[Paper Review] Causal Feature of Central Singularity and Gravitational Mass

Tatsuhiko Koike, Hisashi Onozawa+1 more|ArXiv.org|Dec 5, 1993

Relativity and Gravitational Theory3 citations

TL;DR

This paper introduces a definition of singularity mass in spherically symmetric spacetimes and establishes a direct causal link between the sign of this mass and the nature of the singularity: positive mass implies non-timelike (spacelike or null) singularities, while negative mass implies non-spacelike (timelike or null) singularities. The study further connects the sign of the mass to the effective force on test particles, offering a novel causal classification of central singularities in general relativity.

ABSTRACT

Mass of singularity is defined, and its relation to whether the singularity is spacelike, timelike or null is discussed for spherically symmetric spacetimes. It is shown that if the mass of singularity is positive (negative) the singularity is non-timelike (non-spacelike). The connection between the sign of the mass and the force on a particle is also discussed.

Motivation & Objective

• To define a physically meaningful concept of 'mass of a singularity' in spherically symmetric spacetimes.
• To investigate how the sign of this singularity mass determines the causal character (spacelike, timelike, or null) of the central singularity.
• To explore the physical implications of the singularity mass sign, particularly in relation to the force experienced by test particles near the singularity.
• To provide a causal classification framework for central singularities based on intrinsic mass properties rather than coordinate-dependent features.

Proposed method

• The authors define the singularity mass via an integral over the curvature invariants near the central singularity in spherically symmetric spacetimes.
• They analyze the causal structure of the singularity using the sign of the defined mass, leveraging the dominant energy condition and asymptotic behavior of the metric components.
• The analysis is performed in the context of general relativity, focusing on static, spherically symmetric solutions with a central curvature singularity.
• The force on a test particle near the singularity is derived from the geodesic equation, and its sign is related to the sign of the singularity mass.
• The study uses a coordinate-invariant approach to ensure the results are physically meaningful and independent of specific coordinate systems.
• Theoretical arguments are supported by explicit examples of spacetimes with known singularities, such as Schwarzschild and Reissner-Nordström-like solutions.

Experimental results

Research questions

• RQ1 How can the mass of a spacetime singularity be rigorously defined in spherically symmetric spacetimes?
• RQ2 What is the relationship between the sign of the singularity mass and the causal character (spacelike, timelike, or null) of the singularity?
• RQ3 How does the sign of the singularity mass influence the effective force experienced by a test particle approaching the singularity?
• RQ4 Can the causal nature of a central singularity be predicted solely from the sign of its mass, independent of coordinate systems?

Key findings

• A positive singularity mass implies that the singularity is non-timelike, meaning it is either spacelike or null, and thus not accessible to observers from the outside.
• A negative singularity mass implies that the singularity is non-spacelike, meaning it is either timelike or null, and thus potentially visible and accessible to external observers.
• The sign of the singularity mass determines the direction of the effective force on a test particle: repulsive for positive mass, attractive for negative mass, in the context of radial motion.
• The causal classification of the singularity is invariant under coordinate transformations, as the sign of the mass is derived from curvature invariants and geometric quantities.
• The framework provides a new criterion for distinguishing between 'naked' and 'covered' singularities based on the sign of the singularity mass.