Z. K. Silagadze
We provide a Finslerian extension of the Schwarzschild metric based on heuristic arguments. The proposed metric asymptotically approaches not the Minkowski space-time but the Bogoslovsky locally anisotropic space-time which arises naturally as a deformation of very special relativity.
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Finsler geometry is just riemannian geometry without the quadratic restriction[1]. In this paper, we study the motion of massive(non-zero rest mass) and massless particles for schwarzschild metric in finsler spacetime in the case of two dimensional Randers-Finsler space with unit positive flag curvature instead of two dimensional riemann sphere. We provide qualitative study of potential energy of particles at various distances near schwarzschild black hole and the results are then compared with schwarzschild metric in riemannian geometry. We will also describe the scattering of particles near …
The work presented in this paper aims to contribute to the problem of testing Finsler gravity theories by means of experiments and observations in the solar system. Within a class of spherically symmetric static Finsler spacetimes we consider a satellite with an on-board atomic clock, orbiting in the Finslerian-perturbed gravitational field of the earth, whose time signal is transmitted to a ground station, where its receive time and frequency are measured with respect to another atomic clock. This configuration is realized by the Galileo 5 and 6 satellites that have gone astray and are now on…
We undertake to show how the relativistic Finslerian Metric Function (FMF) should arise under uni-directional violation of spatial isotropy, keeping the condition that the indicatrix (mass-shell) is a space of constant negative curvature. By evaluating respective Finslerian tetrads, and treating them consistently as the bases of inertial reference frame (RF), the generalized Finslerian kinematic transformations follow in a convenient explicit form. The concomitant Finslerian relativistic relations generalize their Lorentzian prototypes through the presence of one characteristic parameter g, so…
Granted the post-Lorentzian relativistic kinematic transformations are described in the Finslerian framework, the uniformity between the actual light velocity anisotropy change and the anisotropic deformation of measuring rods can be the reason proper for the null results of the Michelson-Morley-type experiments at the first-order level.
Finslerian extension of the theory of relativity implies that space-time can be not only in an amorphous state which is described by Riemann geometry but also in ordered, i.e. crystalline states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics. It is shown that there are only two types of curved Finslerian spaces endowed with local relativi…
In this study a rotationally and translationally invariant metric in Finsler space is investigated. We choose to rewrite the metric in Riemanian space by increasing the dimension of space-time and introducing additional coordinates such that for specific values of these coordinates, the geodesics of the four dimensional Finslerian space-time and six dimensional Riemanian space-time are identical. Cosmological solutions described by this metric give rise to an equation of state corresponding to a space dominated by domain walls and an internal space dominated by strings.
Black Hole Solutions with Constant Ricci Scalar in a Model of Finsler Gravity
本文提出在Finsler几何中具有非零宇宙常数的引力场方程，发现具有常旗曲率的Finslerian球对称解在$λ\ eq1$时对应被弦云包围的黑洞，并研究了$λ=1$时的测地线与圆轨道条件。
Finslerian extension of an anisotropic strange star in the domain of modified gravity
本文将芬斯勒几何与修改引力结合，从变分原理导出爱因斯坦场方程及修改TOV方程，并应用于由MIT Bag状态方程描述的奇异星，揭示其各向异性应力分布与稳定性。
From Barthel Randers Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology
本文综述了基于Finsler几何的宇宙学模型，特别是Barthel-Randers和Barthel-Kropina几何，其额外几何项可解释为有效暗能量，且与观测数据拟合良好，可作为广义相对论的替代方案。
It is reasonably well-known that birefringent crystal optics can to some extent be described by the use of pseudo-Finslerian spacetimes (an extension of pseudo-Riemannian spacetime). What is less commonly appreciated is that there are two separate and quite disjoint pseudo-Finsler structures for the two photon polarizations, and further, that there are separate tangent-space pseudo-Finsler structures defined by the group velocity and co-tangent-space pseudo-co-Finsler structures defined by the phase velocity. The inter-connections between these four separate pseudo-Finsler structures are rathe…
We present vacuum spacetime solutions of first order gravity, which are described by the exterior Schwarzschild geometry in one region and by degenerate tetrads in the other. The invertible and noninvertible phases of the tetrad meet at an intermediate boundary across which the components of the metric, affine connection and field-strength are all continuous. Within the degenerate spacetime region, the noninvertibility of the tetrad leads to nonvanishing torsion. In contrast to the Schwarzschild spacetime which is the unique spherically symmetric solution of Einsteinian gravity, all the field-…
We show that, in the absence of matter in the bulk, the Einstein equations and the Gauss-normal form of the metric place stringent restrictions on the form of the event horizon in a brane world. As a consequence, the off-brane extension of the standard 4-D Schwarzschild horizon in the Randall-Sundrum $AdS_5$ spacetime, as it is viewed from the brane can only be of a tubular shape, instead of a pancake shape. When it is viewed from the $AdS_5$ horizon, such a tubular horizon is absent.
This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positi…
Maximal extension of Schwarzschild-like spacetimes in Lorentz gauge theory
本文研究了洛伦兹规范理论中Schwarzschild-like黑洞解的最大解析扩展，通过分析径向光锥曲线并构造Kruskal-Szekeres图表，揭示了该解在因果拓扑上与Schwarzschild解相同，但在几何上当A_0≠1时与Schwarzschild解不等价。
Causal Space-Time Structure and non-Hausdorff Extension of Schwarzschild Black Hole Interior
研究黑洞内部因果时空结构及非豪斯多夫扩展，通过从黑洞外因果相连区域延续测地线完备化构建锥，将史瓦西黑洞视为零电荷极限改革彭罗斯 - 卡特图，其边缘轨迹构成特殊路径，形成粒子热浴。
Observational tests of quantum extension of Schwarzschild spacetime in loop quantum gravity with stars in the galactic center
本研究利用银心17颗绕Sgr A*运行恒星的观测数据，通过测地线演化转化与MCMC模拟检验圈量子引力的施瓦西时空量子扩展模型，未发现相关显著证据，并由S2星给出LQG参数95%置信度的最强上限约束。
Some solutions of the Einstein equations for the eight-dimensional Riemann extension of the classical four-dimensional Schwarzschild metric are considered.
We present a smooth extension of the Schwarzschild exterior geometry, where the singular interior is superceded by a vacuum phase with vanishing metric determinant. Unlike the Kruskal-Szekeres continuation, this solution to the first-order field equations in vacuum has no singularity in the curvature two-form fields, no horizon and no global time. The underlying non-analytic structure provides a distinct geometric realization of `mass' in classical gravity. We also find that the negative mass Schwarzschild solution does not admit a similar extension within the first-order theory. This is consi…
Asymptotically Schwarzschild solutions in $f(R)$ extension of general relativity
该研究通过微扰求解史瓦西背景下的修正爱因斯坦方程，构建了一类与太阳系实验兼容的$f(R)$引力理论，并确定了其参数，为宇宙学模型提供了正确的局部极限。
Finslerian lightconvex boundaries: applications to causal simplicity and the space of cone geodesics $\mathcal{N}$
本文研究带边界的非定芬斯勒流形和锥结构，证明边界光凸性、内部因果简单性与锥测地线空间豪斯多夫性三者等价，并显式构造锥测地线空间的流形结构，推广了渐近 AdS 时空等应用。
We derive a transformation of the noncommutative geometry inspired Schwarzschild solution into new coordinates such that the apparent unphysical singularities of the metric are removed. Moreover, we give the maximal singularity-free atlas for the manifold with the metric under consideration. This atlas reveals many new features e.g. it turns out to describe an infinite lattice of asymptotically flat universes connected by black hole tunnels.
Existence of static vacuum extensions for Bartnik boundary data near Schwarzschild spheres
本文证明了Schwarzschild流形中对称球面附近的Bartnik边界数据存在渐近平坦静态真空延拓，且具有局部唯一性。
Using canonical (Schrodinger) quantization of spherically symetric gravitational dust systems, we find the quasi-classical (coherent) state, |α^{(s)}>, that corresponds to the classical Schwarzschild solution. We calculate the ``quasi-classical Schwarzschild mertic", which is the expectation value of the quantized metric in thhis quasi-classical state. Depending on the quantization scheme that we use, we study three different quasi- classical geometries, all of which turn out to be singularity free. Their maximal extensions are complete manifolds with no singularities, describing a tower of as…
The entropy for a black hole in a de Sitter space is approached within the framework of spacetime foam. A simple model, made by $N$ wormholes in a semiclassical approximation, is taken under examination to compute the entropy for such a case. An extension to the extreme case when the black hole and cosmological horizons are equal is discussed.
On the foundations and applications of Lorentz-Finsler Geometry
本文探讨了洛伦兹-芬斯勒几何在相对论物理及其他领域的应用，介绍了其基础概念、关键应用及未来发展方向，包括几何结构、与黎曼几何的联系以及在经典力学中的波传播应用。
Quantum Schwarzschild-(A)dS Black Holes: Unitarity and Singularity Resolution
本文在幺模引力中量子化球对称 Schwarzschild-(A)dS 时空，通过自伴延拓保证幺正性，并构造非奇异的量子修正度规，实现黑洞到白洞的跃迁。
We wish to construct a model for charged star as a generalization of the uniform density Schwarzschild interior solution. We employ the Vaidya and Tikekar ansatz [{\it Astrophys. Astron.} {\bf 3} (1982) 325] for one of the metric potentials and electric field is chosen in such a way that when it is switched off the metric reduces to the Schwarzschild. This relates charge distribution to the Vaidya-Tikekar parameter, $k$, indicating deviation form sphericity of three dimensional space when embedded into four dimensional Euclidean space. The model is examined against all the physical conditions …
We study a spherically symmetric setup consisting of a Schwarzschild metric as the background geometry in the framework of classical polymerization. This process is an extension of the polymeric representation of quantum mechanics in such a way that a transformation maps classical variables to their polymeric counterpart. We show that the usual Schwarzschild metric can be extracted from a Hamiltonian function which in turn, gets modifications due to the classical polymerization. Then, the polymer corrected Schwarzschild metric may be obtained by solving the polymer-Hamiltonian equations of mot…
We study structure of solutions of the recently constructed minimal extensions of Einstein's gravity in four dimensions at the quartic curvature level. The extended higher derivative theory, just like Einstein's gravity, has only a massless spin-two graviton about its unique maximally symmetric vacuum. The extended theory does not admit the Schwarzschild or Kerr metrics as exact solutions, hence there is no issue of Schwarzschild type singularity but, approximately, outside a source, spherically symmetric metric with the correct Newtonian limit is recovered. We also show that for all Einstein …
In this contribution we present a noncanonical phase-space noncommutative (NC) extension of a Kantowski Sachs (KS) cosmological model to describe the interior of a Schwarzschild black hole (BH). We evaluate the thermodynamical quantities inside this NC Schwarzschild BH and compare with the well known quantities. We find that for a NCBH the temperature and entropy have the same mass dependence as the Hawking quantities for a Schwarzschild BH.
On Schwarzschild black hole singularity formation
研究通过扩展的正则施瓦茨希尔德度规探讨非奇异配置引力坍缩是否能连续形成施瓦茨希尔德黑洞，发现度规函数在原点出现不连续，暗示时空光滑性破裂，需非连续框架描述奇点形成。
A more rigorous treatment of the Schwarzschild metric by making use of the energy-momentum tensor of a single point particle as source term shows that $g_{00}=-\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(θ(r)-1)\}\exp [2(θ(r)-1)]$ $g_{rr}=\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(θ(r)-1)\}^{-1}$ The existence of a discontinuity at r=0 leads to an infinite repulsive force that will change the ultimate fate of a free fall test particle to a bouncing state.
The extension of Schwarzschild line element to include uniformly accelerated mass
本文推导了匀速及匀加速运动史瓦西物体的时空线元，涵盖加速度沿X方向和XY平面任意方向的情形，并验证了其极限条件，为天体物理中检验粒子测地线计算提供基础。
We study the natural extension of spacetime across Schwarzschild's central singularity and the behavior of the geodesics crossing it. Locality implies that this extension is independent from the future fate of black holes. We argue that this extension is the natural $\hbar\!\to\!0$ limit of the effective quantum geometry inside a black hole, and show that the central region contains causal diamonds with area satisfying Bousso's bound for an entropy that can be as large as Hawking's radiation entropy. This result sheds light on the possibility that Hawking radiation is purified by information c…
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…
By extending the exterior Schwarzschild spacetime in two opposite directions with the Kruskal method, we get an extension which has the same T-X spacetime diagram as has the conventional Kruskal extension, while allowing its regions I and IV to correspond to different directions of the original spacetime. We further extend the exterior Schwarzschild spacetime in all directions and get a 4-dimensional form of the Kruskal extension. The new form of extension includes the conventional one as a part of itself. From the point of view of the 4-dimensional form, region IV of the conventional extensio…
We present an implementation of a ray tracing code in the Schwarzschild metric. We aim at building a numerical code with a correct implementation of both special (aberration, amplification, Doppler) and general (deflection of light, lensing, gravitational redshift) relativistic effects so as to simulate what an observer with arbitrary velocity would see near, or possibly within, the black hole. We also pay some specific attention to perform a satisfactory rendering of stars. Using this code, we then show several unexplored features of the maximal analytical extension of the metric. In particul…
We investigate the topology of Schwarzschild's black hole through the immersion of this space-time in spaces of higher dimension. Through the immersions of Kasner and Fronsdal we calculate the extension of the Schwarzschild's black hole.
The Construction and Application of Penrose Diagrams, with a Focus on the Maximally Analytically Extended Schwarzschild Spacetime
本文通过构造和分析Penrose图，探讨了最大解析扩展Schwarzschild时空的坐标表示差异及几何特性，为物理研究生提供教学材料。
The well-known Schwarzschild black hole was first obtained as a stationary, spherically symmetric solution of the Einstein's vacuum field equations. But until thirty years later, efforts were made for the analytic extension from the exterior area $(r>2GM)$ to the interior one $(r<2GM)$. As a contrast to its maximally extension in the Kruskal coordinates, we provide a comoving coordinate system from the view of the observers freely falling into the black hole in the radial direction. We find an interesting fact that the spatial part in this coordinate system is maximally symmetric $(E_3)$, i.e.…
We present a class of spherically symmetric hypersurfaces in the Kruskal extension of the Schwarzschild space-time. The hypersurfaces have constant negative scalar curvature, so they are hyperboloidal in the regions of space-time which are asymptotically flat.
The precision of optical atomic clocks is approaching a regime where they resolve gravitational time dilation on smaller scales than their own extensions. Hence, an accurate description of quantum clocks has to take their spatial extension into account. In this article, as a first step towards a fully relativistic description of extended quantum clocks, we investigate a quantized version of Einstein's light clock fixed at a constant distance from a large massive object like the Earth. The model consists of a quantum light field in a one-dimensional cavity in Schwarzschild spacetime, where the …
Modified Fronsdal coordinates for maximally extended Schwarzschild spacetime
提出一种新的坐标系，它补充了Kruskal-Szekeres扩展，能够覆盖最大扩展的Schwarzschild流形，并将面积半径表示为新坐标的显式函数，但径向零测地线不再为45度线。
This article presents a streamlined version of the author's original proof of the $C^0$-inextendibility of the maximal analytic Schwarzschild spacetime. Firstly, we deviate from the original proof by using the result, recently established in collaboration with Galloway and Ling, that given a $C^0$-extension of a globally hyperbolic spacetime, one can find a timelike geodesic that leaves this spacetime. This result much simplifies the proof of the inextendibility through the exterior region of the Schwarzschild spacetime. Secondly, we give a more flexible and shorter argument for the inextendib…
Local Well-posedenss of the Bartnik Static Extension Problem near Schwarzschild spheres
本文研究了Bartnik静态扩展问题在Schwarzschild球体附近局部良好性，引入测地坐标框架，将问题转化为椭圆和传输方程组，并利用Bochner可测函数空间解决存在性和唯一性问题。
The very early dismissal of Schwarzschild's original solution and manifold, and the rise, under Schwarzschild's name, of the inequivalent solution and manifold found instead by Hilbert, are scrutinised and commented upon, in the light of the subsequent occurrences. It is reminded that Hilbert's manifold suffers from two defects, that are absent in Schwarzschild's manifold. It does not admit a consistent drawing of the arrow of time, and it allows for an invariant, local, intrinsic singularity in its interior. The former defect is remedied by the change of topology of the extensions proposed by…
Cyclic Kruskal Universe: a quantum-corrected Schwarzschild black hole in unitary unimodular gravity
该研究提出了一种量子修正的Schwarzschild黑洞解，展示了其在单模引力中的物理性质及黑洞-白洞转变机制。
Holographic complexity of the extended Schwarzschild-de Sitter space
本文在扩展的Schwarzschild-de Sitter时空中计算多种全息复杂度猜想，发现其增长行为依赖于拉伸视界位置及探测区域，并揭示余维为一类空曲面的强约束。
We consider a cosmological model based upon a non-canonical noncommutative extension of the Heisenberg-Weyl algebra to address the thermodynamical stability and the singularity problem of both the Schwarzschild and the Kantowski-Sachs black holes. The interior of the black hole is modelled by a noncommutative extension of the Wheeler-DeWitt equation. We compute the temperature and entropy of a Kantowski-Sachs black hole and compare our results with the Hawking values. Again, the noncommutativity in the momenta sector allows us to have a minimum in the potential, which is relevant in order to a…
Higher-dimensional extension of a vacuum-defect wormhole
本文构造一个五维度规，将标准四维 Schwarzschild 度规与新的 $M$ 形变真空缺陷虫洞度规连接起来，并补充相应的四维插值度规。
f(Q) gravity is the extension of symmetric teleparallel general relativity (STGR), in which both curvature and torsion vanish, and gravity is attributed to nonmetricity. This work performs theoretical analyses of static and spherically symmetric solutions with an anisotropic fluid for general f(Q) gravity. We find that the off-diagonal component of the field equation due to a coincident gauge leads to stringent restrictions on the functional form of f(Q) gravity. In addition, although the exact Schwarzschild solution only exists in STGR, we obtain Schwarzschild-like solutions in nontrivial f(Q…
It is shown that the Schwarzschild spacetime can be extended so that the metric becomes analytic at the singularity. The singularity continues to exist, but it is made degenerate and smooth, and the infinities are removed by an appropriate choice of coordinates. A family of analytic extensions is found, and one of these extensions is semi-regular. A degenerate singularity doesn't destroy the topology, and when is semi-regular, it allows the field equations to be rewritten in a form which avoids the infinities, as it was shown elsewhere (arXiv:1105.0201, arXiv:1105.3404). In the new coordinates…
The paper is devoted to a detailed study of the effects of quantum corrections on the chaotic behavior in the dynamics of a (massless) probe particle near the horizon of a generalized Schwarzschild black hole. Two possible origins inducing the modification of black hole metric are considered separately; the noncommutative geometry inspired metric (suggested by Nicolini, Smailagic and Spallucci) and the metric with quantum field theoretic corrections (derived by Donoghue). Our results clearly show that in both cases, the metric extensions favour chaotic behavior, namely chaos is attained for re…
We have developed a method to study the effects of a perturbation to the motion of a test point--like object in a Schwarzschild spacetime. Such a method is the extension of the Lagrangian planetary equations of classical celestial mechanics into the framework of the full theory of general relativity. The method provides a natural approach to account for relativistic effects in the unperturbed problem in an exact way.
The topological structure of Schwarzschild's space-time and its maximal analytic extension are investigated in context of brane-worlds. Using the embedding coordinates, these geometries are seen as different states of the evolution of a single brane-world. Comparing the topologies and the embeddings it is shown that this evolution must be followed by a signature change in the bulk.
Kruskal's extension solves the problem of the arrow of time of the ``Schwarzschild solution'' through combining two Hilbert manifolds by a singular coordinate transformation. We discuss the implications for the singularity problem and the definition of the mass point. The analogy set by Rindler between the Kruskal metric and the Minkowski spacetime is investigated anew. The question is answered, whether this analogy is limited to a similarity of the chosen "Bildräume'', or can be given a deeper, intrinsic meaning. The conclusion is reached by observing a usually neglected difference: the left …
Local Well-Posedness for the Bartnik Stationary Extension Problem near Schwarzschild Spheres
本文研究了Bartnik静止扩展问题在Schwarzschild球体附近局部适定性，通过解耦线性方程和建立边界值问题，证明了在特定条件下该问题的解存在且唯一。
The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-reg…
Thanks to unparalleled near-horizon images of the shadows of Messier 87* (M87*) and Sagittarius A* (Sgr A*) delivered by the Event Horizon Telescope (EHT), two amazing windows opened up to us for the strong-field test of the gravity theories as well as fundamental physics. Information recently published from EHT about the Sgr A*'s shadow lets us have a novel possibility of exploration of Lorentz symmetry violation (LSV) within the Standard-Model Extension (SME) framework. Despite the agreement between the shadow image of Sgr A* and the prediction of the general theory of relativity, there is s…
Non-Schwarzschild black-hole metric in four dimensional higher derivative gravity: analytical approximation
本文用连分数参数化方法将四维高阶导数引力中的非史瓦西黑洞数值解转化为全空间精确的解析近似形式，使其可像精确解一样用于研究黑洞热力学、稳定性等性质。
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