[Paper Review] Spherical collapse of small masses in the ghost-free gravity

Valeri P. Frolov, Andrei Zelnikov+1 more|arXiv (Cornell University)|Apr 1, 2015

Cosmology and Gravitation Theories27 references4 citations

TL;DR

This paper investigates the spherical collapse of small masses in ghost-free gravity, a non-local higher-derivative theory that avoids unphysical ghosts by modifying the gravitational propagator via an exponential function of the d'Alembertian. Using heat kernel methods and linearized equations, it demonstrates that the theory regularizes singularities in the gravitational potential and curvature for point masses and collapsing null shells across arbitrary spacetime dimensions, with the mass scale μ providing a UV cutoff that prevents black hole formation below a critical mass gap.

ABSTRACT

We discuss some properties of recently proposed models of a ghost-free gravity. For this purpose we study solutions of linearized gravitational equations in the framework of such a theory. We mainly focus on the version of the ghost-free theory with the exponential modification $\exp(\Box/μ^2)\Box^{-1}$ of the free propagator. The following three problems are discussed: (i) Gravitational field of a point mass; (ii) Penrose limit of a point source boosted to the speed of light; and (iii) Spherical gravitational collapse of null fluid. For the first problem we demonstrate that it can be solved by using the method of heat kernels and obtain a solution in a spacetime with arbitrary number of dimensions. For the second problem we also find the corresponding gyraton-type solutions of the ghost-free gravitational equations for any number of dimensions. For the third problem we obtain solutions for the gravitational field for the collapse of both "thin" and "thick" spherical null shells. We demonstrate how the ghost-free modification of the gravitational equations regularize the solutions of the linearized Einstein equations and smooth out their singularities.

Motivation & Objective

• To examine whether ghost-free gravity, which avoids unphysical ghosts via non-local operators, can resolve spacetime singularities in gravitational collapse.
• To analyze the gravitational field of a point mass in ghost-free gravity using heat kernel techniques in arbitrary dimensions D.
• To investigate the Penrose limit of a boosted point mass, obtaining gyraton-type solutions for ultra-relativistic sources.
• To study spherical collapse of null fluid (thin and thick shells), demonstrating how the theory smooths out singularities present in Einstein gravity.
• To determine whether a mass gap exists for black hole formation in this theory, based on curvature regularity and UV cutoff μ.

Proposed method

• The paper employs the heat kernel method to solve the linearized ghost-free gravitational equations for a point mass in D-dimensional spacetime, using the modified propagator exp(□/μ²)□⁻¹.
• It derives the gravitational potential for a static point mass by solving a modified Poisson-type equation involving the operator (l²△ + 1)△φ = 4πρ, where l is related to μ.
• For boosted sources, the Penrose limit is applied to obtain solutions resembling non-rotating gyratons, with the transverse field governed by a non-local operator a(−△)△φ = 4πρ.
• The method extends to spherical null shells by solving the linearized ghost-free equations for both thin and thick shells, using angular and radial integration techniques.
• The analysis uses coordinate transformations and delta-function constraints to evaluate multi-dimensional integrals over angles and spatial variables.
• It compares results with standard Einstein gravity, showing that ghost-free gravity yields finite potentials and curvatures at r=0, even for small masses.

Experimental results

Research questions

• RQ1 Does the ghost-free gravity model with the exponential modification exp(□/μ²)□⁻¹ yield a finite gravitational potential at r=0 for a point mass in D dimensions?
• RQ2 Can the Penrose limit of a boosted point mass be consistently applied in ghost-free gravity to obtain a gyraton-type solution for ultra-relativistic sources?
• RQ3 How does the spherical collapse of a thin or thick null shell behave in ghost-free gravity compared to Einstein gravity, particularly regarding curvature singularities?
• RQ4 What is the role of the mass scale μ in regularizing the gravitational field and preventing black hole formation for small masses?
• RQ5 Does the ghost-free theory predict a mass gap below which black holes cannot form, based on curvature finiteness?

Key findings

• The gravitational potential for a point mass in D-dimensional ghost-free gravity is finite at r=0 and regular everywhere, with the UV cutoff scale μ determining the degree of smoothing.
• For D ≥ 4, the solution obtained via the heat kernel method remains finite at the origin, confirming regularization of the Newtonian singularity.
• The Penrose limit of a boosted point mass yields a gyraton-type solution in D dimensions, with the transverse field satisfying a non-local equation involving the operator a(−△)△φ = 4πρ.
• The collapse of both thin and thick spherical null shells in ghost-free gravity results in a regular gravitational field, with curvature invariants remaining finite throughout spacetime.
• The theory exhibits a mass gap for black hole formation: for masses M below a critical value determined by μ, the curvature remains uniformly small, suggesting no black hole formation in the linearized regime.
• In D=4, the heat kernel result for the potential exactly matches the earlier Fourier-based solution from Frolov et al. (2014), confirming consistency across methods.