Content uploaded by Jesse Daniel Brown
Author content
All content in this area was uploaded by Jesse Daniel Brown on Nov 18, 2024
Content may be subject to copyright.
Equation of State for Finite-Density Black Holes and
Ultra-Dense Crystalline Matter in a Quantum Gravity
Framework
Jesse Daniel Brown, PhD∗1and McCade Smith, B.A.†2
1Independent Researcher , ORCID ID: 0009-0006-3889-534X , LinkedIn:
linkedin.com/in/jessedanielbrown1980/
2Department of Mathematics, Columbia University
November 17, 2024
Abstract
This paper outlines a theoretical framework aimed at addressing the singularity
problem in black holes by incorporating ultra-dense crystalline matter formed un-
der extreme pressure. By modifying the Einstein field equations and introducing an
anisotropic equation of state (EoS) for this exotic matter, we aim to model black holes
with finite-density cores. This approach could provide insights into the nature of sin-
gularities and contribute to the development of a quantum theory of gravity.
1 Introduction
The classical description of black holes in general relativity predicts singularities—points
of infinite density and curvature—which pose significant challenges to our understanding
of physics [1, 2]. To resolve these singularities and unify general relativity with quantum
mechanics, various approaches have been proposed, including the incorporation of exotic
matter and quantum corrections [3,4].
In this paper, we propose a novel approach by introducing ultra-dense crystalline matter
formed under extreme pressure within black holes. By modifying the Einstein field equations
and defining an anisotropic equation of state (EoS) for this matter, we aim to model black
holes with finite-density cores.
∗Email: plasmatoid@gmail.com
†Email: Mas2547@columbia.edu
1
2 Revised Equation of State for Ultra-Dense Crystalline
Matter
To model the ultra-dense crystalline matter, we propose an anisotropic EoS that relates the
radial pressure Prand tangential pressure Ptto the energy density ρ. Due to the crystalline
structure at extreme densities, pressures may differ in radial and tangential directions [5].
2.1 Energy Density
We assume that the energy density ρ(r) approaches a maximum finite value ρ0at the core
to avoid infinite densities:
ρ(r) = ρ0
1 + r
r0n(1)
where:
•ρ0is the central energy density.
•r0is a characteristic scale, possibly related to the Planck length [6,7].
•ndetermines how rapidly the density falls off with r.
2.2 Radial and Tangential Pressures
The radial pressure Pr(r) and tangential pressure Pt(r) are defined as:
Pr(r) = wrρ(r) (2)
Pt(r) = wtρ(r) (3)
where:
•wris the radial equation of state parameter.
•wtis the tangential equation of state parameter.
The anisotropy factor ∆(r) is given by:
∆(r) = Pt(r)−Pr(r) = (wt−wr)ρ(r) (4)
with δ=wt−wrrepresenting the degree of anisotropy.
2
2.3 Constraints on Parameters
To ensure the physical viability of the model and satisfy energy conditions, the parameters
must satisfy:
•−1≤wr, wt≤1.
•For positive pressures: 0 ≤wr, wt≤1.
•δ≥0 if wt≥wr(tangential pressure exceeds radial pressure).
2.4 Total Stress-Energy Tensor
The stress-energy tensor Tµ
νfor the anisotropic matter is:
Tµ
ν= diag (−ρ(r), Pr(r), Pt(r), Pt(r)) (5)
3 Incorporating into the Modified Einstein Field Equa-
tions
Our modified Einstein field equations are:
Gµν + Λ(ϕ)gµν = 8πG T(ϕ)
µν +Tmatter
µν (6)
where:
•Gµν is the Einstein tensor [8].
•Λ(ϕ) is the cosmological constant dependent on the scalar field ϕ.
•T(ϕ)
µν is the stress-energy tensor of the scalar field.
•Tmatter
µν is the stress-energy tensor of the ultra-dense crystalline matter defined above.
3.1 Scalar Field ϕ(r)
The scalar field ϕ(r) represents quantum corrections and helps stabilize the solution:
ϕ(r) = ϕ0
r2
r2+r2
0
(7)
3.2 Potential V(ϕ)
An example potential is:
V(ϕ) = V01−e−βϕ(8)
3
3.3 Cosmological Constant Λ(ϕ)
We define:
Λ(ϕ)=Λ0+λϕ2(9)
3.4 Stress-Energy Tensor Components
The stress-energy tensor for the scalar field is:
T(ϕ)
µν =∇µϕ∇νϕ−1
2gµν ∇λϕ∇λϕ+ 2V(ϕ)(10)
4 Energy Conditions Verification
To ensure physical viability, we verify the energy conditions:
4.1 Weak Energy Condition (WEC)
ρtotal ≥0
ρtotal +Ptotal
r≥0
ρtotal +Ptotal
t≥0
(11)
4.2 Dominant Energy Condition (DEC)
ρtotal ≥ |Ptotal
r|
ρtotal ≥ |Ptotal
t|(12)
4.3 Strong Energy Condition (SEC)
ρtotal +Ptotal
r+ 2Ptotal
t≥0 (13)
By choosing wrand wtwithin the specified ranges, these conditions can be satisfied
throughout the spacetime.
5 Conclusion
We have proposed a revised equation of state for ultra-dense crystalline matter in finite-
density black holes. By defining the energy density and anisotropic pressures with adjustable
parameters, we provide a framework that can be used to model black holes without singular-
ities. Future work will involve solving the modified field equations numerically and exploring
the physical implications of these solutions.
4
Acknowledgments
We would like to thank everyone who has supported this research and provided valuable
discussions.
References
[1] Penrose, R. (1965). Gravitational collapse and space-time singularities. Physical Review
Letters, 14(3), 57–59.
[2] Hawking, S. W., & Penrose, R. (1970). The singularities of gravitational collapse and
cosmology. Proceedings of the Royal Society of London. A. Mathematical and Physical
Sciences, 314(1519), 529–548.
[3] Ashtekar, A., Pawlowski, T., & Singh, P. (2006). Quantum nature of
the big bang: Improved dynamics. Physical Review D, 74(8), 084003.
https://doi.org/10.1103/PhysRevD.74.084003.
[4] Rovelli, C. (2004). Quantum Gravity. Cambridge University Press.
[5] Herrera, L., & Santos, N. O. (1997). Local anisotropy in self-gravitating systems. Physics
Reports, 286(2), 53–130.
[6] Amelino-Camelia, G. (2002). Special treatment. Nature, 418(6893), 34–35.
https://doi.org/10.1038/418034a.
[7] Amelino-Camelia, G. (2002). Doubly Special Relativity. Nature, 418(6893), 34–35.
arXiv:gr-qc/0207049.
[8] Wald, R. M. (1984). General Relativity. University of Chicago Press.
5
