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Schwarzschild solution in brane induced gravity

Gregory Gabadadze, Alberto Iglesias

TL;DR

The paper derives an exact, nonperturbative Schwarzschild-like solution on the brane for brane-induced gravity (DGP model) and analyzes its bulk extension. It uncovers a mass-screening mechanism and a self-shielding curvature extending to the macroscopic scale $r_*=(r_M r_c^2)^{1/3}$, ensuring weak fields and reproducing 4D gravity with correct tensor structure at observable distances while avoiding the vDVZ discontinuity. The results reveal two branches of bulk behavior (regular and accelerated) and establish a precise relation between 4D Schwarzschild radius $r_M$ and an effective 5D mass parameter $\tilde{r}_M$, demonstrating nonperturbative control over gravity across regimes $r\ll r_*$, $r_*\ll r\ll r_c$, and $r\gg r_c$. The work emphasizes self-consistent nonlinear summation as essential to preserving GR-like phenomenology in modified gravity and discusses implications for cosmology and black hole physics on the brane. Altogether, the findings bolster the viability of brane-induced gravity as a framework for modifying gravity at large distances without triggering strong-coupling pathologies or vDVZ discontinuities.

Abstract

The metric of a Schwarzschild solution in brane induced gravity in five dimensions is studied. We find a nonperturbative solution for which an exact expression on the brane is obtained. We also find a linearized solution in the bulk and argue that a nonsingular exact solution in the entire space should exist. The exact solution on the brane is highly nontrivial as it interpolates between different distance scales. This part of the metric is enough to deduce an important property -- the ADM mass of the solution is suppressed compared to the bare mass of a static source. This screening of the mass is due to nonlinear interactions which give rise to a nonzero curvature outside the source. The curvature extends away from the source to a certain macroscopic distance that coincides with the would-be strong interaction scale. The very same curvature shields the source from strong coupling effects. The four dimensional law of gravity, including the correct tensorial structure, is recovered at observable distances. We find that the solution has no vDVZ discontinuity and show that the gravitational field on the brane is always weak, in spite of the fact that the solution is nonperturbative.

Schwarzschild solution in brane induced gravity

TL;DR

The paper derives an exact, nonperturbative Schwarzschild-like solution on the brane for brane-induced gravity (DGP model) and analyzes its bulk extension. It uncovers a mass-screening mechanism and a self-shielding curvature extending to the macroscopic scale , ensuring weak fields and reproducing 4D gravity with correct tensor structure at observable distances while avoiding the vDVZ discontinuity. The results reveal two branches of bulk behavior (regular and accelerated) and establish a precise relation between 4D Schwarzschild radius and an effective 5D mass parameter , demonstrating nonperturbative control over gravity across regimes , , and . The work emphasizes self-consistent nonlinear summation as essential to preserving GR-like phenomenology in modified gravity and discusses implications for cosmology and black hole physics on the brane. Altogether, the findings bolster the viability of brane-induced gravity as a framework for modifying gravity at large distances without triggering strong-coupling pathologies or vDVZ discontinuities.

Abstract

The metric of a Schwarzschild solution in brane induced gravity in five dimensions is studied. We find a nonperturbative solution for which an exact expression on the brane is obtained. We also find a linearized solution in the bulk and argue that a nonsingular exact solution in the entire space should exist. The exact solution on the brane is highly nontrivial as it interpolates between different distance scales. This part of the metric is enough to deduce an important property -- the ADM mass of the solution is suppressed compared to the bare mass of a static source. This screening of the mass is due to nonlinear interactions which give rise to a nonzero curvature outside the source. The curvature extends away from the source to a certain macroscopic distance that coincides with the would-be strong interaction scale. The very same curvature shields the source from strong coupling effects. The four dimensional law of gravity, including the correct tensorial structure, is recovered at observable distances. We find that the solution has no vDVZ discontinuity and show that the gravitational field on the brane is always weak, in spite of the fact that the solution is nonperturbative.

Paper Structure

This paper contains 10 sections, 62 equations, 2 figures.

Table of Contents

  1. Introduction, Discussions, and Summary
  2. The setup
  3. Structure of the solution
  4. Solution on the brane
  5. Solution in the bulk
  6. The solutions
  7. No vDVZ discontinuity
  8. Weak-field versus $G_N$-expansion
  9. Existence of the bulk solution
  10. Concluding remarks

Figures (2)

  • Figure 1: The solid line depicts $P(r)/r_M$ (on the vertical axis), wher $P$ is defined in (\ref{['Pdef']}), as a function of $r$ on the horizontal axis. The dashed line presents the function $0.28{r_*}/r$; The value of $r_*$ is set to 1 on this graph.
  • Figure 2: The solid line depicts the magnitude of the four-dimensional Ricci scalar curvature (on the vertical axes) as a function of $r$ on the horizontal axes. The dashed line depicts the dependence of the 4D Kretschmann scalar on $r$. The value of $r_*$ is set to 1 on this plot.