Singularity Theorems Of General Relativity Research Articles

The Singularity Theorems of General Relativity and Their Low Regularity Extensions

On the occasion of Sir Roger Penrose’s 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of C^{1}-singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook.

The Return of the Singularities: Applications of the Smeared Null Energy Condition

The classic singularity theorems of General Relativity rely on energy conditions that can be violated in semiclassical gravity. Here, we provide motivation for an energy condition obeyed by semiclassical gravity: the smeared null energy condition (SNEC), a proposed bound on the weighted average of the null energy along a finite portion of a null geodesic. We then prove a semiclassical singularity theorem using SNEC as an assumption. This theorem extends the Penrose theorem to semiclassical gravity. We also apply our bound to evaporating black holes and the traversable wormhole of Maldacena-Milekhin-Popov, and comment on the relationship of our results to other proposed semiclassical singularity theorems.

Geometry of weighted Lorentz–Finsler manifolds I: singularity theorems

We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz-Finsler version of the Bonnet-Myers theorem based on a generalized Bishop inequality.

Of Light and Shadows

Einstein’s genius and penetrating physical intuition led to the general theory of relativity, which incorporates gravity into the geometry of spacetime. However, the theory of general relativity leads to perspectives which go far beyond the vision of its creator. Many of these insights came to light only after Einstein’s death in 1955. These developments were due to a new breed of relativists, like Penrose, Hawking and Geroch, who approached the subject with a higher degree of mathematical sophistication than earlier workers. Some of these insights were made possible because of work by Amal Kumar Raychaudhuri (AKR) who derived an equation which turned out to be a key ingredient in the singularity theorems of general relativity. This article explains AKR’s work in elementary terms.

Singularity Theorems for $$C^1$$-Lorentzian Metrics

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal geodesic completeness is $C^1$-fine stable. This improves a similar older stability result of Beem and Ehrlich where they also used the $C^1$-fine topology to measure closeness but still required smoothness of all metrics. Lastly, we include a brief appendix where we use some of the same techniques in the Riemannian case to give a proof of the classical Myers Theorem for $C^1$-metrics.

EFFECT OF SPIN–TORSION INTERACTION ON RAYCHAUDHURI EQUATION

Raychaudhuri equation is generalized in the parametrized absolute parallelism geometry. This version of absolute parallelism is more general than the conventional one. The generalization takes into account the suggested interaction between the quantum spin of the moving particle and the torsion of the background gravitational field. The generalized Raychaudhuri equation obtained contains some extra terms, depending on the torsion of space–time, that would have some effects on the singularity theorems of general relativity. Under a certain condition, this equation could be reduced to the original Raychaudhuri equation without any need for a vanishing torsion.

HIGHER DIMENSIONAL INHOMOGENEOUS DUST COLLAPSE AND COSMIC CENSORSHIP

The singularity theorems of general relativity predict that gravitational collapse finally ends up in a spacetime singularity1. The cosmic censorship hypothesis (CCH) states that such a singularity is covered by an event horizon2. Despite much effort, there is no rigorous formulation or proof of the CCH. In view of this, examples that appear to violate the CCH and lead to naked singularities, in which non-spacelike curves can emerge, rather than black holes, are important to shed more light on the issue. We have studied several collapse scenarios which can lead to both situations3. In the case of the Vaidya-de Sitter spacetime4, we have shown that the naked singularities that arise are of the strong curvature type. Both types of singularities can also arise in higher dimensional Vaidya and Tolman-Bondi spacetimes, but black holes are favoured in some sense by the higher dimensions. The charged Vaidya-de Sitter spacetime also exhibits both types of singularities5.

Graceful exit and energy conditions in string cosmology

String cosmology solutions are examined in a generalized phase-space including sources representing arbitrary corrections to lowest order string-dilaton-gravity effective action. We find a set of necessary conditions for a graceful exit transition from a dilaton-driven inflationary phase to a radiation dominated era. We show that sources allowing such a transition have to violate energy conditions similar to those appearing in singularity theorems of general relativity. Since familiar classical sources, excepting spatial curvature, obey these energy conditions we conclude that a generic graceful exit in string cosmology requires a new effective phase of matter. Our results clarify and generalize previous analyses and enable us to critically reexamine proposed non-singular cosmologies.

Avoidance of singularities in relativity through two-body interactions.

The strong-energy condition releavant to the singularity theorems of general relativity is examined in the high-density regime. We find that a class of two-body interactions between particles of a fluid can produce a negative pressure sufficient to violate the strong-energy condition. The cosmological implication of this phenomenon is investigated in a closed Robertson-Walker universe. We present a model in which the collapse of the universe into a singularity is avoided with unit probability. The minimum radius of the universe is determined by the strength of the interaction.

Does the electron contain negative mass?

Recent experimental evidence that the diameter of the electron is no larger than 10 -16 cm has important implications for its internal structure. Modelling it as a charged sphere obeying Einstein-Maxwell theory, we find that it must contain some negative rest mass. Moreover, the total active gravitational mass within this sphere is negative, which calls into question one of the assumptions made in singularity theorems of general relativity.

The existence of singularities in general relativity despite isolated failures of geodesic focusing

A key ingredient of the singularity theorems of general relativity is that geodesics will focus under certain conditions. While the conditions that are generally imposed are fairly weak and can be expected to hold for most geodesics in any reasonable space-time, the question arises of what the consequences might be if these conditions happen to fail to hold for isolated geodesics. The standard singularity theorems in the literature do not cover such cases. A result is presented here that demonstrates that a singularity will exist even in cases where geodesic focusing does not occur for some geodesics.

A matter model violating the strong energy condition-the influence of temperature

The influence of temperature on the possibility of violating the strong energy condition (SEC) in the singularity theorems of general relativity is studied in the matter model of massive fermions interacting via scalar meson exchange with the usual interaction Lagrangian Lint=gs psi psi phi . It is found that non-vanishing temperatures weaken the strong energy condition violation (SECV) which already exists at zero temperature. Furthermore there is a maximal dimensionless temperature (i.e. a maximal value for the ratio of thermic energy to fermion rest mass) above which no SECV can occur no matter what values for the model parameters are taken; the thermic contribution to the pressure exceeds any SECV contribution for every set of model parameters. This will have consequences in considerations about avoiding the initial singularity in cosmology with this matter model.

On singularity theorems and curvature growth

It is shown that the proofs of a series of classical singularity theorems of general relativity can be modified such that these theorems also state the maximality of the incomplete nonspacelike geodesics. Since along maximal incomplete nonspacelike geodesics with affine parameter u certain parts of the tidal curvature cannot blow up faster than (ū−u)−2, where ū is the parameter value until which the geodesics cannot be extended, the classical singularity theorems do restrict the behavior of the curvature.

Quantum spinors and the singularity theorems of general relativity

I suggest that space-time singularities may be removed through the quantum effects of fermionic spinor matter. I support this viewpoint on the basis of an effective negative contribution, coming from spinor fields with anticommutative (“quantum”) nature, to the energy condition of the singularity theorems of general relativity.