Relevant scales for the $C$-metric with positive cosmological constant

The $C$-metric with a positive cosmological constant $\Lambda$ is a special case of the Pleb$\acute{\mbox{a}}$nski-Demi$\acute{\mbox{a}}$nski family of metrics Griff0 . It can be obtained from the line element (23) in Griff by setting the rotation parameter $a=0$ and further imposing that the electric and magnetic charges vanish, that is $e=g=0$ and it describes two causally disconnected black holes of mass $M$ each accelerating in opposite direction due to the presence of a force generated by conical singularities located along the axes $\vartheta=0$ and $\vartheta=\pi$ Griff . More precisely, in Boyer-Lindquist coordinates and in geometric units ($c=G_{N}=1$) the line element is expressed as

with

where $\vartheta\in(0,\pi)$, $\varphi\in(-k\pi,k\pi)$ and $r_{H}<r<r_{h}$. Note that $r_{H}$ denotes the event horizon, $\alpha$ is the acceleration parameter and $r_{c}$ is the cosmological horizon. We present a detailed analysis of the horizons and their spatial ordering in terms of the relevant physical parameters in the section. Here, it suffices to mention that while for the case of a $C$-metric the Schwarzschild horizon is smaller than the acceleration horizon whenever $0<2\alpha M<1$, it is not clear a priori if the same condition ensures that $r_{H}<r_{c}$ in the case of a $C$-metric with positive cosmological constant. Furthermore, by adapting the tetrad (7) in Griff to the present case the only non-zero component of the Weyl tensor is

The above expression confirms that the spacetime with line element (1) is of algebraic type D and the only curvature singularity occurs at $r=0$. Hence, the horizons $r_{H}$ and $r_{C}$ are just coordinate singularities. It is interesting to observe that for $\alpha\to 0$ the metric in (1) becomes the metric of a Schwarzschild-de Sitter black hole while in the case of $\Lambda\to 0$ but $\alpha\neq 0$ (1) correctly reproduces the line element associated to the $C$-metric as given in Maha . This fact means that any prediction regarding the bending of light in a manifold described by (1) should reproduce the corresponding results for the Schwarzschild-de Sitter case in the limit of $\alpha\to 0$ as well as the gravitational lensing results for the $C$-metric obtained in Maha when we let $\Lambda\to 0$ with $\alpha$ kept constant. According to Griff , the conical singularity on $\vartheta=0$ can be removed by making the following choice for the parameter $k$ entering in the range of the angular variable $\varphi$

while the conical singularity with constant deficit angle along the half-axis $\vartheta=\pi$ which is computed by means of (22) in Griff as

can be explained in terms of a semi-infinite cosmic string pulling the black hole and/or of a strut pushing it. Similarly as for the $C$-metric, this allows us to think of (1) as of a Schwarzschild-de Sitter-like black hole experiencing an acceleration along the $\vartheta=\pi$ direction due to the presence of a force, i.e. the tension of a cosmic string. Moreover, we observe that the length of interval for the range of the coordinate $\varphi$ can be transformed to its standard value $2\pi$ with the help of the rescaling $\varphi=k\phi$ so that $\phi\in(-\pi,\pi)$. Moreover, Kof was able to provide an interpretation of the string/strut in terms of null dust. In the rest of this paper, we will work with the line element obtained after the aforementioned rescaling is introduced, namely

where

with $F$, $f_{\Lambda}$ and $g$ given in (2).

In the present work, we study the geodesic motion of a massive particle and the light bending in a two black hole metric with positive cosmological constant. For this end, a preliminary study of the behaviour of the null geodesics turns out to be convenient in detecting some features of strong gravity in the aforementioned spacetime. The question of new phenomena arises if we consider a metric which in some limiting case reduces to the Schwarzschild metric (for examples see we1 ). Here, the fate of the circular orbit, already appearing in the Schwarzschild metric, and issues regarding its stability deserve careful attention because they will give us useful insights on how to construct an appropriate impact parameter. While the $C$-metric has been extensively studied in the last decades, the same cannot be said for its counterpart with $\Lambda$. The independence of null geodesics on the cosmological constant was first recognized in a seminal and extremely comprehensive paper on photon surfaces by Claudel where a general class of static spherically symmetric space-times was considered. The case of non spherically symmetric manifolds was addressed by Gibbons . There, among several physically relevant space-times, the case of a $C$-metric with cosmological constant was studied and the authors discovered that a non-spherically symmetric photon surface continues to exist even in that scenario. The conclusion is that, while the Schwarzschild metric exhibits a photon sphere, the same cannot be said for the $C$-metric. More precisely, Gibbons showed that instead of a photon sphere there is a photon surface displaying at least one conical singularity.

Regarding geodesic motion in a $C$-black hole, radial time-like geodesics were analyzed by Far whereas the study of the circular motion of massive and massless particles was undertaken by Pravda . Moreover, Lim ; Bini0 offered an exhaustive treatment of time-like and null geodesics by a mixture of analytical and numerical methods. Finally, Bini probed into the motion of spinning particles around the direction of acceleration of the black hole. Furthermore, Maha determined the coordinate angle of the so-called photon cone and performed a Jacobi stability analysis to show that all circular null geodesic on the photon cone are radially unstable. The shadow of a $C$-black hole was studied in Gr1 ; Gr2 while Frost derived inter alia an exact solution of the light-like geodesic equation by means of Jacobi elliptic functions and determined the angular radius of the shadow. We should also mention that the analysis of the light-like geodesics and the black hole shadow for a rotating $C$-metric have been addressed in ZZ . Moreover, other probed into photon spheres and black hole shadows for dynamically evolving spacetimes. Finally, we refer to QNMs1 ; QNMs2 for the analysis of the QNMs and the stability properties for a $C$-black hole.

Regarding a $C$-black hole in an anti-de Sitter (AdS) or a de Sitter (dS) background, Griff1 ; Griff2 ; Dias1 ; Dias2 ; Krt1 ; Poldo ; Krt2 ; Xu studied in detail the geometric structure and the related properties of these spacetimes. The analysis of the circular motion of massive and massless particles in the $C$-metric with a negative cosmological constant was performed by Chamb where the author proved that the circular null geodesics are unstable whereas Poldo completed the study of Chamb by considering some special characteristics associated to the null geodesics in the aforementioned metric. Recently, Lim in Lim21 classified all possible trajectories for photons in the (A)dS $C$-metric in terms of the particle angular momentum and the energy scaled in units of the Carter constant. Similarly as in Frost , it was possible to construct exact solutions for null geodesics by means of Jacobi elliptic functions. However, the stability problem for such trajectories has not been addressed by Lim21 . We complete the results of Lim21 by concentrating on the interplay of different scales in the potentially observable or physical relevant case. To make a comparison with known results of the Schwarzschild- de Sitter metric more accessible, we work in Boyer-Lindquist coordinates.

Concerning gravitational lensing, detailed studies on light bending in the weak and strong regimes were pioneered by the George Ellis lensing group. Some of their exciting results which have relevance to the present work are Virb ; Virb1 ; Virb2 ; Virb3 ; Virb4 . A nice and thorough review article on this subject has been written by Perl . Finally, Gibbons01 exploited a novel geometrical approach based on the Gauss-Bonnet theorem applied to the optical metric of the gravitational lens in order to derive weak leansing formulae for spherically symmetric metrics generated by certain static, perfect non-relativistic fluids. Regarding the gravitational lensing Frost derived a lens equation and showed that the lens results of Ifti for the rotating $C$-metric with NUT parameter does not contain as a special case the $C$-metric (where both the acceleration and NUT parameters are set equal to zero). Moreover, Maha studied the strong and weak lensing for null rays on the photon cone. To the best of our knowledge, we could not identify any paper studying the bending of light and analyzing the (in)stability problem of null circular orbits for the $C$-metric with positive cosmological constant. We hope to fill this gap with the present work.

The remainder of the paper is structured as follows. In Section II, we analyze the horizon structure of the dS $C$-metric in terms of certain orderings among the Schwarzschild, the cosmological and the acceleration scales. In order to understand which scale orderings are physically relevant, we consider three typical black hole representatives: ultramassive, massive and light. In Section III, we derive the effective potential for massive and massless particles and we show that the null-orbits have the same radius and and take place on the same family of invariant cones as in the $C$-metric, i.e. they do not depend on $\Lambda$. This result is in agreement with Gibbons . Moreover, we find that the circular orbit is due to a saddle point in the effective potential, which requires an additional effort to probe into the associated stability problem. This is addressed in Section IV where we perform the Jacobi (in)stability analysis of the circular orbits. In Section V, since the Sachs optical scalars cannot be used to optically distinguish between $C$- and a dS $C$-black holes, we study the the gravitational lensing in the weak and strong regimes. More precisely, the corresponding deflection angles are analytically computed when the light propagation occurs on a certain family of invariant cones, and their dependence on the observer position is shown. Our formulae correctly reproduce the corresponding ones in the $C$-metric case in the limit of vanishing $\Lambda$ and indicate that the deflection angles may depend on the cosmological constant if the position of the observer is close to the cosmological horizon.

The structure of the horizons for the metric associated to the line element (1) can be unraveled by analyzing the roots of the equation $f_{\Lambda}(r)=0$ with $f_{\Lambda}$ given as in (2). To this purpose, it is convenient to introduce the Schwarzschild, the cosmological and the acceleration length scales defined as $r_{s}=2M$, $r_{\Lambda}=\sqrt{3/\Lambda}$ and $r_{a}=1/\alpha$, respectively. The appearance of several scales in the metric makes a precise study of the horizons a worthwhile undertaking since a priori it is not clear what structure of the horizons will emerge. Apart from that, we recall a curious fact from the Schwarzschild de Sitter metric with two horizons, one dominated by $r_{s}$ and the second one (the cosmological horizon) by $r_{\Lambda}$. The Boyer-Lindquist coordinates which one uses to study this metric are valid within these two horizons where we locate ourselves and the rest of the universe. An observer outside $r_{\Lambda}$ might even claim that we are living inside a black hole. It is interesting to reconsider the unique position for the case of the C-metric as more scales enter the calculation. In terms of these scales the equation $f_{\Lambda}(r)=0$ gives rise to the following cubic equation

with

First of all, we observe that the extrema of the cubic in (8) are located at

with $r_{-}<0$, $r_{+}>0$ and moreover,

Since $P(r)\to-\infty$ as $r\to-\infty$, we conclude that $r_{-}$ is a maximum. The fact that $P(r_{-})>0$ for any positive value of the scales, as it can be immediately seen from (11), together with the observation that $P(0)>0$, allows us to conclude that the cubic (8) admits always a negative root, here denoted by $r_{<}$ while the other two zeroes may be positive and distinct, having algebraic multiplicity $2$ or being complex conjugate of each other. However,the scenario where the two positive roots coincide, i.e. the discriminant of (8) vanishes, is not physically relevant because as the discriminant tends to zero, the region between the two positive roots gets smaller and smaller and it is impossible to define a static observer in most of the spacetime. Furthermore, the case of two complex conjugate roots corresponds to the presence of a naked singularity at $r=0$. Since the cubic (8) depends on the three scales $r_{s}$, $r_{a}$, and $r_{\Lambda}$, it is imperative to understand which orderings among these scales are physically relevant. To this purpose, we consider three typical black hole representatives: ultramassive BHs such as TON618 in Canes Venatici TON whose Schwarzschild radius is $32$ times the distance from Pluto to the Sun (see Table 1), massive BHs like Sagittarius A^∗ at the galactic centre of the Milky Way SAG whose event horizon is approximately $18$ times the sun radius and light BHs such as GW170817 in the shell elliptical galaxy NGC 4993 ABB with an event horizon diameter of $18$ km.

From Table 1, we immediately observe that we can always assume $r_{s}\ll r_{\Lambda}$. Regarding the acceleration parameter $\alpha$, it is important to observe that for $C$-black holes with cosmological constant the only constraint we need to impose on $\alpha$ is that $\alpha>0$. The situation is dramatically different in the case of the $C$-metric where the black hole mass and the acceleration parameter must satisfy the condition $2M\alpha<1$ which is equivalent to require that $r_{s}<r_{a}$. Let us discuss and interpret the roots of (8) for the following cases

1. 1.

   $r_{s}=r_{a}\ll r_{\Lambda}$: in this scenario, given $M$ the acceleration parameter of the black hole in SI units is

   $$\alpha_{s}=\frac{c^{2}}{2G_{N}M}$$

   (12)

   See Table 1 for typical values for $\alpha_{s}$. Note that this case has no corresponding physical counterpart for a $C$-metric because in the limit of $\Lambda\to 0$ we would have a $C$-BH such that the event and acceleration horizons coincide. Setting $r_{s}=r_{a}$ in (8) and introducing the small parameter $\epsilon=r_{s}/r_{\Lambda}$, the discriminant of the reduced cubic is

   $$D_{1}=\frac{\epsilon^{2}r_{s}^{6}(27\epsilon^{2}+32)}{108(1+\epsilon^{2})^{4}},$$

   (13)

   which is clearly positive. This observation together with the remark below equation (11) allows us to conclude that there is one negative root and two complex conjugate roots. Hence, this is the case of a naked singularity at $r=0$ and the coordinate $r$ can be extended up to space-like infinity. In comparison a naked singularity is not possible in the Schwarzschild-de Sitter metric.

2. 2.

   $r_{s}<r_{a}\ll r_{\Lambda}$: if we rewrite (8) as

   $$-\frac{\Lambda}{3}r^{3}+(r-r_{s})\left(1-\frac{r^{2}}{r_{a}^{2}}\right)=0,$$

   (14)

   we see that $\Lambda$ is the small parameter and a straightforward application of perturbation methods for algebraic equations shows that the event horizon $r_{H}$ and the cosmological horizon $r_{h}$ are represented by the following expansions

   	$\displaystyle r_{H}$	$\displaystyle=$	$\displaystyle r_{s}+\frac{r_{s}^{3}r_{a}^{2}}{3(r_{a}^{2}-r_{s}^{2})}\Lambda+\frac{r_{s}^{5}(3r_{a}^{2}-r_{s}^{2})}{(r_{a}^{2}-r_{s}^{2})^{3}}\Lambda^{2}+\mathcal{O}(\Lambda^{3}),$		(15)
   	$\displaystyle r_{h}$	$\displaystyle=$	$\displaystyle r_{a}-\frac{r_{a}^{4}}{6(r_{a}-r_{s})}\Lambda+\frac{r_{a}^{7}(3r_{a}-5r_{s})}{72(r_{a}-r_{s})^{3}}\Lambda^{2}+\mathcal{O}(\Lambda^{2}).$		(16)

   From the point of view of the horizon structure this is an interesting case. Even if $\Lambda$ appears in the corrections, the horizon associated with it disappears and in its place, we encounter $r_{a}$ which we could rightly call the acceleration horizon. As a consequence, we locate our position within the acceleration horizon. Note that the same expansion holds also for $r_{a}<r_{s}\ll r_{\Lambda}$.

3. 3.

   $r_{s}\ll r_{a}=r_{\Lambda}$: in this regime the acceleration parameter is completely determined by the cosmological constant and is given in SI units as

   $$\alpha_{\Lambda}=c^{2}\sqrt{\frac{\Lambda}{3}}\approx 5.2\cdot 10^{-10}~{}\mbox{m/s${}^{2}$}.$$

   (17)

   Since $r_{s}\ll r_{a}$, it must be $2M\alpha_{\Lambda}\ll 1$. As we can see from the last column in Table 1, such a condition is violated by ultramassive and massive black holes. Hence, the present case may be relevant for light black holes such as GW170817. By means of the rescaling $\mathfrak{r}=r/r_{\Lambda}$ and the introduction of the same small parameter $\epsilon$ already defined in 1. we can rewrite (8) as

   $$\frac{\epsilon}{2}(1-\mathfrak{r}^{2})+\mathfrak{r}\left(\mathfrak{r}^{2}-\frac{1}{2}\right)=0.$$

   (18)

   The discriminant of the associated reduced cubic is

   $$D_{3}=-\frac{\epsilon^{4}}{432}+\frac{71}{1728}\epsilon^{2}-\frac{1}{216}.$$

   (19)

   From Fig. 1 we observe that also in this case the discriminant may become positive. More precisely, we have three distinct real roots if $0\leq\epsilon<\epsilon_{0}=\sqrt{142-34\sqrt{17}}/4$ while the naked singularity case occurs when $\epsilon>\epsilon_{0}$. Applying again perturbation methods to find expansions for the positive roots of (18) yields

   

   	$\displaystyle r_{H}$	$\displaystyle=$	$\displaystyle r_{s}+\frac{r_{s}^{3}}{r_{\Lambda}^{2}}+4\frac{r_{s}^{5}}{r_{\Lambda}^{4}}+\mathcal{O}\left(\frac{r_{s}^{7}}{r_{\Lambda}^{6}}\right),$		(20)
   	$\displaystyle r_{h}$	$\displaystyle=$	$\displaystyle\frac{r_{\Lambda}}{\sqrt{2}}-\frac{r_{s}}{4}+\frac{7}{16\sqrt{2}}\frac{r_{s}^{2}}{r_{\Lambda}}-\frac{1}{2}\frac{r_{s}^{3}}{r_{\Lambda}^{2}}-\frac{689}{512\sqrt{2}}\frac{r_{s}^{4}}{r_{\Lambda}^{3}}+\mathcal{O}\left(\frac{r_{s}^{5}}{r_{\Lambda}^{4}}\right).$		(21)

   This case resembles indeed the Schwarzschild-de Sitter order of horizons.

We conclude this section by observing that in general equation (8) can be transformed into the reduced third order polynomial equation

by means of the variable transformation $Y=r+\rho/3$. According to Bron the associated discriminant is

and we have the following classification

1. 1.

   three distinct real roots for $D<0$;

2. 2.

   two real roots where one root has algebraic multiplicity two whenever $D=0$;

3. 3.

   one real and two complex conjugate roots for $D>0$.

Since the first case is physically relevant, we will stick to the condition $D<0$. Then, if we introduce the additional parameter $R=\sqrt{|p|/3}$ and the auxiliary angle $\omega$ defined as $\cos{\omega}=q/(2R^{3})$, then the roots are parametrized with the help of trigonometric functions and their inverses in the following form

We observe that in addition to the inequality $D<0$, there is the additional constraint that $\cos{\omega}\leq 1$. These two constraints are not satisfied for any value of the scales entering in our problem as it can be seen in Figure 2. However, it can be seen that the condition $D<0$ ensures that $\cos{\omega}<1$.

When we turn our attention to the study of the geodesic motion, in addition to $r_{s}$, $r_{\Lambda}$ and $r_{a}$, a new scale $\ell$ associated to the angular momentum of the particle enters the scene. To probe into the interplay of these scales, the method of the effective potential seems most adequate. One might expect that the many new scales as compared to the Schwarzschild-de Sitter metric will result into a richer structure of critical points. It will turn out that this is partly true, but only if we pay careful attention to the emergence of saddle points. To study the motion of a particle in the gravitational field described by (6), we need to analyze the geodesic equations FL

subject to the constraint

with $\epsilon=0$ and $\epsilon=1$ for light-like and time-like particles, respectively. The system of coupled ODEs associated to (25) can be immediately obtained from equations (8)-(11) in Maha by replacing $A$, $B$ and $f$ therein with the functions $A_{\Lambda}$, $B_{\Lambda}$ and $f_{\Lambda}$ defined in (7) and noticing that the functions $C$ and $D$ remain the same. In view of this observation, one can proceed as in Maha and conclude that the dynamics is governed by the following coupled system of ODE

where $\mathcal{E}$ and $\ell$ are the energy per unit mass and the angular momentum per unit mass of the particle, respectively. Moreover, the constraint equation (26) can be cast into the form

where $E=\mathcal{E}^{2}/2$ and the effective potential is given by

At this step, it is gratifying to observe that in the limit of vanishing $\alpha$ and $\Lambda$ equation (29) reproduces correctly equation (25.26) in FL for the Schwarzschild case. Moreover, the functions $A_{\Lambda}$, $B_{\Lambda}$ and $C$ are non negative for $r_{H}\leq r\leq r_{h}$ and therefore, $E-U_{eff}\geq 0$ as in classical mechanics. Finally, the equality, $E=U_{eff}$, corresponds to a circular orbit and a critical point of the effective potential. Since in the present work we are interested in the study of the light bending, we recall that in the case of null geodesics $\epsilon=0$ and hence, the effective potential simplifies as follows

To study the null circular orbits for the potential (31), we need to find its critical points. Imposing that $\partial_{r}\mathfrak{B}=0=\partial_{\vartheta}\mathfrak{B}$ leads to the following equations

At this point a comment is in order. First of all, the above equations does not contain $\Lambda$. This is surprising because the spacetimes described by (6) and the $C$-metric are not conformally related. The same phenomenon occurs when we study the null circular orbits for the Schwarzschild and Schwarzschild-de Sitter black holes, i.e. in both cases the corresponding photon spheres are characterized by a typical radius which is $\Lambda$-independent Claudel . Hence, we can conclude as in Maha that null geodesics admit circular orbits with radius

only for a certain family of orbital cones with half opening angle given by

Note that due to the fact that $\alpha M\in(0,1/2)$ and $\vartheta_{c}$ is a monotonically decreasing function in the variable $\alpha M$, it follows that $\vartheta_{c}$ cannot take every value from $0$ to $\pi$. More precisely, it can only vary on the interval $(\vartheta_{c,min},\pi/2)$ with $\vartheta_{c,min}=\arccos{(1/3)}\approx 70.52^{\circ}$. To classify the critical point of our effective potential, we compute the determinant of the Hessian matrix associated with the effective potential (31) at the critical point $(r_{c},\theta_{c})$. We find that the determinant $\Delta$ of the Hessian matrix is

with $x:=\alpha M$, $d=r_{s}/r_{\Lambda}$, $\tau=\sqrt{1+12x^{2}}$. The functions $S$ and $T$ are the same as those computed in Maha , namely

while the new contribution due to the cosmological constant is encoded in the function $K(d,x)$ which is given by

Note that in the limit of $d\to 0$ equation (35) reproduces correctly (31) in Maha . From the analysis performed in Maha we already know that the function $T$ is always positive for $x\in(0,1/2)$. This signalizes that the sign of (35) is controlled by the term $S(x)+K(d,x)$ which is positive for $x$ in the interval $(0,1/2)$ and

where $\mathfrak{f}$ is the function representing the dotted boundary of the yellow region in Fig. 3. Since black holes of astrophysical interest are characterized by $r_{s}\ll r_{\Lambda}$, this implies that $d\ll 1$. The yellow part in Fig. 3 represents the region in the space of the parameters $x$ and $d$ where the function $S(x)+K(d,x)$ is positive. This is clearly the case for $x\in(0,1/2)$ and $d\ll 1$. Hence, we conclude that the critical point $(r_{c},\vartheta_{c})$ of the effective potential is a saddle point.