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Content uploaded by Jesse Daniel Brown
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All content in this area was uploaded by Jesse Daniel Brown on Nov 17, 2024
Content may be subject to copyright.
Finite-Density Black Holes in a Quantum Gravity Framework
Jesse Daniel Brown, PhD∗McCade 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 [1–3]. 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=Zd4x√−g1
16πG (R−2Λ(ϕ)) −1
2gµν ∇µϕ∇νϕ−V(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 =∇µϕ∇νϕ−gµν 1
2∇λϕ∇λϕ+V(ϕ)(3)
3 Scalar Field Equation of Motion
Varying the action with respect to ϕgives:
□ϕ−dV
dϕ +1
8πG
dΛ
dϕ = 0 (4)
[5, 7]
∗Email: plasmatoid@gmail.com
†Email: Mas2547@columbia.edu
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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
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769–822.
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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.
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[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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