[Paper Review] Is mass conformally invariant?

K. C. K. Chan|ArXiv.org|Jan 14, 1997

Relativity and Gravitational Theory1 references3 citations

TL;DR

This paper investigates whether mass is conformally invariant in general relativity by analyzing static, spherically symmetric black hole solutions using the Hawking-Horowitz reference action. It demonstrates that the asymptotic mass parameter is not invariant under conformal transformations, challenging the assumption of conformal invariance for mass in gravitational systems.

ABSTRACT

By using the Garfinkle, Horowitz and Strominger black hole solutions as examples, we illustrate that, with respect to the reference action functional proposed by Hawking and Horowitz, the asymptotic mass parameter is not invariant between two conformally related static spherically symmetric metrics.

Motivation & Objective

• To assess whether the concept of mass in general relativity is invariant under conformal transformations.
• To investigate the behavior of the asymptotic mass parameter in conformally related spacetimes.
• To test the conformal invariance of mass using explicit black hole solutions as a probe.
• To evaluate the consistency of the Hawking-Horowitz reference action in defining mass under conformal rescaling.
• To clarify foundational issues in defining mass in conformally transformed gravitational systems.

Proposed method

• Utilizes the Garfinkle-Horowitz-Strominger (GHS) black hole solutions as explicit examples of static, spherically symmetric spacetimes.
• Applies the Hawking-Horowitz reference action formalism to compute the asymptotic mass in both the original and conformally related metrics.
• Performs a conformal transformation on the metric while preserving the spherical symmetry and static nature of the solution.
• Compares the computed mass parameter before and after the conformal transformation to assess invariance.
• Analyzes the boundary terms and surface integrals in the action to extract the mass in the asymptotic region.
• Relies on the standard ADM-like mass formula derived from the reference action for stationary, asymptotically flat spacetimes.

Experimental results

Research questions

• RQ1 Is the asymptotic mass parameter invariant under conformal transformations of the spacetime metric?
• RQ2 How does the Hawking-Horowitz reference action behave when applied to conformally related metrics?
• RQ3 Do the GHS black hole solutions exhibit conformal invariance of mass under metric rescaling?
• RQ4 What is the role of boundary terms in the action when computing mass in conformally related spacetimes?
• RQ5 Can mass be considered a conformally invariant quantity in general relativity?

Key findings

• The asymptotic mass parameter is not invariant under conformal transformations of the metric.
• The Hawking-Horowitz reference action yields different mass values in conformally related spacetimes, even when the physical content remains similar.
• The GHS black hole solution provides a concrete counterexample to conformal invariance of mass.
• The mass computed via the reference action depends on the conformal factor, indicating that mass is not a conformal scalar.
• The result implies that mass cannot be treated as a conformally invariant quantity in general relativity.
• The analysis shows that conformal transformations alter the asymptotic structure in a way that affects the definition of mass.