ (12)

We remark that $\Omega$ is determined by an indefinite integral. Therefore $H$
equal to a constant would imply a constant $\Omega$, non necessarily equal to
zero, . Finally, the remaining metric components are

$\displaystyle g_{zz}=g_{rr}=e^{2(k-U)}\equiv e^{\mu},$ (13)

where the function $\mu$ is determined by

$\displaystyle\mu_{,r}=$
$\displaystyle\frac{1}{2r}\left[(g_{tt})_{,r}(g_{\phi\phi})_{,r}-(g_{tt})_{,z}(g_{\phi\phi})_{,z}-((g_{t\phi})_{,r})^{2}+((g_{t\phi})_{,z})^{2}\right],$
(14) $\displaystyle\mu_{,z}=$
$\displaystyle\frac{1}{2r}\left[(g_{tt})_{,z}(g_{\phi\phi})_{,r}+(g_{tt})_{,r}(g_{\phi\phi})_{,z}-2(g_{t\phi})_{,r}(g_{t\phi})_{,z}\right].$
(15)

The role played by the function $\eta$ is clear when we consider that the tri-
velocity of the dust $v$ as measured from Zero Angular Momentum Observers
(ZAMOs444ZAMOs, consistently, satisfy the requirement of zero angular
momentum, i.e., $g_{\phi\phi}\,\chi+g_{\phi t}=0$., see section 3) results to
be

$\displaystyle v(r,z)=\frac{\eta(r,z)}{r},$ (16)

in natural units (so that the speed of light is $c=1$). For this reason it is
also convenient to rewrite the constraint $(\ref{harm})$ directly as a
functional equation for the velocity $v(r,z)$, i.e.,

$\displaystyle 0=$ $\displaystyle
l^{\prime}\left(\frac{1}{v}-v\right)[r^{2}v_{,z}^{2}+(rv_{,r}+v)^{2}]+$ (17)
$\displaystyle+l\left[\left(\frac{1}{v}-v\right)(rv_{,zz}+rv_{,rr}+3v_{,r})-\left(\frac{1}{v}+v\right)\frac{r}{v}(v_{,z}^{2}+v_{,r}^{2})+\frac{2}{r}\right]+$
(18)
$\displaystyle+2\left(v_{,zz}+v_{,rr}+\frac{v_{,r}}{r}-\frac{v}{r^{2}}\right),$
(19)

where $l=H^{\prime}/H$ and $H^{\prime}=dH/d\eta$.

In general, $H$ can be assumed a priory, then eq. 17 is a differential
equation for the field $v(r,z)$. However, in our context it would be relevant
to determine $H$ through, e.g., comparison with observational data. In this
case, the above eq. 17 becomes a (rather cumbersome) functional equation.

Finally, the matter density is

$8\pi G\rho=\left(\frac{v^{2}(2-\eta
l)^{2}-r^{2}l^{2}}{4g_{rr}}\right)\left(\frac{\eta_{,r}^{2}+\eta_{,z}^{2}}{\eta^{2}}\right),$
(20)

and the metric components satisfy the following constraints:

$\displaystyle-\det(g_{ij})_{i,j=t,\phi}:=g_{t\phi}^{2}-g_{tt}g_{\phi\phi}$
$\displaystyle=r^{2}.$ (21)

##  3 Considerations on the assumed metric and on the angular momentum of test
particles

The assumed metric (eq. 1) can be recasted into the following form,

$\displaystyle ds^{2}$
$\displaystyle=\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2\nu}dt^{2}+e^{2\psi}\left(d\phi-\chi
dt\right)^{2}+e^{\mu}\left(dr^{2}+dz^{2}\right),$ (22)

where

$\displaystyle e^{2\nu}$ $\displaystyle=\frac{r^{2}}{g_{\phi\phi}},\quad
e^{2\psi}=g_{\phi\phi},\quad\chi\equiv-\frac{g_{t\phi}}{g_{\phi\phi}}=\Omega+\frac{H\eta}{(r^{2}-\eta^{2})}.$
(23)

allowing us to define a tetrad corresponding to ZAMOs, i.e.,

$\displaystyle e^{0}=\frac{r}{\sqrt{g_{\phi\phi}}}dt,\qquad
e^{1}=\sqrt{g_{\phi\phi}}(d\phi-\chi dt),$ (24) $\displaystyle\qquad
e^{2}=e^{\mu/2}\,dr,\qquad e^{3}=e^{\mu/2}\,dz.$ (25)

The congruence of the wordlines associated to ZAMOs carries an angular
velocity $\chi=\chi(r,z)$ as measured from a Minkowskian observer at infinity.
Equation (23) combined with (16) can be rewritten in the more meaningful way

$\displaystyle r(\Omega-\chi)=-H\frac{v}{1-v^{2}}.$ (26)

The above expression states that $v$ is a measure of the rotation velocity,
$\omega=\Omega-\chi$ represents the relative angular velocity respect to “non
rotating” ZAMOs, so, for weak field ($H\sim-1$) and non relativistic
velocities, it takes the form $r\omega=v$. As an example, in the rigidly
rotating model by Balasin & Grumiller (2008), defined by $H=-1$ and
$\Omega=\Omega_{0}$ as constant, eq. (26) implies that the true physical
motion (relative to ZAMOs) is given by $v(r,z)$. It follows that the
“gravitomagnetic” term $\chi$ is

$\displaystyle\chi(r,z)=\Omega_{0}-\frac{v(r,z)}{r}\frac{1}{1-v^{2}(r,z)}=\Omega_{0}-\frac{v(r,z)}{r}+O(v^{2}).$
(27)

Then, although the angular velocity measured from an asymptotic inertial
observer $\Omega_{0}$ is constant, $v(r,z)$ has a non trivial profile and,
consistently with GR, the motion of matter is not rigid in a strict sense.

Let us now consider the motion of a test mass having four velocity

$\displaystyle\vartheta=\vartheta^{t}\partial_{t}+\vartheta^{r}\partial_{r}+\vartheta^{z}\partial_{z}+\vartheta^{\phi}\partial_{\phi}.$
(28)

In the given background there are two conserved quantities for such motion,
i.e.,

$\displaystyle E=-g_{0\mu}\vartheta^{\mu},\quad M=g_{\phi\mu}\vartheta^{\mu},$
(29)

clearly corresponding to the conservation of energy and of the $z$-component
of the angular momentum. From their expressions, we can then write

$\displaystyle\vartheta^{t}=$
$\displaystyle\frac{Eg_{\phi\phi}+Lg_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}=\frac{g_{\phi\phi}}{r^{2}}\left(E-\chi
M\right),$ (30) $\displaystyle\vartheta^{\phi}=$
$\displaystyle-\frac{Eg_{t\phi}+Lg_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}=\frac{g_{\phi\phi}}{r^{2}}\left(E-\chi
M\right)\chi+\frac{M}{g_{\phi\phi}},$ (31)

which, from $\Omega=\vartheta^{\phi}/\vartheta^{t}$, allows us to write the
physical angular velocity $\omega$ as

$\displaystyle\omega=\Omega-\chi=\frac{H^{2}}{r^{2}(1-v^{2})^{2}}\frac{M}{(E-\chi
M)}.$ (32)

The condition $g(\vartheta,\vartheta)=-1$ implies

$\displaystyle
g_{zz}(\vartheta^{z})^{2}+g_{rr}(\vartheta^{r})^{2}+1+\frac{M}{g_{\phi\phi}}-\frac{g_{\phi\phi}}{r^{2}}.(E-M\chi)^{2}=0,$
(33)

The above equation would be the same for the motion of a particle in an
effective potential $V_{\text{eff}}$ given by (see Herrera-Aguilar et al.,
2013)

$\displaystyle
2V_{\text{eff}}=-1+\frac{HM^{2}}{r^{2}(1-v^{2})}-\frac{1-v^{2}}{H}(E-M\chi)^{2}.$
(34)

If we are interested in circular orbits, we have to impose

$\displaystyle\partial_{r}V_{\text{eff}}=0,\qquad\partial_{z}V_{\text{eff}}=0,$
(35)

which, because of (33), gives

$\displaystyle V_{\text{eff}}=0,$ (36)

hence relating $E$ and $M$ for the given circular motion. The values of the
angular velocity along circular orbits is therefore

$\displaystyle\omega=\frac{rM}{g_{\phi\phi}}\frac{1}{\sqrt{M^{2}+g_{\phi\phi}}}.$
(37)

As expected, in case $M=0$ we have a unique orbit which represent the
trajectory of the ZAMO. Replacing the relation between $\omega$ and $v$ we
obtain the following expression for $v$,

$\displaystyle v=\pm\frac{1}{\sqrt{1+\frac{g_{\phi\phi}}{M^{2}}}},$ (38)

which relates it to the geometry ($g_{\phi\phi}$) and the angular momentum.

##  4 Frequency shift of light from disc galaxies

We will now determine the frequency shift of a photon emitted by a particle of
dust, as detected by asymptotically inertial detectors. The emitters are
supposed to move in stable circular geodesic motions.555This implies that our
approximations are valid only in the regime where the velocity dispersion is
negligible with respect to the mean motion of the stars. If the dispersion is
not too large, then its effect can be simply then taken into account by
estimating the kinematic Doppler effect, without the need to account for
further gravitational corrections. We will assume that, after emission, the
photons propagate along null geodesics, thus neglecting any possible
relativistic aberration. Our results are essentially a generalization of what
is presented in Herrera-Aguilar et al. (2013). In general, the frequency shift
is the combination of gravitational and Doppler effects. After the removal of
the galaxy peculiar motion and Hubble expansion, the observed shift is usually
interpreted in pure kinematic terms. However, we will show that in a genuine
general relativistic galaxy models the gravitational frequency shift can be of
the same order of the kinematic Doppler effect.

Let’s start by considering a disc galaxy with the galactic plane tilted by an
angle $\lambda$ with respect to the line of sight. We assume conventionally
$\lambda>0$ when the galaxy appears rotating counterclockwise. We neglect the
relativistic aberration, since we are assuming that the gravitational fields
are indeed weak. Such an assumption allows us to work in eikonal approximation
Misner et al. (1973). We also assume that Hubble expansion and any possible
galaxy peculiar motion can be removed from the observed shift (see Section
4.1.3). Moreover, we assume that the detector is at rest in an asymptotically
flat region far away from the galaxy, which is assumed to be isolated, so that
the four-velocity of the detector is $\partial_{t}$ in the coordinates of (1).
Since, in a real context, observations are made from inside the Milky Way, to
consistently compare our model to observations one should also remove the
effects associated to the local properties of the Galaxy metric. Because of to
the small angular size of the observed galaxy, that ensures that all collected
photons travelled through similar paths, such correction can be treated as a
“systematic shift".

Now, assume that the photon is emitted at frequency $\omega_{e}$ by a dust
particle, and observed at frequency $\omega_{d}$, with
$1+z=\omega_{e}/\omega_{d}$ the observed redshift. The proper frequencies are

$\omega_{e,d}:=-U^{\mu}_{e,d}k_{\mu},$ (39)

where $U^{\mu}_{d}$ is the four-velocity of the detector, $U^{\mu}_{e}$ is the
four-velocity of the source, and $k_{\mu}$ is the photon’s four-momentum along
its world line. A photon moving freely on a curved background is described by
a gauge field $A^{\nu}(x^{\mu})$ obeying, in the Landau-Lorentz gauge, the
wave equation

$\Box A^{\mu}=R^{\mu}_{\nu}A^{\mu},$ (40)

where $\Box$ is the D’Alembertian acting on vector fields and $R^{\mu}_{\nu}$
is the Ricci tensor. As shown in Misner et al. (1973), for wavelengths much
smaller than the scale length of the spatial curvature, these equations can be
solved in the eikonal approximation

$\Psi(x^{\mu})=\Psi_{0}e^{i\varphi}\quad s.t.\quad d\varphi=k_{\mu}dx^{\mu},$

where we recall that the four-momentum of the photon satisfies
$k^{\mu}k_{\mu}=0.$ The time dependence of the phase is only through the term
$k_{t}x^{0}=-\omega t$, where $\omega:=-k_{t}$ is the energy of the photon
(relative to the coordinate time), which is conserved along the photon’s
trajectory. For the spatial components of the four-momentum one defines an
“effective refraction index” $\alpha$ such that

$|\vec{k}|=:\alpha\omega,$ (41)

where $|\vec{k}|$ is defined by

$|\vec{k}|^{2}=g^{\phi\phi}k_{\phi}^{2}+g^{rr}k_{r}^{2}+g^{zz}k_{z}^{2}=-\frac{g_{tt}}{r^{2}}k_{\phi}^{2}+e^{-\mu}(k_{r}^{2}+k_{z}^{2}),$

It is also convenient to use $\phi$ and $\lambda$ as spherical coordinates
with azimuthal angle $\lambda$, so that

$\displaystyle
k_{\phi}=\omega\alpha\frac{r}{\sqrt{-g_{tt}}}\sin\phi\cos\lambda,$ (42)
$\displaystyle k_{r}=\omega\alpha e^{\mu/2}\cos\phi\cos\lambda,$ (43)
$\displaystyle k_{z}=\omega\alpha e^{\mu/2}\sin\lambda.$ (44)

Finally, we introduce the angle $\theta$ of projection along the direction
$\partial_{\phi}$, such that

$\displaystyle\sin\theta=\sin\phi\cos\lambda.$ (45)

We can now determine the frequency shift $z$. Since the detector four-velocity
is $U^{\mu}_{d}=\partial_{t}$ the detected proper frequency is
$\omega_{d}\equiv\omega$. On the other hand, the four-velocity of the emitter
is given by eq, (4), so we get

$\displaystyle 1+z$
$\displaystyle=\frac{1}{\sqrt{-H}}\left(1-\Omega\frac{k_{\phi}}{\omega}\right).$
(46)

Notice that

$\displaystyle b\equiv\frac{k_{\phi}}{\omega}$ (47)

has the dimensions of a length, and can be seen as an impact parameter. The
light-like condition on the four-momentum

$\displaystyle 0=g^{\mu\nu}k_{\mu}k_{\nu}=$ $\displaystyle
g^{tt}\omega^{2}-2g^{t\phi}\omega k_{\phi}+k^{2}=$ (48) $\displaystyle=$
$\displaystyle\omega^{2}[\alpha^{2}-2g^{t\phi}\alpha\frac{r}{\sqrt{-g_{tt}}}\sin\theta+g^{tt}],$
(49)

solved for $\alpha$, gives

$\frac{k_{\phi}}{\omega}=\alpha\frac{r}{\sqrt{-g_{tt}}}\sin\theta=\frac{g_{t\phi}}{r\sqrt{-g_{tt}}}\sin\theta+\sqrt{\frac{r^{2}-g_{t\phi}^{2}\cos^{2}\theta}{-r^{2}g_{tt}}},$

where the positive sign for the square root has been fixed taking into account
that $\alpha$ is positive.666We are assuming that $\partial_{t}$ is always
timelike, that is, we are considering only solutions where $g_{tt}$ and
$g_{\phi\phi}$ never changes in sign. Hence

$\displaystyle\frac{k_{\phi}}{\omega}$ $\displaystyle=-r\sin\theta\,\times$
(50)
$\displaystyle\left[\sqrt{(\gamma^{2}H)^{2}-(\gamma^{2}Hv+r\Omega)^{2}\cos^{2}\theta}-(\gamma^{2}Hv+r\Omega)\sin\theta\right]^{-1},$
(51)

where we have made use of the explicit expressions of the metric components
and have introduced the Lorentz gamma factor

$\displaystyle\gamma=\frac{1}{\sqrt{1-v^{2}}}.$ (52)

Then, (46) becomes

$\displaystyle 1+$ $\displaystyle z=$ (53)
$\displaystyle\frac{1}{\sqrt{-H}}\left[1+\frac{r\Omega\sin\theta}{\sqrt{(\gamma^{2}H)^{2}-(\gamma^{2}Hv+r\Omega)^{2}\cos^{2}\theta}-(\gamma^{2}Hv+r\Omega)\sin\theta}\right].$
(54)

This is the general formula we were looking for. It explicitly gives a
relation between the photon’s frequency shift and the metric and, through the
solution of Einstein’s equations, to the density and four-velocity of matter.

###  4.1 Particular cases

Equation (53) can be evaluated in specific cases in order to infer interesting
insights and physical interpretations of the quantities appearing in the
metric. As a starting point, it is instructive to consider the particular case
of a rigidly rotating dust. It can be obtained imposing $H(\eta)\equiv-1$ and
$\Omega=\Omega_{0}=$ constant in §2. Moreover, imposing $\Omega_{0}=0$, we
recover the model considered by Balasin & Grumiller (2008) and Crosta et al.
(2020). Most of the terms cancel out in eq. 53), leading to

$z_{H=-1}\equiv 0.$ (55)

Eq. 55 proves that GR corrections can be indeed non negligible since, for the
rigid rotating case considered here, they have exactly the same magnitude of
the special relativistic terms.

We now consider more physical cases. We start with the cases of perfectly
face-on $(\sin\theta=0)$ §4.1.1 and edge-on $(\lambda=0)$ §4.1.2 galaxies.

####  4.1.1 Face-on galaxies

A galaxy that is face-on with respect to the asymptotic detector corresponds
to $\lambda=\pi/2$, that is, $\sin\theta=0$. This corresponds to the fact that
the projection of the four-momentum of the photon along the galactic plane is
negligible. Then, eq. 53 simplifies to

$1+z(\pi/2;\phi,r,z)=\frac{1}{\sqrt{-H}}.$ (56)

The redshift depends only on the differential rotation described by $H$ and,
in the general case where $H$ is not constant, it can show an observable
radial profile, as discussed in section 5.

####  4.1.2 Edge-on galaxies

In the case of an edge-on galaxy, we have $\lambda=0$. Suppose now a star
located at $r=|b|$ along the line of sight, for which $\phi=\pm\pi/2$, where
$b$ is the impact parameter, and the sign of the angle changes from the left
to the right hand side. This corresponds to assume that the photon is emitted
exactly in the $\partial_{\phi}$ direction, so that $\sin\theta=\pm 1$.
Substituting in eq. 53, it gives

$\displaystyle
1+z(0;\pm\pi/2,r,z)=\frac{1}{\sqrt{-H}}\left[1+\frac{r\Omega}{(v\mp
1)\gamma^{2}(-H)-r\Omega}\right].$ (57)

Using now eq. 35, it can be rewritten in the form

$\displaystyle 1+z(0;\pm\pi/2,r,z)=\frac{(\gamma\sqrt{-H})}{\gamma^{2}(-H)\mp
r\chi}\frac{\left(1\pm v\right)}{\sqrt{1-v^{2}}},$ (58)

where the upper and lower sign refers to backward and forward emission,
respectively. We note that for weak field ($H\sim-1$) and negligible non-
diagonal effects $\chi\rightarrow 0$, it holds $r\Omega=v$. Notice that in the
special relativistic limit ($\gamma\sqrt{-H}\rightarrow 1$ and
$\chi\rightarrow 0$) $1+z(0;\pm\pi/2,r,z)\rightarrow\left(1\pm
v\right)/\sqrt{1-v^{2}}$.

####  4.1.3 Redshift of the galactic center

Up to now, we assumed that the galaxy is purely rotating, without any bulk
motion with respect to the detector. This is of course not necessary true for
a real galaxy, which concurs in the Hubble expansion and can have a possible
superimposed peculiar velocity dictated by local inhomogeneities of the large
scale structure. All this sums-up in the measured shift of the galaxy center.
In general, the net shift of the galaxy could affect the solution by non
linear effect. However, it should be a systematical effect, as long as photons
arrive from the observed galaxy through almost the same path, i.e., if the
angular size of the observed galaxy is sufficiently small and gravitational
lensing along the line of sight is absent/negligible. One could expect an
analogous systematical shift due to the detector orbiting within the non-
static axisymmetric metric of the Milky Way.

##  5 Proposal of an empirical check of the GR model

It is almost universally taken that gravitational effects could be neglected
when measuring the frequency shift of the light coming from an external
galaxy. Under such an assumption, the measured redshift $z$ would just be
comparable to the pure kinematic Doppler effect in a Minkowski space-time,
corresponding to a “special-relativistic” velocity $\vec{v}_{SR}$ of the
source, i.e.,

$1+z=\frac{1+v_{SR}^{||}}{\sqrt{1-v_{SR}^{2}}}.$ (59)

Since $\vec{v}_{SR}$ is assumed to be directed along $\partial_{\phi}$, its
projection along the line of sight is $v_{SR}\sin\theta$.

Let us now compare the widely accepted special relativistic description (SR),
which accounts only for the kinematic Doppler shift, to the general
relativistic description (GR), which instead includes the gravitational shift
effect. The expressions for the redshift are

$1+z=\begin{cases}\text{SR}\quad\frac{1+v_{SR}\sin\theta}{\sqrt{1-v_{SR}^{2}}},\\\
\text{GR}\quad\frac{1}{\sqrt{-H}}\left[1+\frac{r\Omega\sin\theta}{\sqrt{(\gamma^{2}H)^{2}-(\gamma^{2}Hv+r\Omega)^{2}\cos^{2}\theta}-(\gamma^{2}Hv+r\Omega)\sin\theta}\right].\end{cases}$
(60)

Here, the quantities $H$ and $\Omega$ describe the four-velocity of the
emitting star $u=(-H)^{-1/2}(\partial_{t}+\Omega\partial_{\phi})$, so that
$\Omega$ is its angular velocity as measured with respect to the Killing
vectors $\partial_{t},\partial_{\phi}$, and $H$ is the normalization factor.
$v$ is the speed of the star as measured by the ZAMO observer, and we refer to
this speed when we write the Lorentz factor $\gamma=(1-v^{2})^{-1/2}$. The
off-diagonal component of the metric, which is responsible for the non-
Newtonian effects, is proportional to $\chi=\Omega+H\gamma^{2}v/r$. For
further details see sections 2 and 3.

Different galaxies corresponds to different velocity profiles $v_{SR}$, as
well as different profiles of $v(r,z)$ and $H(\eta)$. Let us consider eq. 60
for a disc galaxy tilted by a given angle with respect to the line of sight.
Along the minor axis one measures the transverse redshift as (see eq. 56),

$1+z^{\perp}=\begin{cases}\text{SR}\quad\frac{1}{\sqrt{1-v_{SR}^{2}}},\\\
\text{GR}\quad\frac{1}{\sqrt{-H}}.\end{cases}$ (61)

The SR limit is again obtained by $\gamma\sqrt{-H}\rightarrow 1$ and
$\chi\rightarrow 0$, which corresponds to $\frac{1}{\sqrt{-H}}\sim\gamma$.

A measure of the redshift along the major axis, at the same location $(r,z)$,
corresponds to a different value of $\phi$. Through eq. 60 with
$\phi=\pm\pi/2$, we find

$1+z^{||}=\begin{cases}\text{SR}\quad\frac{1\pm
v_{SR}\cos\lambda}{\sqrt{1-v_{SR}^{2}}},\\\
\text{GR}\quad\frac{1}{\sqrt{-H}}\left[1+\frac{\pm
r\Omega\cos\lambda}{\sqrt{(\gamma^{2}H)^{2}-(\gamma^{2}Hv+r\Omega)^{2}\sin^{2}\lambda}-(\gamma^{2}Hv+r\Omega)(\pm\cos\lambda)}\right].\end{cases}$
(62)

We conclude that, in the SR case, the measure of the transverse frequency
shift is sufficient to determine the frequency shift along the major axis
since it does depend only upon $v_{SR}$. Indeed, special relativity implies

$\displaystyle(1+z^{||})=(1+z^{\perp})\left(1\pm\cos\lambda\sqrt{1-\frac{1}{(1+z^{\perp})^{2}}}\right),$
(63)

which is not true in the general relativistic description. This is ultimately
due to the presence of the additional degrees of freedom $H$ and $\chi$.
Generally speaking, by neglecting gravitational effects one chooses a
particular subclass of solutions, with $\gamma(-H)\rightarrow 1$ and
$\chi\rightarrow 0$.

In principle, by measuring $z^{||}(r)$ and $z^{\perp}(r)$, we would be able to
derive $H(\eta)$, $v(r)$ and all the others characteristics of the solution.
In particular, we would have a certain required mass $\rho$ from eq. 20.
Therefore, the proposed comparison would provide a good indicator of the
validity of the usual SR description with respect to the more fundamental
general relativistic counterparts.

###  5.1 Non relativistic velocities

Let us discuss the case of the dust velocity $v\ll 1$. For non relativistic
velocities, we can rewrite eq. 61 as

$\text{SR}\quad 1+z^{\perp}\approx 1+\frac{1}{2}v_{SR}^{2}.$ (64)

On the other hand, eq. 61 is not affected by taking $v\rightarrow 0$, and we
still have

$\displaystyle\text{GR}\quad 1+z^{\perp}=\frac{1}{\sqrt{-H}}.$ (65)

In the same spirit, we can expand the redshift along the major axis as

$\displaystyle 1+z^{||}=\begin{cases}\text{SR}\quad 1\pm
v_{SR}\cos\lambda+O(v_{SR}^{2}),\\\
\text{GR}\quad\frac{1}{\sqrt{-H}}\left[1\pm\frac{r\Omega\cos\lambda}{-H}\right]+O(r^{2}\Omega^{2})+O(r\Omega
v)+O(v^{2})\end{cases}$ (66)

We remember that when we have non negligible diagonal effects $\chi$,
$r\Omega$ is different from $v$. The SR consistency equation eq. 63 in the non
relativistic velocity limit becomes

$\displaystyle(z^{||})^{2}\approx 2z^{\perp}\cos^{2}\lambda,$ (67)

while still there is no analog constraint in the GR model. The measurements of
$z^{\perp}$ and $z^{||}$ fix only $H$ and $r\Omega$, while the function $\chi$
is determined by the equations of motion.

###  5.2  A crude estimation of the observational test feasibility

From eq. 61 we can expect that the order of magnitude of $H+1$ is
$v_{SR}^{2}$. Assuming close to circular orbits (i.e., $v,v_{SR}\sim v_{c}$)
with, e.g., $v_{c}=10^{-3}$ (i.e., 300 km/s), we obtain

$H+1=O(v^{2})\sim v_{c}^{2}=10^{-6}.$ (68)

Since we expect that the $-H$ factors do not carry corrections at the first
order, we can give the estimation of eq.60 as

$\displaystyle 1+z^{||}=\begin{cases}\text{SR}\quad 1\pm
v_{SR}\sin\theta+O(v^{2}),\\\ \text{GR}\quad 1\pm
r\Omega\sin\theta+O(v^{2})\end{cases}$ (69)

returning $v_{SR}\approx r\Omega$ as expected.

Once expressed in the commonly used (but not strictly correct in the case in
exam) velocity terms, the expected corrections are of $\sim 100$ m s-1 for a
rotational velocity of about $300$ km s-1. The (small) magnitude of the
expected corrections to the frequency shift profiles makes this novel test not
straightforward to perform.

Optical instruments potentially able to measure the corrections to the line
shifts predicted by our metric are already available. For example, the ESO
ESPRESSO spectrograph mounted on the VLT proved to be able to measure
frequency shifts down to $\sim 10$ cm s-1 in spectra of stars (Pepe et al.,
2021). Such an accuracy would exceed our requirements, but a few peculiarities
of the proposed test complicate its feasibility. First of all, such exquisite
accuracy in searches for planets, allowing for the detection of frequency
shifts significantly smaller than the intrinsic broadening of the absorption
lines used, is achieved convolving a mask (which models the star photosphere
with the observed spectra) so that the signal from all the photosphere
absorption lines observable in the spectral range (order of few thousands)
contributes to the spectral shift evaluation. In the galactic scenario the
number of lines that could survive the large Doppler broadening due to the
(sub-dominant) dispersion in the velocity field can be significantly smaller
(up to $\sim$ 10 in the best case), implying the need for an even larger
signal to noise ratio (and, therefore, observation time). Furthermore, in the
galactic case the integrated flux would come only from a small region of the
disc (in the opposite case the characterization frequency shift profile would
not be possible). An approximate back of the envelope estimate indicated that,
in order to achieve a (far from sufficient) signal-to-noise ratio of 2 for a
single pointing at the scale radius of Andromeda, an observing time of $100$
hr would be required. More efficient instrument/telescope pairs (even with a
lower spectral resolution) are therefore needed. Possible candidates are the
HIRES spectrograph on the Keck (Vogt et al., 1994) or the ANDES instrument to
be installed on the ELT (Maiolino et al., 2013). We plan to quantify in a next
study the observation time needed to achieve a signal-to-noise ratio
sufficient to perform the proposed test in the optical band.

Although deviations from the Newtonian rotation curves associated to the
baryonic components only are measurable within a few scale radii of the
stellar disc, in particular for low-mass galaxies (see, e.g., Strigari et al.,
2008), the most constraining tests come from the outer regions of the discs
(Rubin et al., 1980; Gentile et al., 2004), where the stellar component
becomes exponentially fainter while measurable emission from the typically
more extended (Bosma, 1981) HI disc is still visible in the 21-cm line
emission. Also in this case the intrinsic line broadening is significantly
larger ($\sim$ few km s-1) than the expected shifts, requiring a high signal
to noise ratios and long observation time. We plan to complement our future
study on the optical tests by performing mock observations using already
online instruments (e.g. large single dish radio telescopes as FAST, Zhu,
2018), as well as those planned for the future (such as, e.g. the next
generation VLA, Selina et al., 2018), to constrain the observation time needed
to achieve the required accuracy.

##  6 Conclusions and perspectives

We employed a general relativistic framework to describe disc galaxies. This
novel description gives rise to various non-Newtonian effects777As an example,
our set up does take in account the effects of gravito-electromagnetism, and,
being an exact solution, it allows to account for more general contributions,
too., which we showed could be important and non negligible in disc galaxies.

As an important application, here we focused on frequency shift of photons
from the emission to the detector. eq. 60 makes explicit the fact that
gravitational effects are, in general, not negligible and that they do need to
be taken into account in interpreting radial velocity measurements. Neglecting
such gravitational effects, which can be of the same order of the special
relativistic Doppler effect, is equivalent to setting to zero different
components of the metric and therefore, in all respects it is equivalent to
selecting a particular subclass of solutions of the GR equations. This cannot
be mathematically and physically justified a priori. Observations are the only
judge by which we can decide which class of solutions is appropriate an
physically motivated. In this sense, we propose the observational consistency
check in eq.63 (or its low energy counterpart, eq. 67 to test whether
gravitational effects are actually negligible, since from a GR perspective
there is no reason a priori why the conditions posed by eq. 63 and eq. 67
should be met.

It must be stressed that our GR galaxy model in its present form is not
sufficient to fully constrain the amount of DM present in disc galaxies. Even
if the less general GR model by Balasin & Grumiller (2008) managed to achieve
the outstanding result of reproducing the observed properties of the MW
(Crosta et al., 2020) without the requirement of any DM, one has to refrain
from invoking Occam’s razor against the DM hypothesis, as multiple and
independent observational arguments in its support do exist, as discussed in
the introduction. If the proposed test were to result in a metric consistent
with the observed distribution of baryonic matter and with the observed
frequency shift and kinematics, it would not be sufficient to reject the
presence of a DM halo on individual galaxies. In order to have such a
stringent -GR motivated- test of the maximum amount of DM present in disc
galaxies, the model needs further improvements. The most important improvement
would be, in our opinion, the self-consistent inclusion of a close-to-
spherical dark component in the metric. Only the application of such extended
model to carefully selected galaxies (characterized by, e.g., dynamically cold
discs and with negligible bulge components) can put statistically significant
upper limits to the amount of DM allowed by observations.

To significantly increase the sample on which to test our model, further
improvements are needed. These include (but are not limited to) the inclusion
of a non negligible velocity dispersion in the stars (and gas) of the disc,
and of a dynamically hot stellar component modelling the bulges often present
in the few central kpcs of massive disc galaxies. Through the use of the
proposed GR test, we could be able to put observational upper limits of the
amount of DM mass present in disc galaxies and, in turn, to put unprecedented
constraints on the very nature of DM itself.

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