Nonlogarithmic divergence of a deflection angle by a marginally unstable photon sphere of the Damour-Solodukhin wormhole in a strong deflection limit

Recently, LIGO and VIRGO Collaborations have reported the direct detection of gravitational waves from black holes Abbott:2016blz ; LIGOScientific:2018mvr and Event Horizon Telescope Collaboration has reported the ring image of supermassive black hole candidates at the center of a giant elliptical galaxy M87 Akiyama:2019cqa . The black holes and the other compact objects with a strong gravitational field described by general relativity will be more important to understand our universe.

It is well known that static, spherically symmetric compact objects have unstable (stable) circular photon orbit called photon (antiphoton) sphere Claudel:2000yi ; Perlick_2004_Living_Rev . The upper bound of the radius of the (anti)photon sphere of the static, spherically symmetric black hole under the weak energy condition is given by $r=3M$, where $M$ is the mass of the black hole Hod:2017xkz . The (anti)photon sphere has important roles in several phenomena in a strong gravitational field: Light rays emitted by a source and reflected by the photon sphere make infinite number of dim images Hagihara_1931 ; Darwin_1959 ; Atkinson_1965 ; Luminet_1979 ; Ohanian_1987 ; Nemiroff_1993 ; Frittelli_Kling_Newman_2000 ; Virbhadra_Ellis_2000 ; Bozza_Capozziello_Iovane_Scarpetta_2001 ; Bozza:2002zj ; Perlick_2004_Phys_Rev_D ; Bozza_2010 , which are named relativistic images in Ref. Virbhadra_Ellis_2000 , on the both sides of the photon sphere and we can survey the compact object with the photon sphere by the images even if the compact objects themselves do not emit light rays. The photon sphere can be observed during a collapsing star to be a black hole Ames_1968 ; Synge:1966okc ; Yoshino:2019qsh . The photon sphere has strong influence on the high-frequency behavior of the photon absorption cross section Sanchez:1977si ; Decanini:2010fz and the high-frequency spectrum of quasinormal modes of compact objects Press:1971wr ; Goebel_1972 ; Raffaelli:2014ola . An observer moving on the photon sphere feels no centrifugal force and no gyroscopic precession Abramowicz_Prasanna_1990 ; Abramowicz:1990cb ; Allen:1990ci ; Hasse_Perlick_2002 and it is the fastest way to circle a static, spherically symmetric black hole for massless particles Hod:2012nk . The photon sphere is correspond to Bondi’s sonic horizon of a radial fluid Mach:2013gia ; Chaverra:2015bya ; Cvetic:2016bxi ; Koga:2016jjq ; Koga:2018ybs ; Koga:2019teu .

Stability of light rings, i.e., circular photon orbits Koga:2019uqd , of compact objects are an important property of the spacetime. Instability of the compact objects with the stable light rings has been concerned since they cause the slow decay of linear waves Keir:2014oka ; Cardoso:2014sna ; Cunha:2017qtt . The numbers of light rings of stationary, axisymmetic compact objects without an event horizon under energy conditions are two at least, they are even number in general, and the inner light ring is stable Cunha:2017eoe . Hod has shown that spherically symmetric compact objects without an event horizon have odd number light rings because of degeneration Hod:2017zpi . We note that we cannot apply the theorem of the number of the light rings for wormholes since a trivial topology has been assumed in Refs. Cunha:2017eoe ; Hod:2017zpi .

General relativity permits the wormholes which can have nontrivial topological structures Visser_1995 ; Morris_Thorne_1988 . The wormholes have a throat which connects two regions of one universe or two universes. Gravitational lensing Schneider_Ehlers_Falco_1992 ; Schneider_Kochanek_Wambsganss_2006 ; Perlick_2004_Living_Rev is used to find dark gravitating objects like wormholes. However, we cannot distinguish the wormhole with a positive mass from other massive objects such as a black hole under a weak-field approximation. Therefore, we must rely on the observation near the throat or the photon sphere in a strong gravitational field such as gravitational lensing of light rays scatter by the throat or the photon sphere Chetouani_Clement_1984 ; Perlick_2004_Phys_Rev_D ; Nandi:2006ds ; Muller:2008zza ; Tsukamoto_Harada_Yajima_2012 ; Perlick:2014zwa ; Tsukamoto:2016qro ; Tsukamoto:2016zdu ; Nandi:2016uzg ; Tsukamoto:2017edq ; Shaikh:2018oul ; Shaikh:2019jfr ; Shaikh:2019itn , visualizations Muller_2004 ; James:2015ima , shadows in an accretion gas Ohgami:2015nra ; Ohgami:2016iqm ; Kuniyasu:2018cgv , wave optics Nambu:2019sqn , and gravitational waves Cardoso:2016rao to distinguish wormhole from the other compact objects.

A Damour-Solodukhin wormhole Damour:2007ap has been suggested as a black hole mimicker. Its metric was created by making a slight modification to the Schwarzschild metric and it can be the simplest metric among black hole mimickers. The difference of the metric from the Schwarzschild spacetime is described by a positive dimensionless parameter $\lambda$. Lemos and Zaslavskii have pointed out that we can distinguish the wormhole from the black hole by a tidal force acting on a body near the throat even if the difference of the metric tensors of the wormhole from the black hole is small, i.e., $\lambda\ll 1$ Lemos:2008cv . Emissions from its accretion disk Karimov:2019qco , quasinormal modes and gravitational waves Bueno:2017hyj ; Volkel:2018hwb , images of its accretion disks Paul:2019trt , shadow Amir:2018pcu , gravitational lensing Nandi:2018mzm ; Ovgun:2018fnk ; Bhattacharya:2018leh ; Ovgun:2018swe ; Ovgun:2018oxk , and particle collision Tsukamoto:2019ihj in the Damour-Solodukhin wormhole spacetime have been investigated.

Gravitational lensing in a strong deflection limit, which is a semianalytic formalism for gravitational lensing of a light ray reflected by the photon sphere, has been investigated by Bozza Bozza:2002zj . The deflection angle $\alpha_{\mathrm{def}}$ of the light ray in the strong deflection limit has the following form

or

where $b$ and $b_{\mathrm{m}}$ are the impact parameter and the critical impact parameter of the light ray, respectively, $\theta$ is an image angle, $\theta_{\infty}\equiv b_{\mathrm{m}}/D_{OL}$ is the image angle of the photon sphere, where $D_{OL}$ is a distance between an observer and a lens object, and $\bar{a}$ is a positive parameter and $\bar{b}$ is a parameter. We can assume that the impact parameter $b$ is non-negative without loss of generality when we treat one light ray in a spherically symmetric spacetime. The analysis in the strong deflection limit has been improved Eiroa:2002mk ; Bozza:2005tg ; Bozza:2006nm ; Bozza:2007gt ; Tsukamoto:2016qro ; Tsukamoto:2016jzh ; Ishihara:2016sfv ; Shaikh:2019itn ; Shaikh:2019jfr and its relations to high-energy absorption cross section Wei:2011zw and to quasinormal modes Stefanov:2010xz have been investigated. Recently, Shaikh et al. have considered the deflection angle of light rays scattered by a photon sphere at the throat of a wormhole Shaikh:2018oul ; Shaikh:2019jfr .

The deflection angle of a light ray scattered by a photon sphere of the Damour-Solodukhin wormhole in the strong deflection limit was obtained by Nandi et al. Nandi:2018mzm , Ovgun Ovgun:2018fnk , and Bhattacharya and Karimov Bhattacharya:2018leh when the wormhole metric is similar to the black hole, i.e., for $\lambda<\sqrt{2}/2$. In Ref. Bhattacharya:2018leh , Bhattacharya and Karimov have pointed out that $\bar{b}$ by Ovgun Ovgun:2018fnk is in error.

In this paper, firstly we reexamine the deflection angle of the light ray scattered by the photon spheres of the Damour-Solodukhin wormhole for $\lambda<\sqrt{2}/2$. We recover Bhattacharya and Karimov’s result Bhattacharya:2018leh and we give a small modification for calculation in Nandi:2018mzm . Secondly we extend the analysis for $\lambda\geq\sqrt{2}/2$. The deflection angles diverge logarithmically in the strong deflection limit if the dimensionless parameter $\lambda\neq\sqrt{2}/2$. Interestingly, we have found that the deflection angle of a light ray scattered by a marginally unstable photon sphere at the throat diverges nonlogarithmically for $\lambda=\sqrt{2}/2$. We construct the strong deflection limit analysis for the deflection angle with a nonlogarithmic divergence in the Damour-Solodukhin wormhole spacetime.

This paper is organized as follows. In Sec. II, we review the Damour-Solodukhin wormhole spacetime. In Secs. III and IV, we investigate the deflection angle and observables, respectively, in the strong deflection limit. We review the gravitational lensing under a weak-field approximation in Sec. V and we summarize our result in Sec. VI. In Appendixes A and B, the Arnowitt-Deser-Misner (ADM) masses and the violation of energy conditions of the Damour-Solodukhin wormhole are shown, respectively. In this paper we use the units in which a light speed and Newton’s constant are unity.

In this section, we review the trajectory of a light ray and its deflection angle $\alpha_{\mathrm{def}}$ in a Damour-Solodukhin wormhole spacetime Damour:2007ap with a line element given by, in coordinates $(\tilde{t},r,\vartheta,\varphi)$,

where $\lambda$ and $\tilde{M}$ are positive parameters and the radial coordinate $r$ is defined in a range $2\tilde{M}\leq r<\infty$. A throat exists at $r=r_{\mathrm{th}}\equiv 2\tilde{M}$. The wormhole spacetime can take different values of the parameters $\tilde{M}$ and $\lambda$ in two asymptotically flat regions. For simplicity, we assume the equal parameters in the both regions. The line element is the same the one in the Schwarzschild spacetime in a limit $\lambda\rightarrow 0$ and it is the same as the static case of a Kerr-like wormhole Bueno:2017hyj . Bozza has considered the deflection angle of a light ray in the strong deflection limit in a general asymptotically flat, static, and spherically symmetric spacetime with its metric tensor behaving

where $t$ is a time coordinate and $M$ is a positive parameter Bozza:2002zj . The Damour-Solodukhin wormhole spacetime is an asymptotically flat, static, and spherically symmetric spacetime but the $(\tilde{t},\tilde{t})$-component of the metric tensor $g_{\tilde{t}\tilde{t}}$ (3) does not satisfy the assumption (4). We notice that $(\tilde{t},\tilde{t})$-component of the metric tensor $g_{\tilde{t}\tilde{t}}$ shown in Eq. (3) behaving asymptotically

is not suitable for a variable $z$ defined in Sec. III. Thus, we introduce a new time coordinate $t$ and a positive parameter $M$ which are defined by

and

respectively, to satisfy the assumption (4). By using $t$ and $M$, the line element (3) is rewritten in

where $A(r)$, $B(r)$, and $C(r)$ are given by

and

respectively. ¹¹1The metric tensor $(\ref{eq:line_element2})$ does not satisfy the condition (5) but it does not give us troubles to define the variable $z$. Notice that $r_{\mathrm{th}}$ is rewritten as $r_{\mathrm{th}}=2M(1+\lambda^{2})$. Since the spacetime is a static, spherically symmetric spacetime, there are time-translational and axial Killing vectors $t^{\mu}\partial_{\mu}=\partial_{t}$ and $\varphi^{\mu}\partial_{\mu}=\partial_{\varphi}$, respectively.

From $k^{\mu}k_{\mu}=0$, where $k^{\mu}\equiv\dot{x}^{\mu}$ is the wave number of a light ray and where the dot denotes the differentiation with respect to an affine parameter, the trajectory of the light ray is obtained as

Here we have set $\vartheta=\pi/2$ without loss of generality. Equation (14) can be expressed as

where the effective potential $\tilde{V}(r)$ is defined by

where $b\equiv L/E$ is the impact parameter of the light ray and the conserved energy $E\equiv-g_{\mu\nu}t^{\mu}k^{\nu}$ and the conserved angular momentum $L\equiv g_{\mu\nu}\varphi^{\mu}k^{\nu}$ of the light ray are constant along the trajectory. Since $\tilde{V}(r)\rightarrow E^{2}>0$ in spatial infinity $r\rightarrow\infty$, the light ray exists there.

By introducing a proper radial distance $l$ from the throat given by

the line element, the equation of trajectory of the light ray, and the effective potential are rewritten in

and

respectively. Note that the proper radial distance $l$ is defined in a range $-\infty<l<\infty$ and the throat is at $l=0$.

From $\tilde{v}=\frac{d\tilde{v}}{dl}=0$, we get the circular light orbits with $b=2M(1+\lambda^{2})^{\frac{3}{2}}/\lambda$ and $b=3\sqrt{3}M$ at $r=r_{\mathrm{th}}=2M(1+\lambda^{2})$ and $r=3M$, respectively. From straightforward calculations, we obtain $\left.\frac{d^{2}\tilde{v}}{dl^{2}}\right|_{r=r_{\mathrm{th}}}>0$ for $\lambda<\sqrt{2}/2$ and $\left.\frac{d^{2}\tilde{v}}{dl^{2}}\right|_{r=r_{\mathrm{th}}}\leq 0$ for $\lambda\geq\sqrt{2}/2$. Therefore, the throat is a photon (antiphoton) sphere for $\lambda>\sqrt{2}/2$ ($\lambda<\sqrt{2}/2$) and the photon sphere is marginally unstable for $\lambda=\sqrt{2}/2$. Figure 1 shows the dimensionless effective potential $v\equiv\tilde{v}/E^{2}$ for the (marginally unstable) photon sphere and antiphoton sphere.

The circular orbit of a light ray with $b=3\sqrt{3}$ at $r=3M$ is unstable for $\lambda\leq\sqrt{2}/2$, i.e., it is a photon sphere. Notice that the wormhole has two asymptotically flat regions and it has two photon spheres at $r=3M$ for $\lambda<\sqrt{2}/2$. The photon spheres at $r=3M$ and the antiphoton sphere at the throat $r=r_{\mathrm{th}}=2M(1+\lambda^{2})$ degenerate to be a marginally unstable photon sphere at the throat just for $\lambda=\sqrt{2}/2$. The wormhole has only one photon sphere at the throat for $\lambda>\sqrt{2}/2$.

As shown Fig. 2, we can classify the light ray which coming from a spatial infinity into a falling case with $b<b_{\mathrm{m}}$, a critical case with $b=b_{\mathrm{m}}$, and a scattered case with $b<b_{\mathrm{m}}$. Here the critical impact parameter $b_{\mathrm{m}}$ is defined by

where $r_{0}$ is the closest distance of the light ray and hereafter subscript $m$ denotes quantities of the photon sphere at $r=r_{\mathrm{m}}$.

We concentrate on the scattered case. In this case, a light ray comes from a spatial infinity, it is deflected by the wormhole at a reflection point $r=r_{0}$, and it goes back to the same spatial infinity. The reflection point is obtained as the largest positive solution of the equation $v(l)=0$ or $V(r)=0$. At the reflection point $r=r_{0}$, the equation of the trajectory (14) gives

where the subscript $0$ denotes the quantity at $r=r_{0}$. The impact parameter $b=b(r_{0})$ is expressed by

Here we have used Eq. (22).

From Eq. (14), the deflection angle $\alpha_{\mathrm{def}}(r_{0})$ of the light ray as a function of the reflection point $r_{0}$ is obtained as

where $I(r_{0})$ is defined by

where $V(r)\equiv\tilde{V}(r)/E^{2}$ is a dimensionless effective potential in the radial coordinate $r$ and where $l_{0}\equiv\left.l(r)\right|_{r=r_{0}}$ is the position of the reflection point in the radial coordinate $l$.

In this section, we investigate the deflection angle in the strong deflection limit $r_{0}\rightarrow r_{\mathrm{m}}$ or $b\rightarrow b_{\mathrm{m}}$. We treat it in the cases for $\lambda<\sqrt{2}/2$, $\lambda>\sqrt{2}/2$, and $\lambda=\sqrt{2}/2$ in this order. In the strong deflection limit $b\rightarrow b_{\mathrm{m}}$, the deflection angle of the light ray in a strong deflection is expressed by a following form Bozza:2002zj : ²²2The subleading term is $O(b-b_{\mathrm{m}})$ in Ref. Bozza:2002zj but we should read it as $O\left(\left(\frac{b}{b_{\mathrm{m}}}-1\right)\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)\right)$. See Refs. Tsukamoto:2016jzh ; Tsukamoto:2016qro ; Iyer:2006cn .

We introduce a variable $z$ Bozza:2002zj defined by