Singularity Theorems and Their Consequences (1801.04912v1)

Abstract: This is a 20-year old review on singularities and singularity theorems. The main reason to submit it now is -apart from increasing its availability- to correct a very strange error that appears in the journal's online version: it contains wrong headers and footers in its 1st page. This has led to many wrong citations, and to some confusion. The pdf file here contains the right information. ORIGINAL ABSTRACT: A detailed study of the singularity theorems is presented. I discuss the plausibility and reasonability of their hypotheses, the applicability and implications of the theorems, as well as the theorems themselves. The consequences usually extracted from them, some of them without the necessary rigour, are widely and carefully analysed with many clarifying examples and alternative views.

• The paper reviews singularity theorems in general relativity, analyzing their theoretical basis, key assumptions, and derived implications while critiquing traditional interpretations and methodologies.
• It discusses landmark results by Penrose and Hawking, detailing the necessary energy, causality, and boundary conditions, and highlights the distinction between geodesic incompleteness and physical singularities.
• The review challenges the common reliance on the Strong Energy Condition and advocates for more precise definitions of singularities, suggesting avenues for future research particularly concerning quantum gravity scenarios.

Summary of "Singularity Theorems and Their Consequences" by Jose M. M. Senovilla

Jose M. M. Senovilla's paper presents an in-depth review of singularity theorems, exploring both their theoretical foundations and implications within the field of general relativity. The paper critically examines the assumptions underlying these theorems, considers their applicability, and explores the consequences deduced from them, often emphasizing the need for rigorous interpretation.

Key Elements of Singularity Theorems

At the heart of Senovilla's exposition are the classic results of Penrose, Hawking, and others, which necessitate the presence of singularities under certain conditions in relativistic spacetimes. The singularity theorems typically require the satisfaction of key assumptions:

1. Energy Condition: Usually the Strong Energy Condition (SEC) or its averaged form is necessary to ensure the focusing of geodesics. The weaker null convergence condition is also frequently considered.
2. Causality Condition: Most theorems necessitate a condition like the strong causality or chronology condition to prevent pathological causal structures, though later works have relaxed these to accommodate causality violations.
3. Boundary Condition: This usually manifests as the existence of some trapped surface or a condition signifying reconvergence of light rays, serving as a precursor to gravitational collapse.

Notable Theorems and Their Implications

Senovilla recaps landmark results, such as Penrose's theorem concerning trapped surfaces and Hawking's theorems addressing both cosmological models and black holes. These results generally propound that under the stated conditions, spacetimes are geodesically incomplete. However, the implications of these theorems are nuanced, particularly regarding what constitutes a "singularity," given that singularities are not explicit regions of spacetime but rather an indication of some pathological incompleteness.

Singularities and Their Realizations

Singularities inferred by these theorems pose interpretative challenges, especially regarding their physical realization and whether they epitomize truly pathological spacetime regions. The paper discusses several models and examples, such as the Schwarzschild and flrw universes, illustrating potential misunderstandings regarding the nature and characteristics of singularities. For instance, examples like Misner's taub-nut metric show that incompleteness doesn't always equate to a physical singularity.

Critique and Areas for Further Exploration

Senovilla critiques the traditional reliance on SEC, suggesting that it's physically less justified compared to more observationally validated conditions like the Weak Energy Condition (WEC) or the Dominant Energy Condition (DEC).

Moreover, the paper argues for more rigorous definitions surrounding singularities, suggesting that concepts such as the causal boundary or geodesic boundary hold potential for deeper insights into their mathematical structure.

Conclusion and Forward-Looking Remarks

Senovilla's review highlights both the elegance and limitations of singularity theorems. While they underscore fundamental insights into Einstein's theory, they also point to areas where greater mathematical rigor and reinterpretation could yield richer understanding, particularly when exploring potential quantum-gravitational scenarios or more generalized forms of relativity. Speculation about non-singular quantum cosmologies or regularized models serves as a potent reminder of the incompleteness of our current classical theories when approaching these profound questions.

In essence, the paper provides a thorough academic synthesis of singularity theorems, challenging, and inviting researchers to refine and expand upon them, better aligning theoretical predictions with physical realism.