Testing the black hole "no-hair" hypothesis

TL;DR

This review analyzes the no-hair hypothesis for black holes in General Relativity, detailing why Kerr geometry, specified by mass and spin, is expected to describe astrophysical BHs and how this can be tested. It covers both dynamical (gravitational-wave ringdown and inspiral) and non-dynamical (multipole moments and electromagnetic observations) tests, emphasizing the role of quasinormal modes, multipole structure, and horizon-proximity signals. While numerous hairy BH solutions exist in extended theories, most are either dynamically unstable or observationally subdominant, keeping Kerr as the leading description for real BHs; nonetheless, forthcoming third-generation GW detectors and high-resolution EM observatories will push tests to the regime where deviations, if present, could be detected. The work highlights practical strategies to constrain non-Kerr parameters, assess environmental effects, and leverage multiple observational channels (stars, pulsars, disks, shadows) to map the strong-field spacetime around BHs and probe fundamental physics.

Abstract

Black holes in General Relativity are very simple objects. This property, that goes under the name of "no-hair," has been refined in the last few decades and admits several versions. The simplicity of black holes makes them ideal testbeds of fundamental physics and of General Relativity itself. Here we discuss the no-hair property of black holes, how it can be measured in the electromagnetic or gravitational window, and what it can possibly tell us about our universe.

Figures (15)

• Figure 1: Four different physical processes leading to substantial quasinormal ringing. In all of them, quasinormal ringing is clearly visible. The upper-left panel (adapted from Ref. Berti:2007fi) is the signal from two equal-mass BHs initially on quasi-circular orbits, inspiralling towards each other due to the energy loss induced by GW emission, merging and forming a single final BH. The upper-right panel shows gravitational waveforms from numerical simulations of two equal-mass BHs, colliding head-on with $v/c=0.94$ in the center-of-mass frame: as the center-of-mass energy grows (i.e., as the speed of the colliding BHs tends to the speed of light) the waveform is more and more strongly ringdown-dominated Sperhake:2008ga. The bottom-left panel shows the gravitational waveform (or more precisely, the dominant, $l=2$ multipole of the Zerilli function) produced by a test particle of mass $\mu$ falling from rest into a Schwarzschild BH Davis:1971gg: the shape of the initial precursor depends on the details of the infall, but the subsequent burst of radiation and the final ringdown are universal features. The bottom-right panel (reproduced from Ref. Baiotti:2008ra) shows GWs emitted by two massive neutron stars with a polytropic equation of state, inspiralling and eventually collapsing to form a single BH. With the exception of the infalling-particle case (where $M$ is the BH mass, $\mu$ the particle's mass and $\psi_2$ the Zerilli wavefunction), $\psi_{22}$ is the $l=m=2$ multipolar component of the Weyl scalar $\Psi_4$, $M$ denotes the total mass of the system and $r$ the extraction radius (see e.g. Ref. Berti:2007fi). Taken from Ref. Berti:2009kk.
• Figure 2: Frequencies and quality factors for the fundamental modes with $l=2,~3,~4$ and different values of $m$. Solid lines refer to $m = l, .., 1$ (from top to bottom), the dotted line to m = 0, and dashed lines refer to $m = -1, ..,-l$ (from top to bottom). Quality factors for the higher overtones are lower than the ones we display here. Taken from Ref. Berti:2009kk.
• Figure 3: Evolution of a Gaussian profile of a massless scalar field with width $w=2~M$ centered at $r_0=12~M$ around a Schwarzschild (left panel) and a Kerr BH with $a/M = 0.99$ (right panel). We depict the $l=m=0$ (solid black line) and $l=m=1$ (red dashed line) multipoles. The multipolar components of the field were extracted at $r_{\rm ex}=10$. The waveform displays an early transient followed by an exponentially decaying sinusoid (QNM ringdown) and a power-law tail at late times. The late-time power-law tail has the form $t^{p}$ for the monopole, with $p=-3.08$ for $a/M=0$ and $p=-3.07$ for $a/M=0.99$ in good agreement with the prediction $p=-3$ obtained from the low-frequency expansion of the wave equation (\ref{['tail_massless']}). From Ref. Witek:2012tr.
• Figure 4: Evolution of (the dipole component of) a scalar Gaussian wavepacket in a background Kerr geometry, with $a=0.99M$. The scalar has mass parameter $M\mu_S = 0.42$. From Ref. Witek:2012tr.
• Figure 5: The gravitational field of the Earth (known as the Potsdam Potato), based on data from the LAGEOS, GRACE, and GOCE satellites and surface data. Gravitational field strength is represented by elevation and color. Credit: CHAMP CHAMP, GRACE GRACE, Research Center for Geophysics (GFZ) GFZ, NASA NASA, DLR DLR.