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PreprintCC BY 4.0arXiv

An Entropy-Based Criterion for Geodesic Incompleteness

  • Rohit Dhormare
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Abstract

We develop an entropy-based formulation of gravitational focusing in Lorentzian geometry by associating a scalar entropy functional to timelike geodesic congruences. Rewriting the Raychaudhuri equation as an entropy production law, we show that entropy accumulation governs the evolution of the expansion scalar and drives finite-time focusing. We prove that when the integrated entropy exceeds a critical threshold set by the initial expansion, geodesic incompleteness necessarily follows. This result provides a quantitative refinement of the classical singularity theorems, replacing qualitative geometric conditions with an explicit entropy criterion. Our approach offers a thermodynamic interpretation of gravitational collapse and suggests that singularities arise as endpoints of irreversible entropy growth in spacetime.

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Cite

BibTeXRIS
BibTeX
@misc{dhormare2026entropybased,
  title = {{An Entropy-Based Criterion for Geodesic Incompleteness}},
  author = {Rohit Dhormare},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02905},
  eprint = {2609.02905},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02905},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Entropy-Based Criterion for Geodesic Incompleteness
AU  - Rohit Dhormare
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.02905
UR  - https://arxiv.org/abs/2609.02905
AB  - We develop an entropy-based formulation of gravitational focusing in Lorentzian geometry by associating a scalar entropy functional to timelike geodesic congruences. Rewriting the Raychaudhuri equation as an entropy production law, we show that entropy accumulation governs the evolution of the expansion scalar and drives finite-time focusing. We prove that when the integrated entropy exceeds a critical threshold set by the initial expansion, geodesic incompleteness necessarily follows. This result provides a quantitative refinement of the classical singularity theorems, replacing qualitative geometric conditions with an explicit entropy criterion. Our approach offers a thermodynamic interpretation of gravitational collapse and suggests that singularities arise as endpoints of irreversible entropy growth in spacetime.
ER  -