Since its first introduction, the Schwarzschild metric has been written in various coordinate systems. This has been done primarily to understand the nature of the coordinate singularity at the event horizon. However, very often, the mathematics of a coordinate system does not provide a clear physical interpretation. In Schwarzschild's original work, the origin of the radial coordinate was at the event horizon. Hence, there was no black hole. The generally accepted current definition of the radial coordinate has an origin beyond the horizon. This necessitates the discussion of black holes. Here, some well-known and some not-so-well-known coordinate systems will be visited in search of a physical interpretation. It will be noted that they all agree at large radial distances. However, the location of the origin of the radial coordinate can be different for different systems. Some "natural" coordinate systems can exclude the entirety of the interior of a black hole from the physical manifold. Such coordinate systems maybe physically more acceptable as they avoid the issue of the metric signature change across the horizon. Mathematically, the metric signature alone can distinguish time from space coordinates. Hence, coordinate systems that include the interior of the black hole need to switch the physical meanings of time and one of the space coordinates.
In this paper we construct the Fermi coordinates along any arbitrary line in simple analytical way without use the orthogonal frames and their parallel transport. In this manner we extend the Eddington approach to the construction of the Fermi metric in terms of the Riemann tensor. In the second part of the present article we show how the proposed approach works practically by applying it for deriving the Fermi coordinates for the static observer in the Schwarzschild spacetime.
Action-angle coordinates are a tool commonly used in celestial mechanics to systematically parametrize and store general solutions of the equations of motion of astrophysical bodies. I perturbatively construct action-angle coordinates for bound test particle motion in static, spherically symmetric space-times using a post-circular expansion. Then I specialise the expressions to the motion in the gravitational fields of Schwarzschild black holes and give explicit formulas for the Hamiltonian to the 10th power in the radial action (20th power in eccentricity), and the transformation to angle coo…
The Kottler spacetime in isotropic coordinates is known where the metric is time-dependent. In this paper, the Kottler spacetime is given in isotropic static coordinates (i.e., the metric components are time-independent). The metric is found in terms of the Jacobian elliptic functions through coordinate transformations from the Schwarzschild-(anti-)de Sitter metric. In canonical coordinates, it is known that the unparameterized spatially projected null geodesics of the Kottler and Schwarzschild spacetimes coincide. We show that in isotropic static coordinates, the refractive indices of Kottler…
Schwarzschild-de Sitter Metric and Inertial Beltrami Coordinates
在考虑坐标条件下，得到了具有宇宙学常数$Λ$的爱因斯坦场方程的Schwarzschild-Beltrami-deSitter（S-BdS）度规解.本文对de Sitter不变狭义相对论（dS-SR）和de Sitter广义相对论（dS-GR，或GR with a $Λ$）作了简要的评述。Beltrami度规$B_{μν}$为dS时空提供了惯性参考系。通过对1918年以来文献中的Schwarzschild-de Sitter（S-dS）度规$g_{μν}^{(M)}$的考察，发现已有的S-dS度规$g_{μν}^{(M)}$描述的是引力和惯性力的混合效应，而不是dS-GR中“太阳质量”$g_{μν}^{(M)}|_{M\rightarrow 0}=B_{μν}$引起的纯引力效应。这样，我们找到了dS-GR的S-BdS解，写在惯性Beltrami坐标中。这是S-dS度量的一种新形式。讨论了它的物理意义和可能的应用。
We present analogue models for the Schwarzschild and Reissner-Nordström (RN) spacetimes based on unidirectional hydrodynamic flows. We show that, with appropriate coordinate transformations, sound waves in a moving fluid propagate as scalar fields at the equatorial sections of Schwarzschild and RN black holes. The coordinate associated with the direction of the flow plays the role of the radial coordinate in standard Schwarzschild/RN coordinates, while the transversal spatial coordinate is related to the angular coordinate on each equatorial slice. The magnitude of the flow velocity is related…
In this report, we discuss the dynamics of photons in a five-dimensional Schwarzschild black hole. We find that the corresponding Eddington-Finkelstein-like coordinate admits two time-coordinate singularities for a particular condition. This condition is due to the existence of a non-zero angular momentum related to the extra spatial dimension. By introducing a new Regge-Wheeler tortoise coordinate, both singularities can be removed from the Eddington-Finkelstein-like coordinate. It is identified that the additional singularity is related to the photon sphere in a hyperplane, and we analyze it…
The Construction and Application of Penrose Diagrams, with a Focus on the Maximally Analytically Extended Schwarzschild Spacetime
Argelander Institute for Astronomy - University of Bonn（波恩大学阿尔兰德天文研究所）；Lichtenberg Group for History and Philosophy of Physics - Faculty of Philosophy, University of Bonn（波恩大学哲学学院利希滕贝格物理历史与哲学研究组）
Matching Radial Geodesics in Two Schwarzschild Spacetimes (e.g. Black-to-White Hole Transition) or Schwarzschild and de Sitter Spacetimes (e.g. Interior of a Non-singular Black Hole)
The Painleve-Gullstrand coordinates are extended to describe the black hole in the cosmological environment: the Schwarzschild-de-Sitter black hole, which has two horizons. The extension is made using the Arnowitt-Deser-Misner formalism. In this extension, which describes the metric in the whole range of radial coordinates $0<r < \infty$, there is the point $r=r_0$ at which the shift function (velocity) changes sign. At this point the observer is at rest, while the observers at $r<r_0$ are free falling to the black hole and the observers at $r>r_0$ are free falling towards the cosmological hor…
