Theorem In General Relativity Research Articles

The influence of Penrose’s singularity theorem in general relativity

Penrose’s crucial contributions to General Relativity, symbolized by his 1965 singularity theorem, received (half of) the 2020 Nobel prize in Physics. A renewed interest in the ideas and implications behind that theorem, its later developments, and other Penrose’s ideas improving our understanding of the gravitational field thereby emerged. In this paper I highlight some of the advancements motivated by the theorem that were developed over the years. I also identify some common misconceptions about the theorem’s implications. A modern perspective on the concept of closed trapped submanifolds, based on the mean curvature vector, is advocated.

The Friedmann–Lemaître–Robertson–Walker metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric used to describe the cosmic spacetime is based on the cosmological principle, which assumes homogeneity and isotropy throughout the Universe. It also adopts free-fall conditions via the selection of a constant lapse function, [Formula: see text], regardless of whether or not the chosen energy–momentum tensor [Formula: see text] produces an accelerated expansion. This is sometimes justified by arguing that one may shift the gauge, if necessary, transforming the time [Formula: see text] to a new coordinate [Formula: see text], thereby re-establishing a unitary value for [Formula: see text]. Previously, we have demonstrated that this approach is inconsistent with the Friedmann equations derived using comoving coordinates. In this paper, we advance this discussion significantly by using the Local Flatness Theorem in general relativity to prove that [Formula: see text] in FLRW is inextricably dependent on the expansion dynamics via the expansion factor [Formula: see text], which itself depends on the equation-of-state in [Formula: see text]. One is therefore not free to choose [Formula: see text] arbitrarily without ensuring its consistency with the energy–momentum tensor. We prove that the use of FLRW in cosmology is valid only for zero active mass, i.e. [Formula: see text], where [Formula: see text] and [Formula: see text] are, respectively, the total energy density and pressure in the cosmic fluid.

Wave four-tensor of a plane light wave in free space with application to the positive mass theorem in general relativity

The positive mass theorem in general relativity states that in an asymptotically flat spacetime, if the momentum–energy tensor is divergence-free and satisfies a dominant energy condition, then a total momentum–energy four-vector can be formed, of which the energy component is nonnegative. In this paper, we take the wave four-tensor of a plane light wave in free space as a counterexample to show that there is no guarantee that a total four-vector can be formed. Thus the theoretical framework for the positive mass theorem is flawed. In addition, it is also shown as well that the Lorentz covariance of Dirac wave equation is not compatible with Einstein mass–energy equivalence.

Relative depolarization of the black hole photon ring in GRMHD models of Sgr A* and M87*

ABSTRACT Using general relativistic magnetohydrodynamic simulations of accreting black holes, we show that a suitable subtraction of the linear polarization per pixel from total intensity images can enhance the photon ring feature. We find that the photon ring is typically a factor of ≃2 less polarized than the rest of the image. This is due to a combination of plasma and general relativistic effects, as well as magnetic turbulence. When there are no other persistently depolarized image features, adding the subtracted residuals over time results in a sharp image of the photon ring. We show that the method works well for sample, viable GRMHD models of Sgr A* and M87*, where measurements of the photon ring properties would provide new measurements of black hole mass and spin, and potentially allow for tests of the ‘no-hair’ theorem of general relativity.

Black hole spectroscopy in the next decade

Gravitational wave observations of the ringdown of the remnant black hole in a binary black hole coalescence provide a unique opportunity of confronting the black hole no-hair theorem in general relativity with observational data. The most robust tests are possible if multiple ringdown modes can be observed. In this paper, using state-of-the-art Bayesian inference methods and the most up-to-date knowledge of binary black hole population parameters and ringdown mode amplitudes, we evaluate the prospects for black hole spectroscopy with current and future ground based gravitational wave detectors over the next 10 years. For different population models, we estimate the likely number of events for which the subdominant mode can be detected and distinguished from the dominant mode. We show that black hole spectroscopy could significantly test general relativity for events seen by the proposed LIGO Voyager detectors.

A no-hair test for binary black holes

One of the consequences of the black-hole "no-hair" theorem in general relativity (GR) is that gravitational radiation (quasi-normal modes) from a perturbed Kerr black hole is uniquely determined by its mass and spin. Thus, the spectrum of quasi-normal mode frequencies have to be all consistent with the same value of the mass and spin. Similarly, the gravitational radiation from a coalescing binary black hole system is uniquely determined by a small number of parameters (masses and spins of the black holes and orbital parameters). Thus, consistency between different spherical harmonic modes of the radiation is a powerful test that the observed system is a binary black hole predicted by GR. We formulate such a test, develop a Bayesian implementation, demonstrate its performance on simulated data and investigate the possibility of performing such a test using previous and upcoming gravitational wave observations.

Self-anisotropizing inflationary universe in Horndeski theory and beyond

As opposed to Wald's cosmic no-hair theorem in general relativity, it is shown that the Horndeski theory (and its generalization) admits anisotropic inflationary attractors if the Lagrangian depends cubically on the second derivatives of the scalar field. We dub such a solution as a self-anisotropizing inflationary universe because anisotropic inflation can occur without introducing any anisotropic matter fields such as a vector field. As a concrete example of self-anisotropization we present the dynamics of a Bianchi type-I universe in the Horndeski theory.

Black hole topology in gravity

Hawking’s topology theorem in general relativity restricts the cross-section of the event horizon of a black hole in 3 + 1 dimension to be either spherical or toroidal. The toroidal case is ruled out by the topology censorship theorems. In this article, we discuss the generalization of this result to black holes in gravity in 3 + 1 and higher dimensions. We obtain a sufficient differential condition on the function , which restricts the topology of the horizon cross-section of a black hole in gravity in 3 + 1 dimension to be either S2 or . We also extend the result to higher dimensional black holes and show that the same sufficient condition also restricts the sign of the Yamabe invariant of the horizon cross-section.

Nonextensive kinetic theory and [formula omitted]-theorem in general relativity

The nonextensive kinetic theory for degenerate quantum gases is discussed in the general relativistic framework. By incorporating nonadditive modifications in the collisional term of the relativistic Boltzmann equation and entropy current, it is shown that Tsallis entropic framework satisfies a H-theorem in the presence of gravitational fields. Consistency with the 2nd law of thermodynamics is obtained only whether the entropic q-parameter lies in the interval q∈[0,2]. As occurs in the absence of gravitational fields, it is also proved that the local collisional equilibrium is described by the extended Bose–Einstein (Fermi–Dirac) q-distributions.

On the initial state of the universe in the theory of inflation

A toy quantum model of the inflationary universe is considered in which the role of cosmic time is played by the inflaton scalar field (its logarithm). Based on a variant of the positive energy theorem in General Relativity for the case of a closed universe, a strictly positive energy of space is introduced. The principle of minimum of the energy of space is proposed which determines a ground state as well as the excited states of the universe in quantum cosmology. According to this principle, the Beginning of the universe does exist as a state of minimal excitation of the energy of space. The initial proto-inflation quantum state of the universe is defined as a state of minimal excitation of the energy of space provided that the potential energy of the inflaton scalar field is large at the Beginning. Simultaneously, quanta of space energy excitation are introduced and the expansion of the universe can be considered as the birth of these quanta. Quantum birth of the ordinary matter becomes significant when the potential energy of the inflaton scalar field comes down to zero value.

Black hole collapse and democratic models

We study the evolution of black hole entropy and temperature in collapse scenarios, finding three generic lessons. First, entropy evolution is extensive. Second, at large times, entropy and temperature ring with twice the frequency of the lowest quasinormal mode. Third, the entropy oscillations saturate black hole area theorems in general relativity. The first two features are characteristic of entanglement dynamics in `democratic' models. Solely based on general relativity and Bekenstein-Hawking entropy formula, our results point to democratic models as microscopic theories of black holes. The third feature can be taken as a prediction for democratic models coming from black hole physics.

Entanglement conservation, ER=EPR, and a new classical area theorem for wormholes

We consider the question of entanglement conservation in the context of the ER=EPR correspondence equating quantum entanglement with wormholes. In quantum mechanics, the entanglement between a system and its complement is conserved under unitary operations that act independently on each; ER=EPR suggests that an analogous statement should hold for wormholes. We accordingly prove a new area theorem in general relativity: for a collection of dynamical wormholes and black holes in a spacetime satisfying the null curvature condition, the maximin area for a subset of the horizons (giving the largest area attained by the minimal cross section of the multi-wormhole throat separating the subset from its complement) is invariant under classical time evolution along the outermost apparent horizons. The evolution can be completely general, including horizon mergers and the addition of classical matter satisfying the null energy condition. This theorem is the gravitational dual of entanglement conservation and thus constitutes an explicit characterization of the ER=EPR duality in the classical limit.

Black magic session of concordance: Regge mass spectrum from Casson’s invariant

Recently, there had been a great deal of interest in obtaining and describing all kinds of knots in links in hydrodynamics, electrodynamics, non-Abelian gauge field theories and gravity. Although knots and links are observables of the Chern–Simons (CS) functional, the dynamical conditions for their generation lie outside the scope of the CS theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S3are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space is thus far attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of nonnegative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of nonnegative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is a specialization of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson’s surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew–Frautschi (CF) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding CF plots. The physics behind Casson’s surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons.

Recent progress on the positive energy theorem

We give a short review of recent progress on the positive energy theorem in general relativity, especially for spacetimes with nonzero cosmological constant.

Space-time singularities in Weyl manifolds

We extend one of the Hawking-Penrose singularity theorems in general relativity to the case of some scalar-tensor gravity theories in which the scalar field has a geometrical character and space-time has the mathematical structure of a Weyl integrable space-time (WIST). We adopt an invariant formalism, so that the extended version of theorem does not depend on a particular frame.

A Birkhoff theorem for shape dynamics

Shape dynamics is a theory of gravity that replaces refoliation invariance for spatial Weyl invariance. Those solutions of the Einstein equations that have global, constant mean curvature slicings, are mirrored by solutions in shape dynamics. However, there are solutions of shape dynamics that have no counterpart in general relativity (GR), just as there are solutions of GR that are not completely foliable by global constant mean curvature slicings (such as the Schwarzschild spacetime). It is therefore interesting to analyze directly the equations of motion of shape dynamics in order to find its own solutions, irrespective of properties of known solutions of GR. Here we perform a first study in this direction by utilizing the equations of motion of shape dynamics in a spherically symmetric, asymptotically flat ansatz to derive an analogue of the Birkhoff theorem. There are two significant differences with respect to the usual Birkhoff theorem in GR. The first regards the construction of the solution: we heavily use the shape synamics spatial Weyl gauge freedom to simplify the problem, and require boundary conditions. In fact the derivation is simpler than the usual Birkhoff theorem as no Christoffel symbols are needed. The second, and most important difference is that the solution obtained is uniquely the isotropic wormhole solution, in which no singularity is present, as opposed to maximally extended Schwarzschild. This provides an explicit example of the breaking of the duality between GR and shape dynamics, and exhibits some of its consequences.

TESTS OF GENERAL RELATIVITY IN THE STRONG-GRAVITY REGIME BASED ON X-RAY SPECTROPOLARIMETRIC OBSERVATIONS OF BLACK HOLES IN X-RAY BINARIES

Although General Relativity (GR) has been tested extensively in the weak gravity regime, similar tests in the strong gravity regime are still missing. In this paper we explore the possibility to use X-ray spectropolarimetric observations of black holes in X-ray binaries to distinguish between the Kerr metric and the phenomenological metrics introduced by Johannsen and Psaltis (2011) (which are not vacuum solutions of Einstein's equation) and thus to test the no-hair theorem of GR. To this end, we have developed a numerical code that calculates the radial brightness profiles of accretion disks and parallel transports the wave vector and polarization vector of photons through the Kerr and non-GR spacetimes. We used the code to predict the observational appearance of GR and non-GR accreting black hole systems. We find that the predicted energy spectra and energy dependent polarization degree and polarization direction do depend strongly on the underlying spacetime. However, for large regions of the parameter space, the GR and non-GR metrics lead to very similar observational signatures, making it difficult to observationally distinguish between the two types of models.

On the status of the geodesic principle in Newtonian and relativistic physics

A theorem due to Geroch and Jang (1975) provides a sense in which the geodesic principle has the status of a theorem in General Relativity. I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton–Cartan theory) (Weatherall, J.O., 2011). Here I compare the interpretations of these two theorems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case.

Positive gravitational energy in arbitrary dimensions

We present a streamlined, complete proof, valid in arbitrary space dimension n, and using only spinors on the oriented Riemannian space ( M n ; g ) , of the positive energy theorem in General Relativity.

Almost Birkhoff theorem in general relativity

We extend Birkhoff's theorem for almost LRS-II vacuum spacetimes to show that the rigidity of spherical vacuum solutions of Einstein's field equations continues even in the perturbed scenario.