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Über das Gravitationsfeld eines Massenpunktes nach der EINSTEINschen Theorie
... The Einstein field equations find their quintessential solution in the Schwarzschild metric, which delineates the spacetime curvature around a static, spherically symmetric black hole devoid of charge. While the external Schwarzschild geometry has been thoroughly analyzed and validated by observational evidence [1], the internal structure presents a more nuanced picture. The interior solution, formulated through the Tolman-Oppenheimer-Volkoff (TOV) equations [2,3], models the black hole's internal composition as a perfect fluid sphere under hydrostatic equilibrium. ...
... In pure general relativity, the Schwarzschild metric [1] is the unique spherically symmetric black hole solution without a cosmological constant: ...
... In general spacetimes, the natural basis associated with the coordinate system may not be the most suitable for straightforward application of the energy condition inequalities. To address this, we introduce a natural orthonormal basis of one-forms, (0) , (1) , (2) , (3) , in the spacetime described by metric (47), following the approach outlined in [39]. This orthonormal basis, defined as: ...
We present a novel approach for the construction of interior solutions for the Kerr metric, extending J. Ovalle’s foundational work through ellipsoidal coordinate transformations. By deriving a physically plausible interior solution that smoothly matches the Kerr exterior metric, we analyze the energy conditions across various rotation parameters. Our findings reveal anisotropic fluid properties and energy condition behaviors in specific space-time regions, providing insights into the strong-field regime of rotating black holes. The proposed solution offers a more realistic description of rotating black hole interiors, with implications for understanding compact astrophysical objects.
... The first discovered non-trivial exact solution for the Einstein equations was the Schwarzschild spacetime [4]. For every M > 0, the exterior region of the Schwarzschild spacetime of mass M may be covered by so-called "Schwarzschild" coordinates (t , r, θ, φ) ∈ R × (2M , ∞) × S 2 : ...
... One natural question left open by all of these various results on trapped surface formation, is whether anything can be said about the short pulse spacetimes as the parameter δ → 0. This question has been addressed in [53] where the authors employ their renormalization techniques to understand the connection between certain types of high-frequency limits of the Einstein equations and null dust solutions 4 . In particular, they show that one may take the δ → 0 (weak) limit of the short pulse spacetimes and obtain a solution to the Einstein-null dust system which also exhibits the formation of a trapped surface. ...
The weak cosmic censorship conjecture posits that, generically, all singularities in General Relativity arising from regular asymptotically flat initial data should have a complete future null infinity. While this conjecture remains wide open, it has inspired many mathematical works concerning topics such as trapped surface formation and the construction of naked singularities. In this article we will review some of these works and attempt to emphasize their interconnectedness.
... The usual (Schwarzschild [11]) catchy picture of a funnel-like space deformation (which seems to suggest an inevitable fall of all objects towards the source, Fig.2) next paragraph) is replaced by the standard familiar one i.e. the distance between two concentric spheres is simply the Euclidean one Fig.1). ...
... The Schwarzschild [11] solution of GR is given by ...
... In the Painleve-Gullstrand (PG) coordinate frame, the Schwarzschild black hole solution [1,2] has a form such that the description of matter resembles the description of liquid motion [3,4]. Such similar reference frames also apply to the charged (Reissner-Nordstrom) and to the rotated (Kerr) black holes [5,6]. ...
... In the present paper, we rely on the description of the black hole in Einstein's theory of gravity, which enables the use of the Painleve-Gullstrand reference frame, where the metric has the especially simple form of Equation (1). Quantum fields may be considered on the same grounds in a similar reference frame for a rotating black hole (given in [5]). ...
In the Painleve--Gullstrand (PG) reference frame, the description of elementary particles in the background of a black hole (BH) is similar to the description of non-relativistic matter falling toward the BH center. The velocity of the fall depends on the distance to the center, and it surpasses the speed of light inside the horizon.~Another analogy to non-relativistic physics appears in the description of the massless fermionic particle. Its Hamiltonian inside the BH, when written in the PG reference frame, is identical to the Hamiltonian of the electronic quasiparticles in type~II Weyl semimetals (WSII) that reside in the vicinity of a type~II Weyl point. When these materials are in the equilibrium state, the type II Weyl point becomes the crossing point of the two pieces of the Fermi surface called Fermi pockets. {It was previously stated} that there should be a Fermi surface inside a black hole in equilibrium. In real materials, type II Weyl points come in pairs, and the descriptions of the quasiparticles in their vicinities are, to a certain extent, inverse. Namely, the directions of their velocities are opposite. In line with the mentioned analogy, we propose the hypothesis that inside the equilibrium BH there exist low-energy excitations moving toward the exterior of the BH. These excitations are able to escape from the BH, unlike ordinary matter that falls to its center. The important consequences to the quantum theory of black holes follow.
... Black holes are sufficiently massive objects that create "a region of no escape", i.e., a region from where no object can escape, not even light [1]. The first and simplest black hole solution was obtained by Schwarzschild in 1916 by solving the Einstein fields equations for a static and spherically symmetric gravitating body [2]. Since then, black holes have attracted the attention of scientists and have been extensively studied. ...
... As should be expected, in the limit k = 0, the solution is reduced to the Letelier solution with quintessence [11], and if k = 0 and a = 0, we recover the Kiselev solution [25]. If k = 0 and α = 0, we obtain the Letelier solution [6] and, finally, we obtain the Schwarzschild solution [2] for k = 0, a = 0, and α = 0. It is worth calling attention to the fact that, for r >> k, the obtained solution reduces to the Reissner-Nordström with cloud of strings and quintessence as should be expected [67]. ...
We obtain exact black hole solutions to Einstein gravity coupled with a nonlinear electrodynamics field, in the presence of a cloud of strings and quintessence, as sources. The solutions have four parameters, namely m, k, a, and α, corresponding to the physical mass of the black hole, the nonlinear charge of a self-gravitating magnetic field, the cloud of strings, and the intensity of the quintessential fluid. The consequences of these sources on the regularity or singularity of the solutions, on their horizons, as well as on the energy conditions, are discussed. We study some aspects concerning the thermodynamics of the black hole, by taking into account the mass, Hawking temperature, and heat capacity and show how these quantities depend on the presence of the cloud of strings and quintessence, in the scenario considered.
... and k = 1 − AB = 0.999999920123 (10) are determined by the basic parameters A and B, as needed. On the other hand, A 0 and B 0 can be related to the perihelion (a) and aphelion (b) of the trajectory given by (8): 3 An important point is that (8) is, in fact, the exact solution of Binet's equation for a central potential of a different (nonrelativistic) type. ...
The nonlinear trajectory equation (Binet's equation) for a particle in a relativistic force field can only be solved numerically or, alternatively, by using a perturbational solution scheme. The lattter approach has been successfully applied by Albert Einstein in 1915, to deduce the celebrated formula which explains the anomalous precession of the perihelion of Mercury. In this article, the Binet's equation for Mercury is solved numerically to a high degree of accuracy (16 decimal digits). This is a necessary comparison basis for the main goal of this work-to deduce a simple analytic formula which perfectly reproduces the real relativistic trajectory. Several analytical models are proposed, and the main goal has been indeed achieved. Moreover, the fitting parameters for the model D described in Section 3 can be obtained independent of the solution of the Binet's equation. Thus, we can say that the highly accurate relativistic trajectory (the largest discrepancy being about 30 cm) can be obtained without actually solving the nonlinear differential equation for this trajectory.
... The Black Hole Information Paradox exemplifies this discord between QM and GR [1]. Classically, black holes are regions from which nothing can escape after crossing the event horizon, not even light [2,3]. Black holes appear to be defined solely by their mass, charge, and angular momentum, with no trace of the information contained in the matter that formed them, as stated in the no-hair theorem [4]. ...
We present the Quantum Memory Matrix (QMM) hypothesis, which addresses the longstanding Black Hole Information Paradox rooted in the apparent conflict between Quantum Mechanics (QM) and General Relativity (GR). This paradox raises the question of how information is preserved during black hole formation and evaporation, given that Hawking radiation appears to result in information loss, challenging unitarity in quantum mechanics. The QMM hypothesis proposes that space-time itself acts as a dynamic quantum information reservoir, with quantum imprints encoding information about quantum states and interactions directly into the fabric of space-time at the Planck scale. By defining a quantized model of space-time and mechanisms for information encoding and retrieval, QMM aims to conserve information in a manner consistent with unitarity during black hole processes. We develop a mathematical framework that includes space-time quantization, definitions of quantum imprints, and interactions that modify quantum state evolution within this structure. Explicit expressions for the interaction Hamiltonians are provided, demonstrating unitarity preservation in the combined system of quantum fields and the QMM. This hypothesis is compared with existing theories, including the holographic principle, black hole complementarity, and loop quantum gravity, noting its distinctions and examining its limitations. Finally, we discuss observable implications of QMM, suggesting pathways for experimental evaluation, such as potential deviations from thermality in Hawking radiation and their effects on gravitational wave signals. The QMM hypothesis aims to provide a pathway towards resolving the Black Hole Information Paradox while contributing to broader discussions in quantum gravity and cosmology.
... Even if built on Euclidean space and absolute time, it represents a non-Minkowskian space-time, due to the crossed term. The P-G metric is equivalent to the Schwarzschild [17] metric , the relation being given by a change of the time variable (see SEction4.2). ...
The physical principles at the basis of an "elementary derivation" of the General Relativity (GR) effects in a static centrally symmetric field are reexamined. We propose a theoretical framework in which all the GR results follow from the EP, local SR and Newton law in intrinsic coordinates.
... It also serves as a theoretical laboratory to inspect the fundamental behaviour of the Nature. The first BH solution, the Schwarzschild solution [1], came as a unique static vacuum solution to Einstein's field equations, G ab = 8πT ab = 0. This solution gives the 4D metric ds 2 = −F (r)dt 2 + F −1 (r)dr 2 + r 2 dΩ 2 , with F (r) = (1 − 2m/r), for a black hole of ADM mass m, and dΩ 2 is the line element on a 2-sphere of unit radius. ...
Classical general relativity predicts a singularity at the center of a black hole, where known laws of physics break down. This suggests the existence of deeper, yet unknown principles of Nature. Among various theoretical possibilities, one of the most promising proposals is a transition to a de Sitter phase at high curvatures near the black hole center. This transition, originally proposed by Gliner and Sakharov, ensures the regularity of metric coefficients and avoids the singularity. In search for such a regular black hole solution with finite curvature scalar, we propose a metric function that exhibits a de Sitter-like core in the central region. An appealing feature of this metric is the existence of a single event horizon resembling the Schwarzschild black hole. Furthermore, the entire spacetime geometry is determined by the black hole mass alone, in agreement with the Isarel-Carter theorem for a charge-less, non-rotating black hole. To determine the gravitational action consistent with such a solution, we consider a general Lagrangian density in place of the Einstein-Hilbert action. By numerically solving the resulting field equation, we find that, in addition to the Einstein-Hilbert term, a Pad\'e approximant in the Ricci scalar can produce such regular black hole solutions. To assess the physical viability of these black hole solutions, we verify that the proposed metric satisfies the principal energy conditions, namely, the dominant energy condition, weak energy condition, and null energy condition, throughout the entire spacetime. Furthermore, in agreement with Zaslavskii's regularity criterion, the metric satisfies the strong energy condition in the range , where is the event horizon.
... Where the first term is the same with the Schwarzschild solution [86] and the remaining terms are the HL corrections depending upon the parameter ω. Now, utilizing the Eqs. ...
We study the structure and basic physical properties of non-rotating dark energy stars in Hořava–Lifshitz (HL) gravity. The interior of proposed stellar structure is made of isotropic matter obeys extended Chaplygin gas EoS. The structure equations representing the state of hydrostatic equilibrium i.e., generalize TOV equation in HL gravity is numerically solved by using chosen realistic EoS. Next, we investigate the deviation of physical features of dark energy stars in HL gravity as compared with general relativity (GR). Such investigation is depicted by varying a parameter ω , whereas for ω → ∞ HL coincide with GR. As a result, we find that necessary features of our stellar structure are significantly affected by ω in HL gravity specifically on the estimation of the maximum mass and corresponding predicted radius of the star. In conclusion, we can predict the existence of heavier massive dark energy stars in the context of HL gravity as compared with GR with not collapsing into a black hole. Moreover, we investigate the stability of our proposed stellar system. By integrating the modified perturbations equations in support of suitable boundary conditions at the center and the surface of the stellar object, we evaluate the frequencies and eigenfunctions corresponding to six lowest excited modes. Finally, we find that physically viable and stable dark energy stars can be successfully discussed in HL gravity by this study.
... A key tenet of this model is that time is treated as a fully independent scalar magnitude, orthogonal to space and unaffected by gravity. This opposes relativity's foundational principle, which allows time to be bent by gravitational fields [2,5]. In contrast, this framework removes the "dead time" introduced by light propagation delays effectively projecting objective events onto virtual time surfaces. ...
This paper introduces a conceptual 4-dimensional space time model that departs fundamentally from Einstein’s relativity. Unlike observer dependent systems, this framework emphasizes true simultaneity, distinguishing objective reality from perceived events. It defines a 4D orthogonal vector coordinate system, combining 3D Cartesian space with a fourth dimension of virtual time surfaces, which represent instantaneous temporal slices across space. These time surfaces are curved within an otherwise flat 3D space, forming a dynamic, evolving “present” enclosed at the boundary of the space time system. Time is modeled as an independent scalar magnitude, making it fully orthogonal to spatial dimensions and immune to external influences like gravity. This redefines the role of time as a pure sequence of events, without the curvature or distortion proposed in general relativity. By projecting 3D space onto this new virtual topology, the model offers a unique geometric view of space time. It challenges conventional gravitational space time interactions and repositions time as an unlinked, autonomous coordinate within a unified but orthogonal framework.
... Later, general relativity (GR) was introduced, defying the absoluteness of spacetime, as it becomes curved in the presence of, now equaled, energy-matter. An example of an unobvious feature of GR is black holes [15], inaccessible regions of spacetime with infinite curvature, now well evidenced [16]. ...
The measurement problem in quantum mechanics (QM) is related to the inability to include learning about the properties of a quantum system by an agent in the formalism of quantum theory. It includes questions about the physical processes behind the measurement, uniqueness, and randomness of obtained outcomes and an ontic or epistemic role of the state. These issues have triggered various interpretations of quantum theory. They vary from refusing any connection between physical reality and a measurement process to insisting that a collapse of the wave-function is real and possibly involves consciousness. On the other hand, the actual mechanism of a measurement is not extensively discussed in these interpretations. This essay attempts to investigate the quantum measurement problem from the position of the scientific consensus. We begin with a short overview of the development of sensing in living organisms. This is performed for the purpose of stressing the relation between reality and our experience. We then briefly present different approaches to the measurement problem in chosen interpretations. We then state three philosophical assumptions for further consideration and present a decomposition of the measurement act into four stages: transformation, conversion, amplification and broadcasting, and, finally, perception. Each of these stages provides an intuition about the physical processes contributing to it. These conclusions are then used in a discussion about, e.g., objectivity, the implausibility of reversing a measurement, or the epistemic status of the wave-function. Finally, we argue that those in favor of some of the most popular interpretations can find an overlap between their beliefs and the consequences of considerations presented here.
... Despite being much less known than the ⇤-CDM model, such cosmological models continue to be actively discussed to this day [2][3][4][5][6][7][8][9][10][11]. For example, the Schwarzschild [12,13] radius, given by r s = 2GM c 2 , is well known to be identical to the Hubble radius when applying the Friedmann [14] critical mass to the universe. The critical Friedmann mass is: ...
We calculate the viscous force of the Hubble sphere in a black hole RH = ct universe. We then prove that for the viscous force to be equal to the gravitational force of the Hubble sphere, the speed of the fluid must be the speed of light c. This supports the idea that the Hubble sphere vacuum energy is a kind of superfluid that is indeed a form of energy, which, in turn, can be interpreted as dark energy. Our analysis demonstrates that the superfluid, which is likely dark energy, must move at a velocity of c to counteract the gravitational force of the entire cosmos and prevent a black hole universe from collapsing into a central singularity. For the universe to undergo accelerating expansion, this velocity must be greater than c, which brings us closer to the Λ-CDM model, where we must account for the expansion of space itself.
... BHs are incredibly dense celestial bodies with intense gravitational attraction, anticipated by general relativity (GR). BHs gained significant attention when Karl Schwarzschild found the first exact solution of Einstein field equations in 1916 [1]. BHs give rise to various astronomical phenomena, such as GWs [2], oscillations in spacetime caused by accelerating masses [3], gravitational lensing [4], shadows [5], etc. ...
This article explores the motion of massive particles in the gravitational field of a modified gravity (MOG) black hole (BH), characterized by the parameter α. Using the Hamiltonian formalism, the geodesic equations and the effective potential governing particle trajectories are derived. Key features, including the innermost stable circular orbit (ISCO) and the innermost bound circular orbit (IBCO), are analyzed, revealing their dependence on the particle's energy , angular momentum, and the MOG parameter. In the extremal case, where α = −1, the event horizon merges with the Cauchy horizon, forming a distinctive BH configuration. Numerical methods are employed to compute periodic orbits in this spacetime, with a comparison drawn to the Schwarzschild BH. The findings indicate that for α > 0, periodic orbits around Schwarzschild-MOG BH exhibit lower energy requirements than those in Schwarzschild space-time, whereas for −1 < α < 0, the energy requirements are higher. Precessing orbits near periodic trajectories are also examined, offering insights into their complex dynamical behavior. Finally, the gravitational wave (GW) radiation from the periodic orbits of a test particle around the Schwarzschild-MOG BH is examined, generating intricate waveforms that provide insights into the gravitational structure of the system.
... Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations [1] just a few months after their publication in 1915 [2]. However, it took several decades for the scientific community to fully understand the physical implications of this solution [3]. ...
The study of regular black holes and black hole mimickers as alternatives to standard black holes has recently gained significant attention, driven both by the need to extend general relativity to describe black hole interiors, and by recent advances in observational technologies. Despite considerable progress in this field, significant challenges remain in identifying and characterizing physically well-motivated classes of regular black holes and black hole mimickers. This report provides an overview of these challenges, and outlines some of the promising research directions -- as discussed during a week-long focus programme held at the Institute for Fundamental Physics of the Universe (IFPU) in Trieste from November 11th to 15th, 2024.
... BHs are incredibly dense celestial bodies with intense gravitational attraction, anticipated by general relativity (GR). BHs gained significant attention when Karl Schwarzschild found the first exact solution of Einstein field equations in 1916 [1]. BHs give rise to various astronomical phenomena, such as GWs [2], oscillations in spacetime caused by accelerating masses [3], gravitational lensing [4], shadows [5], etc. ...
This article explores the motion of massive particles in the gravitational field of a modified gravity (MOG) black hole (BH), characterized by the parameter . Using the Hamiltonian formalism, the geodesic equations and the effective potential governing particle trajectories are derived. Key features, including the innermost stable circular orbit (ISCO) and the innermost bound circular orbit (IBCO), are analyzed, revealing their dependence on the particle's energy, angular momentum, and the MOG parameter. In the extremal case, where , the event horizon merges with the Cauchy horizon, forming a distinctive BH configuration. Numerical methods are employed to compute periodic orbits in this spacetime, with a comparison drawn to the Schwarzschild BH. The findings indicate that for , periodic orbits around Schwarzschild-MOG BH exhibit lower energy requirements than those in Schwarzschild spacetime, whereas for , the energy requirements are higher. Precessing orbits near periodic trajectories are also examined, offering insights into their complex dynamical behavior. Finally, the gravitational wave (GW) radiation from the periodic orbits of a test particle around the Schwarzschild-MOG BH is examined, generating intricate waveforms that provide insights into the gravitational structure of the system.
... The Lorentzian manifold (M ext , g ext ) is called the exterior of a d + 1 dimensional Schwarzschild black hole with mass m. It was introduced in 1916 in [19] by Schwarzschild for d = 3, where he also showed that it is a solution to the vaccum Einstein equations Ric(g) = 0. In 1963, Tangherlini generalised Schwarzschild's metric to higher dimensions, cf. ...
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-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.
... 24 CPAE), less than a month after he had published his Mercury paper, Karl Schwarzschild wrote to Einstein that he had found the exact solution that Einstein's solution to the linearized field equations approximated. Schwarzschild [1916b] had found the unique spherically symmetric, static and asymptotically flat exact solution to Einstein's vacuum field equations. 53 For a mathematician of Hermann Weyl's calibre, the discovery of the Schwarzschild solution must have offered an immediate and clearly formulated challenge. ...
In this paper I describe the genesis of Einstein's early work on the problem of motion in general relativity (GR): the question of whether the motion of matter subject to gravity can be derived directly from the Einstein field equations. In addressing this question, Einstein himself always preferred the vacuum approach to the problem: the attempt to derive geodesic motion of matter from the vacuum Einstein equations. The paper first investigates why Einstein was so skeptical of the energy-momentum tensor and its role in GR. Drawing on hitherto unknown correspondence between Einstein and George Yuri Rainich, I then show step by step how his work on the vacuum approach came about, and how his quest for a unified field theory informed his interpretation of GR. I show that Einstein saw GR as a hybrid theory from very early on: fundamental and correct as far as gravity was concerned but phenomenological and effective in how it accounted for matter. As a result, Einstein saw energy-momentum tensors and singularities in GR as placeholders for a theory of matter not yet delivered. The reason he preferred singularities was that he hoped that their mathematical treatment would give a hint as to the sought after theory of matter, a theory that would do justice to quantum features of matter.
... The latter is known also as Shapiro time delay effect (or gravitational time delay effect) and was proposed in 1964 [1]. Upon assuming a static and spherically symmetric body, its exterior gravitational field can be described by the Schwarzschild metric [2], and this makes it possible to measure the increase in time delay between the transmission of an electromagnetic signal towards either of the inner planets and the detection of the echoes. Such a phenomenon is mainly due to the well know relativistic result according to which the speed of light depends on the strength of the gravitational field encountered along its path, whereas the contribution arising from the deflection of its trajectory is negligible, being of order O(G 2 M 2 /c 6 ) (see Eq. (1.4)). ...
We examine quantum corrections of time delay arising in the gravitational field of a spinning oblate source. Low-energy quantum effects occurring in Kerr geometry are derived within a framework where general relativity is fully seen as an effective field theory. By employing such a pattern, gravitational radiative modifications of Kerr metric are derived from the energy-momentum tensor of the source, which at lowest order in the fields is modelled as a point mass. Therefore, in order to describe a quantum corrected version of time delay in the case in which the source body has a finite extension, we introduce a hybrid scheme where quantum fluctuations affect only the monopole term occurring in the multipole expansion of the Newtonian potential. The predicted quantum deviation from the corresponding classical value turns out to be too small to be detected in the next future, showing that new models should be examined in order to test low-energy quantum gravity within the solar system.
... Einstein [44] solved that problem by successive approximations. The exact vacuum solution was obtained one year later by Schwarzschild [77] and, independently, Droste [78]; their results are virtually indistinguishable from those of Einstein. Relevant simplifications were introduced one year later by Weyl [79], who used cartesian coordinates instead of spherical ones, and worked on the basis of the action principle instead of recurring to the differential equations for the field g. ...
The present Editorial introduces the Special Issue dedicated by the journal Universe to the General Theory of Relativity, the beautiful theory of gravitation of Einstein, a century after its birth. It reviews some of its key features in a historical perspective, and, in welcoming distinguished researchers from all over the world to contribute it, some of the main topics at the forefront of the current research are outlined.
... on the WEB.3 Here we refer to the so-called gravitoelectric[3], static component of the gravitational field[28] yielding well known general relativistic phenomena like the geodetic ...
LETSGO (LEnse-Thirring Sun-Geo Orbiter) is a proposed space-based mission involving the use of a spacecraft moving along a highly eccentric heliocentric orbit perpendicular to the ecliptic. It aims to accurately measure some important physical properties of the Sun and to test some post-Newtonian features of its gravitational field by continuously monitoring the Earth-probe range. Preliminary sensitivity analyses show that, by assuming a cm-level accuracy in ranging to the spacecraft, it would be possible to detect, in principle, the Lense-Thirring effect on it at a 10^-3-10^-4 level over a timescale of 2 yr, while the larger Schwarzschild component of the solar gravitational field may be sensed with a relative accuracy of about 10^-8-10^-9 during the same temporal interval. The competing range perturbation due to the non-sphericity of the Sun would be a source of systematic error, but it turns out that all the three dynamical features of motion examined affect the Earth-probe range in different ways, allowing for a separation in data analyses. The high eccentricity would help in reducing the impact of the non-gravitational perturbations whose impact would certainly be severe when LETSGO would approach the Sun at just a few solar radii. Further studies should be devoted to investigate both the consequences of the non-conservative forces and the actual measurability of the effects of interest by means of extensive numerical data simulations, parameter estimations and covariance analyses. Also an alternative, fly-by configuration is worth of consideration.
... The visual distortion that will be described here would be caused by gravitation in the Schwarzschild metric. 14 Einstein's general relativity 15 is not the only gravitational theory that admits the Schwarzschild metric as an exterior solution for a spherically symmetric, non-rotating gravitational field, but it is the preferred theory, and the theory that will be assumed implicitly here. The Schwarzschild metric is ds 2 = −(1 − R S /r)c 2 dt 2 + (1 − R S /r) −1 dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 . ...
The visual distortion effects visible to an observer traveling around and descending to the surface of an extremely compact star are described. Specifically, trips to a ``normal" neutron star, a black hole, and an ultracompact neutron star with extremely high surface gravity, are described. Concepts such as multiple imaging, red- and blue-shifting, conservation of surface brightness, the photon sphere, and the existence of multiple Einstein rings are discussed in terms of what the viewer would see. Computer generated, general relativistically accurate illustrations highlighting the distortion effects are presented and discussed. A short movie (VHS) depicting many of these effects is available to those interested free of charge.
... 443) in Kip Thorne's 1994 book on the history of black holes. [63] However, GR1916's adoption of the SR shift relationships, expressed via the Schwarzschild solution, [80] [81] makes relative horizons impossible and absolute horizons compulsory. [82] [83] Gravitational theories incorporating the SR relationships are not compatible with quantum mechanics, and since the incompatibility exists at the level of the Doppler equations, the two blocks of theory cannot coexist as part of a larger system (such as a theory of quantum gravity) without contradictions. ...
This is the reformatted, 2025 version of the article that was on Wikipedia between September and November 2024.
It gives an overview of the deep and unresolved problems that exist in Einstein's failed 1916 attempt to construct a theory around a combination of the general principle of relativity and special relativity, and includes many useful contemporary quotes and references.
... The Black Hole Information Paradox exemplifies this discord between QM and GR [1]. Classically, black holes are regions from which nothing can escape after crossing the event horizon, not even light [2,3]. Black holes appear to be defined solely by their mass, charge, and angular momentum, with no trace of the information contained in the matter that formed them, as stated in the no-hair theorem [4]. ...
We present the Quantum Memory Matrix (QMM) hypothesis, which addresses the longstanding Black Hole Information Paradox rooted in the apparent conflict between Quantum Mechanics (QM) and General Relativity (GR). This paradox raises the question of how information is preserved during black hole formation and evaporation, given that Hawking radiation appears to result in information loss, challenging unitarity in quantum mechanics. The QMM hypothesis proposes that space–time itself acts as a dynamic quantum information reservoir, with quantum imprints encoding information about quantum states and interactions directly into the fabric of space–time at the Planck scale. By defining a quantized model of space–time and mechanisms for information encoding and retrieval, QMM aims to conserve information in a manner consistent with unitarity during black hole processes. We develop a mathematical framework that includes space–time quantization, definitions of quantum imprints, and interactions that modify quantum state evolution within this structure. Explicit expressions for the interaction Hamiltonians are provided, demonstrating unitarity preservation in the combined system of quantum fields and the QMM. This hypothesis is compared with existing theories, including the holographic principle, black hole complementarity, and loop quantum gravity, noting its distinctions and examining its limitations. Finally, we discuss observable implications of QMM, suggesting pathways for experimental evaluation, such as potential deviations from thermality in Hawking radiation and their effects on gravitational wave signals. The QMM hypothesis aims to provide a pathway towards resolving the Black Hole Information Paradox while contributing to broader discussions in quantum gravity and cosmology.
... 1. Schwarzschild's solution [22] (B = 0, r s ̸ = 0), which describes an isolated spherically symmetric black hole with the event horizon ...
This paper investigates the trajectories of neutral particles in the Schwarzschild-Melvin spacetime. After reduction by cyclic coordinates this problem reduces to investigating a two-degree-of-freedom Hamiltonian system that has no additional integral. A classification of regions of possible motion of a particle is performed according to the values of the momentum and energy integrals. Bifurcations of periodic solutions of the reduced system are analyzed using a Poincare map.
... Flat spacetime is an approximation of curved spacetime like Earth's surface appears flat to us even though it is indeed curved as it is the surface of the spherical-shaped Earth. In 1916, the first exact solution to the Einstein field equations of the theory of general relativity was found by German physicist and astronomer Karl Schwarzschild (Karl Schwarzschild 1916). The solution assumes that the central mass is charge neutral (no electric charge) and non-rotating (no angular momentum). ...
The paper reviews the theoretical formulae of different astrophysical conditions to describe spacetime and connects theory with observational evidence. The spacetime is governed by gravity, which is well-explained by the theory of General Relativity. The paper starts from the simplest version of spacetime, that is, flat spacetime which has no gravitational influence. This spacetime is described by the Minkowski metric. Then the paper goes to the properties of spacetime in the presence of gravity, which creates curved spacetime. The Schwarzschild metric defines this spacetime. Although these phenomena are well-established by experimental proof, the most intricate characteristic of spacetime has not been discovered until very recently. That is the spacetime around a rotating massive body. The paper will present the mathematical expressions for describing such spacetime, the Kerr metric, and finally will end with the observational evidence of the effect of a spinning heavy body around it. Some particular exotic effects such as “frame-dragging" and “ergosphere" will be presented in brief.
... A radical view in which gravity is equivalent to a space acceleration. Later in 1916, Karl Schwarzschild [2] [3] took a further step presenting an intuitive equivalence to an escaping speed; i.e., a simpler expression of Einstein´s general relativity for the case of a non-charged, non-rotating spherically symmetric body. Presenting a compacted mass confined to the smallest radius Rs forming an event horizon at its surface. ...
This paper proposes a novel approach to gravitational potential energy, deriving it from impediments to the flow of space. Building on a previous theory, the study demonstrates that gravitational energy is sourced from interactions that prevent objects from following space flow. It further explores time dilation by recognizing kinetic and gravitational contributions, presenting an alternative to conventional negative energy interpretation. In 1760, Lagrange included a minus sign to the gravitational potential energy to compensate for this misunderstanding. A time dilation due to an increment in energy is observed in nature, i.e., a time dilation in cell phones which makes GPS accurate. This theory is based on relativistic and quantum contribution, taking a step forward in explaining the gravitational effect and posits a universe of positive energy, challenging many-world hypotheses and offering an experimental approach to test this theory considering gravitational and kinetic time dilation during free-fall.
Standard General Relativity, based on the analysis of the Schwarzschild solution for a non-rotating, uncharged black hole, predicts extreme spacetime distortion inside the event horizon radius (r s). A detailed analysis of this metric in its native Schwarzschild coordinates reveals a fundamental breakdown of standard spatial properties: the radial coordinate r assumes a time-like character, preventing static positions or outward motion , and the standard spatial volume element becomes imaginary, precluding conventional volume measurements. We present a formal definition of "normal" 3-Dimensional space, derived from fundamental requirements for spatial measurement and mobility that are compliant with the theory of General Relativity. Applying these criteria, we demonstrate that the Schwarzschild interior comprehensively fails this definition. We argue that this mathematical breakdown, particularly evident in the coordinate system naturally adapted to the problem's spherical symmetry, signifies not merely a coordinate pathology resolvable by abstract transformations (like Kruskal-Szekeres coordinates), but reflects a fundamental change in the physical nature of spacetime. This suggests an incompatibility with the existence of conventional 3D space within the event horizon, offering a distinct perspective on the black hole interior.
The study of regular black holes and black hole mimickers as alternatives to standard black holes has recently gained significant attention, driven both by the need to extend general relativity to describe black hole interiors, and by recent advances in observational technologies. Despite considerable progress in this field, significant challenges remain in identifying and characterizing physically well-motivated classes of regular black holes and black hole mimickers. This paper provides an overview of these challenges, and outlines some of the promising research directions — as discussed during a week-long focus program held at the Institute for Fundamental Physics of the Universe (IFPU) in Trieste from November 11th to 15th, 2024.
In this study, we explore the thermodynamic and geodesic stability of Reissner–Nordström (RN) black holes within the framework of Bumblebee gravity (BG), a Lorentz-violating extension of general relativity (GR). By examining the black hole’s mass distribution and horizon thermodynamics, we establish that although charged black holes in BG exhibit local stability, they remain globally unstable. This conclusion is substantiated by the positive-definite Gibbs free energy and the negative-definite Hessian matrix, which collectively indicate a fundamental instability in the system’s global equilibrium. We utilized the effective potential to derive the Lyapunov exponents for both time-like and null geodesics, assessing their stability. We found that the circular orbits beyond the innermost stable circular orbits in the Schwarzschild regime are stable, which contrasts with the RN black hole solution in GR. Furthermore, our findings demonstrate that a strong Bumblebee field induces unstable orbits. We establish that extreme RN black holes in Einstein gravity do not possess quasi-normal frequencies and instead exhibit stable, constant perturbations. In the non-extremal case, perturbations settle into a steady state, maintaining a constant amplitude. This behavior parallels observations made in the context of the Bumblebee effect.
Gravitational Lensing)是廣義相對論的重要預測，通常被解釋為由時空曲率引起的 光線偏折。本研究提出了一種新的詮釋框架，假設光的偏折來自於局部時空膨脹速率的變化，而非時空 彎曲。我們建立了一個基於時空膨脹速率梯度的模型，並推導出對應的重力透鏡公式。在強重力透鏡的 分析中，我們成功復現了愛因斯坦的透鏡公式，並透過 Abell 1689、SDSS J1004+4112 等天體的觀測 數據驗證了模型的有效性，計算結果與觀測值的誤差小於 1%。此外，我們進一步推導出標準弱重力透 鏡的會聚(κ) 、形變(γ)與角度偏折(α)公式，並證明它們與廣義相對論的對應公式一致。最後，我 們將該模型應用於子彈星團(Bullet Cluster)的重力透鏡效應，發現我們的計算結果與實際觀測的透鏡 偏折角相符，顯示此理論可能為暗物質的必要性提供了新的思考方向。本研究的結果表明，時空膨脹速 率的不均勻性可能提供了一種替代性機制來解釋重力透鏡現象，未來將進一步探討該模型在宇宙大尺度 結構、宇宙微波背景(CMB)透鏡效應及星系旋轉曲線等領域的應用。
Gravitational lensing is a fundamental prediction of general relativity, commonly interpreted as the deflection of light due to spacetime curvature. In this study, we propose an alternative theoretical framework in which light deflection arises from variations in the local spacetime expansion rate rather than curvature. We construct a model based on the gradient of the spacetime expansion rate and derive the corresponding gravitational lensing formulae. For strong gravitational lensing, our model successfully reproduces Einstein's lensing equation, and its validity is confirmed through observational data from celestial objects such as Abell 1689 and SDSS J1004+4112, with computational results differing from observed values by less than 1%. Furthermore, we derive the standard weak gravitational lensing convergence (κ), shear (γ), and deflection angle (α) equations and demonstrate their consistency with those in general relativity. Finally, we apply our model to the gravitational lensing effect of the Bullet Cluster and find that our calculations are consistent with the observed lensing deflection angle. This suggests that the necessity of dark matter may warrant reconsideration. Our findings indicate that inhomogeneities in the spacetime expansion rate may provide an alternative mechanism to explain gravitational lens-ing. Future work will explore the implications of this model in large-scale cosmic structures, the gravitational lensing effect of the cosmic microwave background (CMB), and galactic rotation curves.
本研究提出了一個新的假設：如果所有時空都在膨脹，則重力可能僅是時空膨脹速率變化的結果，而非傳統的吸引力。本研究基於「萬物膨脹假設」，推導出牛頓萬有引力定律，並分析該理論與廣義相對論的關聯性。此外，我們利用此理論重新計算哈勃常數，成功解釋了 CMB 測量與超新星測量的差異。本研究的結果提供了一種可能的理論框架，統一重力、宇宙學紅移與黑洞性質，並提出可供未來實驗驗證的預測。\\
This study proposes a new hypothesis: if all spacetime is expanding, gravity might merely be a consequence of variations in the spacetime expansion rate, rather than a traditional attractive force. Based on this "Universal Expansion Hypothesis," we derive Newton's law of universal gravitation and examine its compatibility with general relativity. Furthermore, we apply this theory to recalculate the Hubble constant, successfully resolving the discrepancy between the values measured from Cosmic Microwave Background (CMB) and supernovae observations. Our results offer a potential theoretical framework to unify gravity, cosmological redshift, and black hole phenomena, while providing predictions that can be tested in future experiments.
This study proposes a new hypothesis: if all spacetime is expanding, gravity might merely be a consequence of variations in the spacetime expansion rate, rather than a traditional attractive force. Based on this "Universal Expansion Hypothesis," we derive Newton's law of universal gravitation and examine its compatibility with general relativity. Furthermore, we apply this theory to recalculate the Hubble constant, successfully resolving the discrepancy between the values measured from Cosmic Microwave Background (CMB) and supernovae observations. Our results offer a potential theoretical framework to unify gravity, cosmological redshift, and black hole phenomena, while providing predictions that can be tested in future experiments.
Addressing the black hole information paradox necessitates the exploration of various hypotheses and theoretical frameworks. Among these, the proposition to utilize quantum entanglement, as introduced by Don N. Page, shows great promise. This study builds upon Page’s theoretical foundation and proposes a simplified model for elucidating the evolution of black hole von Neumann entropy. This model simulates the process of Hawking radiation using entangled photon pairs. Our experiment suggests that quantum entanglement may offer a plausible avenue for resolving the paradox, thereby lending support to Page’s proposal. The results suggest that this model may contribution to the exploration of one of the most profound puzzles in theoretical physics.
Black hole perturbation theory on spherically symmetric backgrounds has been instrumental in establishing various aspects about the gravitational dynamics close to black holes, and continues to be an interesting avenue to confront current challenges in gravitational physics. In this paper, we present an approach to perturbation theory in spherical symmetry that addresses simultaneously some conceivably inconvenient aspects of the traditional methods. In particular, focusing on Schwarzschild's background we are able to derive a decoupled wave equation, for a single complex variable, by simply computing one component of the curvature wave equation satisfied by a complex self-dual version of the Riemann tensor. The real and imaginary parts of the variable consist only of even and odd pieces of the metric fluctuation, respectively, and both satisfy the Regge-Wheeler equation. Besides providing a systematic derivation of decoupled equations, an immediate corollary of our results is the isospectrality between even and odd sectors. We conclude by discussing potential extensions of our formalism to include matter and higher orders in perturbation theory.
We propose a prescription for a new series expansion of the periapsis shift. The prescription formulates the periapsis shift in various spacetimes analytically without using special functions and provides simple and highly accurate approximate formulae. We derive new series representations for the periapsis shift in the Kerr and the Chazy–Curzon spacetimes by using the prescription, where the expansion parameter is defined as the eccentricity divided by the non-dimensional quantity that vanishes in the limit of the innermost stable circular orbit (ISCO). That is to say, the expansion parameter characterizes both the eccentricity of the orbit and its proximity to the ISCO. The smaller the eccentricity, the higher the accuracy of the formulae that are obtained by truncating the new series representations up to a finite number of terms. If the eccentricity is sufficiently small, the truncated new representations have higher accuracy than the post-Newtonian (PN) expansion formulae even in strong gravitational fields where the convergence of the PN expansion formula is not guaranteed. On the other hand, even if the orbit is highly eccentric, the truncated new representations have comparable or higher accuracy than the PN expansion formulae if the semi-major axis is sufficiently large. An exact formula for the periapsis shift of the quasi-circular orbit in the Chazy–Curzon spacetime is also obtained as a special case of the new series representation.
An exact solution of the vacuum Einstein field equations over a nonsimply-connected manifold is presented. This solution is spherically symmetric and has no curvature singularity. It can be considered to be a regularization of the Schwarzschild solution over a simply-connected manifold, which has a curvature singularity at the center. Spherically symmetric collapse of matter in R^4 may result in this nonsingular black-hole solution, if quantum-gravity effects allow for topology change near the center.
We discuss an apparent information paradox that arises in a materialist's description of the Universe if we assume that the Universe is 100% quantum. We discuss possible ways out of the paradox, including that Laws of Nature are not purely deterministic, or that gravity is classical. Our observation of the paradox stems from an interdisciplinary thought process whereby the Universe can be viewed as a "quantum computer". Our presentation is intentionally nontechnical to make it accessible to as wide a readership base as possible.
We present a historical review of Einstein's 1917 paper 'Cosmological Considerations in the General Theory of Relativity' to mark the centenary of a key work that set the foundations of modern cosmology. We find that the paper followed as a natural next step after Einstein's development of the general theory of relativity and that the work offers many insights into his thoughts on relativity, astronomy and cosmology. Our review includes a description of the observational and theoretical background to the paper; a paragraph-by-paragraph guided tour of the work; a discussion of Einstein's views of issues such as the relativity of inertia, the curvature of space and the cosmological constant. Particular attention is paid to little-known aspects of the paper such as Einstein's failure to test his model against observation, his failure to consider the stability of the model and a mathematical oversight concerning his interpretation of the role of the cosmological constant. We recall the response of theorists and astronomers to Einstein's cosmology in the context of the alternate models of the universe proposed by Willem de Sitter, Alexander Friedman and Georges Lemaitre. Finally, we describe the relevance of the Einstein World in today's 'emergent' cosmologies.
The quest for a Theory of Everything (ToE) represents the pinnacle of scientific ambition, seeking to unify the fundamental forces of nature and provide a complete understanding of the universe at all scales. While remarkable progress has been made in physics and mathematics, bridging quantum mechanics and general relativity remains an elusive challenge. A true ToE may require moving beyond traditional mathematical paradigms, embracing novel frameworks that account for phenomena spanning the sub-Planck scale to the vastness of cosmic structures. This paper explores speculative mathematical concepts that could underpin a ToE, ranging from higher-dimensional algebra, dynamic topology, and information-theoretic foundations to emergent fractal logic, non-linear arithmetic, and hyper-complex systems. Additionally, we consider sub-Planck informational geometry and metaphysical dimensions as potential components of a unified framework. By synthesizing these innovative ideas, we aim to inspire interdisciplinary collaboration and encourage the reimagining of mathematical and physical principles. While speculative, these approaches hold the potential to transform our understanding of reality and provide profound insights into the fabric of existence.
Keywords: Theory of Everything, speculative mathematics, quantum mechanics, general relativity, higher-dimensional algebra, dynamic topology, information theory, fractal geometry, non-linear arithmetic, hyper-complex systems, sub-Planck scale, emergent phenomena, metaphysical dimensions, scale invariance, unified physics. 54 pages.
La Termodinámica Geométrica (TG), una rama emergente de la Física, ha ganado relevancia recientemente debido a su capacidad para realizar numerosas predicciones. En esta tesis se examinan dos sistemas: el gas ideal y el sistema de Van der Waals. Se presenta un análisis detallado de la formulación geométrica de estos sistemas, las implicaciones de su estudio en el marco de la TG, y la métrica asociada a cada uno. Además, se interpreta la curvatura y se discuten las divergencias que surgen en ella. Se reproduce la curva de coexistencia para el sistema de Van der Waals utilizando el método conocido como R-Crossing y se evalúa la precisión de este método en comparación con los cálculos tradicionales en Termodinámica.
We demonstrate that at the rim of the photon sphere of a black hole, the quantum statistics transition takes place in any multi-particle system of indistinguishable particles, which passes through this rim to the inside. The related local departure from Pauli exclusion principle restriction causes a decay of the internal structure of collective fermionic systems, including the collapse of Fermi spheres in compressed matter. The Fermi sphere decay is associated with the emission of electromagnetic radiation, taking away the energy and entropy of the falling matter without unitarity violation. The spectrum and timing of the related e-m radiation agree with some observed short giant gamma-ray bursts and X-ray components of the luminosity of quasars and of short transients powered by black holes. The release of energy and entropy when passing the photon sphere rim of a black hole significantly modifies the premises of the information paradox at the falling of matter into a black hole.
Black holes, among the most fascinating predictions of Einstein’s general relativity, have captivated scientists and the public alike. These enigmatic objects, with gravitational fields so intense that nothing—not even light—can escape, have been supported by a wealth of observational evidence, including gravitational wave detections and the imaging of black hole shadows. Despite these successes, classical black hole theory presents significant challenges, such as the existence of singularities, where physical laws break down, and event horizons, which raise unresolved paradoxes about information loss. These issues highlight the need for a more complete model. The Dark Star Analog (DSA) model offers a reimagined framework for black holes, replacing singularities with finite-density cores and event horizons with gravitational surfaces. This approach resolves theoretical inconsistencies while remaining consistent with observational data. By integrating principles of general relativity, quantum mechanics, and astrophysical phenomena, the DSA model bridges key gaps in our understanding of the universe’s most extreme objects and provides testable predictions for future exploration.
Keywords: black holes, dark star analog, general relativity, quantum gravity, singularity resolution, gravitational waves, event horizon replacement, astrophysics, information paradox, spacetime curvature.
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