Black Hole Spectroscopy: Testing General Relativity through Gravitational Wave Observations

TL;DR

This work proposes a definitive test of general relativity in the strong-field regime by exploiting the Kerr black hole ringdown spectrum, which consists of damped sinusoids with complex frequencies $\omega_{n\ell m}$ determined uniquely by the black hole’s mass $M$ and spin parameter $a$. By observing two or more QNMs and requiring a single $(a,M)$ pair to explain all modes, the method tests the no-hair theorem and GR; the authors develop a frequentist confidence-interval framework and demonstrate its viability with a numerical LISA-like example, including false alarm $\alpha$ and false dismissal $\beta$ analyses. The results show that, given sufficient signal-to-noise, LISA could identify or rule out Kerr BHs across cosmological volumes, enabling robust discrimination between black holes and exotic compact objects. This approach provides a direct, measurable test of strong-field GR and the BH no-hair conjecture, with practical applicability to future gravitational-wave observations.

Abstract

Assuming that general relativity is the correct theory of gravity in the strong field limit, can gravitational wave observations distinguish between black hole and other compact object sources? Alternatively, can gravitational wave observations provide a test of one of the fundamental predictions of general relativity? Here we describe a definitive test of the hypothesis that observations of damped, sinusoidal gravitational waves originated from a black hole or, alternatively, that nature respects the general relativistic no-hair theorem. For astrophysical black holes, which have a negligible charge-to-mass ratio, the black hole quasi-normal mode spectrum is characterized entirely by the black hole mass and angular momentum and is unique to black holes. In a different theory of gravity, or if the observed radiation arises from a different source (e.g., a neutron star, strange matter or boson star), the spectrum will be inconsistent with that predicted for general relativistic black holes. We give a statistical characterization of the consistency between the noisy observation and the theoretical predictions of general relativity, together with a numerical example.

Figures (7)

• Figure 1: The dimensionless, complex QNM frequencies $\Omega_{n{\ell}m}$ for rotating, uncharged black holes. Each family of curves corresponds to one $n{\ell}$ pair, and each branch to a possible value of $m$. The large black dot at the base of each family is the Schwarzschild ($a=0$) limit, where the frequencies are degenerate in $m$. This degeneracy is broken for $a\neq0$, and the curves emanating from the dots give the QNM frequencies for Kerr black holes as a function of positive $a$ for different $m$. In this figure $a$ ranges from $0$ to $0.9958$, with the small diamonds marking the QNM frequencies for $a=0.4, 0.6, 0.8, 0.9,$ and $0.98$. On this figure, an observation, corresponding to a (complex) frequency $\omega$, is represented by the line $\Omega=M\omega$, parameterized by the (unknown) black hole mass $M$. Each intersection of this line with a QNM curve in dimensionless $\Omega$ represents a candidate $n{\ell}m$, $M$ and $a$ for the mode.
• Figure 2: Here we show, in schematic form, several $\Omega_{n{\ell}m}(a)$ curves and their intersection with the lines $M\omega_i$, $M>0, i=1,2$, corresponding to two observed modes. These two lines we denote by $+$ and $\times$ respectively. (b) The candidate $(a,M)$ pairs determined in figure (a) are plotted here in the $(a,M)$--plane. The pairs belonging to $\omega_1$ are denoted by $+$, the ones belonging to $\omega_2$ by $\times$. There is only one candidate $(a,M)$ consistent with both observations --- indicated by the overlapping $+$ and $\times$ --- and this is the actual mass and angular momentum of the underlying black hole.
• Figure 3: A reformulation of the consistency criterion. A set of quasi-normal modes ${\cal Q}=\{(n_k,l_k,m_k) : k=1,\ldots,N\}$ corresponds to a surface in the $(2N + 2)$-dimensional space depicted in this figure. A measurement $\boldsymbol{\omega} = (\omega_1,\ldots,\omega_N)$ is consistent with general relativity if the constant surface that is obtained by ranging over all $(a,M)$ while keeping the frequencies $\boldsymbol{\omega}$ fixed intersects at least one of the surfaces corresponding to one of the sets $\cal Q$. This intersection is indicated in this figure by a dot.
• Figure 4: The construction of classical confidence intervals. A sampling distribution $P(x\vert\mu)$, an ordering principle, and a probability $p$ are needed to construct a confidence interval. The ordering principle is used to find the intervals $J(\mu)$ such that $\int_{J(\mu)} dx\; P(x\vert\mu) = p$. The classical confidence interval $\mathcal{R}$ is then given by the set of $\mu$ for which $J(\mu)$ contains the measured value $x_0$.
• Figure 5: The construction of classical confidence intervals generalized to higher dimensions. Given a sampling distribution $P$, an ordering principle, and a probability $p$ one can construct classical confidence regions $R$ just as in the one-dimensional case. The difference here is that we are now trying to determine a small number of parameters $(a,M)$ from a larger number of observations $\boldsymbol{\omega} = (\omega_1,\ldots,\omega_N)$. There are thus additional consistency conditions that need to be satisfied to obtain a non-empty confidence region $R$. This is the basis of our proposed test.