Gravitational Lensing by Lemaître–Tolman–Bondi Wormholes in a Friedmann Universe

Abstract

The Lemaître–Tolman–Bondi (LTB) solution to the Einstein equations describes the dynamics of a self-gravitating spherically symmetric dust cloud with an arbitrary density profile and any distribution of initial velocities, encoded in three arbitrary functions $f\left(R\right)$, $F\left(R\right)$, and ${\tau }_0\left(R\right)$, where R is a radial coordinate in the comoving reference frame. A particular choice of these functions corresponds to a wormhole geometry with a throat defined as a sphere of minimum radius at a fixed time instant. In this paper we explore LTB wormholes and discuss their possible observable appearance, studying in detail the effects of gravitational lensing by such objects. For this aim, we study photon motion in wormhole space-time inscribed in a closed Friedmann dust-filled universe and find the wormhole shadow as it could be seen by a distant observer. Because the LTB wormhole is a dynamic object, we analyze the dependence of its shadow size on the observation time and on the initial size of the wormhole region. We reveal that the angular size of the shadow exhibits a non-monotonic dependence on the observation time. At early times, the shadow size decreases as photons with smaller angular momentum gradually reach the observer. At later times, the expansion of the Friedmann universe becomes a dominant factor that leads to an increase in the shadow size.

1. Introduction

Wormholes are hypothetical astrophysical objects, resembling tunnels that connect two different regions of the same space-time or two different space-times. A basic problem of wormhole physics, at least in the case of static configurations in general relativity (GR), is that a wormhole requires exotic matter which violates the null energy condition (NEC) and prevents the throat from collapsing [1,2].

There have been numerous attempts to circumvent the theorems on NEC violation, either by invoking extensions of GR (which are desirable for many reasons but are quite unnecessary as regards the empirical situation on the macroscopic level [3]) or by abandoning the assumption on the static nature of a wormhole. A minimal way to do this is to consider stationary systems invoking spin or rotation; indeed, such examples of wormhole solutions in GR have been obtained, in particular those with classical spinor fields [4,5,6] and with rotating cylindrical sources [7,8]. It may be remarked, however, that such matter sources without exotic matter, being of evident theoretical interest, still look rather unrealistic from an observational viewpoint. One can also mention some ways of wormhole construction that try to join massive objects in two parallel universes (modeled by two branes) [9] or in opposite regions of de Sitter space [10] by a thin shell that has positive energy density.

However, it looks evidently more promising to obtain wormhole models without exotic matter by considering manifestly dynamic systems. In such cases, wormholes in the framework of GR cannot exist eternally [11], but their lifetime may be sufficiently long from any practical viewpoint. Thus, examples of wormholes existing against a cosmological background, sourced by some special examples of nonlinear electromagnetic fields, were found in [12,13]. It has also turned out that dynamic wormhole configurations can even be obtained with such a familiar source of gravity as evolving dust clouds, as follows from some recent papers [14,15,16]. Such models were found there as particular cases of the Lemaître–Tolman–Bondi (LTB) solution, obtained in GR by Lemaître and Tolman in 1933–1934 [17,18] and studied by Bondi in 1947 [19], as well as its extensions to a nonzero cosmological constant and an electromagnetic field. (One more recent paper [20] also claimed to study LTB wormholes but actually considered mixtures of fluids with nonzero pressure in a cosmological background. On the other hand, an exact solution for a black hole embedded in a nonstatic dust-filled universe was considered on the basis of the LTB solution in [21]).

The LTB solution describes the evolution of a spherical dust cloud. It contains three arbitrary functions $f\left(R\right)$, $F\left(R\right)$, and ${\tau }_0\left(R\right)$, where R is a radial coordinate in the comoving reference frame. A particular choice of these functions corresponds to a wormhole geometry with a throat defined as a sphere of minimum radius at a fixed time instant. It should be stressed that dust matter, being the only material source of the LTB solution, satisfies all standard energy conditions including the NEC.

The normal vector to a throat of a dynamic wormhole is timelike; hence, a throat is in general located in a T-region of space-time. Thus, if such a dust cloud is placed between two empty Schwarzschild space–time regions, the whole configuration is a black hole rather than a wormhole. However, dust clouds with throats can be inscribed into closed isotropic cosmological models filled with dust to form wormholes which exist for a finite period of time and experience expansion and contraction together with the corresponding cosmology.

In Ref. [16], we studied in detail evolving wormholes able to exist in a closed Friedmann dust-filled universe. In particular, we have shown that the lifetime of wormhole throats is much shorter than that of the whole wormhole region in the universe (which coincides with the lifetime of the universe as a whole). Nevertheless, studying radial null geodesics, i.e., radial photons paths, we established the possible traversability of the LTB wormhole configurations.

In this paper, we continue to explore the LTB wormholes, and now discuss their possible observable appearance by studying in detail the effects of gravitational lensing by such objects.

One can note that gravitational lensing by wormholes and photon orbits therein are rather widely discussed in the literature, but mostly for static or stationary wormhole models; see, e.g., [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] and references therein. The problem under consideration here is much more complicated due to the essentially dynamic nature of the wormholes. A similar problem was recently discussed in the black hole context [37], but the present study is completely independent. On the other hand, the main difference between shadows of dynamic and static or stationary wormholes is that in the dynamic case the shadow parameters are time-dependent, as demonstrated in Section 4.

The rest of this paper is organized as follows: in Section 2, we briefly describe the class of solutions under study; Section 3 is devoted to an analysis of photon motion in wormhole space-time; In Section 4, we describe the procedure of inscribing LTB wormholes into a closed Friedmann dust-filled universe and find their shadows as they could be seen by distant observers; finally, Section 5 provides a conclusion.

2. The Lemaître–Tolman–Bondi Solution and Wormholes

Consider the Lemaître–Tolman–Bondi (LTB) solution to the Einstein equations [17,18,19], describing the dynamics of a spherically symmetric dust cloud with an arbitrary density profile and any distribution of initial velocities. In a reference frame comoving to the dust particles, the metric can be written in the form [38]

$$d{s}^2=d{\tau }^2-e^{2\lambda (R,\tau )}d{R}^2-r^2(R,\tau )(d{\theta }^2+{sin}^2 \theta d{\varphi }^2),$$

(1)

where $\tau$ is physical time measured by clocks attached to dust particles, while R is the radial coordinate whose values correspond to fixed Lagrangian spheres, i.e., such a sphere of particles does not change its particular value of R during its motion.

Here, we are interested in dynamic wormholes described by the metric (1), which according to [15,16] are only possible in the elliptic branch of the LTB solution. Therefore, we consider only this branch here, for which (assuming the cosmological constant $\mathrm{\Lambda }=0$) we can write the following first integrals of the Einstein equations [16]:

$$\begin{array}{ccc}\hfill e^{2\lambda (R,\tau )}& =& {\displaystyle \frac{r^{\prime 2}(R,\tau )}{1-h\left(R\right)}},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\dot{r}}^2(R,\tau )& =& {\displaystyle \frac{F\left(R\right)}{r(R,\tau )}}-h\left(R\right),\hfill \end{array}$$

(3)

where $r^{\prime }\equiv \partial r/\partial R$, $\dot{r}\equiv \partial r/\partial \tau$, and $F\left(R\right)$ and $h\left(R\right)$ are arbitrary functions such that $F\left(R\right)$ is responsible for the mass distribution and $h\left(R\right)$ (such that $0<h\left(R\right)<1$) for the distribution of initial velocities. Note that Equation (2) has been obtained by integrating the $\left(tR\right)$ component of the Einstein equations and Equation (3) by subsequent integration of the $\left(RR\right)$ component, taking into account the relevant conservation law ${\nabla }_{\nu }{T}_0^{\nu }=0$.

Integrating Equation (3), we obtain an implicit expression for the radius $r(R,\tau )$

$$\pm [\tau -{\tau }_0\left(R\right)]={\displaystyle \frac{1}{h}}\sqrt{Fr-h{r}^2}+{\displaystyle \frac{F}{2h^{3/2}}}arcsin{\displaystyle \frac{F-2hr}{F}},$$

(4)

where ${\tau }_0\left(R\right)$ is another arbitrary function, responsible for clock synchronization between different Lagrangian spheres. It is more convenient to rewrite this solution in a parametric form in terms of the parameter $\eta$ [38]:

$$\begin{array}{ccc}& & r={\displaystyle \frac{F}{2h}}(1-cos\eta ),\hfill \\ & & \pm [\tau -{\tau }_0\left(R\right)]={\displaystyle \frac{F}{2h^{3/2}}}(\eta -sin\eta ).\hfill \end{array}$$

(5)

In wormhole solutions, the arbitrary functions must satisfy certain conditions; thus, at a throat, defined as a minimum of r at fixed $\tau$, we must have [15,16]

$$\begin{array}{ccc}& & h=1,h^{\prime }=0,h^{″}<0,\hfill \\ & & F^{\prime }=0,r^{\prime }=0,{\displaystyle \frac{h^{\prime }}{r^{\prime }}}<0,{\displaystyle \frac{F^{\prime }}{r^{\prime }}}>0.\hfill \end{array}$$

(6)

It is also supposed that the size r of a fixed-time section of space-time is much larger on both sides of the throat than the size $r{|}_{\mathrm{th}}$ of the throat itself.

Here, let us once more remind the reader that static wormholes need exotic matter to prevent their collapse. Exotic matter, by definition, violates the null energy condition (NEC), i.e., $k^{\mu }{k}^{\nu }{T}_{\mu \nu }<0$, where $k^{\mu }$ is a null vector; as a consequence, all other energy conditions, including the weak and strong energy conditions, are also violated. In our case, the dust filling the wormhole is ordinary matter satisfying the NEC. Indeed, the stress–energy tensor of dust (in a comoving frame) is $T_{\mu }^{\nu }=\mathrm{diag}(\rho ,0,0,0)$, where $\rho >0$ is the energy density. A (radial) null vector in the metric (1) can be taken as $k^{\mu }=(1,-e^{-\lambda },0,0)$. Then, manifestly, $k^{\mu }{k}^{\nu }{T}_{\mu \nu }=\rho >0$. Therefore, the NEC does hold. The cost of not violating the NEC is that we get a dynamic wormhole configuration collapsing during the Universe’s evolution.

In what follows, we will solve geodesic equations in the wormhole branch of the solution under study in the approximation of small $\eta$; thus, we consider only the beginning of wormhole evolution and assume ${\tau }_0\left(R\right)=0$. Then, Equation (5) in the linear approximation in $\eta$ is rewritten as

$$\tau \approx {\displaystyle \frac{F{\eta }^3}{12h^{3/2}}},r\approx {\displaystyle \frac{F{\eta }^2}{4h}}\Rightarrow r\approx {\left({\displaystyle \frac{3}{2}}\right)}^{2/3}{F}^{1/3}{\tau }^{2/3}.$$

(7)

The next terms in the $\eta$ expansions of $\tau$ and r are respectively

$$-{\displaystyle \frac{F{\eta }^5}{240h^{3/2}}}\mathrm{and}-{\displaystyle \frac{F{\eta }^4}{48h}}.$$

Their ratios to the first terms (7) may be considered as relative errors of our approximation at given $\eta$. We will construct photon trajectories for $\eta \le 0.1$, meaning that the relative errors in the quantities $\tau$ and r will not exceed $1/2000$ and $1/1200$, respectively. Thus, we can hope that the results of all further calculations will be correct up to $\sim {10}^{-3}$.

3. Photon Motion in Wormhole Space-Times

3.1. Null Geodesic Equations

The geodesic equations ${\displaystyle \frac{d^2{x}^i}{d{s}^2}}+{\mathrm{\Gamma }}_{kl}^i{\displaystyle \frac{d{x}^k}{ds}}{\displaystyle \frac{d{x}^l}{ds}}=0$, where ${\mathrm{\Gamma }}_{kl}^i$ are Christoffel symbols and s the affine parameter, have the following form for the metric (1):

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}{\displaystyle \frac{d^2\tau }{d{s}^2}}& =& -\dot{\lambda }{e}^{2\lambda }{\left({\displaystyle \frac{dR}{ds}}\right)}^2-r\dot{r}\left[{\left({\displaystyle \frac{d\theta }{ds}}\right)}^2+{sin}^2\theta {\left({\displaystyle \frac{d\varphi }{ds}}\right)}^2\right],\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{d^{2}R}{d{s}^2}}& =& -2\dot{\lambda }{\displaystyle \frac{d\tau }{ds}}{\displaystyle \frac{dR}{ds}}-{\lambda }^{\prime }{\left({\displaystyle \frac{dR}{ds}}\right)}^2+r{r}^{\prime }{e}^{-2\lambda }\left[{\left({\displaystyle \frac{d\theta }{ds}}\right)}^2+{sin}^2\theta {\left({\displaystyle \frac{d\varphi }{ds}}\right)}^2\right],\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}{\displaystyle \frac{d^2\theta }{d{s}^2}}& =& -2{\displaystyle \frac{\dot{r}}{r}}{\displaystyle \frac{d\theta }{ds}}{\displaystyle \frac{d\tau }{ds}}-2{\displaystyle \frac{r^{\prime }}{r}}{\displaystyle \frac{d\theta }{ds}}{\displaystyle \frac{dR}{ds}}+sin\theta cos\theta {\left({\displaystyle \frac{d\varphi }{ds}}\right)}^2,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d^2\varphi }{d{s}^2}}& =& -2{\displaystyle \frac{\dot{r}}{r}}{\displaystyle \frac{d\varphi }{ds}}{\displaystyle \frac{d\tau }{ds}}-2{\displaystyle \frac{r^{\prime }}{r}}{\displaystyle \frac{d\varphi }{ds}}{\displaystyle \frac{dR}{ds}}-2cot\theta {\displaystyle \frac{d\theta }{ds}}{\displaystyle \frac{d\varphi }{ds}}.\hfill \end{array}$$

(11)

Due to spherical symmetry, as usual, without loss of generality we can consider particle motion in the equatorial plane $\theta =\pi /2$. Let us also try to reduce the system to first-order differential equations. Because $\varphi$ is a cyclic coordinate, there is an integral of motion of the form $p_{\varphi }=r^2{\displaystyle \frac{d\varphi }{ds}}=\mathrm{const}=L$, where L is the azimuthal angular momentum of a particle.

Let us write the Lagrangian for photon motion:

$$2\mathcal{L}={\left({\displaystyle \frac{d\tau }{ds}}\right)}^2-e^{2\lambda }{\left({\displaystyle \frac{dR}{ds}}\right)}^2-r^2{\left({\displaystyle \frac{d\varphi }{ds}}\right)}^2=0.$$

(12)

Its derivative in the affine parameter s is actually a combination of Equations (8) and (9); hence, we can replace one of them with (12). Then, with Equation (2), the geodesic equations for photons moving in the equatorial plane can be rewritten as

$$\begin{array}{ccc}\hfill \hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\displaystyle \frac{d^2\tau }{d{s}^2}}& =& -{\displaystyle \frac{{\dot{r}}^{\prime }}{r^{\prime }}}{\left({\displaystyle \frac{d\tau }{ds}}\right)}^2+{\displaystyle \frac{L^2}{r^2}}\left({\displaystyle \frac{{\dot{r}}^{\prime }}{r^{\prime }}}-{\displaystyle \frac{\dot{r}}{r}}\right),\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\displaystyle \frac{dR}{ds}}& =& \pm \sqrt{{\displaystyle \frac{1-h}{r^{\prime 2}}}\left[{\left({\displaystyle \frac{d\tau }{ds}}\right)}^2-{\displaystyle \frac{L^2}{r^2}}\right]},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d\varphi }{ds}}& =& {\displaystyle \frac{L}{r^2}}.\hfill \end{array}$$

(15)

3.2. Geodesics in the Small $\eta$ Approximation

Let us begin with Equation (13) and notice that ${\dot{r}}^{\prime }/r^{\prime }=\dot{r}/r=2/\left(3\tau \right)$ (see Equation (7)), i.e., in our approximation L disappears from Equation (13), which now takes the following form:

$${\displaystyle \frac{d^2\tau }{d{s}^2}}=-{\displaystyle \frac{2}{3\tau }}{\left({\displaystyle \frac{d\tau }{ds}}\right)}^2.$$

(16)

This equation is easily integrated, giving

$$\tau \left(s\right)={\left(Cs+s_0\right)}^{3/5},$$

(17)

where $s_0$ is related to the photon launching time, $s_0=\tau {\left(0\right)}^{5/3}$, and C is related to the initial photon energy $E_0$ such that $C=(5/3)\tau {\left(0\right)}^{2/3}{(d\tau /ds)|}_{s=0}=(5/3)\tau {\left(0\right)}^{2/3}{p}_{\tau }\left(0\right)=(5/3)\tau {\left(0\right)}^{2/3}{E}_0$.

Let us now address Equation (14). Taking into account (17) and substituting the expression (7) for r, we can rewrite Equation (14) as

$${\displaystyle \frac{dR}{ds}}=\pm {\displaystyle \frac{C{\left(18F\right)}^{2/3}}{5|F^{\prime }|}}{(1-h)}^{1/2}\sqrt{1-{\displaystyle \frac{L^2}{C^2}}{\left({\displaystyle \frac{500}{243F}}\right)}^{2/3}}{\tau }^{-4/3}.$$

(18)

It is important to notice here that the right-hand side in (18) can turn to zero if the expression under the square root vanishes. This is possible at a point $R=R_t$ such that

$$F\left(R_t\right)={\displaystyle \frac{500}{243}}{\left|{\displaystyle \frac{L}{C}}\right|}^3.$$

(19)

The condition ${dR/ds|}_{R=R_t}=0$ means that a photon trajectory with given L has a turning point. If one supposes that the photon moves initially inward, i.e., from spheres with larger to smaller radial coordinates R, then the turning point is a sphere with a minimum coordinate $R_t$, where the inward motion of the photon stops and changes to the outward one.

Integrating Equation (18), we obtain

$$\pm {⨏}_{R\left(0\right)}^R{\displaystyle \frac{|F^{\prime }|d\tilde{R}}{{12}^{1/3}{F}^{2/3}\sqrt{(1-h)\left[1-{\displaystyle \frac{L^2}{C^2}}{\left({\displaystyle \frac{500}{243F}}\right)}^{2/3}\right]}}}=3{\left(Cs+s_0\right)}^{1/5},$$

(20)

where ${⨏}_{R\left(0\right)}^R={\int }_{R\left(0\right)}^{R_t}+{\int }_{R_t}^R$.

3.3. Photon Paths in Dynamic Wormhole Space-Time

Thus far, we have not specified the functions $F\left(R\right)$ and $h\left(R\right)$. Now, following Ref. [16], we choose them as follows:

$$F=2b{(1+R^2)}^k,h={\displaystyle \frac{1}{1+R^2}},$$

(21)

with $b,k=\mathrm{const}>0$. The function $h\left(R\right)$ is taken in this form without loss of generality due to arbitrariness of R parametrization (but according to the wormhole existence conditions (6)), whereas the choice of $F\left(R\right)$ is significant: the parameter b is the maximum size of the throat, while k is responsible for the wormhole density. Then the function $r(R,\tau )$ in the small $\eta$ approximation is

$$r(R,\tau )\approx {\left(4.5b\right)}^{1/3}{\left(1+R^2\right)}^{k/3}{\tau }^{2/3}.$$

(22)

Under this choice, the metric in (1) describing the space-time of a dynamic wormholeis symmetric with respect to its throat, $R=0$. The turning point of a photon path with specified L and C is provided by

$$R_t=\pm \sqrt{{\left({\displaystyle \frac{250}{243b}}\right)}^{1/k}{\left|{\displaystyle \frac{L}{C}}\right|}^{3/k}-1}.$$

(23)

Hereafter, the photon trajectories have been calculated by numerical integration of the system of geodesic equations in (15), (17) and (18) in Wolfram Mathematica using the NDSolveValue function with its default adaptive step-size Runge–Kutta 4(5) order method. Figure 1 and Figure 2 present examples of photon paths $R\left(\tau \right)$ and $R\left(\varphi \right)$ in a wormhole with the parameters $b=1,k=0.1$. Dashed lines mark the photon motion in the region $R<0$. All photons are launched at $R_0=-10,{\tau }_0\approx 0.005$ with initial energy $E_0\approx 0.3(C\approx 0.017)$. With such parameters, photons with $\left|L\right|>0.0165$ have a turning point; hence, they cannot cross the throat and remain in the region $R<0$.

4. A Friedmann Universe with a Dynamic Wormhole

In what follows, we will study photon paths in a Friedmann universe containing a dynamic wormhole; therefore, let us first consider null geodesics in the Friedmann metric.

The metric of a closed isotropic Friedmann universe filled with dustlike matter can be obtained from the LTB metric (1) by choosing the arbitrary functions h and F as [38]

$$F\left(\chi \right)=2a_0{sin}^3\chi ,h\left(\chi \right)={sin}^2\chi ,a_0=\mathrm{const},$$

(24)

where the coordinate $R\equiv \chi$ is the radial angle. Then, in terms of the parameter $\eta$, assuming ${\tau }_0\left(R\right)=0$ in Equation (4), we can write the functions r and $\tau$ in the form

$$\begin{array}{ccc}& & r(\eta ,\chi )=a\left(\eta \right)sin\chi ,\hfill \\ & & a\left(\eta \right)=a_0(1-cos\eta ),\hfill \\ & & \tau \left(\eta \right)=a_0(\eta -sin\eta ),\hfill \end{array}$$

(25)

where $a\left(\eta \right)$ is the cosmological scale factor and the Friedmann metric can be written as

$$d{s}^2=a^2\left(\eta \right)\left[d{\eta }^2-d{\chi }^2-{sin}^2\chi \left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)\right].$$

(26)

4.1. Null Geodesics in a Friedmann Universe

The geodesic equations for photons in a Friedmann universe can be obtained by substituting the functions $r(\eta ,\chi )$ and $\tau \left(\eta \right)$ into the geodesic Equations (13)–(15) for the LTB metric. Then, we can integrate the equation for $\tau$ and write the geodesic equations as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d\eta }{ds}}& =& {\displaystyle \frac{\sqrt{K}}{a^2\left(\eta \right)}},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\displaystyle \frac{d\chi }{ds}}& =& \pm {\displaystyle \frac{1}{a^2\left(\eta \right)}}\sqrt{K-{\displaystyle \frac{L^2}{{sin}^2\chi }}},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\displaystyle \frac{d\varphi }{ds}}& =& {\displaystyle \frac{L}{a^2\left(\eta \right){sin}^2\chi }}={\displaystyle \frac{L}{r^2}}.\hfill \end{array}$$

(29)

The constant K in these equations is related to the initial photon energy $E_0$ such that $\sqrt{K}=a_0[1-cos\eta \left(0\right)]E_0$ and L is the azimuthal angular momentum of a photon. Next, when constructing images, we will assume $K=1$.

Note that the root expression in Equation (28) should not be negative; therefore, particles with a nonzero angular momentum have a turning point ${\chi }_t$:

$$\begin{array}{c}\hfill {\chi }_t=arcsin\left(\sqrt{{\displaystyle \frac{L^2}{K}}}\right).\end{array}$$

(30)

4.2. A Wormhole in a Friedmann Universe

When matching the dynamic wormhole solution with the one for a Friedman universe, on the boundary determined by some values ${\chi }_{*}$ and $R_{*}>0$ of the radial coordinates, the functions $F\left(R_{*}\right)$ and $h\left(R_{*}\right)$ in the wormhole must coincide with $F\left({\chi }_{*}\right)=2a_0{sin}^3{\chi }_{*}$ and $h\left({\chi }_{*}\right)={sin}^2{\chi }_{*}$ in the Friedmann universe (see Equation (24)). Then, we have

$$h_{*}={sin}^2{\chi }_{*},F_{*}=2a_0{sin}^3{\chi }_{*}.$$

(31)

With our choice of the arbitrary functions $F\left(R\right)$ and $h\left(R\right)$ in (21), these conditions lead to the equalities

$$R_{*}=cot{\chi }_{*},b=a_0{(sin{\chi }_{*})}^{3+2k}.$$

(32)

The size and other parameters of a wormhole in a Friedmann universe are estimated in

Table 1. Estimates of the radius $r_{*}$ of the wormhole region, the boundary values ${\chi }_{*}$ and $R_{*}$ of the radial coordinates, and the angular size of the shadow $d_{\mathrm{sh}}$ (for an observer at a point with ${\chi }_{\mathrm{obs}}=1,{\eta }_{\mathrm{obs}}=1.1$) for various throat radii $r_{\mathrm{th}}$ in the case with $k=0.1$.

${\mathit{r}}_{\mathbf{th}}{|}_{\mathit{\eta }=\mathit{\pi }}=2\mathit{b}$	${\mathit{r}}_{*}{|}_{\mathit{\eta }=\mathit{\pi }}$	${\mathit{\chi }}_{*}$	${\mathit{R}}_{*}$	${\mathit{d}}_{\mathbf{sh}}$
1 km	${10}^{21}$ cm = 338 pc	$5.2\times {10}^{-8}$	$1.9\times {10}^7$	${0.004}^{″}$
10 km (neutron star)	700 pc	$1.1\times {10}^{-7}$	$9.3\times {10}^6$	${0.009}^{″}$
$6.4\times {10}^3$ km (Earth)	5.1 Kpc	$8.1\times {10}^{-7}$	$1.2\times {10}^6$	${0.08}^{″}$
$2.3\times {10}^5$ km	16 Kpc (Milky Way)	$2.5\times {10}^{-6}$	$4\times {10}^5$	${0.3}^{″}$
$695\times {10}^3$ km (Sun)	23 Kpc	$3.5\times {10}^{-6}$	$2.9\times {10}^5$	${0.4}^{″}$
$1.2\times {10}^7$ km (SgrA*)	56 Kpc	$8.7\times {10}^{-6}$	$1.2\times {10}^5$	$1^{″}$
$2\times {10}^{10}$ km (M87*)	557 Kpc	$8.6\times {10}^{-5}$	$1.2\times {10}^4$	${12}^{″}$
1 pc	5.7 Mpc	$8.6\times {10}^{-4}$	1165	${2.3}^{\prime }$
6.5 pc	10 Mpc (galaxy cluster)	0.0015	648.8	${4.3}^{\prime }$
10 Kpc	100 Mpc (void)	0.015	65.5	$0.8°$

. Here and henceforth we assume $a_0\sim {10}^{28}$ cm, which corresponds to the size of the observable universe. For the observer at a point with ${\chi }_{\mathrm{obs}}=1,{\eta }_{\mathrm{obs}}=1.1$, we have

$$\begin{array}{ccc}\hfill \tau (\eta =1.1)& \approx & 2.2\times {10}^9\mathrm{ly},\hfill \\ \hfill a(\eta =1.1)& \approx & 5.5\times {10}^{25}\mathrm{km},\hfill \\ \hfill r(\eta =1.1,\chi =1)& \approx & 4.9\times {10}^9\mathrm{ly}.\hfill \end{array}$$

(33)

Here, it is necessary to additionally comment on the results listed in Table 1. The angular size of the wormhole shadow $d_{\mathrm{sh}}$ presented in the last column of the table is discussed below, and the expression for $d_{\mathrm{sh}}$ is provided by Equation (37). The numerical values of the shadow size are given for hypothetical astrophysical wormholes of different scales. These values are calculated from the point of view of a very distant observer having the dimensionless parameters ${\chi }_{\mathrm{obs}}=1$ and ${\eta }_{\mathrm{obs}}=1.1$. The corresponding dimensional parameters provided by Equation (33) are ${\tau }_{\mathrm{obs}}\approx 2.2\times {10}^9\mathrm{y}$ and $r_{\mathrm{obs}}\approx 4.9\times {10}^9\mathrm{y}$. Obviously, such an observer observes wormholes which emerge at very early times, $\sim 2.2\times {10}^9\mathrm{ly}$, and situated at very long distances, $\sim 4.9\times {10}^9\mathrm{ly}$. For this reason, the results presented in the table cannot be directly applied to such relatively close (from a cosmological point of view) astrophysical objects as the supermassive black holes SgrA* and M87*. At the same time, we can apply our estimates to such distant objects as voids. The last line of Table 1 shows that if one assumes that a void is a dynamic wormhole with the typical size of $\sim 100\mathrm{Mpc}$, then its shadow has the size $\sim 0.8°$. This result is in good agreement with astronomical observations in which the voids are certainly not quite empty but are regions of substantially smaller density of visible matter seen directly.

4.3. Shadow of a Dynamic Wormhole

One method to detect a wormhole is by observing its shadow. When an observer looks at a wormhole with a luminous background behind it, he/she sees a dark spot. This is the so-called shadow, which emerges due to the fact that some of the photons from the luminous background are captured by the wormhole. We assume that light sources exist only in the observer’s Friedmann universe and that there are no light sources inside the wormhole. We also do not consider photons arriving from the other universe, as our small parameter $\eta$ approximation is not applicable in that case.

In stationary scenarios, the boundary of a wormhole shadow is determined by photons moving along cyclic orbits. However, in our dynamic case, such orbits do not exist, and the shadow boundary is determined by photons with the smallest angular momentum L that reach the observer at the point ${\chi }_{\mathrm{obs}}$ at the observation time ${\tau }_{\mathrm{obs}}$.

Figure 3 shows a schematic representation of photon motion from a light source to the observer through the wormhole. All photons are emitted simultaneously. Photons 3 and 4 (red trajectories) have higher angular momentum L and reach the observer by the observation time. Photons 1 and 2 (brown trajectories) have lower angular momenta L and do not reach the observer by that time. Thus, the shadow boundary at this moment is formed by photon 3. However, after some time, photon 2 will eventually reach the observer and it will then determine the shadow boundary, meaning that the shadow size decreases. Hence, the observation time is the first factor affecting the size of the wormhole shadow.

The second factor affecting the shadow size is the expansion of the Friedmann universe. Figure 4 provides a numerical simulation of photon motion in a Friedmann universe containing a dynamic wormhole. The images have been constructed for the case of photons that began to move parallel to the x axis from the point ${\chi }_{\mathrm{obs}}$ (blue line), ${\tau }_{\mathrm{obs}}$ to the left, i.e., back in time. Some of the photons (black paths) fall into the wormhole and do not have time to get out of it by $\tau =0$. Another part of the photons (red paths) either move only in the Friedmann universe or fall into the wormhole but manage to get out of it into the observer’s universe. It should be noted that when photons are in the wormhole space-time, the value of the parameter $\eta$ does not exceed $0.1$, which is consistent with our small $\eta$ approximation. Thus, if we “invert” the paths and consider them from left to right, it turns out that the photons begin their motion at time $\tau \approx 0$ and fly to the right towards the observer. In this case, the “red” photons reach the observer ${\chi }_{\mathrm{obs}}=1$ by the observation time ${\tau }_{\mathrm{obs}}$, while the “black” ones do not reach the observer and form a wormhole shadow. It is easy to see that the shadow in the lower image (${\tau }_{\mathrm{obs}}/b\approx 38$) is larger than in the upper one (${\tau }_{\mathrm{obs}}/b\approx 27$). This occurs due to the expansion of the Friedmann universe, which makes the shadow increase in size.

To calculate the size of the shadow seen by the observer, it is necessary to obtain the coordinates of the light ray on the observer’s instantaneous celestial sphere. Assuming that space-time is flat near the observer at any fixed moment of the observer’s proper time, they can be found in the following way [39]:

$${\alpha }_i=-r_{\mathrm{obs}}^{2}sin{\theta }_{\mathrm{obs}}{\left.{\displaystyle \frac{d\phi }{dr}}\right|}_{r_{\mathrm{obs}}},{\beta }_i=r_{\mathrm{obs}}^2{\left.{\displaystyle \frac{d\theta }{dr}}\right|}_{r_{\mathrm{obs}}},$$

(34)

where $r_{\mathrm{obs}}$ and ${\theta }_{\mathrm{obs}}$ are the observer’s coordinates.

Because the space-times under consideration are spherically symmetric, the boundary of the shadow will be a circle with radius ${\alpha }_{\mathrm{sh}}$ formed by the photons that reach the observer and have the lowest angular momentum $L_{\mathrm{sh}}$. Assuming ${\theta }_{\mathrm{obs}}=\pi /2$, we obtain the shadow radius as

$${\alpha }_{\mathrm{sh}}={\left.\left[r^2{\displaystyle \frac{d\phi }{ds}}{\left({\displaystyle \frac{dr}{ds}}\right)}^{-1}\right]\right|}_{r_{\mathrm{obs}},L_{\mathrm{sh}}}={\left.\left[r^2(\eta ,\chi ){\displaystyle \frac{d\phi }{ds}}{\left({\displaystyle \frac{\partial r}{\partial \eta }}{\displaystyle \frac{d\eta }{ds}}+{\displaystyle \frac{\partial r}{\partial \chi }}{\displaystyle \frac{d\chi }{ds}}\right)}^{-1}\right]\right|}_{{\chi }_{\mathrm{obs}},{\eta }_{\mathrm{obs}},L_{\mathrm{sh}}}.$$

(35)

Substituting the geodesic Equations (27)–(29) and the expression for $r(\eta ,\chi )$ (25), we derive the final formula for ${\alpha }_{\mathrm{sh}}$:

$${\alpha }_{\mathrm{sh}}={\displaystyle \frac{a_0{L}_{\mathrm{sh}}{\left(1-cos\left({\eta }_{\mathrm{obs}}\right)\right)}^2}{cos\left({\chi }_{\mathrm{obs}}\right)\left(1-cos\left({\eta }_{\mathrm{obs}}\right)\right)\sqrt{K-{\displaystyle \frac{L_{\mathrm{sh}}^2}{{sin}^2\left({\chi }_{\mathrm{obs}}\right)}}}+sin\left({\chi }_{\mathrm{obs}}\right)sin\left({\eta }_{\mathrm{obs}}\right)\sqrt{K}}}.$$

(36)

The angular size of the shadow is provided by

$$d_{\mathrm{sh}}=2arctan\left({\displaystyle \frac{{\alpha }_{\mathrm{sh}}}{r_{\mathrm{obs}}}}\right).$$

(37)

Figure 5 shows the dependence of the angular size of the wormhole shadow $d_{\mathrm{sh}}$ for ${\chi }_{*}=0.015(R_{*}=65.5)$ on the observation time. As already mentioned, the size of the shadow of a given wormhole is determined by both the parameter L of the photons that reach the observer and by the expansion of the universe itself. Therefore, the observation time dependence of the shadow size looks somewhat unusual. At the beginning of the observation, the shadow size decreases. This occurs because photons with lower angular momentum $L_{\mathrm{sh}}$ gradually reach the observer, and this decrease is not immediately compensated by the expansion of the universe. However, at later stages of observation, the angular momentum of the photons slowly decreases to the minimum value at which there is a turning point, and the shadow size increases due to the expansion of the Friedmann universe.

We will also obtain the dependence of the shadow size $d_{\mathrm{sh}}$ on the boundary value ${\chi }_{*}$. To do this, it is sufficient to find the minimum value $|L_{min}|$ of photons having a turning point for each wormhole, since precisely they form the shadow boundary at the observation time ${\eta }_{\mathrm{obs}}=1.1$ if the observer is at the point ${\chi }_{\mathrm{obs}}=1$. From the expression (23), we get

$$|L_{min}|={\displaystyle \frac{3C}{5}}{\left({\displaystyle \frac{9b}{2}}\right)}^{1/3}.$$

(38)

From the conditions (32), we obtain that the parameter b is related to the wormhole size as $b=a_0{(sin{\chi }_{*})}^{3+2k}$. Because ${\chi }_{*}\ll 1$, $b\approx a_0{\chi }_{*}^{3+2k}$ and $|L_{min}|=(3C/5){\left(9a_0/2\right)}^{1/3}{\chi }_{*}^{1+2k/3}$. From the equality of the values $d\tau /ds$ on the junction surface as calculated in the Friedmann universe and in the wormhole, we can find that $\sqrt{K}\approx (3C/5){\left(9a_0/2\right)}^{1/3}$ for all photons because of the small $\eta$ approximation. Thus, we obtain

$$|L_{min}|\approx \sqrt{K}{\chi }_{*}^{1+2k/3}.$$

(39)

Now, we should substitute this expression to Equation (36). However, let us first consider the term with $L_{\mathrm{sh}}$ in the denominator. Replacing $L_{\mathrm{sh}}$ with $L_{min}$, we obtain

$$\sqrt{K-{\displaystyle \frac{L_{min}^2}{{sin}^2\left({\chi }_{\mathrm{obs}}\right)}}}=\sqrt{K}\sqrt{1-{\chi }_{*}^{2+4k/3}{sin}^{-2}\left({\chi }_{\mathrm{obs}}\right)}\approx \sqrt{K}.$$

Now, the quantity $L_{\mathrm{sh}}$ remains only in the numerator, and in the approximation of small angles we arrive at

$$d_{\mathrm{sh}}\approx {\displaystyle \frac{2(1-cos{\eta }_{\mathrm{obs}})}{sin{\chi }_{\mathrm{obs}}\left[cos{\chi }_{\mathrm{obs}}(1-cos{\eta }_{\mathrm{obs}})+sin{\chi }_{\mathrm{obs}}sin{\eta }_{\mathrm{obs}}\right]}}{\chi }_{*}^{1+2k/3}\sim {\chi }_{*}^{1+2k/3}.$$

(40)

Figure 6 shows the dependence of the angular size of the wormhole shadow $d_{\mathrm{sh}}$ on the boundary value ${\chi }_{*}$ at the observation time ${\eta }_{\mathrm{obs}}=1.1$. The observer is located at the point ${\chi }_{\mathrm{obs}}=1$. The orange dots represent the values of ${\chi }_{*}$ and $d_{\mathrm{sh}}$ taken from Table 1. The black line represents the power-law dependence we have obtained (Equation (40)).

5. Conclusions

In this paper, we have studied in detail the shadow of a dynamic traversable wormhole inscribed into a closed dust-filled Friedmann universe. We have shown that the shadow is determined by photons with the minimum angular momentum $L_{\mathrm{sh}}$ that reach a particular distant observer by the observation time ${\tau }_{\mathrm{obs}}$.

Supposing $\eta \ll 1$, where $\eta$ is an auxiliary parameter used to express the solution $r(R,\tau )$ in a parametric form (see (5)), we have derived an expression for the size of the wormhole shadow and investigated the dependence of the shadow size on the observation time as well as on the boundary value ${\chi }_{*}$ of the wormhole, which characterizes the size of the wormhole region.

We find that the angular size of the shadow $d_{\mathrm{sh}}$ exhibits a non-monotonic dependence on the observation time. At early times, the shadow size decreases as photons with smaller angular momentum gradually reach the observer. At later times, the expansion of the Friedmann universe becomes the dominant factor that leads to an increase in the shadow size. We have also derived a power-law dependence of the shadow size $d_{\mathrm{sh}}$ on the value ${\chi }_{*}$ of the wormhole region boundary: $d_{\mathrm{sh}}\sim {\chi }_{*}^{1+2k/3}$. This analytical result was confirmed by our numerical calculations.

It should be noted that these results are incomplete, since we performed all calculations in the small $\eta$ approximation and assumed ${\tau }_0\left(R\right)=0$. That is, we considered only the initial stage of the wormhole evolution for a wormhole that appears simultaneously with the whole universe. Additionally, due to the small $\eta$ approximation, we could not consider photons that come to the observer from another universe, since the parameter $\eta$ significantly increases near the wormhole throat. Thus, in a future work it would be interesting to abandon the small $\eta$ approximation as well as to consider wormhole models with different ${\tau }_0\left(R\right)$ (that is, wormholes emerging at different cosmic times). In particular, we can expect a similar time dependence of the wormhole shadow size for different emergence times ${\tau }_0$; however, to confirm such expectations it will be necessary to carry out a new study under much more general assumptions.

In addition, as noticed in [16], the density of dust matter at the outskirts of wormhole regions is much smaller than the current mean density of the surrounding Friedmann universe, which allowed us to guess that such wormhole regions could be related to the observed voids in the distribution of matter in our Universe. Such a relationship could be relevant even for wormholes born together with the Universe: their rapidly evolving throats may have disappeared long ago, while the outer volumes of wormhole regions survive till nowadays. This relationship can be one more promising subject of future studies.