Positive energy from timelike singularities

Technical Summary: Positive Energy from Timelike Singularities

Problem Statement
The paper addresses the challenge of defining and computing gravitational energy in spacetimes containing timelike singularities, specifically within asymptotically Anti-de Sitter (AdS) and asymptotically flat solutions to the Einstein equations. While it is widely accepted that singularities require resolution via ultraviolet physics, their presence in classical general relativity poses issues for stability. Specifically, solutions such as black holes with negative masses or spacetimes with naked timelike singularities can yield arbitrarily negative energy if the singularity is smoothed out. This raises concerns regarding the existence of a stable ground state in any corresponding quantum theory. The central problem is to determine whether a consistent definition of energy exists for these spacetimes that accounts for the singularity's contribution, thereby preventing unphysical negative energy states.

Methodology
The author proposes a method to compute the total gravitational energy of time-translational invariant spacetimes with timelike singularities by modifying the on-shell Hamiltonian definition originally formulated by Hawking and Horowitz [1]. This definition, which aligns with the Abbott-Deser [2] and ADM [3] expressions, is adapted to handle the singularity through a regularization procedure:

1. Regulation: A timelike regulating cut-off boundary, $\Sigma_\epsilon$, is introduced to enclose the singularity. This surface is compatible with a foliation of the spacetime into constant-time slices $\{\Sigma_t\}$.
2. Boundary Decomposition: The regulated spacetime boundary consists of the asymptotic boundary ($\Sigma_\infty$), the initial and final time slices, and the regulating boundary near the singularity ($\Sigma_\epsilon$).
3. Energy Calculation: The total energy $E$ is computed as the sum of two distinct contributions:
   • $E_\infty$: The contribution from the asymptotic boundary.
   • $E_{sing}$: The contribution from the boundary near the singularity ($\Sigma_\epsilon$).
4. Limiting Procedure: Calculations are performed at a finite value of the regulator $\epsilon$. The final energy is obtained by taking the limit where the regulator approaches the singularity (e.g., $\epsilon \to z_0$ or $\epsilon \to \infty$ depending on the coordinate system).

The method is applied to three specific families of solutions:

• Asymptotically AdS metrics with Kasner-type timelike singularities (AdS-Kasner eons).
• Asymptotically AdS spaces including planar black holes with "negative masses."
• Schwarzschild metrics with positive and negative masses in asymptotically flat space.

Key Contributions and Results

1. AdS-Kasner Solutions:

   • The study analyzes a family of static radial solutions in $n+1$ dimensions characterized by Kasner parameters $p_a$.
   • It is found that the asymptotic contribution $E_\infty$ can be arbitrarily negative for certain ranges of the parameter $p_0$.
   • Crucially, the contribution from the singularity, $E_{sing}$, is always finite, positive, and well-defined (independent of the specific regulating surface used).
   • For all solutions in this family that possess a timelike singularity, the total energy $E = E_\infty + E_{sing}$ is strictly positive.
   • The AdS soliton and the AdS planar black hole are identified as the only non-singular limits in this family; the soliton retains negative energy relative to AdS space, while the black hole has positive energy.

2. AdS Black Holes with "Negative Masses":

   • For a family of AdS solutions with $f(z) = 1 + \bar{z}^n$ (representing negative mass parameters), the asymptotic contribution $E_\infty$ is again found to be arbitrarily negative.
   • The singularity contribution $E_{sing}$ is positive and exactly compensates for the negative asymptotic term.
   • The minimal energy solution corresponds to the black hole with negative mass, which possesses the same total energy as the pure AdS background (zero relative energy).

3. Schwarzschild Metrics (Positive and Negative Mass):

   • In the asymptotically flat case, the positive mass Schwarzschild black hole yields a positive energy relative to Minkowski space, derived solely from $E_\infty$.
   • For the negative mass case ($m < 0$), there is no event horizon, only a timelike singularity at $r=0$.
   • Here, both $E_\infty$ and $E_{sing}$ contribute. The asymptotic term is negative, while the singularity term is positive.
   • The sum $E = E_\infty + E_{sing}$ vanishes, implying that a Schwarzschild black hole with negative mass has the same energy as Minkowski space. Unlike the AdS cases, the reference background contribution is necessary here to ensure both terms remain finite.

Significance and Claims
The paper claims that the energy contribution from a timelike singularity is finite and positive, dependent on the local geometry, and sufficient to render the total gravitational energy non-negative in all examined cases containing such singularities. In most instances, this results in a strictly positive total energy; however, specific minimal energy solutions (such as the negative mass AdS black hole and the negative mass Schwarzschild black hole) yield a total energy of zero relative to their respective backgrounds. This result supports the view that timelike singularities are necessary to eliminate unphysical regular negative energy solutions.

The author emphasizes that without including the singularity's contribution, one would encounter gravitational fields with arbitrarily negative energy, which would likely preclude a stable ground state in a quantum theory. The analysis positions gravitational energy as a probe for singularities, drawing a parallel with recent work on holographic complexity in the AdS/CFT correspondence, where similar regularization techniques are used to show that singularities contribute positively to action complexity. The paper concludes that geometries with timelike singularities are not candidates for stable ground states in the absence of these positive energy contributions, reinforcing the necessity of the singularity for the consistency of the classical theory.