[Paper Review] Evolution of Black Holes in Brans-Dicke Cosmology

Nobuyuki Sakai, John D. Barrow|ArXiv.org|Dec 19, 2000

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates black hole evolution in Brans-Dicke cosmology using a modified Swiss-cheese model, defining black hole size via the Misner-Sharp mass. It shows that in both dust and vacuum universes, the Misner-Sharp radius and mass decrease over time during cosmic expansion, indicating black holes shrink as the universe expands, regardless of spatial curvature or initial conditions.

ABSTRACT

We consider a modified ``Swiss cheese'' model in the Brans-Dicke theory, and discuss the evolution of black holes in the expanding universe. We define the black hole radius by the Misner-Sharp mass and find the time evolution for dust and vacuum universes.

Motivation & Objective

• To investigate how black holes evolve in an expanding universe within the framework of Brans-Dicke scalar-tensor gravity.
• To resolve the debate on whether black holes retain 'gravitational memory' of their formation G-value or if G evolves dynamically.
• To extend Saida and Soda's cell-lattice model to a Swiss-cheese model for a more realistic inhomogeneous cosmological setup.
• To define and analyze the time evolution of black hole size using the Misner-Sharp mass as a geometric measure.
• To determine whether black hole mass and radius increase or decrease in different cosmological backgrounds (dust vs. vacuum, k=0, +1, -1).

Proposed method

• Construct a modified Swiss-cheese model in Brans-Dicke theory, replacing spherical regions in FRW spacetime with vacuum black hole interiors.
• Use Israel's junction conditions for a singular hypersurface to derive equations of motion for the boundary between FRW and black hole spacetimes.
• Define the black hole radius via the Misner-Sharp mass: $ R_{MS} = R^3 (H^2 + k/a^2) $, where $ R $ is the comoving radius.
• Solve the field equations for dust and vacuum FRW universes in Brans-Dicke theory, using exact solutions for $ a(t) $, $ \phi(t) $, and $ H(t) $.
• Evolve the Misner-Sharp radius and mass $ M_{MS} = \phi R_{MS}/2 $ using time-dependent solutions for $ a(t) $, $ \phi(t) $, and $ H(t) $.
• Analyze the time evolution of $ R_{MS} $ and $ M_{MS} $ in flat, open, and closed universes, focusing on asymptotic behavior and curvature dependence.

Experimental results

Research questions

• RQ1 Does the Misner-Sharp mass of a black hole in Brans-Dicke cosmology remain constant or evolve over time in an expanding universe?
• RQ2 How does the time evolution of black hole size depend on the spatial curvature (k=0, +1, -1) of the background universe?
• RQ3 Does the black hole mass increase or decrease in a scalar-field-dominated (vacuum) universe, and how does this compare to the dust universe?
• RQ4 Is the assumption of a Schwarzschild-like metric for the black hole interior consistent with the Misner-Sharp mass definition in this model?
• RQ5 Does the black hole exhibit 'gravitational memory' of its initial G-value, or does the effective G evolve dynamically with the cosmological scalar field?

Key findings

• In the $ k=0 $ dust universe, the Misner-Sharp radius $ R_{MS} $ and mass $ M_{MS} $ both decrease with time, indicating black holes shrink as the universe expands.
• For the $ k=0 $ dust model, $ M_{MS} $ remains constant only in the special case $ t_+ = t_- $, corresponding to the asymptotic power-law solution.
• In vacuum universes with $ \omega > 500 $, both $ R_{MS} $ and $ M_{MS} $ decrease over time, except during the contracting phase of the closed ($ k=+1 $) universe.
• In the $ k=+1 $ vacuum case, $ R_{MS} $ and $ M_{MS} $ exhibit oscillatory behavior during the contraction phase, but still decrease overall in the expanding phase.
• The Misner-Sharp mass $ M_{MS} $ is consistent with Saida and Soda's earlier results using a Schwarzschild-like metric, validating their ansatz as physically meaningful.
• The time evolution of $ v $ and $ \sigma $ from zero remains zero if initially zero, ensuring stability of the boundary conditions in the dust case, which supports the model's consistency.