Testing general relativity’s no-hair theorem with x-ray observations of black holes

Despite its success in the weak gravity regime, general relativity (GR) has yet to be verified in the regime of strong gravity. In this paper, we present the results of detailed ray-tracing simulations aiming at clarifying if the combined information from x-ray spectroscopy, timing, and polarization observations of stellar mass and supermassive black holes can be used to test GR’s no-hair theorem. The latter states that stationary astrophysical black holes are described by the Kerr family of metrics, with the black hole mass and spin being the only free parameters. We use four “non-Kerr metrics,” some phenomenological in nature and others motivated by alternative theories of gravity, and study the observational signatures of deviations from the Kerr metric. Particular attention is given to the case when all the metrics are set to give the same innermost stable circular orbit in quasi–Boyer-Lindquist coordinates. We give a detailed discussion of similarities and differences of the observational signatures predicted for black holes in the Kerr metric and the non-Kerr metrics. We emphasize that even though some regions of the parameter space are nearly degenerate even when combining the information from all observational channels, x-ray observations of very rapidly spinning black holes can be used to exclude large regions of the parameter space of the alternative metrics. Although it proves difficult to distinguish between the Kerr and non-Kerr metrics for some portions of the parameter space, the observations of very rapidly spinning black holes like Cyg X-1 can be used to rule out large regions for several black hole metrics.

Figure 1

Diagram showing the thermal and coronal lamp-post emission surrounding a black hole.

Figure 2

$r_{\mathrm{ISCO}}$ as a function of deviation parameter for the JP metric (a) and the KN metric (b) with different spins illustrating the degeneracies within the metrics. The regions shaded grey indicate the portion of the parameter space excluded by observations of Cyg X-1 where the observed $r_{\mathrm{ISCO}}$ [56] is represented by the horizontal dashed black line.

Figure 3

Radial flux [Eq. (11)] (a) and power (b) for the JP, KN, and Kerr metrics, which all give $r_{\mathrm{ISCO}}=2.32r_g$. The bottom panels show the comparison of the alternative metrics to the Kerr metric.

Figure 4

Radial flux [Eq. (11)] (a) and power (b) for the GB, PMCC, and Kerr metrics, which all give $r_{\mathrm{ISCO}}=5.33r_g$. The bottom panels show the comparison of the alternative metrics to the Kerr metric.

Figure 5

Flux (top panel), polarization fraction (middle panel), and polarization degree (bottom panel) of the thermal disk emission for the JP, KN, and Kerr metrics (a) and corresponding comparisons of the alternative metrics with respect to the Kerr metric (b).

Figure 6

Flux (top panel), polarization fraction (middle panel), and polarization degree (bottom panel) for the reflected spectrum showing the $\mathrm{Fe}\text{−}\mathrm{K}\alpha$ line of an AGN with $h=3r_g$ for the JP, KN, and Kerr metrics (a) and corresponding comparisons of the alternative metrics with respect to the Kerr metric (b).

Figure 7

Orbital periods (top panels) for the metrics and the percent difference (bottom panels) for the JP and KN metrics compared to the Kerr metric (a) and the GB and PMCC metrics (b).

Figure 8

Lag-frequency spectrum (top panel) of an AGN with $h=3r_g$ for the JP, KN, and Kerr metrics along with the percent difference (bottom panel) of the KN and JP metrics when compared with the Kerr metric. A positive lag corresponds to the reflected emission lagging behind the direct emission where the direct emission is given in the 1 to 2 keV band and the reflected emission is in the 2–10 keV band.

Figure 9

Intensity-polarization maps of the three rapidly spinning metrics in Kerr (top left panel), JP (top right panel), KN (bottom left panel) for a black hole at a high inclination along with a comparison of their shadows (bottom right panel).

Figure 10

Photon rings of the Kerr and KN metrics as seen by an observer at both $i=25°$ (a) and $i=75°$ (b).