The dynamics of the leading self-force (0SF) order, corresponding to the geodesic motion of a light body in the exact background of a heavy body, are explicitly derived for the first time using a flat space scattering amplitude-based approach. This is achieved by utilising the cubic formulation of Einstein's general relativity coupled to the effective worldline action of massive point particles, which was employed to derive the Schwarzschild black hole metric in arXiv:2405.14421.
This paper performs a systematic investigation of geodesic motion in Euclidean Schwarzschild geometry, which is studied in the equatorial plane. The explicit form of geodesic motion is obtained in terms of incomplete elliptic integrals of first, second and third kind. No elliptic-like orbits exist in Euclidean Schwarzschild geometry, unlike the corresponding Lorentzian pattern. Among unbounded orbits, only unbounded first-kind orbits are allowed, unlike general relativity where unbounded second-kind orbits are always allowed.
We have studied the null geodesics of the Schwarzschild black hole surrounded by quintessence matter. Quintessence matter is a candidate for dark energy. Here, we have done a detailed analysis of the geodesics and exact solutions are presented in terms of Jacobi-elliptic integrals for all possible energy and angular momentum of the photons. The circular orbits of the photons are studied in detail. As an application of the null geodesics, the angle of deflection of the photons are computed.
We study the time-like geodesic congruences, in the space-time geometry of a Schwarzschild black hole surrounded by quintessence. The nature of effective potential along with the structure of the possible orbits for test particles in view of the different values of quintessence parameter are analysed in detail. %wherever possible it is done analytically otherwise numerically. An increase in quintessence parameter is seen to set the particles from farther distance into motion around black hole. The effect of quintessence parameter is investigated analytically wherever possible otherwise we perf…
We study the geodesic equations in the space-time of a Schwarzschild black hole pierced by an infinitely thin cosmic string and give the complete set of analytical solutions of these equations for massive and massless particles, respectively. The solutions of the geodesic equations can be classified according to the particle's energy and angular momentum, the ratio between the component of the angular momentum aligned with the axis of the string and the total angular momentum, the deficit angle of the space-time and as well the horizon radius (or mass) of the black hole. For bound orbits of ma…
Causal radial geodesics with a positive interval in the Schwarzschild metric include a subset of trajectories completely confined under a horizon, which compose a thermal statistical ensemble with the Hawking-Gibbons temperature. The Bekenstein--Hawking entropy is given by an action at corresponding geodesics of particles with a summed mass equal to that of black hole in the limit of large mass.
Recently, an analytical study of radial and circular orbits for null and time-like geodesics that propagate in the spacetime produced by a Schwarzschild black hole associated with cloud of strings, in a universe filled by quintessence, has been done in Ref. \cite{Mustafa:2021}. In this paper, we complete the aforementioned study by investigating possible analytical solutions to the equations of motion for other types of bound orbits, besides taking into account the cases of unbound orbits. This requires an extensive study of the corresponding effective potentials that categorize the test parti…
In this paper we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both $M/r_0$ and $M/b$ where $M$ is the mass of the black hole, $r_0$ the distance of closest approach of the light ray and $…
A Geodesic Route toward Holography beyond AdS: Tessellating the Schwarzschild Black Hole
School of Physics and Shing-Tung Yau Center, Southeast University（东南大学物理学院与邵逸夫数学中心）；College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）
The Landau-Lifshitz decomposition of spacetime, or (1+3)-split, determines the three-dimensional velocity and acceleration as measured by static observers. We use these quantities to analyze the geodesic particles in Schwarzschild and Kerr spacetimes. We show that in both cases there is no room for a positive acceleration (repulsion). We also compute the escape and terminal speeds. The escape speed in the case of a static black hole coincides with the Newton result. For the Kerr spacetime, the escape speed depends on the polar angle, showing that a particle needs less energy to escape in the d…
We present novel methods to numerically address the problem of characterizing the response of particle detectors in curved spacetimes. These methods allow for the integration of the Wightman function, at least in principle, in rather general backgrounds. In particular we will use this tool to further understand the nature of conformal massless scalar Hawking radiation from a Schwarzschild black hole in anti-de Sitter space. We do that by studying an Unruh-DeWitt detector at rest above the horizon and in circular geodesic orbit. The method allows us to see that the response rate shows peaks at …
Causal Space-Time Structure and non-Hausdorff Extension of Schwarzschild Black Hole Interior
Landau Phystech-School, Moscow Institute of Physics and Technology（兰道物理技术学院，莫斯科物理技术学院）；Institute for High Energy Physics of National Research Centre "Kurchatov Institute"（库尔恰托夫国家研究中心高能物理研究所）
We consider a wide class of static spherically symmetric black holes of arbitrary dimension with a photon sphere (a hypersurface on which a massless particle can orbit the black hole on unstable circular null geodesics). This class includes various spacetimes of physical interest such as Schwarzschild, Schwarzschild-Tangherlini and Reissner-Nordström black holes, the canonical acoustic black hole or the Schwarzschild-de Sitter black hole. For this class of black holes, we provide general analytical expressions for the Regge poles of the $S$-matrix associated with a massless scalar field theory…
Kehagias-Sfetsos black hole in Hořava-Lifshitz gravity is probed through particle geodesics. Gravitational force of KS black hole becomes weaker than that of Schwarzschild around horizon and interior space. Particles can be always scattered or trapped in new closed orbits, unlike those falling forever in Schwarzschild black. The properties of null and timelike geodesics are classified with values of coupling constants. The precession rates of the orbits are evaluated. The time trajectories are also classified under different values of coupling constants for both null and timelike geodesics. Ph…
Analytical Fluxes from Generic Schwarzschild Geodesics
Racah Institute of Physics, Hebrew University of Jerusalem（希伯来大学拉卡物理研究所）；School of Mathematics and Statistics, University College Dublin（都柏林大学学院数学与统计学院）
We derive the static Schwarzschild-Tangherlini metric by extracting the classical contributions from the multi-loop vertex functions of a graviton emitted from a massive scalar field. At each loop orders the classical contribution is proportional to a unique master integral given by the massless sunset integral. By computing the scattering amplitudes up to three-loop order in general dimension, we explicitly derive the expansion of the metric up to the fourth post-Minkowskian order $O(G_N^4)$ in four, five and six dimensions. There are ultraviolet divergences that are cancelled with the introd…
We present a general framework with which the Schwarzschild-Tangherlini metric of a point particle in arbitrary dimensions can be derived from a scattering amplitude to all orders in the gravitational constant, $G_N$, in covariant gauge (i.e. $R_ξ$-gauge) with a generalized de Donder-type gauge function, $G_σ$. The metric is independent of the covariant gauge parameter $ξ$ and obeys the classical gauge condition $G_σ=0$. We compute the metric with the generalized gauge choice explicitly to second order in $G_N$ where gravitational self-interactions become important and these results verify the…
For the radial motion of massive particles with large angular momenta in Schwarzschild geometry and that of massive charged particles with large angular momenta or energy in a particular range in Reissner--Nordström geometry, there exist classically forbidden regions on the outside of the respective event horizons which scatter certain infalling geodesics or screen some of the black holes' evaporation by reflecting the emitted particles back into the black holes. Quantum tunneling across this forbidden regions is studied.
We study the timelike geodesics and geodesic deviation for a two-dimensional stringy blackhole spacetime in Schwarzschild gauge. We have analyzed the properties of effective potential along with the structure of the possible orbits for test particles with different settings of blackhole parameters. The exactly solvable geodesic deviation equation is used to obtain corresponding deviation vector. The nature of deviation and tidal force is also examined in view of the behavior of corresponding deviation vector.The results are also compared with an another two-dimensional stringy blackhole spacet…
Using the analytical solution for the Schwarzschild metric containing supertranslation field, we consider two main ingredients of calculation of the thermal Hawking black hole radiation: solution for eigenmodes of the d'Alambertian and solution of the geodesic equations for null geodesics. For calculation of Hawking radiation it is essential to determine the behavior of both the eigenmodes and geodesics in the vicinity of horizon. The equation for the eigenmodes is solved, first, perturbatively in the ratio $O(C)/M$ of the supertranslation field to the mass of black hole, and, next, non-pertur…
We investigate the timelike geodesics and the periapsis precession of orbits in the Fisher-Janis-Newman-Winicour-Wyman spacetime. This spacetime represents the naked singularity spacetime in the Einstein-massless scalar system. We revisit the results in the previous studies and relax the assumptions about the eccentricity of a bound orbit and the size of a semilatus. We find that the negative periapsis precession occurs when the spacetime sufficiently deviates from the Schwarzschild spacetime. In particular, for the small eccentric orbits, we show the negative periapsis precession occurs for $…
We study the detector's response when moving along an ingoing null geodesic. The backgrounds are chosen to be black hole spacetimes ($(1+1)$ dimensional Schwarzschild metric and near horizon effective metric for any stationary black hole in arbitrary dimensions) as well as Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe. For black holes the trajectories are defined in Schwarzschild coordinates and the field modes are corresponding to {\it Boulware vacuum}. Whereas for FLRW case, the detector is moving along the path defined in original {\it cosmic time} and the field modes are related to {…
We study geodesics in the Schwarzschild space-time affected by an uncertainty in the mass parameter described by a Gaussian distribution. This study could serve as a first attempt at investigating possible quantum effects of black hole space-times on the motion of matter in their surroundings as well as the role of uncertainties in the measurement of the black hole parameters.
We find a specific coordinate system that goes from the Painlevé-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass $E$. $E$ is in the range $1\leq E<\infty$ with the limit $E=\infty$ yielding the Kruskal-Szekeres maximal extension. So, thro…
We study thermodynamic evaporation of Schwarzschild-de Sitter black holes in terms of a low energy perturbation theory. A small black hole which is far from the cosmological horizon and observers at the spacelike hypersurface where black hole attraction and expansion of cosmological horizon balance exactly are considered. In the low energy perturbation, scalar field equations are solved in both regions of the hypersurface and scattering amplitudes are derived. And then the desired thermal temperatures from the two horizons are obtained as a ``minimal'' value of the statistical thermal temperat…
We explicitly prove that the Weyl conformal symmetry solves the black hole singularity problem, otherwise unavoidable in a generally covariant local or non-local gravitational theory. Moreover, we yield explicit examples of local and non-local theories enjoying Weyl and diffeomorphism symmetry (in short co-covariant theories). Following the seminal paper by Narlikar and Kembhavi, we provide an explicit construction of singularity-free spherically symmetric and axi-symmetric exact solutions for black hole spacetimes conformally equivalent to the Schwarzschild or the Kerr spacetime. We first che…
In this paper we derive the leading quantum gravitational corrections to the geodesics and the equations of motion for a scalar field in the spacetime containing a constant density star. It is shown that these corrections can be calculated in quantum gravity reliably and in a model independent way. Furthermore, we find that quantum gravity gives rise to an additional redshift that results from the gradient instead of the amplitude of the density profile.
