###### Abstract

The general theory of relativity (GR) is symmetric under smooth coordinate
transformations, also known as diffeomorphisms. The general coordinate
transformation group has a linear subgroup denoted as the Lorentz group of
symmetry, which is also maintained in the weak field approximation to GR. The
dominant operator in the weak field equation of GR is thus the d’Alembert
(wave) operator, which has a retarded potential solution. Galaxies are huge
physical systems with dimensions of many tens of thousands of light years.
Thus, any change at the galactic center will be noticed at the rim only tens
of thousands of years later. Those retardation effects are neglected in the
present day galactic modelling used to calculate rotational velocities of
matter in the rims of the galaxy and surrounding gas. The significant
differences between the predictions of Newtonian instantaneous action at a
distance and observed velocities are usually explained by either assuming dark
matter or by modifying the laws of gravity (MOND). In this paper, we will show
that, by taking general relativity seriously without neglecting retardation
effects, one can explain the radial velocities of galactic matter in the M33
galaxy without postulating dark matter. It should be stressed that the current
approach does not require that velocities $v$ are high; in fact, the vast
majority of galactic bodies (stars, gas) are substantially subluminal—in other
words, the ratio of $\frac{v}{c}\ll 1$. Typical velocities in galaxies are 100
km/s, which makes this ratio $0.001$ or smaller. However, one should consider
the fact that every gravitational system, even if it is made of subluminal
bodies, has a retardation distance, beyond which the retardation effect cannot
be neglected. Every natural system, such as stars and galaxies and even
galactic clusters, exchanges mass with its environment, for example, the sun
loses mass through solar wind and galaxies accrete gas from the intergalactic
medium. This means that all natural gravitational systems have a finite
retardation distance. The question is thus quantitative: how large is the
retardation distance? For the M33 galaxy, the velocity curve indicates that
the retardation effects cannot be neglected beyond a certain distance, which
was calculated to be roughly 14,000 light years; similar analysis for other
galaxies of different types has shown similar results. We demonstrate, using a
detailed model, that this does not require a high velocity of gas or stars in
or out of the galaxy and is perfectly consistent with the current
observational knowledge of galactic and extra galactic material content and
dynamics.

###### keywords:

spacetime symmetry; relativity of spacetime; Lorentz symmetry group;
retardation; galactic rotation curves

xx 1 5 Received: 29 September 2020; Accepted: 12 October 2020; Published: date
Lorentz Symmetry Group, Retardation, Intergalactic Mass Depletion and
Mechanisms Leading to Galactic Rotation Curves Asher Yahalom 1,2 Asher Yahalom

##  1 Introduction

The general theory of relativity (GR) is known to be symmetric under general
coordinate modifications (diffeomorphism). This group of general coordinate
transformations has a Lorentz subgroup, which is maintained even in the weak
field approximation. This is manifested through the field equations containing
the d’Alembert (wave) operator, which has a retarded potential solution.

From an observational point of view, it is well known that GR is verified by
many observations. However, at the present time, the standard Newton–Einstein
gravitational theory stands at something of a crossroads. It simultaneously
has much in its favor observationally, while, at the same time, it has some
very disquieting challenges. The observational successes that it has achieved
in both astrophysics and cosmology have to be tempered by the fact that the
theory needs to appeal to two as yet unconfirmed ingredients, dark matter and
dark energy, in order to achieve these successes. The dark matter problem has
not only been with us since the 1920s and 1930s (when it was initially known
as the missing mass problem), but it has also become more and more severe as
more and more dark matter has had to be introduced on larger and larger
distance scales as new data have come online. Moreover, extensive—now
40-year—underground and accelerator searches have failed to find any of it or
establish its existence. The dark matter situation has become even more dire
in the last few years as the Large Hadron Collider has failed to find any
super symmetric particles, not only of the community’s preferred form of dark
matter, but also the form of it that is required in string theory, a theory
that attempts to provide a quantized version of Newton–Einstein gravity.

While things may still eventually work out in favor of the standard dark
matter paradigm, the situation is disturbing enough to warrant consideration
of the possibility that the standard paradigm might at least need to be
modified in some way if not outright replaced. The present proposal sets out
to seek such a modification. Unlike other approaches such as Milgrom’s MOND,
Mannheim’s Conformal Gravity or Moffat’s MOG, the present approach is, in a
sense, the minimalist one adhering strictly to the basic scientific principle
dictated by Occam’s razor. It seeks to replace dark matter by effects within
standard General Relativity itself.

In 1933, Fritz Zwicky noticed that the velocities of a set of Galaxies within
the Comma Cluster are much higher than those predicted by the virial
calculation that assumes Newtonian theory zwicky . He calculated that the
amount of matter required to account for the velocities could be 400 times
greater with respect to that of visible matter, which led to suggesting dark
matter throughout the entire cluster. Volders, in 1959, indicated that stars
in the outer rims of the nearby spiral galaxy M33 do not move “correctly”
volders . It is the result of the virial theorem coupled with Newtonian
Gravity which implies that $MG/r\sim Mv^{2}$, that is to say, the expected
rotation curve should at some point decrease as $1/\sqrt{r}$. This was well
established during the seventies when Rubin and Ford rubin1 ; rubin2
demonstrated that, for a large sample of spiral galaxies, this behavior can be
considered a general feature: velocities at the our rim of the galaxies do not
decrease—rather, in a general case, they attain a plateau at some velocity,
which is different for each galaxy. In what follows, we will show that such
effects can be deduced from GR if retardation effects are not neglected. The
derivation of the retardation force described in previous publications YaRe1 ;
ge ; YaRe2  is repeated for completeness. However, a fit of the theory to the
M33 rotation galaxy is given for the first time. It should be stressed that
the current approach does not require that velocities $v$ are high; in fact,
the vast majority of galactic bodies (stars, gas) are substantially
subluminal—in other words, the ratio of $\frac{v}{c}\ll 1$. Typical velocities
in galaxies are 100 km/s, which makes this ratio $0.001$ or smaller. However,
one should consider the fact that every gravitational system, even if it is
made of subluminal bodies, has a retardation distance, beyond which the
retardation effect cannot be neglected. Every natural system, such as stars
and galaxies and even galactic clusters, exchanges mass with its environment.
For example, the sun loses mass through solar wind and galaxies accrete gas
from the intergalactic medium. This means that all natural gravitational
systems have a finite retardation distance. The question is thus quantitative:
how large is the retardation distance? The change in the mass of the sun is
quite small and thus the retardation distance of the solar system is quite
large, allowing us to neglect retardation effects within the solar system.
However, for the M33 galaxy, the velocity curve indicates that the retardation
effects cannot be neglected beyond a certain distance smaller than 14,000
light years; similar analyses for other galaxies of different types have shown
similar results Wagman ; Wagman2 . We demonstrate, in Section 6, using a
detailed model, that this does not require a high velocity of gas or stars in
or out of the galaxy and is perfectly consistent with the current
observational knowledge of galactic and extra galactic material content and
dynamics.

##  2 General Relativity

The general theory of relativity is based on two fundamental equations, the
first being Einstein equations Narlikar ; Weinberg ; MTW ; Edd :

$G_{\mu\nu}=-\frac{8\pi G}{c^{4}}T_{\mu\nu}$ (1)

$G_{\mu\nu}$ stands for the Einstein tensor, $T_{\mu\nu}$ indicates the
stress–energy tensor, $G\simeq 6.67\ 10^{-11}\
\text{m}^{3}\text{kg}^{-1}\text{s}^{-2}$ is the universal gravitational
constant and $c\simeq 3\ 10^{8}\ \text{m s}^{-1}$ indicates the velocity of
light in the absence of matter (Greek letters are indices in the range $0-3$).
The second fundamental equation that GR is based on is the geodesic equation:

$\frac{d^{2}x^{\alpha}}{ds^{2}}+\Gamma^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}=\frac{du^{\alpha}}{ds}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}=0$
(2)

$x^{\alpha}(s)$ are the coordinates of the particle in spacetime, $s$ is a
typical parameter along the trajectory that for massive particles is chosen to
be the length of the trajectory, $u^{\mu}=\frac{dx^{\mu}}{ds}$ is the $\mu$-th
component of the 4-velocity of a massive particle moving along the geodesic
trajectory $s$ (increment of $x$ per unit length $s$) and
$\Gamma^{\alpha}_{\mu\nu}$ is the affine connection (Einstein summation
convention is assumed). The stress–energy tensor of matter is usually taken in
the form:

$T_{\mu\nu}=(p+\rho c^{2})u_{\mu}u_{\nu}-p\ g_{\mu\nu}$ (3)

In the above, $p$ is the pressure and $\rho$ is the mass density. We remind
the reader that lowering and raising indices is done through the metric
$g_{\mu\nu}$ and inverse metric $g^{\mu\nu}$, such that
$u_{\mu}=g_{\mu\nu}u^{\nu}$. The same metric serves to calculate $s$:

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu},$ (4)

and the affine connection:

$\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}\left(g_{\beta\mu,\nu}+g_{\beta\nu,\mu}-g_{{\mu\nu},\beta}\right),\qquad
g_{\beta\mu,\nu}\equiv\frac{\partial g_{\beta\mu}}{\partial x^{\nu}}$ (5)

The affine connection serves to calculate the Riemann and Ricci tensors and
the curvature scalar:

$R^{\mu}_{\nu{\alpha\beta}}=\Gamma^{\mu}_{\nu\alpha,\beta}-\Gamma^{\mu}_{\nu\beta,\alpha}+\Gamma^{\sigma}_{\nu\alpha}\Gamma^{\mu}_{\sigma\beta}-\Gamma^{\sigma}_{\nu\beta}\Gamma^{\mu}_{\sigma\alpha},\quad
R_{{\alpha\beta}}=R^{\mu}_{{\alpha\beta}\mu},\quad
R=g^{\alpha\beta}R_{{\alpha\beta}}$ (6)

which, in turn, serves to calculate the Einstein tensor:

$G_{{\alpha\beta}}=R_{{\alpha\beta}}-\frac{1}{2}g_{\alpha\beta}R.$ (7)

Hence, the given matter distribution determines the metric through Equation
(1) and the metric determines the geodesic trajectories through Equation (2).
Those equations are well known to be symmetric under smooth coordinate
transformations (diffeomorphism).

$x^{\prime}_{\alpha}=x^{\prime}_{\alpha}(x_{\mu}).$ (8)

##  3 Linear Approximation of GR

Only in extreme cases of compact objects (black holes and neutron stars) and
the primordial reality or the very early universe does one need not consider
the solution of the full non-linear Einstein Equation YaRe1 . In typical cases
of astronomical interest (the galactic case included) one can use a linear
approximation to those equations around the flat Lorentz metric
$\eta_{{\mu\nu}}$ such that (Private communication with the late Professor
Donald Lynden-Bell):

$g_{{\mu\nu}}=\eta_{{\mu\nu}}+h_{{\mu\nu}},\quad\eta_{{\mu\nu}}\equiv\ {\rm
diag}\ (1,-1,-1,-1),\qquad|h_{{\mu\nu}}|\ll 1$ (9)

One then defines the quantity:

$\bar{h}_{\mu\nu}\equiv h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h,\quad
h=\eta^{{\mu\nu}}h_{{\mu\nu}},$ (10)

$\bar{h}_{\mu\nu}=h_{\mu\nu}$ for non diagonal terms. For diagonal terms:

$\bar{h}=-h\Rightarrow
h_{\mu\nu}=\bar{h}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\bar{h}.$ (11)

The general coordinate transformation symmetry of Equation (8) has a subgroup
of infinitesimal transformations which are manifested in the gauge freedom of
$h_{\mu\nu}$ in the weak field approximation. It can be shown (Narlikar  page
75, exercise 37, see also Edd ; Weinberg ; MTW ) that one can choose a gauge
such that the Einstein equations are:

$\Box{\bar{h}}_{{\mu\nu}}\equiv{\bar{h}}_{{\mu\nu},\alpha}{}^{\alpha}=-\frac{16\pi
G}{c^{4}}T_{\mu\nu},\qquad{\bar{h}}_{\mu\alpha,}{}^{\alpha}=0.$ (12)

The d’Alembert operator $\Box$ is clearly invariant under the Lorentz symmetry
group (another subgroup of the general coordinate transformation symmetry
described by Equation (8)), of which the Newtonian Laplace operator
$\vec{\nabla}^{2}$ is not, but this comes with the price that "action at a
distance" solutions are forbidden and only retarded solutions are allowed. The
$T_{\mu\nu}$ stress energy tensor should be calculated at the appropriate
frame and thus for the rotation curve frame of reference for galaxies, matter
is approximately at rest.

Equation (12) can always be integrated to take the form Jackson  (For reasons
why the symmetry between space and time is broken, see Yahalom ; Yahalomb ):

$\displaystyle{\bar{h}}_{{\mu\nu}}(\vec{x},t)=-\frac{4G}{c^{4}}\int\frac{T_{\mu\nu}(\vec{x}^{\prime},t-\frac{R}{c})}{R}d^{3}x^{\prime},$
$\displaystyle t$ $\displaystyle\equiv$
$\displaystyle\frac{x^{0}}{c},\quad\vec{x}\equiv x^{a}\quad a,b\in[1,2,3],$
$\displaystyle\vec{R}$ $\displaystyle\equiv$
$\displaystyle\vec{x}-\vec{x}^{\prime},\quad R=|\vec{R}|.$ (13)

The factor before the integral is small: $\frac{4G}{c^{4}}\simeq 3.3\times
10^{-44}$; hence, in the above calculation one can take $T_{\mu\nu}$, which is
zero order in $h_{\alpha\beta}$. Let us now calculate the affine connection in
the linear approximation:

$\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}\eta^{\alpha\beta}\left(h_{\beta\mu,\nu}+h_{\beta\nu,\mu}-h_{{\mu\nu},\beta}\right).$
(14)

The affine connection has only first order terms in $h_{\alpha\beta}$; hence,
to the first order $\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}$ appearing in the
geodesic, $u^{\mu}u^{\nu}$ is of the zeroth order. In the zeroth order:

$u^{0}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},u^{a}=\vec{u}=\frac{\frac{\vec{v}}{c}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\vec{v}\equiv\frac{d\vec{x}}{dt},\quad
v=|\vec{v}|.$ (15)

For non relativistic velocities:

$u^{0}\simeq 1,\qquad\vec{u}\simeq\frac{\vec{v}}{c},\qquad u^{a}\ll
u^{0}\qquad{\rm for}\quad v\ll c.$ (16)

Hence, we will not be considering a post-Newtonian approximation in this
paper, in which matter travels at nearly relativistic speeds, but we will be
considering the retardation effects and finite propagation speed of the
gravitational field. We underline that taking $\frac{v}{c}<<1$ is not the same
as taking $\frac{R}{c}<<1$ (with $R$ being the typical size of a galaxy)
since:

$\frac{R}{c}=\frac{v}{c}\frac{R}{v}$ (17)

and, in galaxies, $\frac{R}{v}$ is a very big number ($\frac{R}{v}\simeq
10^{15}$ seconds); thus, $\frac{v}{c}$ can be neglected but not $\frac{R}{c}$,
in which $\frac{R}{c}\simeq 10^{12}$ seconds. By inserting Equations (14) and
(16) in the geodesic equation, we arrive at the approximate form:

$\frac{dv^{a}}{dt}\simeq-c^{2}\Gamma^{a}_{00}=-c^{2}\left(h^{a}_{0,0}-\frac{1}{2}h_{00,}{}^{a}\right)$
(18)

Let us now look at Equation (3). We assume $\rho c^{2}\gg p$ and, taking into
account Equation (16), we arrive at $T_{00}=\rho c^{2}$, while other tensor
components are significantly smaller. Thus, $\bar{h}_{00}$ is significantly
larger than other components of $\bar{h}_{\mu\nu}$. One should notice that it
is not possible to deduce from the magnitudes of quantities that such a
difference exists between their derivatives. In fact, by the gauge condition
in Equation (12):

$\bar{h}_{\alpha 0,}{}^{0}=-\bar{h}_{\alpha
a,}{}^{a}\qquad\Rightarrow\bar{h}_{00,}{}^{0}=-\bar{h}_{0a,}{}^{a},\quad\bar{h}_{b0,}{}^{0}=-\bar{h}_{ba,}{}^{a}.$
(19)

Thus, the zeroth derivative of $\bar{h}_{00}$ (which contains a $\frac{1}{c}$)
is the same order as the spatial derivative of $\bar{h}_{0a}$ and the zeroth
derivative of $\bar{h}_{0a}$ (which appears in Equation (18)) is the same
order as the spatial derivative of $\bar{h}_{ab}$. However, it we can compare
spatial derivatives of $\bar{h}_{00}$ and $\bar{h}_{ab}$ and conclude that the
former is larger than the later. Using Equation (11) and taking into account
the above consideration, we may write Equation (18) as:

$\frac{dv^{a}}{dt}\simeq\frac{c^{2}}{4}\bar{h}_{00,}{}^{a}\Rightarrow\frac{d\vec{v}}{dt}=-\vec{\nabla}\phi=\vec{F},\qquad\phi\equiv\frac{c^{2}}{4}\bar{h}_{00}$
(20)

Thus, $\phi$ is the gravitational potential of the motion which can be
calculated using Equation (13):

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{c^{2}}{4}\bar{h}_{00}=-\frac{G}{c^{2}}\int\frac{T_{00}(\vec{x}^{\prime},t-\frac{R}{c})}{R}d^{3}x^{\prime}$
(21) $\displaystyle=$
$\displaystyle-G\int\frac{\rho(\vec{x}^{\prime},t-\frac{R}{c})}{R}d^{3}x^{\prime}$

and $\vec{F}$ is the force per unit mass. If the mass density $\rho$ is
static, we are in the realm of the Newtonian instantaneous action at a
distance. We point out that it is unlikely that $\rho$ is static, as a galaxy
will obtain mass from the intergalactic medium.

##  4 Beyond the Newtonian Approximation

The retardation time $\frac{R}{c}$ may be a few tens of thousands of years,
but can be considered short in comparison to the time taken for the galactic
density to change significantly. Thus, we can write a Taylor series for the
density:

$\rho(\vec{x}^{\prime},t-\frac{R}{c})=\sum_{n=0}^{\infty}\frac{1}{n!}\rho^{(n)}(\vec{x}^{\prime},t)(-\frac{R}{c})^{n},\qquad\rho^{(n)}\equiv\frac{\partial^{n}\rho}{\partial
t^{n}}.$ (22)

By inserting Equations (22) into Equation (21) and keeping the first three
terms, we will obtain:

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle-G\int\frac{\rho(\vec{x}^{\prime},t)}{R}d^{3}x^{\prime}+\frac{G}{c}\int\rho^{(1)}(\vec{x}^{\prime},t)d^{3}x^{\prime}$
(23) $\displaystyle-$ $\displaystyle\frac{G}{2c^{2}}\int
R\rho^{(2)}(\vec{x}^{\prime},t)d^{3}x^{\prime}$

The Newtonian potential is the first term, the second term has null
contribution, and the third term is the lower order correction to the
Newtonian theory:

$\phi_{r}=-\frac{G}{2c^{2}}\int
R\rho^{(2)}(\vec{x}^{\prime},t)d^{3}x^{\prime}$ (24)

The total force per unit mass:

$\displaystyle\vec{F}$ $\displaystyle=$ $\displaystyle\vec{F}_{N}+\vec{F}_{r}$
$\displaystyle\vec{F}_{N}$ $\displaystyle=$
$\displaystyle-\vec{\nabla}\phi_{N}=-G\int\frac{\rho(\vec{x}^{\prime},t)}{R^{2}}\hat{R}d^{3}x^{\prime},\qquad\hat{R}\equiv\frac{\vec{R}}{R}$
$\displaystyle\vec{F}_{r}$ $\displaystyle\equiv$
$\displaystyle-\vec{\nabla}\phi_{r}=\frac{G}{2c^{2}}\int\rho^{(2)}(\vec{x}^{\prime},t)\hat{R}d^{3}x^{\prime}$
(25)

While the Newtonian force $\vec{F}_{N}$ is always attractive, we notice that
the retardation force $\vec{F}_{r}$ can be attractive or repulsive. The
Newtonian force decreases as $\frac{1}{R^{2}}$, but the retardation force is
independent of distance as long as the Taylor approximation given in Equation
(22) is valid. Below a certain distance, the Newtonian force will be dominant,
but for higher distances the retardation force becomes larger. Newtonian force
should be neglected for distances significantly larger than the retardation
distance, defined as:

$R\gg R_{r}\equiv c\Delta t$ (26)

$\Delta t$ is the typical duration in which the density $\rho$ changes; this
will be defined more accurately later. On the other hand, for $R\ll R_{r}$,
the retardation effect can be neglected and only Newtonian forces should be
considered; this is probably the situation in the solar system. It should be
stressed that the existence of $R_{r}$ does not require that velocities $v$
are high; in fact, the vast majority of galactic bodies (stars, gas) are
substantially subluminal. In other words, the ratio of $\frac{v}{c}\ll 1$.
Typical velocities in galaxies are $100$ km/s, which makes this ratio $0.001$
or smaller. However, one should consider the fact that every gravitational
system, even if it is made of subluminal bodies, has a retardation distance,
beyond which the retardation effect cannot be neglected. Every natural system,
such as stars and galaxies and even galactic clusters, exchanges mass with its
environment. For example, the sun loses mass through solar wind and galaxies
accrete gas from the intergalactic medium. This means that all natural
gravitational systems have a finite retardation distance. The question is thus
quantitative: how large is the retardation distance? This question will be
further addressed in Section 5. For large distances,
$r=|\vec{x}|\rightarrow\infty$, such that
$\hat{R}\simeq\frac{\vec{x}}{|\vec{x}|}\equiv\hat{r}$; thus, we obtain:

$\vec{F}_{r}=\frac{G}{2c^{2}}\hat{r}\int\rho^{(2)}(\vec{x}^{\prime},t)d^{3}x^{\prime}=\frac{G}{2c^{2}}\hat{r}\ddot{M},\qquad\ddot{M}\equiv\frac{d^{2}M}{dt^{2}}.$
(27)

As the galaxy attracts intergalactic gas, its mass becomes larger and
therefore $\dot{M}>0$; however, as the intergalactic gas is depleted, the rate
at which the mass is accumulated must decrease and therefore $\ddot{M}<0$.
Thus, in the galactic case:

$\vec{F}_{r}=-\frac{G}{2c^{2}}|\ddot{M}|\hat{r}$ (28)

and the retardation force is attractive.

##  5 Rotation Curves

To calculate the non-asymptotic rotation curve, we note that the square of the
azimuthal (orbital) velocity, $v_{\theta}$ divided by ${\bar{r}}$, is
proportional to the derivative of the gravitational potential by ${\bar{r}}$,
as shown in Equation (68). We use the cylindrical coordinates
${\bar{r}},\theta,z$, in which $z=0$ is the galactic plane. The gravitational
potential will be evaluated using Equation (23), which is composed of both the
Newtonian $\phi_{N}$ and retardation $\phi_{r}$ parts, such that:

$\frac{v_{\theta}^{2}}{{\bar{r}}}=\frac{\partial\phi}{\partial{\bar{r}}}=\frac{\partial\phi_{N}}{\partial{\bar{r}}}+\frac{\partial\phi_{r}}{\partial{\bar{r}}},$
(29)

To do this, we first need a density profile describing the mass distribution
in and near the galactic plane.

###  5.1 General Considerations

We assume a density distribution of the form:

$\rho(\vec{x},t)=\rho_{a}(\vec{x})+\rho_{b}(\vec{x})g^{\prime}(t)$ (30)

Although the density profiles $\rho_{a}$ and $\rho_{b}$ need not have similar
forms, we will assume, for simplicity, that they do. Hence,
$\rho_{a}=\rho_{b}$ and, by defining $g(t)=1+g^{\prime}(t)$, we obtain a
density distribution of the form:

$\rho(\vec{x},t)=g(t)\rho_{a}(\vec{x});$ (31)

hence,

$M(t)=\int d^{3}x^{\prime}\rho(\vec{x}^{\prime},t)=g(t)\int
d^{3}x^{\prime}\rho_{a}(\vec{x}^{\prime})$ (32)

and

$\ddot{M}(t)=\ddot{g}(t)\int d^{3}x^{\prime}\rho_{a}(\vec{x}^{\prime}),$ (33)

which leads to the result:

$|\ddot{g}(t)|=\frac{|\ddot{M}(t)|}{M(t)}g(t)=\frac{c^{2}}{R_{r}^{2}}g(t);$
(34)

here, we define:

$t_{r}\equiv\sqrt{\frac{M}{|\ddot{M}|}},\qquad\Delta t=t_{r}\Rightarrow
R_{r}\equiv ct_{r}.$ (35)

Inserting Equations (31) and (34) into Equation (24) leads to the following
form of the retardation potential:

$\phi_{r}=-\frac{G}{2c^{2}}\ddot{g}(t)\int
R\rho_{a}(\vec{x}^{\prime})d^{3}x^{\prime}=\frac{G}{2R_{r}^{2}}\int
R\rho(\vec{x}^{\prime},t)d^{3}x^{\prime}.$ (36)

Now, we introduce the dimensionless quantities:

$\tilde{\rho}\equiv\frac{\rho}{\rho_{c}},\qquad\tilde{x}\equiv\frac{\vec{x}}{R_{s}}$
(37)

in terms of a typical density $\rho_{c}$ and a typical scale $R_{s}$. From now
on, we will consider all time-dependent quantities as being given at the time
the galaxy is being observed. This has nothing to do with the present
situation in a certain galactic system that may be millions or billions year
in the future with respect to the data available to us, which of course is
also the result of retardation. The dependence of the temporal variable $t$
will thus be omitted. Using Equations (37) and (32) will take the form:

$M=\int d^{3}x^{\prime}\rho(\vec{x}^{\prime})=\rho_{c}R_{s}^{3}\int
d^{3}\tilde{x}^{\prime}\tilde{\rho}(\tilde{x}^{\prime})$ (38)

We now define the dimensionless constant:

$\Lambda=\int d^{3}\tilde{x}^{\prime}\tilde{\rho}(\tilde{x}^{\prime})$ (39)

In terms of which:

$M=\Lambda\rho_{c}R_{s}^{3}$ (40)

Hence, $\phi_{r}$ can be written as:

$\phi_{r}=\frac{GM}{2R_{r}^{2}}r\
\chi,\qquad\chi\equiv\frac{1}{\Lambda}\int\frac{R}{r}\tilde{\rho}(\tilde{x}^{\prime})d^{3}\tilde{x}^{\prime}$
(41)

$\chi$ is a dimensionless function that satisfies:

$\lim_{r->\infty}\chi=1$ (42)

Similarly, one can write the Newtonian potential as:

$\phi_{N}=-\frac{GM}{r}\
\psi,\qquad\psi\equiv\frac{1}{\Lambda}\int\frac{r}{R}\tilde{\rho}(\tilde{x}^{\prime})d^{3}\tilde{x}^{\prime}$
(43)

$\psi$ is a dimensionless function that also satisfies:

$\lim_{r->\infty}\psi=1$ (44)

###  5.2 M33 Density Profile

In order to calculate the retardation and Newtonian gravitational potentials
and hence the rotation curve through Equation (68), we must know the density
distribution in the galactic plane. This is usually done as follows: for the
radial distribution, we will use the luminosity distribution and assume a
proportionality constant known as the mass to light ratio. Such radial density
distribution is given in Figure 1, which is based on the work of Corbelli
Corbelli2  (see also Rega and Vogel REV ) and its logarithm is fitted to a
polynomial of order five, as described in Equation (45):

$\displaystyle\rho({\bar{r}})$ $\displaystyle\sim$ $\displaystyle
e^{kp({\bar{r}})},$ $\displaystyle kp({\bar{r}})$ $\displaystyle=$
$\displaystyle
6.21207-1.05618{\bar{r}}+0.137599{\bar{r}}^{2}-0.0149017{\bar{r}}^{3}$ (45)
$\displaystyle+$ $\displaystyle
0.000865154{\bar{r}}^{4}-0.0000209513{\bar{r}}^{5}$

Figure 1: The M33 density radial distribution, the dots are data points
Corbelli2  the solid line is a fit Equation (45).

For the $z$ direction distribution orthogonal to the galactic plane, we will
assume a Gaussian profile with a typical width of $\sigma=0.2$ kpc. The mass
distribution can be written as:

$\rho({\bar{r}},z)=\rho_{c}e^{kp({\bar{r}})}e^{-\frac{z^{2}}{\sigma^{2}}},$
(46)

in which we assume cylindrical symmetry. The three-dimensional galactic mass
distribution is depicted in Figure 2.

Figure 2: The M33 three dimensional mass density distribution model.

###  5.3 M33 Rotation Curve

To obtain the rotation curve we need to evaluate the functions
$\psi({\bar{r}})$ and $\chi({\bar{r}})$, which describe the non-trivial
contribution of the Newtonian and retardation potentials, respectively, see
Equations (41) and (43). We will assume, for simplicity, that the galaxy is
cylindrically symmetric and that rotation curves are evaluated at the galactic
plane. Thus, according to Equation (43), we need to numerically evaluate the
integral:

$\psi=\frac{1}{\Lambda}\int_{0}^{2\pi}d\theta^{\prime}\int_{0}^{\infty}d{\bar{r}}^{\prime}{\bar{r}}^{\prime}\int_{-\infty}^{\infty}dz^{\prime}\frac{{\bar{r}}}{R}\tilde{\rho},\qquad\tilde{\rho}=e^{kp({\bar{r}})}e^{-\frac{z^{2}}{\sigma^{2}}},$
(47)

in the above:

$\Lambda=\int_{0}^{2\pi}d\theta^{\prime}\int_{0}^{\infty}d{\bar{r}}^{\prime}{\bar{r}}^{\prime}\int_{-\infty}^{\infty}dz^{\prime}\tilde{\rho}=0.721007$
(48)

and:

$\frac{{\bar{r}}}{R}=\frac{1}{\sqrt{1+\left(\frac{{\bar{r}}^{\prime}}{{\bar{r}}}\right)^{2}-2\left(\frac{{\bar{r}}^{\prime}}{{\bar{r}}}\right)\cos(\theta^{\prime})+\left(\frac{z^{\prime}}{{\bar{r}}}\right)^{2}}}.$
(49)

The result of the numerical evaluation of $\psi({\bar{r}})$ is given in Figure
3; we notice that, due to the cylindrical symmetry, it is enough to evaluate
the function for the azimuthal angle $\theta=0$ as the result is similar for
each azimuthal angle. As can be easily seen, the $\psi$ function of M33
converges to one for large distances, as expected from Equation (44).

Figure 3: The $\psi$ function of M33, the function converges to one for large
distances as expected from Equation (44).

If we ignore the retardation potential contribution Equation (29), then
Equation (43) will lead to the rotation curve depicted in Figure 4, which
decreases for large distances. In the above, we assumed a galactic (baryonic)
mass of $10^{10}$ solar masses ($2\times 10^{40}$ kg) Corbelli2 .

Figure 4: The Newtonian rotation curve of M33; the velocity decreases at large
distances.

As the true galactic rotation curve does not decrease, we need to evaluate the
retardation potential Equation (41) as well. For this, we evaluate the $\chi$
function as follows:

$\chi\equiv\frac{1}{\Lambda}\int_{0}^{2\pi}d\theta^{\prime}\int_{0}^{\infty}d{\bar{r}}^{\prime}{\bar{r}}^{\prime}\int_{-\infty}^{\infty}dz^{\prime}\frac{R}{r}\tilde{\rho}$
(50)

The result of the evaluation is given in Figure 5, where it is obvious that
the $\chi$ function approaches unity for large distances from the galactic
center. The retardation potential and thus the retardation contribution to the
velocity cannot be calculated without knowledge of the retardation distance
$R_{r}$.

Figure 5: The $\chi$ function of M33; the function converges to one for large
distances, as expected from Equation (42).

However, this can be obtained easily by fitting the observational galactic
rotation curve, as demonstrated in Figure 6, which yields a best fit for
$R_{r}=4.54$ kpc and a retardation time of $t_{r}=14,818.7$ years. Looking at
Figure 6, this seems quite reasonable.

Figure 6: Rotation curve for M33. The observational points were supplied by
Dr. Michal Wagman, a former PhD student at Ariel University, under my
supervision, using Corbelli2 ; the full line describes the complete rotation
curve, which is the sum of the dotted line, describing the retardation
contribution, and the dashed line, which is the Newtonian contribution.

We deduce from the above and from Equation (35) that the galactic mass second
derivative is as follows:

$|\ddot{M}|=\frac{M}{t_{r}^{2}}\simeq 9.12\times 10^{16}\ {\rm kg/s^{2}}$ (51)

We underline that this is based only on the current estimation of the second
derivative of mass; we make no claim about the past or future values of
$\ddot{M}$, nor is there any claim in this section on the value of $\dot{M}$
at any time or the value of $M$ in the past or the future. It is obvious that
such questions involve an understanding of the mass exchange between the
galaxy and the intergalactic medium, as described in Section 6.

##  6 A Dynamical Model

The mass accumulation model described in the previous section is based on a
fitting of the second derivative of the galactic mass to the galactic rotation
curve. It is intuitively obvious that, as mass is accumulated in the galaxy,
it must be depleted in the intergalactic medium. This is due to the fact that
the total mass is conserved; still, it is of interest to see if this intuition
is compatible with a model of gas dynamics. For simplicity, we assume that the
gas is a barotropic ideal fluid and its dynamics are described by the Euler
and continuity equations as follows:

$\frac{\partial{\rho}}{\partial t}+\vec{\nabla}\cdot(\rho\vec{v})=0$ (52)
$\frac{d\vec{v}}{dt}\equiv\frac{\partial\vec{v}}{\partial
t}+(\vec{v}\cdot\vec{\nabla})\vec{v}=-\frac{\vec{\nabla}p(\rho)}{\rho}-\vec{\nabla}\phi$
(53)

where the pressure $p(\rho)$ is assumed to be a given function of the density,
$\frac{\partial}{\partial t}$ is a partial temporal derivative, $\vec{\nabla}$
has its standard meaning in vector analysis and $\frac{d}{dt}$ is the material
temporal derivative. We have neglected viscosity terms due to the low gas
density. For simplicity, we assume axial symmetry; hence, all variables are
independent of $\theta$. Moreover, the mass influx coming from above and below
the galaxy is much more significant compared to the influx coming from the
galactic edge. This is due to the large difference in the galaxy surfaces
perpendicular to the z axis compared to the area of its edge. The area of the
surface of the galaxy which is perpendicular to the $z$ axis is:

$S_{z}=S_{z+}+S_{z-}=\pi r_{m}^{2}+\pi r_{m}^{2}=2\pi r_{m}^{2}$ (54)

in which $S_{z}$ is the total surface area of the galaxy perpendicular to the
$z$ axis, $S_{z+}$ is the upper area of the surface of the galaxy
perpendicular to the $z$ axis, $S_{z-}$ is the lower area of the surface of
galaxy perpendicular to the $z$ axis and $r_{m}$ is the galactic radius (see
Figure 7).

Figure 7: An idealized cylindrical galaxy from different perspectives. (a)
From above; (b) tilted edge perspective.

The area of the surface of the galactic edge with thickness $\Delta z$ is:

$S_{e}=2\pi r_{m}\Delta z.$ (55)

Thus, the ratio of the surface area is:

$\frac{S_{e}}{S_{z}}=\frac{\Delta z}{r_{m}}.$ (56)

Typical values of $\Delta z$ are about $0.4$ kilo parsec and $r_{m}$ is about
$17$ kilo parsec (for M33), giving an area ratio of about $1\%$. In such
circumstances, the edge mass influx is less important and we can assume a
velocity field of the form:

$\vec{v}=v_{z}({\bar{r}},z,t)\hat{z}+v_{\theta}({\bar{r}},z,t)\hat{\theta}.$
(57)

$\hat{z}$ and $\hat{\theta}$ are unit vectors in the $z$ and $\theta$
directions, respectively. The influx is described schematically in Figure 8.

Figure 8: A schematic view of the galactic influx from the side.

In this case, the continuity Equation (52) will take the form:

$\frac{\partial{\rho}}{\partial t}+\frac{\partial{(\rho v_{z})}}{\partial
z}=0$ (58)

By defining the quantity:

$\gamma\equiv\rho v_{z}\Rightarrow\rho=\frac{\gamma}{v_{z}}$ (59)

and using the above definition, Equation (58) takes the form:

$\frac{\partial{(\frac{\gamma}{v_{z}})}}{\partial
t}+\frac{\partial{\gamma}}{\partial z}=0$ (60)

Assuming, for simplicity, that $v_{z}$ is stationary and defining the
auxiliary variable $t_{z}$:

$t_{z}\equiv\int\frac{dz}{v_{z}}$ (61)

we arrive at the equations:

$\frac{\partial{\gamma}}{\partial t}+\frac{\partial{\gamma}}{\partial
t_{z}}=0.$ (62)

This equation can be solved easily, as follows:

$\gamma({\bar{r}},z,t)=f(t-t_{z}),\qquad
f(-t_{z})=\gamma({\bar{r}},z,0)=v_{z}\rho({\bar{r}},z,0)$ (63)

for the function $f(x)$, which is fixed by the density initial conditions and
the velocity profile.

Let us now turn our attention to the Euler Equation (53); for stationary
flows, it takes the form:

$(\vec{v}\cdot\vec{\nabla})\vec{v}=-\frac{\vec{\nabla}p(\rho)}{\rho}-\vec{\nabla}\phi$
(64)

According to Equation (57):

$\vec{v}\cdot\vec{\nabla}=v_{z}\frac{\partial}{\partial
z}+\frac{v_{\theta}}{{\bar{r}}}\frac{\partial}{\partial\theta}$ (65)

Now, by writing Equation (64) in terms of its components, we arrive at the
following equations:

$v_{z}\frac{\partial v_{z}}{\partial z}=-\frac{1}{\rho}\frac{\partial
p}{\partial z}-\frac{\partial\phi}{\partial z}$ (66)
$-\frac{v_{\theta}^{2}}{{\bar{r}}}=-\frac{1}{\rho}\frac{\partial
p}{\partial{\bar{r}}}-\frac{\partial\phi}{\partial{\bar{r}}},\qquad\left(\frac{\partial\hat{\theta}}{\partial\theta}=-\hat{{\bar{r}}}\right).$
(67)

It is usually assumed that the radial pressure gradients are negligible with
respect to the gravitational forces and thus we arrive at the equation:

$\frac{v_{\theta}^{2}}{{\bar{r}}}\simeq\frac{\partial\phi}{\partial{\bar{r}}},$
(68)

As for the $z$-component part of the Euler Equation (64), it can be easily
written in terms of the specific enthalpy $w(\rho)=\int\frac{dP}{\rho}$ in the
form:

$\frac{\partial}{\partial
z}\left(\frac{1}{2}v_{z}^{2}+w(\rho)+\phi\right)=0\Rightarrow\frac{1}{2}v_{z}^{2}+w(\rho)+\phi=C(r,t).$
(69)

We recall that $\rho$ depends on $v_{z}$ through Equations (59) and (63):

$\rho(r,z,t)=\frac{\gamma}{v_{z}}=\frac{f(t-\int\frac{dz}{v_{z}})}{v_{z}}$
(70)

As both the specific enthalpy and the gravitational potential are dependent on
the density, Equation (69) turns into a rather complicated nonlinear integral
equation for $v_{z}$. However, many galaxies are flattened structures; hence,
it can thus be assumed that the pressure $z$ gradients are significant as one
approaches the galactic plane. We will thus assume, for the sake of
simplicity, that the pressure gradients balance the gravitational pull of the
galaxy and thus $v_{z}$ is just a function of $r$, in which case the
convective derivative of $v_{z}$ vanishes. The above assumption holds below
and above the galactic plane, but not at the galactic plane itself. This
suggests the following simple model for the velocity $v_{z}$ (see Figure 8):

$v_{z}=\left\\{\begin{array}[]{cc}-|v_{z}|&z>0\\\ |v_{z}|&z<0\\\
\end{array}\right.$ (71)

in which $|v_{z}|$ is a known function of ${\bar{r}}$. The velocity field is
discontinuous at the galactic plane due to our simplification assumptions,
but, of course, need not be so in reality. We also assume, for simplicity,
that the velocity field $|v_{z}|$ is constant for ${\bar{r}}<r_{m}$ and
vanishes for ${\bar{r}}>r_{m}$. According to Equation (63), the time-dependent
density profile is fixed by the initial density conditions. In this section,
we will deal with the density profile outside the galactic plane and will
leave the discussion of the density profile in and near the galactic plane to
the previous section. We consider an initial density profile as follows:

$\displaystyle\rho_{o}({\bar{r}},z,0)$ $\displaystyle=$ $\displaystyle
re(z)\left[\rho_{1}({\bar{r}})+\rho_{2}({\bar{r}})e^{k|z|}\right],$
$\displaystyle re(z)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}1&|z|<z_{i}\\\ 0&|z|\geq~{}z_{i}\\\
\end{array}\right.$ (74)

in which the rectangular function $re(z)$ keeps the exponential function from
diverging. The density profile is depicted in Figure 9.

Figure 9: An initial density profile outside the galactic plane, in which
$\rho_{0}=\rho_{1}+\rho_{2},\frac{\rho_{2}}{\rho_{1}}=-0.2$ and $z_{i}=5$
(kpc) and $k=0.32\ \rm(kpc^{-1})$.

We assume that $\rho_{2}$ is negative and thus the density becomes dilute at
distances far from the galactic plane. As $v_{z}$ is constant both above and
below the galactic plane, $t_{z}=\frac{z}{v_{z}}$ up to a constant. Therefore,
it is easy to deduce from Equation (63) the functional form of $f(\beta)$
($\beta=-t_{z}$):

$f(\beta)=v_{z}re(-v_{z}\beta)[\rho_{1}+\rho_{2}e^{k|v_{z}\beta|}]$ (75)

Hence, according to Equation (70), the time-dependent density function for
matter outside the galactic plane is obtained:

$\rho_{o}({\bar{r}},z,t)=\frac{\gamma}{v_{z}}=re(z-v_{z}t)[\rho_{1}({\bar{r}})+\rho_{2}({\bar{r}})e^{k|z-v_{z}t|}]$
(76)

The density of matter outside the galactic plane will vanish for
$t>t_{m}=\frac{z_{i}}{|v_{z}|}$; hence, we will discuss only the duration of
$t<t_{m}$. Let us look at the mass contained in the cylinder defined by the
galaxy (see Figure 10) and let us assume that the total mass in that cylinder
is $M_{T}$. Now, the mass outside the galactic disk will be:

$\displaystyle M_{o}(t)$ $\displaystyle=$ $\displaystyle
2\pi\left[\int_{-z_{i}}^{-\frac{1}{2}\Delta
z}dz\int_{0}^{r_{m}}d{\bar{r}}{\bar{r}}\rho_{o}({\bar{r}},z,t)\right.$ (77)
$\displaystyle+$ $\displaystyle\left.\int_{\frac{1}{2}\Delta
z}^{z_{i}}dz\int_{0}^{r_{m}}d{\bar{r}}{\bar{r}}\rho_{o}({\bar{r}},z,t)\right]$

Hence, the mass in the galactic disk is:

$M(t)=M_{T}-M_{o}(t)$ (78)

Moreover, the galactic mass derivatives are:

$\dot{M}(t)=-\dot{M_{o}}(t),\qquad\ddot{M}(t)=-\ddot{M_{o}}(t)$ (79)

Figure 10: The mass column defined by the galaxy: (a) side view; (b) three-
dimensional view.

By inserting Equation (76) into Equation (77), we may calculate $M_{o}(t)$:

$\displaystyle M_{o}(t)$ $\displaystyle=$ $\displaystyle
2\left[\lambda_{1}\left(z_{i}-|v_{z}|t-\frac{1}{2}\Delta z\right)\right.$ (80)
$\displaystyle+$
$\displaystyle\left.\frac{\lambda_{2}}{k}\left(e^{kz_{i}}-e^{k(|v_{z}|t+\frac{1}{2}\Delta
z)}\right)\right]$

in which:

$\lambda_{1}\equiv
2\pi\int_{0}^{r_{m}}d{\bar{r}}{\bar{r}}\rho_{1}({\bar{r}}),\qquad\lambda_{2}\equiv
2\pi\int_{0}^{r_{m}}d{\bar{r}}{\bar{r}}\rho_{2}({\bar{r}}).$ (81)

Then, calculating the second derivative of $M_{o}(t)$ and using Equation (79)
leads to the result:

$\ddot{M}(t)=-\ddot{M_{o}}(t)=2k|v_{z}|^{2}e^{\frac{1}{2}\Delta
zk}\lambda_{2}e^{k|v_{z}|t}.$ (82)

We then denote:

$\alpha\equiv
k|v_{z}|,\qquad\tau\equiv\frac{1}{\alpha},\qquad\ddot{M}(0)=2k|v_{z}|^{2}e^{\frac{1}{2}\Delta
zk}\lambda_{2}.$ (83)

Thus:

$\ddot{M}(t)=\ddot{M}(0)e^{\frac{t}{\tau}}.$ (84)

It should be stressed that the current approach does not require that the
velocity $|v_{z}|$ is high; in fact, the vast majority of galactic bodies
(stars, gas) are substantially subluminal—in other words, the ratio of
$\frac{|v_{z}|}{c}\ll 1$. The typical velocities in galaxies are 100 km/s,
which makes this ratio $0.001$ or smaller. However, we stress again the fact
that every gravitational system, even if it is made of subluminal bodies, has
a finite retardation distance, beyond which the retardation effect cannot be
neglected. As demonstrated above, a galaxy exchanges mass with its
environment. This means that a galaxy has a finite retardation distance. The
question is thus quantitative: how large is the retardation distance? For the
M33 galaxy, the velocity curve indicates that retardation effects cannot be
neglected beyond a certain distance, which is calculated in Section 5 to be
roughly 14,000 light years; similar analyses for other galaxies of different
types have shown similar results Wagman . We demonstrate here, using a
detailed model, that this does not require a high velocity of gas or stars, in
or out of the galaxy, and is perfectly consistent with the current
observational knowledge of galactic and extragalactic material content and
dynamics. Equation (83) means that we must have $\lambda_{2}<0$ according to
Equation (28) in order to assure an attractive force. Next, we calculate
$\dot{M}(t)$; using Equations (79) and (80), we obtain:

$\dot{M}(t)=-\dot{M_{o}}(t)=2|v_{z}|\lambda_{1}+2|v_{z}|e^{\frac{1}{2}\Delta
zk}\lambda_{2}e^{k|v_{z}|t}.$ (85)

Hence:

$\dot{M}(0)=2|v_{z}|\lambda_{1}+2|v_{z}|e^{\frac{1}{2}\Delta
zk}\lambda_{2}=2|v_{z}|\lambda_{1}+\frac{\ddot{M}(0)}{\alpha}$ (86)

Thus, $\lambda_{1}$ is:

$\lambda_{1}=\frac{1}{2|v_{z}|}\left[\dot{M}(0)-\frac{\ddot{M}(0)}{\alpha}\right]$
(87)

Inserting Equations (87) and (83) into Equation (85) leads to:

$\dot{M}(t)=\dot{M}(0)+\tau\ddot{M}(0)\left(e^{\frac{t}{\tau}}-1\right)=\dot{M}(0)-\tau|\ddot{M}(0)|\left(e^{\frac{t}{\tau}}-1\right).$
(88)

Finally, by combining Equations (78), (80), (83) and (87) and noticing that:

${M}(0)=M_{T}-\frac{z_{i}-\frac{1}{2}\Delta
z}{|v_{z}|}\left(\dot{M}(0)-\frac{\ddot{M}(0)}{\alpha}\right)-\frac{\ddot{M}(0)}{\alpha^{2}}\left(e^{k(z_{i}-\frac{1}{2}\Delta
z)}-1\right)$ (89)

we arrive at:

$M(t)=M(0)+\left(\dot{M}(0)-\tau\ddot{M}(0)\right)t+\tau^{2}\ddot{M}(0)\left(e^{\frac{t}{\tau}}-1\right),\quad\tau>0.$
(90)

##  7 The Mass Formula

A naive observer may speculate that $\ddot{M}$, given in Equation (51), is
constant over time; however, this is contradicted both by the dynamical model
as presented in Equation (90) and by the following simple argument.

Let us assume that $\ddot{M}$ is constant. This leads to:

$\dot{M}(t)=\dot{M}(0)+t\ddot{M}=\dot{M}(0)-t|\ddot{M}|$ (91)

Let us assume that there is no significant mass accumulation at the current
time $T$, that is, $\dot{M}(T)=0$. In this case:

$\dot{M}(0)=T|\ddot{M}|$ (92)

Assuming that the age of the galaxy is about the age of the universe: $T\simeq
13\times 10^{9}\ {\rm years}=4.1\times 10^{17}{\rm s}$, the initial mass
accumulation rate is about $\dot{M}(0)\simeq 3.74\times 10^{34}{\rm~{}kg\
s^{-1}}$. By integrating Equation (91), we arrive at an equation for the mass
of the galaxy at any time:

$M(t)=M(0)+\dot{M}(0)t+\frac{1}{2}\ddot{M}t^{2}$ (93)

Assuming that, initially, there was no significant amount of mass $M(0)=0$ and
taking into account Equation (92) (as for Equation (92), it should be once
again recalled that the second derivative of $M$ by $t$ is negative) we arrive
at:

$M(t)=\ddot{M}t(\frac{1}{2}t-T)\Rightarrow
M(T)=-\frac{1}{2}\ddot{M}T^{2}=\frac{1}{2}|\ddot{M}|T^{2}$ (94)

By plugging in the mass accumulation decrease rate, we arrive at $M(T)\simeq
7.66\times 10^{51}$ kg, which is clearly 11 orders of magnitude greater than
the known mass of the galaxy. On the other hand, if we assume that $M(T)=M$ is
the current mass of the galaxy, this will lead to $|\ddot{M}|\simeq 2.37\times
0^{5}$ kg ${\rm{s}^{-2}}$, which is 11 orders of magnitude less than what is
needed to explain the current galactic rotation curves. This indicates that
$|\ddot{M}|$ has increased considerably over time due to the depletion of the
surrounding gas and that retardation forces are less significant in the early
stages of galactic formation. Indeed, in young galaxies, the effect of
retardation seems insignificant Dokkum , while, for older galaxies, it seems
like somebody is pressing hard on the brakes of mass accumulation. The
changes, by many orders of magnitude, in $|\ddot{M}|$ suggest exponential
growth in agreement with Equation (90).

If we assume $\dot{M}(t)>0$, it follows from Equation (88) that there is a
maximal duration for the above model to be valid:

$t_{max1}=\tau\ln\left(\frac{\dot{M}(0)}{\tau|\ddot{M}(0)|}+1\right).$ (95)

This is the time it takes the intergalactic gas to deplete (see Section 6) and
thus mass accumulation stops. We shall now partition the mass into a linear
and exponential growing part as follows:

$M(t)=M_{l}(t)+M_{e}(t)=M_{l}(t)-|M_{e}(t)|>0\Rightarrow M_{l}(t)>|M_{e}(t)|.$
(96) $\displaystyle M_{l}(t)$ $\displaystyle=$ $\displaystyle
M(0)+\tau^{2}|\ddot{M}(0)|+\left(\dot{M}(0)+\tau|\ddot{M}(0)|\right)t>0,$
$\displaystyle M_{e}(t)$ $\displaystyle=$
$\displaystyle\tau^{2}\ddot{M}(t)=-\tau^{2}|\ddot{M}(0)|e^{\frac{t}{\tau}}<0.$
(97)

By dividing Equation (96) by $|\ddot{M}(t)|$ and using the definition of
Equation (35), we have:

$t_{r}^{2}=\frac{M_{l}(t)}{|\ddot{M}(t)|}-\tau^{2}.$ (98)

We define the slow retardation time as:

$t_{rs}(t)\equiv\sqrt{\frac{M_{l}(t)}{|\ddot{M}(0)|}};$ (99)

the square of this retardation time is non-linearly dependent on time. Hence:

$t_{r}^{2}=t_{rs}^{2}e^{-\frac{t}{\tau}}-\tau^{2}$ (100)

It is obvious that $t_{rs}>\tau$ and it is also obvious that for any
$t<t_{max1}$, then $t_{rs}e^{-\frac{t}{2\tau}}>\tau$ because the total mass is
not reduced, only mass accumulation slows down (in other words, we assume that
the galaxy does not eject matter, thus reducing its mass). Equation (100)
shows that there is no simple relation between the retardation time of the
galaxy and the typical time $\tau$ associated with the second derivative of
mass.

We conclude this section by deriving constraints from the assumption that the
apparent luminosity and rotation curve have not changed in a detectable way
since observations of the M33 galaxy have begun. If the mass to light ratio is
assumed to be constant, this means that the mass of the galaxy has not changed
significantly during the said period. Furthermore, the retardation time, which
has a significant effect on the galactic rotation curve, is a square root of
the ratio of the galactic mass by the second derivative of the same mass. This
means that if the retardation time (and retardation distance) have not changed
considerably during the duration of observations, then the second derivative
must also be approximately constant during that period. The M33 Triangulum
Galaxy was probably discovered by the Italian astronomer Giovanni Battista
Hodierna before 1654. In his work, De systemate orbis cometici; deque
admirandis coeli caracteribus (“About the systematics of the cometary orbit,
and about the admirable objects of the sky”), he listed it as a cloud-like
nebulosity or obscuration and gave the cryptic description, “near the Triangle
hinc inde”. This is in reference to the constellation of Triangulum as a pair
of triangles. The magnitude of the object matches M33, so it is most likely a
reference to the Triangulum galaxy Fodera . This observation took place 366
years ago; however, there was no measurement of the apparent or real
luminosity of M33 and of course no measurement of Doppler shifts were
available at that time. It was only Hubble that established the distance to
this object and thus declared it an independent galaxy far away from the Milky
Way Van , thus enabling the calculation of the real luminosity and mass of
M33. Doppler shifts of spectral lines were obtained even later, thus enabling
the deduction of rotation curves. In what follows, we shall assume an
observation duration of:

$\delta t={\rm 400\ years}$ (101)

As $\ddot{M}(t)$ is approximately constant during the said duration, it
follows that:

$|\frac{d^{3}M(t)}{dt^{3}}\delta t|\ll|\ddot{M}(t)|.$ (102)

Taking into account Equation (84) gives a lower bound to $\tau$:

$\delta t\ll\tau.$ (103)

This is not a strong bound as $\tau$ can be anything from 100,000 years to ten
times the age of the universe or even larger. Similarly, the fact that the
mass of the galaxy has not changed considerably during the observation
duration indicates that:

$|\dot{M}(t)|\delta
t|\ll|M(t)|\Rightarrow\dot{M}(t)\ll\dot{M}_{max}\equiv\frac{M(t)}{\delta t}$
(104)

The upper bound on the mass derivative is thus about $\dot{M}_{max}\simeq
1.6\times 10^{30}$ kg/s, in which we have taken the baryonic mass of the M33
galaxy to be about $2\times 10^{40}$ kg Corbelli2 . Using Equation (88), we
arrive with an upper bound to $\dot{M}(0)$:

$\dot{M}(0)\ll\dot{M}_{max}+\tau\left(|\ddot{M}(t)|-|\ddot{M}(0)|\right).$
(105)

Notice, however, that without a handle on $\tau$ and $|\ddot{M}(0)|$, this
does not yield any definite information.

##  8 Conclusions

The need to satisfy the Lorentz symmetry group prevents the weak field
approximation of GR from allowing action at distance potentials and thus only
retarded solutions are allowed. Retardation is manifested more strongly when
large distances and large second derivatives are involved. It should be
stressed that the current approach does not require that velocities, $v$ are
high; in fact, the vast majority of galactic bodies (stars, gas) are
substantially subluminal—in other words, the ratio of $\frac{v}{c}\ll 1$. The
typical velocities in galaxies are $100{\rm\frac{km}{s}}$ (see Figure 6),
which makes this ratio $0.001$ or smaller. However, one should consider the
fact that every gravitational system, even if it is made of subluminal bodies,
has a retardation distance, beyond which the retardation effect cannot be
neglected. Every natural system, such as a star or a galaxy and even a
galactic cluster, exchanges mass with its environment. For example, the sun
loses mass through solar wind and galaxies accrete gas from the intergalactic
medium. This means that all natural gravitational systems have a finite
retardation distance. The question is thus quantitative: how large is the
retardation distance? The change in the mass of the sun is quite small and
thus the retardation distance of the solar system is huge, allowing us to
neglect retardation effects within the solar system. However, for the M33
galaxy, the velocity curve indicates that the retardation effects cannot be
neglected beyond a certain distance, which was calculated in Section 5 to be
roughly $R_{r}=4.54$ kpc; similar analyses for other galaxies of different
types have shown similar results Wagman ; Wagman2 . We demonstrated, using a
detailed model, in Section 6, that this does not require a high velocity of
gas or stars in or out of the galaxy and is perfectly consistent with the
current observational knowledge of galactic and extragalactic material content
and dynamics.

We point out that, if the mass outside the galaxy is still abundant (or
totally consumed), $\ddot{M}\simeq 0$ and retardation force should vanish.
This was reported Dokkum  for the galaxy NGC1052-DF2.

We note that the same terms in the gravitation equation that are responsible
for the gravitational radiation recently discovered are also responsible for
the rotation curves of galaxies. The expansion given in Equation (23), being a
Taylor series expansion up to the second order, is only valid for limited
radii:

$R<c\ T_{max}\equiv R_{max}$ (106)

This means that current expansion is related to the near field case; this is
acceptable since the extension of the rotation curve in galaxies is the same
order of magnitude as the size of the galaxy itself. An opposite case in which
the size of the object is much smaller than the distance to the observer will
result in a different approximation to Equation (13), leading to the famous
quadruple equation of gravitational radiation, as predicted by Einstein
Einstein2  and verified indirectly in 1993 by Russell A. Hulse and Joseph H.
Taylor, Jr., for which they received the Nobel Prize in Physics. The discovery
and observation of the Hulse–Taylor binary pulsar offered the first indirect
evidence of the existence of gravitational waves Taylor . On 11 February 2016,
the LIGO and Virgo Scientific Collaboration announced that they had made the
first direct observation of gravitational waves. The observation was made five
months earlier, on 14 September 2015, using Advanced LIGO detectors. The
gravitational waves originated from the merging of a binary black hole system
Castelvecchi . Thus, the current paper involves a near-field application of
gravitational radiation while previous works discuss far- field results.

We regret that direct measurement of the second temporal derivative of the
galactic mass is not available. What is available is the remarkable fit
between the retardation model and the galactic rotation curve, as can be seen
in Figure 6, which constitutes indirect evidence of the galactic mass second
derivative. The reader is reminded that competing theories like dark matter do
not supply any observational evidence either. Despite the work of thousands of
people and the investment of billions of dollars, there is still no evidence
of dark matter. Occam’s razor dictates that when two theories compete, the one
that makes less assumptions has the upper hand. In the case of retardation
theory, only baryonic matter and a large second temporal derivative of mass
are assumed.

Problems that inflict dark matter theory, such as the cuspy halo problem (also
known as the core-cusp problem) refers to a discrepancy between the inferred
dark matter density profiles of low-mass galaxies and the density profiles
predicted by cosmological N-body simulations. Nearly all simulations form dark
matter halos, which have “cuspy” dark matter distributions, with density
increasing steeply at small radii, while the rotation curves of most observed
dwarf galaxies suggest that they have flat central dark matter density
profiles (“cores”). This problem does not occur in the retardation model which
denies the existence of dark matter. One cannot discuss flat or sharp profiles
of dark matter if dark matter does not exist. The inherent problems with dark
matter’s dynamics further strengthen the claim of this work that dark matter
does not exist and the rotation curve characteristics attributed to dark
matter should be attributed to retardation.

How can one reach an erroneous conclusion regarding the existence of “dark
matter”? By ignoring retardation effects and assuming that radial velocities
are a result of some mysterious substance. We obtain for a spherically
symmetric distribution Binney :

$-\frac{v_{c}^{2}}{r}\hat{r}=\vec{F}_{d}=-\frac{GM_{d}(r)}{r^{2}}\hat{r}$
(107)

where $v_{c}$ is the speed of a particle of unchanging radius $r$ and
$M_{d}(r)$ is the dark matter within the radius $r$. Comparing Equations (107)
and (28), we deduce that the "dark matter" mass can be calculated as follows:

$M_{d}(r)=\frac{r^{2}|\ddot{M}|}{2c^{2}}$ (108)

Then, since:

$M_{d}(r)=4\pi\int_{0}^{r}r^{\prime
2}\rho_{d}(r^{\prime})dr^{\prime},\qquad\frac{dM_{d}(r)}{dr}=4\pi
r^{2}\rho_{d}(r)$ (109)

it follows that:

$\rho_{d}(r)=\frac{|\ddot{M}|}{4\pi c^{2}r}$ (110)

This is consistent with the observational data of Corbelli  who concluded that
the “dark matter” density decreases as $r^{-1.3}$ for M33.

An additional method to explaining galactic rotation curves is to postulate
that either the laws of dynamics or the laws of gravitation (GR) should be
changed. This is the case in an approach championed by Milgrom (modifying the
laws of gravity (MOND)—modified Newtonian dynamics) Mond . In this approach,
the classical law of gravity is modified:

$\vec{F}_{M}=-\frac{GM}{\mu(\frac{a}{a_{0}})r^{2}}\hat{r}$ (111)

In the above $\mu$, is the interpolation function that should be one for
$a_{0}\ll a$. Let us assume:

$\mu(x)=\frac{x}{\sqrt{1+x^{2}}}\quad\Rightarrow\quad\mu(\frac{a}{a_{0}})=\frac{1}{\sqrt{1+(\frac{a_{0}}{a})^{2}}}$
(112)

if $a_{0}\gg a$, $\mu\simeq\frac{a}{a_{0}}$. A particle revolving in a
unchanging radius will have a centrifugal acceleration of $a=\frac{v^{2}}{r}$
and thus:

$\vec{F}_{M}=-\frac{GMa_{0}}{v^{2}\ r}\hat{r}$ (113)

For the constant $v$ at a far away distance, this expression is similar to the
retardation force and thus:

$|\ddot{M}|=\frac{2Ma_{0}c^{2}}{v^{2}\ r}.$ (114)

Milgrom found $a_{0}=1.2\times 10^{-10}{\rm ms^{-2}}$ to be most fitting to
the data. The velocity at $15.33\ {\rm kpc}$ from the center of the galaxy is
135,640 $\ {\rm ms^{-1}}$. We thus obtain $|\ddot{M}|\simeq 4.94\times
10^{16}\ {\rm kgs^{-2}}$ and a retardation time of:

$t_{r}=\sqrt{\frac{M}{|\ddot{M}|}}\simeq 6.35\ 10^{11}\ {\rm s}$ (115)

This amounts to a typical accumulation acceleration time scale of
$t_{r}\simeq$20,129 years and a retardation distance of:

$R_{r}=ct_{r}\simeq 20,129{\rm\ light\ years},$ (116)

which seems reasonable according to our fitting estimates. Hence, despite the
fact that modified Newtonian dynamics theory is phenomenological and is in
contradiction to general relativity it can still serve as tool for estimating
retardation theory quantities.

We have thus shown that “dark matter” and “MOND” effects can be explained in
the framework of standard GR as effects due to retardation, without assuming
any exotic matter or modifications of the theory of gravity.

To conclude this section, we would like to mention the remarkable theory of
conformal gravity put forward by Mannheim Mannheim1 ; Mannheim2 . The current
retardation approach leads to a Newtonian potential plus a linear potential.
Such potential types can also be derived from the completely different
theoretical considerations of conformal gravity. Indeed, on purely
phenomenological grounds, such fits have essentially already been published in
the literature. While those fits looked very good, they had to treat the
coefficient of the linear potential as a variable that changed from galaxy to
galaxy; this element is not in line with conformal gravity in which the
coefficient of the linear potential is a new universal constant of nature.
This can be explained much more easily in the framework of retardation theory,
in which a variable $\ddot{M}/M$ seems reasonable depending on the dynamical
conditions of various galaxies. Indeed, $\ddot{M}/M$ has a theoretical basis,
while a pure phenomenological approach does not. As far as we understand the
work of Mannheim Mannheim0 ; Mannheim1 ; Mannheim2 , it is related to
conformal gravity, which is different from GR, and thus has to justify other
results of GR (Big Bang Cosmology, etc.). We also underline that retardation
theory does not contradict conformal gravity and, in reality, both effects may
exist, although Occam’s razor forbids us to add new universal constants if the
existing ones suffice to explain observations.

Retardation theory’s approach is minimalistic (in the sense that it satisfies
the Occam’s razor rule), does not affect observations that are beyond the
near-field regime and, therefore, does not clash with GR theory and its
observations (nor with Newtonian theory, as the retardation effect is
negligible for small distances). We underline that a perfect fit to the
rotation curve is achieved with a single parameter and we do not adjust the
mass to light ratio in order to improve our fit as other authors do.

Retardation effects in electromagnetic theory were discussed in Tuval ;
YahalomT ; Yahalom3 .

Finally the paper does not discuss dark matter in a cosmological context or in
a gravitational lensing scenario; this is left for future works. Another
important application is the study of galaxy clusters. This issue led Zwicky
zwicky  to suggest dark matter in the first place.

This paper has a single author, which has done all the work presented.

This research received no external funding.

###### Acknowledgements.

The author wishes to thank his former student, Michal Wagman, for supplying
the data points for the rotation curve of the M33 galaxy. This work is a
result of more than twenty five years of thinking (discontinuously) on the
dark matter problem, which was first suggested to me by the late Jacob
Bekenstein during my stay at the Hebrew University of Jerusalem. The current
retardation solution arose from my discussions with the late Donald Lynden-
Bell of Cambridge University, and the late Miron Tuval. Other people with
which I discussed this work and offered important feedback are Lawrence
Horwitz of Tel-Aviv University and Marcelo Shiffer of Ariel University.
Recently, this work benefitted from discussions with James Peebles, Neta
Bachall and Sam Cohen, all from Princeton University. Special thanks are due
to Jiri Bicak for inviting me to present the theory at Charles University in
Prague. I would like to thank Philip Mannheim and James Obrien for our
discussions during the recent IARD meetings and for supplying some relevant
data. I have benefited from discussions with a very long list of distinguished
scientists and ask their forgiveness for not mentioning them all. The author
declares no conflict of interest. References

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