[Paper Review] Source integrals for multipole moments in static and axially symmetric spacetimes

Norman Gürlebeck|arXiv (Cornell University)|Jul 18, 2012

Black Holes and Theoretical Physics4 references4 citations

TL;DR

This paper derives source integrals for multipole moments in static and axially symmetric spacetimes, expressing asymptotically defined Geroch-Hansen multipole moments via quasi-local volume or surface integrals over regions enclosing all sources. The key contribution is a systematic construction of these integrals using the linear system underlying the Einstein equations in this symmetry class, enabling direct linkages between interior source distributions and exterior gravitational multipole moments.

ABSTRACT

In this article, we derive source integrals for multipole moments in axially symmetric and static spacetimes. The multipole moments can be read off the asymptotics of the metric close to spatial infinity in a hypersurface, which is orthogonal to the timelike Killing vector. Whereas for the evaluation of the source integrals the geometry needs to be known in a compact region of this hypersurface, which encloses all source, i.e. matter as well as singularities. The source integrals can be written either as volume integrals over such a region or in quasi-local form as integrals over the surface of that region.

Motivation & Objective

• To establish a direct connection between asymptotic multipole moments (e.g., Geroch-Hansen) and the distribution of mass and energy in the source region of static, axially symmetric spacetimes.
• To overcome the nonlinearity of the Einstein equations by exploiting the existence of a linear system in this symmetry class, enabling the construction of source integrals where none were previously known.
• To provide a framework for computing multipole moments from interior matter and geometry, applicable to equilibrium configurations like relativistic stars or isolated black holes.
• To enable comparison between numerical, analytical, and approximate solutions by extracting their multipole moments via these integrals.
• To clarify the relationship between these new source integrals and existing definitions, such as those for isolated horizons or Komar masses.

Proposed method

• Derives source integrals for Weyl multipole moments using the linear system associated with static, axially symmetric spacetimes, which allows reformulation of the Einstein equations in a linear form.
• Expresses multipole moments as quasi-local volume integrals over a compact region enclosing all sources (matter and singularities), or as surface integrals over the boundary of that region.
• Utilizes the inverse scattering technique and the existence of a linear system to ensure integrability and consistency of the derived expressions.
• Applies the formalism to recover known results, such as the Komar mass, by showing that the first multipole moment reduces to the Komar integral in vacuum or when $ W = \rho $.
• Demonstrates the method’s utility by proving the non-existence of static, axially symmetric dust configurations, as their multipole moments vanish identically.
• Establishes that the source integrals are constructed to exactly reproduce the asymptotically defined Geroch-Hansen moments by design, ensuring consistency with established definitions.

Experimental results

Research questions

• RQ1 Can asymptotic multipole moments in static, axially symmetric spacetimes be expressed as integrals over a compact source region, despite the nonlinearity of the Einstein equations?
• RQ2 How do the derived source integrals relate to existing definitions such as Komar mass, Geroch-Hansen moments, and isolated horizon multipole moments?
• RQ3 What conditions must be satisfied for the source integrals to reduce to Newtonian-like forms, and when do non-matter contributions arise due to spacetime curvature?
• RQ4 Can the formalism be generalized to electrovacuum or stationary spacetimes, and how does it handle strong-field configurations with caustics or singularities?
• RQ5 To what extent can source integrals be used to constrain the equation of state of relativistic stars from observed multipole moments?

Key findings

• Source integrals for all multipole moments in static, axially symmetric spacetimes can be derived using the underlying linear system, overcoming the nonlinearity of the Einstein equations.
• The first multipole moment (mass) derived via the volume integral reduces exactly to the Komar integral when $ W = \rho $, confirming consistency with standard definitions.
• The formalism allows the construction of exterior solutions from known interior solutions by computing their multipole moments via source integrals.
• The method proves the non-existence of static, axially symmetric dust configurations, as their multipole moments vanish identically, implying flat spacetime in vacuum.
• The source integrals are constructed to exactly reproduce the asymptotically defined Geroch-Hansen moments, ensuring compatibility with established asymptotic definitions.
• Non-matter contributions in the integrals arise when $ W \neq \rho $, reflecting the influence of spacetime curvature on multipole moments, even in vacuum.