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Abstract

We propose a theoretical framework addressing the singularity problem in black holes by introducing finite-density cores composed of ultra-dense matter. By modifying Einstein field equations with a scalar field representing quantum corrections and defining an anisotropic equation of state. This work explores the implications of ultra-dense structures formed under extreme pressures near the Planck scale for black hole physics, quantum gravity, and potential observational signatures. Analogous to neutron stars, where matter is compressed to nuclear densities, we hypothesize that black holes compress matter even further.
Finite-Density Black Holes in a Quantum Gravity Framework
Jesse Daniel Brown, PhDMcCade Smith, B.A.
November 17, 2024
Introduction
This document outlines a theoretical framework aimed at addressing the singularity problem in black holes
by incorporating quantum corrections through a scalar field [13]. The modifications to the Einstein field
equations are derived from an action principle to ensure consistency and avoid double counting of scalar field
contributions [4, 5].
To resolve these singularities and unify general relativity with quantum mechanics, various approaches
have been proposed, including loop quantum gravity [6,7].
This work has been developed in collaboration with McCade Smith, a brilliant mathematician from
Columbia University who helped resolve the doubling of the scalar field.
1 Action Integral
We start with the action:
S=Zd4xg1
16πG (R2Λ(ϕ)) 1
2gµν µϕνϕV(ϕ) + Lmatter(1)
[5, 8]
2 Modified Einstein Field Equations
Varying the action with respect to the metric gµν yields:
Gµν + Λ(ϕ)gµν = 8πG T(ϕ)
µν +Tmatter
µν (2)
[9]
where:
T(ϕ)
µν =µϕνϕgµν 1
2λϕλϕ+V(ϕ)(3)
3 Scalar Field Equation of Motion
Varying the action with respect to ϕgives:
ϕdV
+1
8πG
dΛ
= 0 (4)
[5, 7]
Email: plasmatoid@gmail.com
Email: Mas2547@columbia.edu
1
4 Cosmological Constant as a Function of ϕ
We consider:
Λ(ϕ)=Λ0+f(ϕ) (5)
[10, 11]
5 Total Stress-Energy Tensor
Tµν =T(ϕ)
µν +Tmatter
µν (6)
[4, 5]
6 Objectives
Choose appropriate forms for V(ϕ) and f(ϕ) to achieve finite-density black hole solutions.
Solve the modified field equations for static, spherically symmetric spacetimes.
Extend the model to rotating black holes.
Ensure the modified equations respect energy-momentum conservation.
Explore potential observational implications.
Conclusion
Any insights or suggestions on the above aspects would be greatly appreciated.
References
[1] Einstein, A. (1916). The foundation of the general theory of relativity. Annalen der Physik, 354(7),
769–822.
[2] Penrose, R. (1965). Gravitational collapse and space-time singularities. Physical Review Letters, 14(3),
57–59.
[3] Hawking, S. W., & Penrose, R. (1970). The singularities of gravitational collapse and cosmology. Pro-
ceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 314(1519), 529–548.
[4] Brans, C., & Dicke, R. H. (1961). Mach’s principle and a relativistic theory of gravitation. Physical
Review, 124(3), 925.
[5] Faraoni, V. (2004). Cosmology in Scalar-Tensor Gravity. Springer.
[6] Ashtekar, A. (1986). New variables for classical and quantum gravity. Physical Review Letters, 57(18),
2244–2247.
[7] Ashtekar, A., Pawlowski, T., & Singh, P. (2006). Quantum nature of the big bang: An analytical and
numerical investigation. Physical Review D, 73(12), 124038.
[8] Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. (2012). Modified gravity and cosmology. Physics
Reports, 513(1-3), 1–189.
[9] Wald, R. M. (1984). General Relativity. University of Chicago Press.
2
[10] Peebles, P. J. E., & Ratra, B. (1988). Cosmology with a time-variable cosmological ’constant’. The
Astrophysical Journal, 325, L17–L20.
[11] Wetterich, C. (1988). Cosmology and the fate of dilatation symmetry. Nuclear Physics B, 302(4),
668–696.
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The foundation of the general theory of relativity
  • A Einstein
Einstein, A. (1916). The foundation of the general theory of relativity. Annalen der Physik, 354(7), 769-822.