Vacuum Spacetime With Multipole Moments: The Minimal Size Conjecture, Black Hole Shadow, and Gravitational Wave Observables

TL;DR

This paper constructs explicit vacuum solutions with multipole moments, proposes a minimal size conjecture for such objects, and explores implications for black hole shadows and gravitational wave signals to aid spacetime characterization.

Contribution

It introduces a new explicit multipole spacetime solution, proposes a minimal size scale for objects with large multipole moments, and analyzes observational signatures in shadows and gravitational waves.

Findings

Minimal size scales as (M_n)^{1/(n+1)} for multipole moments.

Black hole shadow shapes vary with multipole moments, offering observational constraints.

Gravitational wave phase shifts depend on multipole moments, aiding spacetime mapping.

Abstract

In this work, we explicitly construct the vacuum solution of Einstein's equations with prescribed multipole moments. By observing the behavior of the multipole spacetime metric at small distances, we conjecture that for a sufficiently large multipole moment, there is a minimal size below which no object in nature can support such a moment. The examples we have investigated suggest that such minimal size scales as $(M_n)^{1/(n+1)}$ (instead of $(M_n/M)^{1/n}$), where $M$ is the mass and $M_n$ is the $n$th order multipole moment. With the metric of the "multipole spacetime", we analyze the shape of black hole shadow for various multipole moments and discuss the prospects of constraining the moments from shadow observations. In addition, we discuss the shift of gravitational wave phase with respect to those of the Kerr spacetime, for a test particle moving around an object with this set of multipole moments. These phase shifts are required for the program of mapping out the spacetime multipole moments based on gravitational wave observations of extreme mass-ratio inspirals.