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Light Propagation in a Plasma on Kerr Spacetime: Separation of the Hamilton-Jacobi Equation and Calculation of the Shadow

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Light Propagation in a Plasma on Kerr Spacetime: Separation of the Hamilton-Jacobi Equation and Calculation of the Shadow Light propagation in a plasma on Kerr spacetime: Separation of the Hamilton-Jacobi equation and calculation of the shadow Volker Perlick1, ∗ and Oleg Yu. Tsupko2, 3, y 1ZARM, University of Bremen, 28359 Bremen, Germany 2Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia 3National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia (Dated: May 16, 2017) We consider light propagation in a non-magnetized pressureless plasma around a Kerr black hole. We find the necessary and sufficient condition the plasma electron density has to satisfy to guarantee that the Hamilton-Jacobi equation for the light rays is separable, i.e., that a generalized Carter constant exists. For all cases where this condition is satisfied we determine the photon region, i.e., the region in the spacetime where spherical light rays exist. A spherical light ray is a light ray that stays on a sphere r = constant (in Boyer-Lindquist coordinates). Based on these results, we calculate the shadow of a Kerr black hole under the influence of a plasma that satisfies the separability condition. More precisely, we derive an analytical formula for the boundary curve of the shadow on the sky of an observer that is located anywhere in the domain of outer communication. Several examples are worked out. PACS numbers: 04.20.-q { 98.35.Jk { 98.62.Mw { 98.62.Sb I. INTRODUCTION and time-independent plasma density on the light de- flection in the Schwarzschild spacetime was calculated For most applications of general relativity light rays and discussed by Perlick [4]. In this study neither the can be considered as lightlike geodesics of the spacetime gravitational nor the plasma deflection is assumed to be metric, i.e., the influence of a medium on the light rays small. Later, the influence of plasma effects on grav- can be neglected. However, in the radio frequency range itational lensing was investigated, with different meth- this is not always true. A well-known example is the ods, by Bisnovatyi-Kogan and Tsupko [5{7]. In the Kerr influence of the Solar corona on the travel time and on spacetime, light deflection in the equatorial plane was cal- the deflection angle of radio signals that come close to culated by Perlick [4] for the case of a rotationally sym- the Sun. This influence is routinely observed since the metric and time-independent plasma density. Within the 1960s. In this case one may assume that the medium is approximation of small deflection, lensing of light rays a non-magnetized pressureless plasma and for the gravi- off the equatorial plane was considered by Morozova et tational field the linearized theory is sufficient. The rel- al. [8] who assumed a slowly rotating Kerr black hole. evant equations have been determined by Muhleman et The influence of a plasma on the multiple imaging prop- al. [1, 2]. Gravitational lensing in this approximation erties in the strong-bending regime was investigated by was firstly discussed by Bliokh and Minakov [3]. In these Bisnovatyi-Kogan and Tsupko [7] for the Schwarzschild works both the gravitational and the plasma deflection spacetime and recently extended to Kerr spacetime by are assumed to be small and they are calculated sepa- Liu, Ding and Jing [9]. For some possible astrophysical rately from each other. observations of plasma effects near compact objects we There is good reason to assume that also black holes refer to Rogers [10{12] and for plasma effects in strong and other compact objects are surrounded by a plasma lens systems to Er and Mao [13]. For a review see [14]. and it is an interesting question to investigate if the arXiv:1702.08768v2 [gr-qc] 15 May 2017 Here we want to concentrate on the influence of a plasma could have an observable effect on radio signals plasma on the shadow of a black hole. Roughly speaking, that come close to such a compact object. In such cases the shadow is the black disk an observer sees in the sky if the linearized theory is not sufficient. For a spherically a black hole is viewed against a backdrop of light sources symmetric uncharged black hole, one has to consider that are distributed around the black hole but not be- the Schwarzschild metric and for a rotating uncharged tween the observer and the black hole. For constructing black hole one has to consider the Kerr metric. Some the shadow we have to consider all past-oriented light results on light propagation in a non-magnetized pres- rays that issue from a chosen observer position. Each of sureless plasma on the Schwarzschild or Kerr spacetime these light rays corresponds to a point on the observer's are known. The influence of a spherically symmetric sky. We assign darkness to a point if the correspond- ing light ray goes to the horizon and brightness other- wise. The idea is that there are light sources distributed in the spacetime, but not between the observer and the ∗ [email protected] black hole and not inside a possible white-hole extension y [email protected] of the spacetime from where future-oriented light rays 2 could be sent across the horizon towards the observer. ciated with them is of crucial relevance for deriving the Without a plasma, the shadow of a Schwarzschild black boundary curve of the shadow on the sky of an observer hole was calculated by Synge [15] and the shape of the anywhere in the domain of outer communication of the shadow of a Kerr black hole, for an observer at infin- black hole. This will be done, in close analogy to the ity, was calculated by Bardeen [16]. In an earlier paper procedure in Grenzebach et al. [18, 19], in Sec. V. In the [17] we have generalized Synge's formula to the case of a rest of the paper we determine the photon regions and spherically symmetric and static plasma distribution on the shadow for some special cases where the separability a spherically symmetric and static spacetime. As par- condition is satisfied. ticular examples, we have worked out the results for the Our conventions are as follows. We use the summation case that the underlying spacetime geometry is (a) that convention for greek indices that take the values 0,1,2,3. of a Schwarzschild black hole and (b) that of an Ellis Our choice of signature is (−; +; +; +). We raise and wormhole. In the present paper we want to derive a sim- lower greek indices with the spacetime metric. G is the ilar result for a plasma around a Kerr black hole. More gravitational constant and c is the vacuum speed of light. precisely, it is our goal to derive an analytical formula We use units such that ~ = 1, i.e, energies have the same for the boundary curve of the shadow on the sky of an unit as frequencies and momentum vectors have the same observer at an arbitrary position outside of the horizon of unit as wave vectors. the black hole (i.e., in contrast to Bardeen we do not con- sider an observer at infinity). Without the influence of a plasma, such a formula has been derived by Grenzebach II. HAMILTON FORMALISM FOR LIGHT et al. [18, 19] for the case of a black hole of the Pleba´nski- RAYS IN A PLASMA ON KERR SPACETIME Demia´nskiclass (which includes the Kerr black hole as a special case). Their method made use of the fact that the Light propagation in a non-magnetized pressureless equation of motion for vacuum light rays admits a fourth plasma can be characterized by the Hamiltonian constant of motion, the Carter constant, in addition to the ones that follow from the symmetry. Therefore, an 1 H(x; p) = gµν (x)p p + ! (x)2 : (1) important first step in our analysis will be to find out 2 µ ν p under what conditions a Carter constant exists for light µν propagation in a plasma on a Kerr spacetime. Of course, Here g are the contravariant components of the space- this result is of interest not only for the calculation of the time metric tensor and !p is the plasma electron fre- shadow but also for other problems. quency which equals, up to a scalar factor, the electron The perspectives of actually observing the influence of density, ∗ a plasma on the shadow, e.g. for Sgr A or for M87, have 2 been discussed in our earlier paper [17]. 4πe !p(x) = Ne(x) (2) The paper is organized as follows. In Sec. II we re- me view the Hamilton formalism for light rays in a non- magnetized, pressureless plasma on a general-relativistic where e and me are the electron charge and mass, re- spectively, and Ne is the electron number density. x = spacetime and we specialize the relevant equations to the 0 1 2 3 Kerr metric. In Sec. III we derive the necessary and suffi- (x ; x ; x ; x ) are the spacetime coordinates and p = cient condition on the plasma electron density that guar- (p0; p1; p2; p3) are the canonical momentum coordinates. antees separability of the Hamilton-Jacobi equation for The light rays are the solutions to Hamilton's equations light rays, i.e., that guarantees the existence of a Carter dxµ @H dp @H constant1. In Sec. IV we determine the photon region = ; µ = − ;H(x; p) = 0 ; (3) ds @p ds @xµ for a plasma density around a Kerr black hole that satis- µ fies the separability condition. The photon region is the where s is a curve parameter which has no direct phys- spacetime region filled with spherical light rays, i.e., with ical or geometrical meaning.
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