# Evolving wormhole geometry from dark matter energy density

Farook Rahaman<EMAIL_ADDRESS>Department of Mathematics,
Jadavpur University, Kolkata 700032, West Bengal, India  Bikramarka S
Choudhury<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata 700032, West Bengal, India

###### Abstract

We analyse traversable wormholes defined by the dynamic line elements that
asymptotically approaches Friedmann-Robertson-Walker (FRW) universe. This
dynamical wormholes is supported by the galactic dark matter as well as
perfect isotropic fluid. We will discuss several evolving Lorentzian wormholes
comprising with different perfect isotropic fluids in addition to various
scale factors. We will speculate the various significance, features and throat
energy conditions for these evolving traversable Lorentzian wormholes.

###### pacs:

04.40.Nr, 04.20.Jb, 04.20.Dw

Keywords :  Evolving wormhole ; Dark matter energy density ; Friedmann-
Robertson-Walker universe

##  I Introduction:

According to Einstein’s theory PartilcleProblem  , the spacetime fabric and
geometry due to the presence of matter is not inflexible but is adaptable and
deformable. Heavier condensed object produces strong the curvature of space
003 , which fundamentally leads to the concepts of black holes 001 ; 002
containing the curvature singularity. Due to the presence of singularity, one
can not travel through it, however, in some way, if a structure could be
constructed that consists of a throat then it might be possible to travel
through the structure. Fortunately, Einstein field equations einstein1915field
may have such solutions under certain conditions where the spacetime geometry
is distorted such a manner that creates a tunnel like structure in the
galactic fabric. These theoretical solutions, initially proposed by Albert
Einstein and Nathan Rosen, in the year 1935, are known as Einstein-Rosen
bridges or Wormholes FRWwormhole ; Wormhole_collapse . This structure connects
two different points in the spacetime fabric in a way that is analogous to a
shortcut that can ultimately minimize travel time and make it seem like the
travel speed is even greater than that of light. But to be able to travel
through a wormhole, certain conditions for the wormhole must be satisfied, for
example the throat of the wormhole structure should be stable enough for
enabling one to travel through it traversable , also the wormhole must have a
structure that allows the size of the traveler’s body to pass through it.
Wormholes that satisfy such conditions are called traversable wormholes
Travworm_state ; CosmoModel . We understand that the throat of the wormhole is
one essential thing for the structure and a necessary condition for
construction of throat is presence of exotic matter which violates the Null-
Energy condition 004  . Now a big concern is whether such matter exists. In
answer to that we would like to point to the recent finding of the accelerated
expansion of the universe observational . In various attempts to explain this
phenomenon, the arguments have led to the possible existence of matter with a
large negative pressure. These are referred to as dark matter or dark energy
cosmology ; observational . The dark matter or dark energy, which are
basically exotic matter, play a pivotal role in the formation of wormhole
structure Wormhole_darkEnergyVladimir . Now, since the universe is expanding
and wormholes are structures inside the geometry of the 4-dimensional
spacetime manifold, so the idea is that with the expansion of universe, there
is a corresponding effect on the geometry of the manifold as the universe is
expanding from every point and hence it must have an effect on the wormhole
structure as well. The wormholes that exhibit this property are called
evolving wormholes Revisiting_perspec ; Phatom_singularities ; EvoLorentzian ;
EvolvingTopologically  i.e. the wormholes are evolving with time. We know that
the scale factor is a function of time. So we have attempted to determine
corresponding changes in the wormhole structures as the time varies.

In this paper we have discussed several aspects of the evolving wormholes. In
the former sections, we briefly describe metric and Einstein field equations
regarding wormhole in FRW universe. Then we dive into solutions regarding
wormholes as well as scale factors and density functions for different cases.
After that we study the energy conditions and embedding space regarding the
wormhole and next, we calculate the condition for traversability through the
wormhole and finally we calculate the proper length between two distances.

##  II  basic equations of WORMHOLE EMBEDDED IN frw universe:

Let us consider the spacetime metric of a dynamic traversable wormhole in a
Friedmann-Robertson-Walker universe as

$ds^{2}=-e^{2\phi(r)}dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}-\frac{b(r)}{r}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right].$ (1)

Here b(r) and $\phi(r)$ are known as shape function ( describes the nature of
the wormhole throat which indicates the surface of minimum radius ) and
redshift function ( which is finite everywhere to ensure without event horizon
) respectively and $a(t)$ is the scale factor of the universe describing the
size of the universe. $k$ represents the sign of the curvature of spacetime
with values: $+1,0,-1$. Note that the metric (1) becomes static Morris-Thorne
wormhole when $a(t)\rightarrow~{}constant$ and $k\rightarrow~{}0$. When
$b(r)~{}and~{}\phi(r)\rightarrow 0$, the the spacetime metric (1) coincides
with the FRW metric.

In this study, we assume an inhomogeneous and anisotropic fluid matter source
comprising the wormhole with a diagonal energy-momentum tensor as
$(\rho(r,t),p_{r}(r,t),p_{t}(r,t),p_{t}(r,t))$. The quantities
$\rho(r,t),p_{r}(r,t),p_{t}(r,t)$ stand for the mass energy density, radial
pressure and transverse pressure as measured by observers sited at constant
$r,u,\phi$.

With these considerations, the non trivial Einstein’s equations are obtained
as ( $8\pi G=1$ )

$\rho(r,t)=\frac{3\dot{a}^{2}}{a^{2}}e^{-2\phi}+\frac{3k}{a^{2}}+\frac{b^{\prime}}{a^{2}r^{2}},$
(2)
$p_{r}(r,t)=-\frac{2\ddot{a}}{a}e^{-2\phi}-\frac{\dot{a}^{2}}{a^{2}}e^{-2\phi}-\frac{k}{a^{2}}-\frac{b}{a^{2}r^{3}}+\frac{2}{a^{2}r}\phi^{\prime}\left(1-kr^{2}-\frac{b}{r}\right),$
(3)
$p_{t}(r,t)=-\frac{2\ddot{a}}{a}e^{-2\phi}-\frac{\dot{a}^{2}}{a^{2}}e^{-2\phi}-\frac{k}{a^{2}}+\frac{b-rb^{\prime}}{2a^{2}r^{3}}+\frac{1}{a^{2}r}\phi^{\prime}\left(1-kr^{2}-\frac{b}{r}\right)\\\
+\frac{1}{a^{2}}\left[{\phi^{\prime}}^{2}\left(1-kr^{2}-\frac{b}{r}\right)+\phi^{\prime\prime}\left(1-kr^{2}-\frac{b}{r}\right)-\frac{1}{2}\phi^{\prime}\left(2kr+\frac{rb^{\prime}-b}{r^{2}}\right)\right],$
(4)
$T_{tr}=\frac{\dot{a}}{a^{2}}e^{-\phi}{\phi^{\prime}\left(1-kr^{2}-\frac{b}{r}\right)}^{1/2}.$
(5)

Here, a prime and an overdot denote differentiation with respect to r and t,
respectively. $T_{tr}$ stands for the outward energy flow.

Conservation of energy equation, $T_{\mu;\nu}^{\nu}=0$, yields

$\frac{\partial\rho}{\partial t}+\frac{\dot{a}}{a}(3\rho+p_{r}+2p_{t})=0,$ (6)
$p_{r}^{\prime}-(p_{t}-p_{r})\frac{2}{r}=0.$ (7)

##  III  evolving wormhole solutions :

We have assumed no outward energy flow, i.e. $T_{tr}=0$.

Equation (5) implies either $\phi^{\prime}=0$ or $\dot{a}=0$. If $\dot{a}=0$,
then dynamism of wormhole will be lost. So, we reject this case. Now,
$\phi^{\prime}=0$ implies $\phi=constant=0$.

Now the field equations (2)-(4) will be modified as

$\rho(r,t)=\frac{3(\dot{a}^{2}+k)}{a^{2}}+\frac{b^{\prime}}{a^{2}r^{2}},$ (8)
$p_{r}(r,t)=-\frac{2\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}-\frac{k}{a^{2}}-\frac{b}{a^{2}r^{3}},$
(9)
$p_{t}(r,t)=-\frac{2\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}-\frac{k}{a^{2}}+\frac{b-rb^{\prime}}{2a^{2}r^{3}}.$
(10)

Following Sung-Won Kim [1], we use the method of separation of variables to
solve these equations, and assume

$\displaystyle\begin{split}a^{2}(t)\rho(r,t)&=a^{2}(t)\rho^{c}(t)+\rho^{w}(r),\\\
a^{2}(t)p_{r}(r,t)&=a^{2}(t)p^{c}(t)+p_{r}^{w}(r),\\\
a^{2}(t)p_{t}(r,t)&=a^{2}(t)p^{c}(t)+p_{t}^{w}(r).\end{split}$ (11)

The energy density, radial and transverse pressures are taken as separable
forms with superscripts $c$ and $w$ indicating the cosmological part (
function of t only ) and the wormhole part ( function of r only )
respectively. The cosmological part is characterized by an isotropic pressure
$p^{c}$.

Now, the field equations (8) - (10) will be separated as

$a^{2}\left[\rho^{c}-\frac{3(\dot{a}^{2}+k)}{a^{2}}\right]=\frac{b^{\prime}}{r^{2}}-\rho^{w}=L,$
(12)
$a^{2}\left[p^{c}+\frac{2\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right]=-\frac{b}{r^{3}}-p_{r}^{w}=M,$
(13)
$a^{2}\left[p^{c}+\frac{2\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right]=-\frac{b-rb^{\prime}}{2r^{3}}-p_{t}^{w}=M,$
(14)

where L and M are separation constants. In Equations (13) and (14), the
cosmological parts are equal and that is why the separation constants are the
same. Here both the separation constants should not be zero simultaneously as
it play the role of connection between the cosmological part and wormhole
part.

Cosmological part, equations (13) $\&$ (14) yield

$\dot{\rho}_{c}+\frac{3\dot{a}}{a}(\rho_{c}+p_{c})=(L+3M)\frac{\dot{a}}{a^{3}}.$
(15)

[ One can also obtain the above equation from conservation of energy equation,
$T_{\mu;\nu}^{\nu}=0$ ]

From equation (12), we can get the cosmological part as

$\rho_{c}=\frac{L}{a^{2}}+\frac{3(\dot{a}^{2}+k)}{a^{2}}.$ (16)

For cosmological part i.e. evolution part, equation (15) $\&$ (16) are two
master equations with three unknowns namely $a,p_{c},\rho_{c}$.
For wormhole part, we have three master equations with four unknowns as

$\frac{b^{\prime}}{r^{2}}-\rho^{w}=L,$ (17)
$\frac{b-b^{\prime}r}{2r^{3}}-p_{t}^{w}=M,$ (18)
$-\frac{b}{r^{3}}-p_{r}^{w}=M.$ (19)

###  III.1  Wormhole solutions supported by dark matter:

In recent past, Yoshiaki SOFUE so  proposed a new density profile of the dark
matter distribution in the spiral galaxy known as the exponential density
profile and is given by

$\rho^{w}=\rho_{s}e^{-\frac{r}{r_{s}}},$ (20)

where $r_{s}$ is the scale radius and $\rho_{s}$ is the central density.

In the context of evolving wormhole studies, we assume that the wormhole part
is supported by above exponential density profile (20).

Now from equation (17)

$b^{\prime}=Lr^{2}+\rho_{s}e^{-\frac{r}{r_{s}}}r^{2},$

we obtain

$b=\frac{L}{3}r^{3}-\rho_{s}e^{-\frac{r}{r_{s}}}[r_{s}r^{2}+2r_{s}^{2}r+2r_{s}^{3}]+C,$
(21)

where C is an integration constant. At the throat radius $r=r_{0}$,
$b(r_{0})=r_{0}$ which implies

$C=r_{0}-\frac{L}{3}r_{0}^{3}+\rho_{s}.e^{-\frac{r_{0}}{r_{s}}}[r_{s}r_{0}^{2}+2r_{s}^{2}r_{0}+2r_{s}^{3}].$

Now, one can find flaring out condition at the throat as
$b^{\prime}(r_{0})<1$, which yields

$Lr_{0}^{2}+r_{0}^{2}\rho_{s}e^{-\frac{r_{0}}{r_{s}}}<1$

i.e.,

$L<\frac{1}{r_{0}^{2}}-\rho_{s}e^{-\frac{r_{0}}{r_{s}}}.$ (22)

The condition for the asymptotic flatness of the wormhole is
$\lim_{r\rightarrow\infty}\frac{b(r)}{r}\rightarrow 0$. Therefore, to satisfy
the asymptotically flatness condition, one needs to take $L=0$. However, for
asymptotically non flat wormhole, one will have to match the wormhole
spacetinme at some junction interface with the Schwarzschild spherically
symmetric static vacuum solution according to Birkhoff theorem.

Putting the value of $b$ in (18) and (19), we can obtain the expressions of
$p_{t}^{w}$, $p_{r}^{w}$ as

$p_{t}^{w}=-\frac{L}{3}-\rho_{s}e^{-\frac{r}{r_{s}}}\left[\frac{1}{2}+\frac{r_{s}}{2r}+\frac{r_{s}^{2}}{r^{2}}+\frac{r_{s}^{3}}{r^{3}}\right]+\frac{C}{2r^{3}}-M,$
(23)
$p_{r}^{w}=-\frac{L}{3}+\rho_{s}e^{-\frac{r}{r_{s}}}\left[\frac{r_{s}}{r}+\frac{2r_{s}^{2}}{r^{2}}+\frac{2r_{s}^{3}}{r^{3}}\right]-\frac{C}{r^{3}}-M.$
(24)

###  III.2  Solutions for scale factor :

Now we will explore the cosmological part with different criteria as follows:

Case 1: Assume power law form of scale factor

$\displaystyle a(t)$
$\displaystyle=t^{n},~{}~{}~{}n~{}is~{}an~{}arbitrary~{}constant.$ (25)

Here, we assume the evolution of cosmological wormhole is described by the
expansion of the scale factor given in equation (24).

For this scale factor, equations (15) and (16) yield energy density and
pressure ( both cosmological part ) as

$\rho^{c}=Lt^{-2n}+3t^{-2n}(n^{2}t^{2n-2}+k),$ (26)
$p^{c}=t^{-2n}(M-k)+t^{-2}(2n-3n^{2}).$ (27)

The total energy density and pressures are given by

$\rho=Lt^{-2n}+3t^{-2n}(n^{2}t^{2n-2}+k)+t^{-2n}\left[\rho_{s}e^{-\frac{r}{r_{s}}}\right],$
(28)
$p_{r}=t^{-2n}(M-k)+t^{-2}(2n-3n^{2})+t^{-2n}\left[-\frac{L}{3}+\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{r_{s}}{r}+\frac{2r_{s}^{2}}{r^{2}}+\frac{2r_{s}^{3}}{r^{3}}\right)-\frac{C}{r^{3}}-M\right],$
(29)
$p_{t}=t^{-2n}(M-k)+t^{-2}(2n-3n^{2})+t^{-2n}\left[-\frac{L}{3}-\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{1}{2}+\frac{r_{s}}{2r}+\frac{r_{s}^{2}}{r^{2}}+\frac{r_{s}^{3}}{r^{3}}\right)+\frac{C}{2r^{3}}-M\right].$
(30)

Case 2: Assume exponential form of scale factor

$a(t)=e^{\omega t}~{},~{}~{}~{}\omega~{}is~{}an~{}arbitrary~{}constant.$ (31)

This is known as the de-Sitter universe which indicates a non singular
continuously expanding model of the universe.

For this choice, we obtain the energy density and pressure component as

$\rho^{c}=3\omega^{2}+(L+3k)e^{-2\omega t},$ (32) $p^{c}=e^{-2\omega
t}(M-k)-3\omega^{2}.$ (33)

In this case, the total energy density and pressures are given by

$\rho=3\omega^{2}+(L+3k)e^{-2\omega t}+e^{-2\omega
t}\left[\rho_{s}e^{-\frac{r}{r_{s}}}\right],$ (34) $p_{r}=e^{-2\omega
t}(M-k)-3\omega^{2}+e^{-2\omega
t}\left[-\frac{L}{3}+\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{r_{s}}{r}+\frac{2r_{s}^{2}}{r^{2}}+\frac{2r_{s}^{3}}{r^{3}}\right)-\frac{C}{r^{3}}-M\right],$
(35) $p_{t}=e^{-2\omega t}(M-k)-3\omega^{2}+e^{-2\omega
t}\left[-\frac{L}{3}-\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{1}{2}+\frac{r_{s}}{2r}+\frac{r_{s}^{2}}{r^{2}}+\frac{r_{s}^{3}}{r^{3}}\right)+\frac{C}{2r^{3}}-M\right].$
(36)

Case 3: $\rho^{c}=\beta a^{-n_{1}}(t)$, $\beta,~{}n_{1}$ are constants.

Here we have assume cosmological density is inversely proportional to
polynomial function of the scale factor. With this form of $\rho_{c}$,
equation (16) yields

$\beta a^{2-n_{1}}=(L+3k)+3\dot{a}^{2}.$

This implies

$\int\frac{da}{\sqrt{a^{2-n_{1}}-(\frac{L+3k}{\beta})}}=t\sqrt{\frac{\beta}{3}}+C_{1},$
(37)

where, $C_{1}$ is an integration constant.

Subcase - 3.1: $n_{1}=1$ :

$a=\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}$
(38)

For this choice, we obtain the energy density and pressure component as

$\rho^{c}=\beta\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-1},$
(39)
$p^{c}=(L+3M)\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-2}-\frac{2\beta}{3}\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-1}.$
(40)

In this case, the total energy density and pressures are given by

$\rho=\beta\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-1}+\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-2}\left[\rho_{s}e^{-\frac{r}{r_{s}}}\right],$
(41)
$p_{r}=(L+3M)\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-2}-\frac{2\beta}{3}\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-1}$
$+\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-2}\left[-\frac{L}{3}+\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{r_{s}}{r}+\frac{2r_{s}^{2}}{r^{2}}+\frac{2r_{s}^{3}}{r^{3}}\right)-\frac{C}{r^{3}}-M\right],$
(42)
$p_{t}=(L+3M)\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-2}-\frac{2\beta}{3}\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-1}$
$+\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{-2}\left[-\frac{L}{3}-\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{1}{2}+\frac{r_{s}}{2r}+\frac{r_{s}^{2}}{r^{2}}+\frac{r_{s}^{3}}{r^{3}}\right)+\frac{C}{2r^{3}}-M\right].$
(43)

Subcase - 3.2: $n_{1}=2$ :

$a=t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}$ (44)

For this choice, we obtain the energy density and pressure component as

$\rho^{c}=\beta\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2},$ (45)
$p^{c}=\left(l+3M-\frac{\beta}{3}\right)\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}.$
(46)

In this case, the total energy density and pressures are given by

$\rho=\beta\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}+\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}\left[\rho_{s}e^{-\frac{r}{r_{s}}}\right],$
(47)

$p_{r}=\left(l+3M-\frac{\beta}{3}\right)\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}$
$+\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}\left[-\frac{L}{3}+\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{r_{s}}{r}+\frac{2r_{s}^{2}}{r^{2}}+\frac{2r_{s}^{3}}{r^{3}}\right)-\frac{C}{r^{3}}-M\right],$
(48)
$p_{t}=\left(l+3M-\frac{\beta}{3}\right)\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}$
$+\left[t\sqrt{\frac{1}{3}(\beta-L-3k)}+C_{1}\right]^{-2}\left[-\frac{L}{3}-\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{1}{2}+\frac{r_{s}}{2r}+\frac{r_{s}^{2}}{r^{2}}+\frac{r_{s}^{3}}{r^{3}}\right)+\frac{C}{2r^{3}}-M\right].$
(49)

Case 4: $a(t)=t^{\alpha}exp(\gamma t)$, $\gamma$ and $\alpha$ are constants:

This is known as hybrid model of the universe comprising with power law form
as well as the exponential form of the scale factor.

For this choice, we obtain the energy density and pressure component as

$\rho_{c}=(L+3k)t^{-2\alpha}e^{-2\gamma
t}+3\left(\frac{\alpha}{t}+\gamma\right)^{2},$ (50)
$p_{c}=(m-k)t^{-2\alpha}e^{-2\gamma t}+2\alpha
t^{-2}-3\left(\frac{\alpha}{t}+\gamma\right)^{2}).$ (51)

In this case, the total energy density and pressures are given by

$\rho=(L+3k)t^{-2\alpha}e^{-2\gamma
t}+3\left(\frac{\alpha}{t}+\gamma\right)^{2}+t^{-2\alpha}e^{-2\gamma
t}\left[\rho_{s}e^{-\frac{r}{r_{s}}}\right],$ (52)
$p_{r}=(m-k)t^{-2\alpha}e^{-2\gamma t}+2\alpha
t^{-2}-3\left(\frac{\alpha}{t}+\gamma\right)^{2})+t^{-2\alpha}e^{-2\gamma
t}\left[-\frac{L}{3}+\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{r_{s}}{r}+\frac{2r_{s}^{2}}{r^{2}}+\frac{2r_{s}^{3}}{r^{3}}\right)-\frac{C}{r^{3}}-M\right],$
(53) $p_{t}=(m-k)t^{-2\alpha}e^{-2\gamma t}+2\alpha
t^{-2}-3\left(\frac{\alpha}{t}+\gamma\right)^{2})+t^{-2\alpha}e^{-2\gamma
t}\left[-\frac{L}{3}-\rho_{s}e^{-\frac{r}{r_{s}}}\left(\frac{1}{2}+\frac{r_{s}}{2r}+\frac{r_{s}^{2}}{r^{2}}+\frac{r_{s}^{3}}{r^{3}}\right)+\frac{C}{2r^{3}}-M\right].$
(54)

The choice of scale factors corresponds to certain case studies. Note that
finding solutions to arbitrary scale factors is quite cumbersome. During our
case studies we also encountered certain scale factors which we could not
solve by existing methods of calculations. Also the choice of the scale factor
for the case studies are very standard forms such as exponential form, power
law form etc. For these scale factors, we have obtained consistent solutions
of the other physical parameters.

##  IV embedding space :

The radial coordinate ’r’ extends from its minimum value at $r_{0}$,
representing the wormhole throat, to infinity, increasing in range. We possess
an asymptotically flat evolving wormhole characterized by a positive energy
density. From the metric (1) one can see that wormholes at spatial infinity
$(r\rightarrow\infty)$ assume the following asymptotic metric :

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1a)$

The metric exhibits slices where t remains constant, constituting spaces of
constant curvature. Consequently, the asymptotic metric (1a) is organized into
foliations of spaces with constant curvature. Since the wormhole described by
equation (1) evolves over time, each slice at a fixed instant will vary for
different time values.

The shape function actually determines the profile picture of a wormhole.
Consider a slice of the wormhole at a fixed instant of time ,
$t=constant=t_{0}$ and $\theta=\frac{\pi}{2}.$ Here location of $t_{0}$ within
the interval during which the wormhole exists. Present wormhole is non-static
i.e. it evolves with time, therefore, one can get different slices with
different values of time. As a result, scale factor $a(t)$ regulates the shape
of the wormhole. Nevertheless, it can be demonstrated that the wormhole
structure remains unaltered over time through the utilization of an embedding
procedure.

The three dimensional spatial hyper surface given by $t=t_{0}$ of our
spherically symmetric space-time takes the form

$d\sigma^{2}=g_{rr}dr^{2}+r^{2}a^{2}(t_{0})(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$

It can be embedded in a four dimensional space as

$d\sigma^{2}=d\bar{z}^{2}+d\bar{r}^{2}+\bar{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (55)

Let us assume, $\bar{r}=a(t_{0})r$, $d\bar{r}^{2}=a^{2}(t_{0})dr^{2}$.
In the equatorial place, $\theta=\frac{\pi}{2}$, we have

$\displaystyle\begin{split}d\sigma^{2}&=\frac{a^{2}(t_{0})(t)dr^{2}}{1-kr^{2}-\frac{b(r)}{r}}+r^{2}a^{2}(t_{0})(t)d\theta^{2},\\\
&=d\bar{z}^{2}+d\bar{r}^{2}+\bar{r}^{2}d\phi^{2},\\\
&=\frac{d\bar{r}^{2}}{1-\frac{\bar{b}}{\bar{r}}}+\bar{r}^{2}d\phi^{2}.\end{split}$
(56)

This yields

$\frac{d\bar{z}}{d\bar{r}}=\pm\left(\frac{\bar{r}}{\bar{b}}-1\right)^{-\frac{1}{2}}=\pm\left(\frac{a(t_{0})r}{a(t_{0})b+ka(t_{0})r^{3}}-1\right)^{-\frac{1}{2}}=\frac{dz}{dr},$
(57) where $\displaystyle\bar{b}$ $\displaystyle=a(t_{0})b+ka(t_{0})r^{3}.$
(58) Note that here $\displaystyle\bar{z}$ $\displaystyle=\pm a(t_{0})z$ .
(59)

The flare-out condition for consisting of wormhole i.e. to maintain the shape
of the traversable wormhole, the flare-out condition
$\frac{d^{2}\bar{r}}{d\bar{z}^{2}}>0$ should be satisfied.

Now, $\displaystyle\frac{d^{2}\bar{r}}{d\bar{z}^{2}}$
$\displaystyle=\frac{\bar{b}-\bar{b}^{\prime}\bar{r}}{2\bar{b}^{2}}=\frac{1}{a(t_{0})}\frac{b-b^{\prime}r-2kr^{3}}{2(b+kr^{3})^{2}}=\frac{1}{a(t_{0})}\frac{d^{2}r}{dz^{2}}>0$
. (60)

[ Here, $\bar{b}^{\prime}=\frac{d\bar{b}}{d\bar{r}}$ ].

Note that for $k=0$, $a(t_{0})=1$, this flare-out condition reduces to flare-
out condition of static wormhole. Also one can notice that the evolving
wormhole will continue the identical size in the $\bar{z},\bar{r},\phi$
coordinates. We can draw the graph of the embedded curve $z=z(r)$ as well as
the entire imagining of the surface generated by the rotation of the embedded
curve about the vertical z axis ( figure - 1).

Figure 1: Left panel) The embedding diagram for the wormhole spacetime given
in equation (12). (Right panel) The entire imagining of the surface generated
by the rotation of the embedded curve about the vertical z axis.

##  V  ENERGY CONDITIONS:

We have already mentioned that the flaring-out condition has been satisfied
($b^{\prime}(r_{0})<1$). Now at the same time we will try to check whether
null energy condition (NEC) is obeyed or not for constructing the dynamical
wormhole. In static wormhole configuration, one of the fundamental criterion
is that the null energy condition should be violated. However in dynamical
wormhole configuration the situation may change due to extra terms (time
dependent scale factor) in the field equations. In general one can execute the
condition $\rho+p_{r}>0$ for normal matter comprising the wormhole. In the
event of dynamical wormholes it is argued that that there exist wormhole
solutions which obey NEC. So, we will search some Lorentzian dynamical
wormhole geometries with no requirement of the matters that violate the NEC.
Now, we calculate null energy condition expression as

$\displaystyle\begin{split}\rho+p_{r}&=3\frac{\dot{a}^{2}}{a^{2}}+3\frac{k}{a^{2}}-2\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}-\frac{k}{a^{2}}-\frac{b}{a^{2}r^{3}}+\frac{b^{\prime}}{a^{2}r^{2}}\\\
&=2\frac{\dot{a}^{2}}{a^{2}}-2\frac{\ddot{a}}{a}-\frac{(-2kr^{3}-rb^{\prime}+b)}{a^{2}r^{3}}\\\
&=A-\left[\frac{(b+kr^{3})^{2}}{ar^{3}}\right]\frac{d^{2}\bar{r}}{d\bar{z}^{2}},\end{split}$
(61) where, $\displaystyle A$
$\displaystyle=2\frac{\dot{a}^{2}}{a^{2}}-2\frac{\ddot{a}}{a}$ . (62)

Since $\frac{d^{2}\bar{r}}{d\bar{z}^{2}}>0$, so sign of $\rho+p_{r}$ depends
on $A$.

For static case $A=0$, so one gets $\rho+p_{r}$ always negative i.e. matter
distribution should be exotic in nature.

In usual FRW cosmological model of the universe, one can get different scale
factor $a(t)$ which depends on $k$ and some parameters.

If $A>0$, then there exists a possibility to have non exotic matter
$(p_{r}+\rho>0)$ comprising the wormhole.

Here, we calculate A for all cases with subcases.

$\text{Case 1: }A=\frac{2n}{t^{2}},$ (63) $\text{Case 2: }A=0,$ (64)
$\text{Case 3.1:
}A=\frac{\frac{\beta}{12}\left(\sqrt{\frac{\beta}{3}}t+C_{1}\right)^{2}-\frac{L+3k}{3}}{\left[\frac{1}{4}\left(\sqrt{\frac{\beta}{3}}t+C_{1}\right)^{2}+\frac{L+3k}{\beta}\right]^{2}},$
(65) $\text{Case 3.2:
}A={\frac{2\beta-2L-6k}{\left(t\sqrt{3\beta-3L-9k}+\sqrt{3}{C_{1}}\right)^{2}}},$
(66) $\text{Case 4: }A=\frac{2\alpha}{t^{2}}.$ (67)

Note that $A>0$ for positive values of the parameters $n$ , $\alpha$ and
$\beta>L+3k$.

Now we check whether the NEC is violated or not at the throat $r=r_{0}$ for
all t.

$(\rho+p_{r})_{r=r_{0}}=2\frac{\dot{a}^{2}}{a^{2}}-2\frac{\ddot{a}}{a}-\frac{(-2kr_{0}^{3}-r_{0}b^{\prime}(r_{0})+b(r_{0}))}{a^{2}r_{0}^{3}}.$

For $(\rho+p_{r})_{r=r_{0}}>0$, we have ( using $b(r_{0})=r_{0}$) ,

$\dot{a}^{2}-a\ddot{a}>\frac{\delta-kr_{0}^{2}}{r_{0}^{2}},$ (68)

where $2\delta=[1-b^{\prime}(r_{0})]>0$, since $b^{\prime}(r_{0})<1$.

Figure 2:  (left panel) In case 1, during the evolution of the universe, when
$t>t_{0}$, the NEC is obeyed (middle panel) For case 2, the wormhole is always
supported by exotic matter (right panel) In case 3.1, during the evolution of
the universe, when $t>t_{0}$, again the NEC is obeyed.

For case 1, equation (68) implies

$t>\left[\frac{\delta-kr_{0}^{2}}{nr_{0}^{2}}\right]^{\frac{1}{2n-2}}\equiv
t_{0},$

for which $(\rho+p_{r})_{r=r_{0}}>0$. Thus at the evolution of the universe,
when $t>t_{0}$, the NEC is obeyed (see left panel of figure 2). But up to time
$t<t_{0}$, we get wormhole supported by the matter violating NEC i.e. by
exotic matter. Here, one can note that evolution of the universe plays a
crucial role for the formation of wormhole.

For case 2, the wormhole is always supported by exotic matter. Here evolution
of the universe does not affect on wormhole configuration(see middle panel of
figure 2).

For case 3.1, NEC obeys for all time after

$t>\left[\frac{72(\delta-
kr_{0}^{2})}{\beta^{2}r_{0}^{2}}+\frac{12(L+3k)}{\beta^{2}}\right]^{\frac{1}{2}}-C_{1}\sqrt{\frac{3}{\beta}}=t_{0}.$
(69)

But up to time $t<t_{0}$, the wormhole is supported by exotic matter(see right
panel of figure 2).

For case 3.2, NEC obeys if

$\frac{1}{3}(\beta-L-3k)>\frac{\delta-kr_{0}^{2}}{r_{0}^{2}}.$ (70)

Thus through out the evolution of the universe, one gets wormhole without
violating NEC for the above condition (70)(see left panel of figure 3).

For case 4, NEC obeys if

$t^{\alpha-2}e^{2\gamma t}>\frac{\delta-kr_{0}^{2}}{\alpha r_{0}^{2}}.$ (71)

Hence, during the evolution of the universe, when $t>T$, where T is satisfying
the equation

$T^{\alpha-2}e^{2\gamma T}-\frac{\delta-kr_{0}^{2}}{\alpha r_{0}^{2}}=0,$

the NEC is obeyed. But up to time $t<T$, the wormhole is supported by exotic
matter (see right panel of figure 3).

Figure 3:  (left panel) In case 3.2, through out the evolution of the
universe, one gets wormhole without violating NEC (right panel) In case 4,
again the NEC is obeyed after certain time.

The present spacetime metric is describing a dynamical wormhole in FRW
expanding universe. Here, $a(t)\rightarrow\infty$ as $t\rightarrow\infty$.
Also, all A for different proposed models except case 3.1 approach to zero as
$t\rightarrow\infty$ and NEC is violated. So, NEC violations could be desisted
only for finite non zero values of A. Thus we restrict finite interval of
times in which $a(t)$ is finite. The time interval may arbitrary small or
large where $a(t)$ is non zero finite in that interval. However, in the case
3.1, after the time $t=t_{0}$ given in equation (69) one can get a wormhole
supported by non exotic matter for ever. This is possible because, as
$t\rightarrow\infty$, A assumes non zero finite value.

##  VI  traversability conditions:

According to Morris - Thorne, for a convenient journey through the wormhole,
certain conditions should be imposed at the space stations ( initial and final
destinations ) as well as on velocity of the traveler. The traveler feels the
acceleration must not exceed the Earth’s gravitational acceleration,
$g_{\oplus}$. One can calculate the traveler’s four-acceleration in his proper
reference frame as follows: The orthonormal basis
$(e_{\widehat{0}}^{\prime},e_{\widehat{1}}^{\prime},e_{\widehat{2}}^{\prime},e_{\widehat{3}}^{\prime})$
of the traveler’s proper reference is obtained via Lorentz transformation in
terms of the orthonormal basis vectors
$(e_{\widehat{t}},e_{\widehat{r}},e_{\widehat{\theta}},e_{\widehat{\varphi}})$
of the static observers as

$e_{\widehat{0}}^{\prime}=\sigma
e_{\widehat{t}}\mp\sigma\frac{v}{c}e_{\widehat{r}},~{}e_{\widehat{1}}^{\prime}=\mp\sigma
e_{\widehat{r}}+\sigma\frac{v}{c}e_{\widehat{t}},~{}e_{\widehat{2}}^{\prime}=e_{\widehat{\theta}},~{}e_{\widehat{3}}^{\prime}=e_{\widehat{\varphi}},$

where $\sigma=(1-\frac{v^{2}}{c^{2}})^{-\frac{1}{2}}$ , and $v(r)$ be the
velocity of the traveler when he passes $r$ as observed by a static observer
positioned there.

To travel through a wormhole, the tidal gravitational forces experienced by a
traveler must be reasonably small. According to Morris and Thorne, the
acceleration felt by the traveler should not exceed Earth’s gravity. Thus the
tidal accelerations between two parts of the traveler’s body, separated by
several meters, must less than the gravitational acceleration at Earth’s
surface $g_{\oplus}$ . One can calculate the tidal acceleration felt by the
traveler as follows:

The tidal acceleration is given by the formula

$\Delta
a^{\hat{\mu^{\prime}}}=-R_{\hat{\nu^{\prime}}\hat{\alpha^{\prime}}\hat{\beta^{\prime}}}^{\hat{\mu^{\prime}}}U^{\hat{\nu^{\prime}}}\eta^{\hat{\alpha^{\prime}}}U^{\hat{\beta^{\prime}}},$

where $U^{\hat{\mu^{\prime}}}=\delta^{\hat{\mu^{\prime}}}_{\hat{0^{\prime}}}$
is the traveler’s four velocity and $\eta^{\hat{\alpha^{\prime}}}$ is the
separation between two arbitrary parts of his body which is purely spatial in
the traveler’s reference frame. The nonzero components of Riemann tensor
$R_{\hat{2^{\prime}}\hat{0^{\prime}}\hat{2^{\prime}}\hat{0^{\prime}}}$ in the
traveler’s frame from the static observer’s frame can be obtained by using a
Lorentz transformation.

Now the radial tidal constraint is given by

$R_{\hat{1^{\prime}}\hat{0^{\prime}}\hat{1^{\prime}}\hat{0^{\prime}}}=R_{\hat{r}\hat{t}\hat{r}\hat{t}}=\left|\frac{\ddot{a}}{a}\right|\leq\frac{g_{\oplus}}{c^{2}|\eta|}.$
(72)

Note that, for the evolving wormhole, the tidal acceleration depends on time.
More specifically, in this expanding wormhole, the gravitational forces
experienced by an observer at a constant throat radius decreases over time. In
fact, the radial tidal constraint $(72)$ is directly restraining the expansion
of the wormhole. If one considers the size of the traveler’s body is
$|\eta|=2$ m, then

$\frac{g_{\oplus}}{c^{2}|\eta|}\approx\frac{1}{(10^{8}m)^{2}}.$

Now, we calculate the radial tidal acceleration constraint of the evolving
wormholes obtained in this paper with the different scale factors and have
found the restriction on time as follows:

Case 1:

$t\geq c\sqrt{n(n-1)}$ (73)

For, a particular value of $n$, say $n=2$, $c=10^{8}$ $m/sec$, we have

$t\geq 1.414\times 10^{8}\,sec.$

Case 2:

In this case of exponentially expanding wormhole universe, the radial tidal
forces experienced by a traveler is independent of the time evolution. It has
a restriction on the parameter $\alpha$ as

$\omega\geq\frac{1}{\sqrt{c}}.$ (74)

Case 3.1:

$t\geq\left[\left(\frac{2c^{2}\beta}{3}-\frac{4(L+3k)}{\beta}\right)^{\frac{1}{2}}-C_{1}\right]\sqrt{\frac{3}{\beta}}.$
(75)

Here for the following values of the parameters, say
$\beta=1,k=0,L=0,C_{1}=0,c=10^{8}m/sec$, we have

$t\geq 1.414\times 10^{8}\,sec.$

Case 3.2:

In this case, $\ddot{a}=0$, i.e. the radial tidal acceleration is identically
zero and hence satisfies the constraint $(72)$ automatically.

Case 4:

$t\geq\frac{2\alpha\gamma
c^{2}+\sqrt{4\alpha^{2}\gamma^{2}c^{4}+4c^{2}(\alpha^{2}-\alpha)(1-\gamma^{2}c^{2})}}{2(1-\gamma^{2}c^{2})}.$
(76)

For a particular choice of parameters, say $\gamma c=0.2,\alpha=3$ and
$c=10^{8}\,m/sec$, we have the following restriction on time $t$,

$t\geq 8.571\times 10^{7}\,sec.$

Therefore, it is speculated that after a fixed time of the evolution, one may
have a traversable wormhole. However, there is some lateral tidal constraint
also.

The lateral tidal constraint is given by

$R_{\hat{2^{\prime}}\hat{0^{\prime}}\hat{2^{\prime}}\hat{0^{\prime}}}=R_{\hat{3^{\prime}}\hat{0^{\prime}}\hat{3^{\prime}}\hat{0^{\prime}}}=\sigma^{2}R_{\hat{\theta}\hat{t}\hat{\theta}\hat{t}}+\sigma^{2}\frac{v^{2}}{c^{2}}R_{\hat{\theta}\hat{r}\hat{\theta}\hat{r}}+2\sigma^{2}\frac{v}{c}R_{\hat{\theta}\hat{t}\hat{\theta}\hat{r}}$
$=\left|\sigma^{2}\frac{\ddot{a}}{a}-\frac{\sigma^{2}\frac{v^{2}}{c^{2}}}{2a^{2}r^{3}}\left[2\dot{a}^{2}r^{3}-b-kr^{3}+r(b^{\prime}+3kr^{2})\right]\right|\leq\frac{g_{\oplus}}{c^{2}|\eta|}.$
(77)

Lateral tidal constraint gives a restriction on velocity of the traveler as
well as matter distribution comprising the wormhole. We have derived the
limitation for the lateral tidal acceleration and get the restriction of non-
relativistic velocity at the throat (i.e. $v<<c,\,\sigma\approx 1)$ as
follows:

Case 1:

$v\leq\sqrt{2}t^{n}\left[c^{2}n(n-1)t^{-2}+1\right]^{\frac{1}{2}}\left[2n^{2}t^{2n-2}+L+2K+\rho_{s}e^{-\frac{r_{0}}{r_{s}}}-r_{0}^{-2}\right]^{-\frac{1}{2}}.$
(78)

Case 2:

$v\leq\sqrt{2}e^{\omega
t}\left(\omega^{2}c^{2}+1\right)^{\frac{1}{2}}\left[2\omega^{2}e^{2\omega
t}+2K+L+\rho_{s}e^{-\frac{r_{0}}{r_{s}}}-r_{0}^{-2}\right]^{-\frac{1}{2}}.$
(79)

Case 3.1:

$v\leq\sqrt{2}\left[\frac{1}{4}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+\frac{L+3K}{\beta}\right]^{\frac{1}{2}}.\left(\frac{\beta}{6}c^{2}+1\right)^{\frac{1}{2}}.\left[\frac{\beta}{6}\left(t\sqrt{\frac{\beta}{3}}+C_{1}\right)^{2}+L+2K+\rho_{s}e^{-\frac{r_{0}}{r_{s}}}-r_{0}^{-2}\right]^{-\frac{1}{2}}.$
(80)

Case 3.2:

$v\leq\sqrt{2}\left[t\sqrt{\beta-(L+3K)}+\sqrt{3}C_{1}\right]^{\frac{1}{2}}.\left[2\beta+L+3\rho_{s}e^{-\frac{r_{0}}{r_{s}}}-3r_{0}^{-2}\right]^{-\frac{1}{2}}.3^{\frac{1}{4}}.$
(81)

Case 4:

$v\leq\sqrt{2}\left(t^{\alpha}e^{\beta
t}\right)^{\frac{1}{2}}.\left[c^{2}\\{(\alpha^{2}-\alpha)t^{\alpha-2}+2\alpha\beta
t^{\alpha-1}+\beta^{2}t^{\alpha}\\}e^{\beta
t}+1\right]^{\frac{1}{2}}.\left[2\left(\alpha t^{-1}+\beta\right)^{2}e^{2\beta
t}t^{2\alpha}+L+2K+\rho_{s}e^{-\frac{r_{0}}{r_{s}}}-r_{0}^{-2}\right]^{-\frac{1}{2}}.$
(82)

$a=1$, $k=0$ gives the case for static wormhole.

##  VII Proper Length Between Two Distances $r_{1}$, $r_{2}$:

It’s equally intriguing to investigate how changes in the scale factor over
time impact the nature of the proper length or proper circumference of the
throat of the wormhole.

Let us first consider the case of proper circumference of the throat of
wormhole. Consider a slice of the wormhole at a fixed instant of time and
$\theta=\pi/2$. Let $r_{0}$ be the throat radius of the wormhole so that
$r=b(r_{0})=r_{0}$. The proper circumference $(C_{0})$ of the wormhole throat
is

$C_{0}=\int_{0}^{2\pi}\,a(t)r_{0}\,d\phi=a(t)(2\pi r_{0}).$ (83)

This is merely the scale factor times the static wormhole circumference.
Now we calculate the radial proper length $(l(t))$ between any two points
$r_{1}>r_{0}$ and $r_{2}$ as

$l(t)=\pm a(t)\int_{r_{1}}^{r_{2}}\frac{dr}{(1-kr^{2}-\frac{b}{r})^{1/2}},$
(84)

which is just scale factor times of the radial proper separation of the static
wormhole counterpart ( see figure 2).

Figure 4:  The radial proper separation of the static wormhole counterpart.

However, a temporal singularity may occur at $t=t_{0}$ where $a(t_{0})=0$ .

##  VIII concluding remarks:

Within this manuscript, we explore the viability of wormholes containing dark
matter within the framework of classical general relativity against a backdrop
of Friedmann-Robertson-Walker (FRW) cosmology. Our investigation extends to
dynamic wormholes situated in various cosmological contexts, each defined by
distinct criteria such as specific choices of scale factors and cosmological
density. The temporal existence of these geometries, subject to the null
energy condition (NEC) for the involved matter, introduces an intriguing
aspect. Despite the potential perturbations associated with their transient
nature, this dynamical perspective presents an advancement over static
geometries. Notably, wormholes supported by non-exotic matter can be
established within a finite interval of time when the scale factor $a(t)$
assumes a non-zero finite value. The duration of this evolution can span from
arbitrarily small to large intervals. Section 3.1 highlights a scenario where
the evolution period can be infinitely large. After the time $t=t_{0}$ (as
given in equation 69), a wormhole supported by non-exotic matter can endure
indefinitely. Our findings reveal instances where the proper circumference of
the wormhole throat expands due to the temporal evolution of the geometry.
Turning our attention to traversability criteria related to the tidal forces
experienced by a traveler, we adopt a systematic approach. This involves
transitioning from the static observer’s frame in the case of a static
wormhole to the comoving frame for the evolving geometry. Subsequently, we
apply a simple Lorentz transformation to switch to the traveler’s frame.
Riemann tensor components are then derived in this traveler’s frame,
establishing constraints on tidal forces. These constraints necessitate
determining the time at which the inequalities impose the most stringent
conditions through extremization. Subsequent extremization of the traveler’s
velocity is required to obtain the ultimate condition. Satisfying these
conditions renders the wormhole traversable. To quantify traversability, we
determine the maximum speeds at which a traveler could traverse the wormhole,
considering specific values of parameters and the throat radius $r_{0}$. This
ensures compliance with the given constraint.

###  VIII.1 Acknowledgements

We are grateful to the referee for his constructive suggestions which help a
substantial improvement of the manuscript. FR would like to thank the
authorities of the Inter-University Centre for Astronomy and Astrophysics,
Pune, India for providing the research facilities. We are also thankful to
Somi Aktar for her help in checking the solutions. FR and BSC are also
thankful to SERB, DST $\&$ DST FIST programme ( SR/FST/MS-II/2021/101(C)) and
UGC, Govt. of India for financial support respectively.

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11institutetext:  Department of Physics, McGill University, 3600 rue
University, Montreal, QC, Canada

# Conservation Laws and Hydrodynamics

Sangyong Jeon 1133<EMAIL_ADDRESS>
###### Abstract

Recent years have seen much development in analyzing the structure of
relativistic hydrodynamics. In this proceeding, some of the developments are
highlighted including issues related to pseudo-gauge transformations and spin
hydrodynamics.

##  1 Introduction

Hydrodynamics has a long history in describing heavy ion collisions starting
with Landau’s seminal paper Landau1953  and Bjorken’s Bjorken:1982qr . In the
era of RHIC and the LHC, hydrodynamics has become an indispensable tool to
describe the bulk dynamics of relativistic heavy ion collisions.

In this 2022 Strangeness in Quark Matter conference, many of the recent
developments in hydrodynamics were reported. On spin hydrodynamics and
$\Lambda$ polarization, Palermo Palermo:2022lvh , Liao Guo:2021udq  and
Buzzegoli Buzzegoli:2022kyj  presented their work. On the subject of
nonolinear causality conditions, Plumberg Plumberg:2021bme  presented his and
collaborators’ work. On the development of hydro initial states, Shen
Zhao:2022ayk  presented recent developments for ultra-peripheral collisions
and Kanakubo Kanakubo:2022acq  presented her recent work on the core-corona
model. On applying hydrodynamics to systems near the QCD critical point,
Pradeep Pradeep:2022mkf , Wu Wu:2021fjf , and Sogabe Sogabe:2021svv  presented
their work in this conference.

As this is a brief introduction to the recent developments, inevitably only a
limited number of topics can be discussed. Below, I will mainly focus on the
pseudo-gauge transformation and its relevance in developing spin
hydrodynamics. I should emphasize that this is a personal choice dictated by
my familiarity with the subjects Shi:2020htn .

##  2 Is any current density observable?

Hydrodynamics starts with conservation laws

$\partial_{\mu}J^{\mu,q}=0$ (1)

where $q$ is the index for different conserved quantities such as energy,
momentum, electric charge and baryon number. Mathematically, a conserved
current density is not uniquely defined. One can always add a “pseudo-gauge”
term

${J^{\prime}}^{\mu,q}=J^{\mu,q}+\partial_{\lambda}B^{\lambda\mu,q}$ (2)

where $B^{\lambda\mu,q}=-B^{\mu\lambda,q}$ is the pseudo-gauge potential.
Adding $\partial_{\lambda}B^{\lambda\mu,q}$ neither changes the conservation
law $\partial_{\mu}{J^{\prime}}^{\mu,q}=0$, nor the total charge provided that
$B^{i0,q}$ vanishes at the boundary of the volume. But it does change the
local current density since $J^{\mu,q}(x)\neq{J^{\prime}}^{\mu,q}(x)$.

Does this mean that a current density is not observable? That depends on
whether there exist other constraints that may select a particular form to be
physical. For instance, in the classical particle dynamics, a point particle
carries well-defined energy, momentum and other conserved charges. Hence, the
classical current density

$J_{\rm cl}^{\mu,q}(x)=\sum_{s=\rm ptcles}{q_{s}\over
p_{s}^{0}}p^{\mu}_{s}\delta^{(3)}({\bf x}-{\bf x}_{s}(t))$ (3)

is physical and observable as long as a particle is observable. Adding an
arbitrary $\partial_{\lambda}B^{\lambda\mu,q}$ to $J^{\mu,q}_{\rm cl}$ is
still a possibility, but the fact that a point particle carries definite
amounts of energy, momentum and other charges clearly selects Eq.(3) to be the
physical current density.

In reality, due to the uncertainty principle, particles cannot be localized as
in Eq.(3). Particle positions are inevitably smeared so that

$J_{C}^{\mu,q}(x)=\sum_{s=\rm ptcles}{q_{s}\over p_{s}^{0}}p^{\mu}_{s}g({\bf
x}-{\bf x}_{s}(t))$ (4)

with the normalization condition $\int d^{3}x\,g({\bf x}-{\bf x}_{s}(t))=1$
that preserves the total charge. This expression still satisfies
$\partial_{\mu}J_{C}^{\mu,q}=0$ as long as $d{\bf x}_{s}/dt={\bf
p}_{s}/p^{0}_{s}$, but since there can be many choices for the smearing
function, this does not define a unique value of the current density at $x$.
For any change $\delta{J}^{\mu,q}_{C}$ introduced by a change of the smearing
function, one can always find $B^{\lambda\mu,q}$ that satisfies

$\delta{J}^{\mu,q}_{C}=\partial_{\lambda}B^{\lambda\mu,q}$ (5)

since it has the same form as the Maxwell equation. Hence, without any further
constraint on the form of the smearing function, the continuous current
density $J_{C}^{\mu,q}$ can be ambiguous.

I have been so far careful to always refer to $J^{\mu,q}$ as a current
density. This is to distinguish $J^{\mu,q}$ from the current. An electric
current, for instance, is defined as

$I_{S}=\int_{S}d{\bf S}\cdot{\bf J}^{e}$ (6)

where ${\bf S}$ is the surface through which the electric current is measured
and ${\bf J}^{e}$ is the spatial part of the electric current density. The
current itself is a scalar quantity, not a vector quantity. One can easily
show that as long as ${\bf B}$ vanishes at the boundary of the surface,
$I_{S}$ remains unchanged.

Does this mean that no current density is actually observable due to the
pseudo-gauge freedom? In order to answer this question, consider the electric
current density $J_{e}^{\mu}$. As discussed above, the conservation law is not
affected when a pseudo-gauge potential with the property
$B^{\nu\mu}_{e}=-B^{\mu\nu}_{e}$ is added to $J_{e}^{\mu}$. Using the new
current density, The Maxwell equation becomes

$\partial_{\nu}{F^{\prime}}^{\nu\mu}={J^{\prime}_{e}}^{\mu}=J_{e}^{\mu}+\partial_{\nu}B_{e}^{\nu\mu}$
(7)

The electric and magnetic field generated by ${J^{\prime}_{e}}^{\mu}$ is then

${F^{\prime}}^{\nu\mu}=F^{\nu\mu}+B_{e}^{\nu\mu}$ (8)

where $F^{\nu\mu}$ is generated by $J_{e}^{\mu}$. But this doesn’t make much
sense. The electric field and the magnetic field are measurable observables.
They shouldn’t be ambiguous. Hence, there must be a physical electric current
density that generates the measured electric and magnetic fields.

For the energy-momentum tensor, a similar situation arises with the Einstein’s
equation where a symmetric Belinfante energy-momentum tensor (see below)
sources the gravitational field. Since gravitational field is observable and
measurable, the Belinfante tensor must be the physical energy-momentum tensor.

There are two lessons I would take from the discussions above. One is that as
long as a current density generates an observable and measurable field (such
as the electric and magnetic fields), one should be able to identify the
unique physical and observable current density. The second lesson is that even
though unambiguously identifying classical current density using the above
arguement may be possible, identifying quantum current density could be still
ambiguous (c.f. Eq.(5)). However, this ambiguity should represent only the
$O(\hbar)$ correction to the classical current density.

##  3 Stress-energy tensors with spin

Hydrodynamics primarily deals with the energy-momentum tensor. Consider a
field-theory Lagrangian $L(\varphi^{A},\partial_{\mu}\varphi^{A})$ where $A$
is the field index. Using Noether’s theorem, one can show that the conserved
current densities for the translational symmetry in space-time is given by

$\Theta^{\mu\nu}=\Pi_{A}^{\mu}\partial^{\nu}\varphi^{A}-g^{\mu\nu}L(\varphi_{A},\partial_{\mu}\varphi^{A})$
(9)

where $\Pi_{A}^{\mu}=\partial L/\partial(\partial_{\mu}\varphi_{A})$. This
tensor satisfies

$\displaystyle\partial_{\mu}\Theta^{\mu\nu}=\partial^{\nu}\varphi_{A}\left(\partial_{\mu}\Pi^{\mu}_{A}-{\partial
L\over\partial\varphi_{A}}\right)$ (10)

Hence if $\varphi_{A}$ satisfies the Euler-Lagrange equation
$\partial_{\mu}\Pi^{\mu}_{A}=\partial L/\partial\varphi_{A}$, then
$\Theta^{\mu\nu}$ is a conserved energy-momentum current density. The first
index $\mu$ is the space-time index for the current density components and the
second index $\nu$ indicates what is being conserved. Namely, $\nu=0$ is the
label for the energy current and $\nu=1,2,3$ are for the momentum currents
playing the role of the index $q$ in the previous section. This canonical
energy-momentum tensor is not symmetric unless it only contains scalar (spin
$S=0$) particles, and it is not gauge invariant when applied to gauge
theories. As such, it cannot yet be directly interpreted as the measurable
energy-momentum current density.

To see the consequences of having a non-symmetric $\Theta^{\mu\nu}$, define
the orbital angular momentum current density

${\cal
L}_{O}^{\lambda\mu\nu}=x^{\mu}\Theta^{\lambda\nu}-x^{\nu}\Theta^{\lambda\mu}$
(11)

This tensor is in general not conserved $\partial_{\lambda}{\cal
L}_{O}^{\lambda\mu\nu}=\Theta^{\mu\nu}-\Theta^{\nu\mu}$ unless all the fields
have $S=0$. If fields have non-zero spins, then the total angular momentum
current density must include the spin angular momentum part as well as the
orbital angular momentum part

${\cal L}_{\rm
tot}^{\lambda\mu\nu}=x^{\mu}\Theta^{\lambda\nu}-x^{\nu}\Theta^{\lambda\mu}+{\cal
S}^{\lambda\mu\nu}$ (12)

where ${\cal S}^{\lambda\mu\nu}$ is the spin current density. The statement of
the total angular momentum conservation becomes

$\partial_{\lambda}{\cal S}^{\lambda\mu\nu}+2\Theta_{(a)}^{\mu\nu}=0$ (13)

In other words, the anti-symmetric part of the energy-momentum tensor,
$\Theta_{(a)}^{\mu\nu}=(1/2)(\Theta^{\mu\nu}-\Theta^{\nu\mu})$, encodes the
spin angular momentum part of the energy and momentum. That part is then
necessarily of $O(\hbar)$ since the spin current density is $O(\hbar)$.

To find the pseudo-gauge transformation that results in the physical energy-
momentum tensor, let

$\displaystyle
T^{\mu\nu}=\Theta^{\mu\nu}+\partial_{\lambda}B^{\lambda\mu,\nu}$ (14)

where $T^{\mu\nu}$ is the symmetric tensor that we are after. Using
$T^{\mu\nu}-T^{\nu\mu}=0$, we get

$\displaystyle 0$
$\displaystyle=\Theta^{\mu\nu}-\Theta^{\nu\mu}+\partial_{\lambda}(B^{\lambda\mu,\nu}-B^{\lambda\nu,\mu})$
(15)

Using the conservation of total angular momentum, Eq.(13), one can identify

$\displaystyle{\cal S}^{\lambda\mu\nu}=B^{\lambda\mu,\nu}-B^{\lambda\nu,\mu}$
(16)

Equivalently,

$\displaystyle B^{\lambda\mu,\nu}={1\over 2}\left({\cal
S}^{\lambda\mu\nu}-{\cal S}^{\mu\lambda\nu}-{\cal S}^{\nu\lambda\mu}\right)$
(17)

that yields

$\displaystyle T^{\mu\nu}$ $\displaystyle=\Theta^{\mu\nu}_{(s)}-{1\over
2}\partial_{\lambda}\left({\cal S}^{\mu\lambda\nu}+{\cal
S}^{\nu\lambda\mu}\right)$ (18)

after explicit symmetrization. This form of the energy-momentum tensor is
usually referred to as the Belinfante energy-momentum tensor and it can be
shown that this is identical to the gravitational source term in the Einstein
equation obtained by varying the space-time metric. As such, classically, this
is the observable energy-momentum tensor. One should note that this form of
energy-momentum tensor now incorporates the conservation of total angular
momentum since $\partial_{\mu}T^{\mu\nu}=0$ is satiafied only after using both
the equations of motion and the total angular momentum conservation, Eq.(13).
It also means that the energy density obtained from $T^{\mu\nu}$ contains the
energy due to the spins Fukushima:2020ucl .

The the total angular momentum current density is now just given by

$\displaystyle{\cal L}_{\rm
tot}^{\lambda\mu\nu}=x^{\mu}T^{\lambda\nu}-x^{\nu}T^{\lambda\mu}$ (19)

which could be further divided this into the orbital angular momentum term and
the spin term although this division is not so clear-cut.111For instance, for
a spin-1 gauge field, only the sum is gauge-invariant. This also means that
there is no separate spin current density which presents a bit of a problem
when formulating spin hydrodynamics.

##  4 Spin hydrodynamics

STAR has measured $\Lambda$ polarizations that showed a surprising azimuthal
pattern STAR:2017ckg  that may indicate a strong spin-orbit coupling between
the polarization of a hadron and the underlying fluid motion. In order to
figure out where such a strong correlation can originate from, it is clear
that we need a formulation of hydrodynamics that includes the spin degrees of
freedom. This is, however, not straightforward: If one insists that the
energy-momentum tensor must be observable, then it must be the Belinfante
tensor. But that means there is no separate spin current density. On the other
hand, one can naturally get the spin current density via the anti-symmetric
part of the canonical energy-momentum tensor. However, the canonical energy-
momentum tensor is not physical.

There are currently two broad approaches to this problem. One is to approach
the spin hydrodynamics from the semi-classical kinetic theory point of view
Palermo:2022lvh ; Buzzegoli:2022kyj ; Weickgenannt:2022zxs ; Shi:2020htn .
Another approach is to start with the general expression of the non-symmetric
energy-momentum tensor and the spin current density tensor and develope a
gradient expansion Daher:2022xon ; Fukushima:2020ucl . One can then use the
extended thermodynamic identities and the production of entropy to obtain the
first order and the second order form of spin hydrodynamics. The relationships
between various formulations of the spin hydrodynamics, however, is not yet
fully settled and full spin hydrodynamics simulations are yet to be performed.

In all these approaches, a common task is to decompose the non-symmetric
tensor into the physical symmetric part and the $O(\hbar)$ spin dependent
part. This decomposition is necessary because quantities such as the energy
density and the flow velocity must be defined to carry out any hydrodynamic
calculations. In some sense, this largely deals with the pseudo-gauge problem.
Namely, once we have identified the classical energy-momentum tensor
(corresponding to the Belinfonte tensor), then only $O(\hbar)$ pseudo-gauge
potentials are allowed in order not to spoil the classial part of the energy-
momentum tensor.

##  5 Outlook

In this short review, I have tried to highlight some of the issues in
formulating physical hydrodynamics that were raised in recent literatures. The
two related topics I have chosen to highlight here, namely the issue of
pseudo-gauge transformations and spin hydrodynamics, have been studied by
several groups motivated by the $\Lambda$ polarizations measured by the STAR
collaboration at RHIC. Through such efforts, it has become clearer what role
pseudo-gauge transformations can and cannot play in hydrodynamics in general
and in spin hydrodynamics in particular.

Theoretically, the STAR $\Lambda$ polarization measurement presents an
interesting challenge. Although some resolutions of this puzzle222The measured
azimuthal dependence of the $\Lambda$ polarization has the opposite sign
compared to the vorticity tensor. have been suggested Palermo:2022lvh , more
studies, especially realistic spin hydrodynamics simulations, are needed to
firmly settle it.

## Acknowledgement

S.J. is supported in part by the Natural Sciences and Engineering Research
Council of Canada.

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