A note on the Gannon–Lee theorem

We prove a Gannon–Lee theorem for non-globally hyperbolic Lorentzian metrics of regularity $$C^1$$ C 1 , the most general regularity class currently available in the context of the classical singularity theorems. Along the way, we also prove that any maximizing causal curve in a $$C^1$$ C 1 -spacetime is a geodesic and hence of $$C^2$$ C 2 -regularity.

Asymptotically flat black hole solutions in quadratic gravity

Black holes constitute some of the most fascinating objects in our universe. According to Einstein’s theory of general relativity, they are also deceivingly simple: Schwarzschild black holes are completely determined by their mass. Moreover, the singularity theorems by Penrose and Hawking indicate that they host a curvature singularity within their event horizon. The presence of the latter invites the question whether these dead-end points of spacetime can be made regular by considering (quantum) corrections to the classical field equations. In this light, we use the Frobenius method to investigate the phase space of asymptotically flat, static, and spherically symmetric black hole solutions in quadratic gravity. We argue that the only asymptotically flat black hole solution visible in this approach is the Schwarzschild solution.

Effective GUP-modified Raychaudhuri equation and black hole singularity: four models

The classical Raychaudhuri equation predicts the formation of conjugate points for a congruence of geodesics, in a finite proper time. This in conjunction with the Hawking-Penrose singularity theorems predicts the incompleteness of geodesics and thereby the singular nature of practically all spacetimes. We compute the generic corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole, arising from modifications to the algebra inspired by the generalized uncertainty principle (GUP) theories. Then we study four specific models of GUP, compute their effective dynamics as well as their expansion and its rate of change using the Raychaudhuri equation. We show that the modification from GUP in two of these models, where such modifications are dependent of the configuration variables, lead to finite Kretchmann scalar, expansion and its rate, hence implying the resolution of the singularity. However, the other two models for which the modifications depend on the momenta still retain their singularities even in the effective regime.

Errors Related to General Relativity, Repulsive Gravitation and the Question of Black Holes

Galileo and Newton considered gravity to be independent of temperature, while Einstein claimed that the weight of metal will increase as temperature increases. Further, Maxwell maintained that charge is unrelated to gravity. Experiments show, however, that the weight of a metal piece is reduced as its temperature increases. Thus, charge-initiated repulsive gravitation exists. In fact, repulsive gravity has been demonstrated by the use of a charged capacitor hovering over Earth. Further, it is expected that a piece of heated metal would fall more slowly than a feather in a vacuum. Einstein developed an invalid notion of gravitational mass, and failed to establish the unification of gravitation and electromagnetism since he overlooked repulsive gravitation. Moreover, photons are a combination of the gravitational wave and the electromagnetic wave. For electromagnetic energy is invalid, and is in conflict with the Einstein equation. The non-linear Einstein equation has no bounded dynamic solution, Space-time singularity theorems are based on an invalid implicit assumption that all the couplings have a unique sign. Since gravity is no longer always attractive, the existence of black holes is questionable. The fact that Penrose was awarded the 2020 Nobel Prize in Physics for the derivation of black holes is due to that the Nobel Prize Committee for Physics did not sufficiently understand the physics of general relativity. A distinct characteristic of Penrose's work, as usual, is that it is not verifiable.

Singularity Theorems in the Effective Field Theory for Quantum Gravity at Second Order in Curvature

In this paper, we discuss singularity theorems in quantum gravity using effective field theory methods. To second order in curvature, the effective field theory contains two new degrees of freedom which have important implications for the derivation of these theorems: a massive spin-2 field and a massive spin-0 field. Using an explicit mapping of this theory from the Jordan frame to the Einstein frame, we show that the massive spin-2 field violates the null energy condition, while the massive spin-0 field satisfies the null energy condition, but may violate the strong energy condition. Due to this violation, classical singularity theorems are no longer applicable, indicating that singularities can be avoided, if the leading quantum corrections are taken into account.

Some applications

The aim of this chapter is to present key applications of causality theory, as relevant to black-hole spacetimes. For this we need to introduce the concept of conformal completions, which is done in Section 3.1. We continue, in Section 3.2, with a review of the null splitting theorem of Galloway. Section 3.3 contains complete proofs of a few versions of the topological censorship theorems, which are otherwise scattered across the literature, and which play a basic role in understanding the topology of black holes. In Section 3.4 we review some key incompleteness theorems, also known under the name of singularity theorems. Section 3.5 is devoted to the presentation of a few versions of the area theorem, which is a cornerstones of ‘black-hole thermodynamics’. We close this chapter with a short discussion of the role played by causality theory when studying the wave equation.

Elements of causality

A standard part of studies of black holes, and in fact of mathematical general relativity, is causality theory, which is the study of causal relations on Lorentzian manifolds. An essential issue here is understanding the influence of energy conditions on the causality relations. The highlights of such studies include the incompleteness theorems, known also as singularity theorems, of Penrose, Hawking and Geroch, the area theorem of Hawking, and the topology theorems of Hawking and others. The aim of this chapter is to provide an introduction to the subject, with a complete exposition of those topics which are needed for the global treatment of the uniqueness theory of black holes. In particular we provide a coherent introduction to causality theory for metrics which are twice differentiable.