[Paper Review] Vacuum Decay in General Relativity

Thomas C. Bachlechner|arXiv (Cornell University)|Sep 13, 2018

Black Holes and Theoretical Physics22 references3 citations

TL;DR

This paper re-evaluates vacuum decay in general relativity using covariant, observable boundary conditions that fix the mass rather than the global three-geometry, leading to a new vacuum decay rate that is identical for true and false vacua. The key result is that in the weak gravity limit, the decay rate is $ B = \frac{27\pi^{2}\sigma^{4}}{2|\Delta\rho|^{3}} + \mathcal{O}(G) $, and the second variation of the action always has a negative mode, confirming barrier penetration.

ABSTRACT

We provide a novel, concise and self-contained evaluation of true- and false vacuum decay rates in general relativity. We insist on general covariance and choose observable boundary conditions, which yields the well known false-vacuum decay rate and a new true-vacuum decay rate that differs significantly from prior work. The rates of true- and false vacuum decays are identical in general relativity. The second variation of the action has a negative mode for all parameters. Our findings imply a new perspective on cosmological initial conditions and the ultimate fate of our universe.

Motivation & Objective

• To resolve the measure problem in vacuum decay by replacing unobservable global boundary conditions with locally observable, covariant ones.
• To derive a consistent vacuum decay rate in general relativity that does not depend on causally disconnected regions or coordinate choices.
• To show that true vacuum decay is possible and occurs at the same rate as false vacuum decay in general relativity.
• To establish that the second variation of the Euclidean action always has a negative mode, confirming non-perturbative tunneling.

Proposed method

• Imposes covariant boundary conditions by fixing the mass $\mathsf{M}$ and requiring the boundary metric to be in Schwarzschild-(anti)de Sitter form.
• Uses the new gravitational action $S_{\text{G}}$ from tbgravity, which vanishes for isotropic and stationary spacetimes, to ensure a well-posed variational problem.
• Derives the tunneling exponent $B$ via the Euclidean bounce action, with $\Gamma \propto e^{-B/\hbar}$, under these boundary conditions.
• Evaluates the second variation of the action using a parametrization by maximum radius $\hat{\mathsf{R}}(t_{\text{f}})$, showing $\partial^2 B / \partial \hat{\mathsf{R}}^2 < 0$.
• Assumes $\eta_\pi = \text{sgn}(\hat{\mathsf{R}}'_{+}\hat{\mathsf{R}}'_{-})$ to ensure a small tunneling probability and physical consistency.
• Applies the formalism to the weak gravity limit, recovering a rate that matches Coleman–de Luccia for false vacuum decay but extends to true vacuum decay.

Experimental results

Research questions

• RQ1 Does vacuum decay in general relativity yield the same rate for true and false vacua under consistent, observable boundary conditions?
• RQ2 Can true vacuum decay occur in general relativity, and if so, at what rate?
• RQ3 How does the choice of boundary conditions—specifically fixing global geometry versus fixing mass—affect the vacuum decay rate?
• RQ4 Does the second variation of the Euclidean action always possess a negative mode, confirming barrier penetration?
• RQ5 Is the decay rate independent of unobservable, causally disconnected regions or coordinate choices?

Key findings

• The vacuum decay rate is identical for true and false vacua in general relativity, with $\Gamma_{\rho_+ > \rho_-} = \Gamma_{\rho_+ < \rho_-}$.
• In the weak gravity limit, the decay rate exponent is $ B = \frac{27\pi^{2}\sigma^{4}}{2|\Delta\rho|^{3}} + \mathcal{O}(G) $, matching the non-gravitational result for false vacuum decay.
• The new boundary conditions—fixing mass $\mathsf{M}$ and requiring Schwarzschild-(anti)de Sitter form—eliminate dependence on unobservable global geometry and coordinate choices.
• The second variation of the Euclidean action always has a negative eigenvalue, as shown by $ \partial^2 B / \partial \hat{\mathsf{R}}^2 \propto -\eta_\pi / (\hat{\mathsf{R}}'_+ \hat{\mathsf{R}}'_-) < 0 $, confirming barrier penetration.
• True vacuum decay is possible in general relativity and can proceed unimpeded, implying low-entropy initial states for inflation may be naturally attainable.
• The result differs from prior work due to the use of observable, covariant boundary conditions instead of fixed global three-geometry, resolving a longstanding measure problem.