Testing black hole candidates with electromagnetic radiation

Astrophysical black hole candidates are thought to be the Kerr black holes of general relativity, but there is not yet direct observational evidence that the spacetime geometry around these objects is described by the Kerr solution. The study of the properties of the electromagnetic radiation emitted by gas or stars orbiting these objects can potentially test the Kerr black hole hypothesis. This paper reviews the state of the art of this research field, describing the possible approaches to test the Kerr metric with current and future observational facilities and discussing current constraints.

Figure 1

Radial coordinates of the event horizon $r_{+}$, the photon circular orbit $r_{\gamma }$, the marginally bound circular orbit $r_{\mathrm{mb}}$, and the ISCO $r_{\mathrm{ISCO}}$ in the Kerr metric as functions of the spin parameter $a_{*}$. For every radius, the upper curve refers to the counterrotating orbits, and the lower curve to the corotating orbits.

Figure 2

Sketches of 22 binaries with a stellar-mass BH candidate confirmed by dynamical measurements. For every system, the BH accretion disk is on the left and the companion star is on the right. The orientation of the disks indicates the inclination angles $i$ of the binaries. The distorted shapes of the stellar companions are due to the gravitational fields of the BH candidates. The size of the latter should be about 50 km, compared with the distance of the Sun to Mercury of about 50 million km and the radius of the Sun of 0.7 million km (top left corner). From Jerome Orosz.

Figure 3

Contour levels of the (left panel) ISCO radius $r_{\mathrm{ISCO}}$ and the (right panel) Novikov-Thorne radiative efficiency ${\eta }_{\mathrm{NT}}=1-E_{\mathrm{ISCO}}$ in the JP metric with nonvanishing deformation parameter ${\epsilon }_3$. The black dotted lines separate spacetimes with a regular event horizon (on the left of the line) from those with topologically nontrivial event horizons and naked singularities (on the right). See the text for more details.

Figure 4

Corona-disk model with lamppost geometry. The BH is surrounded by a geometrically thin and optically thick accretion disk. The accretion disk radiates like a blackbody locally and as a multicolor blackbody when integrated radially. The hot electron cloud called a corona acts as an x-ray source and is located just above the BH. The power-law component represents the direct radiation from the hot corona. The latter also illuminates the disk, producing a reflection component and some emission lines, the most prominent of which is usually the iron $K\alpha$ line. From Jiachen Jiang.

Figure 5

Impact of the model parameters on the thermal spectrum of a thin disk: (top left panel) mass $M$, (top right panel) mass accretion rate $\dot{M}$, (central left panel) viewing angle $i$, (central right panel) distance $d$, (bottom left panel) spin parameter $a_{*}$, and (bottom right panel) JP deformation parameter ${\epsilon }_3$. When not shown, the values of the parameters are $M=10M_{\odot }$, $\dot{M}=2\times {10}^{18}\mathrm{g}{\mathrm{s}}^{-1}$, $d=10\mathrm{kpc}$, $i=45°$, $a_{*}=0.7$, and ${\epsilon }_3=0$. $M$ is in units of $M_{\odot }$, $\dot{M}$ is in units of ${10}^{18}\mathrm{g}{\mathrm{s}}^{-1}$, $d$ is in kpc, the flux density $N_{E_{\text{obs}}}$ is in photons ${\mathrm{keV}}^{-1}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, and the photon energy $E_{\text{obs}}$ is in keV.

Figure 6

Continuum-fitting constraints on the JP parameter ${\epsilon }_3$ from A0620-00 (top left panel), LMC X-3 (top right panel), XTE J1550-564 (bottom left panel), and 4U 1543-47 (bottom right panel). The spacetimes along the solid red lines cannot be distinguished and represent the central values of the measurement. In every panel, the dash-dotted green lines are the boundaries of the allowed region ($1\sigma$ error) along the solid red line inferred within the analysis from current x-ray measurements. The dashed blue lines are the same boundaries inferred if the best fit were exactly for ${\epsilon }_3=0$. The dotted black curve on the right of each panel separates spacetime with a regular exterior (on the left of the curve) from those with naked singularities (on the right of the curve). From [179].

Figure 7

Continuum-fitting constraints on the JP parameter ${\epsilon }_3$ from $\mathrm{GRS}1915+105$ (left panel) and Cygnus X-1 (right panel). The allowed regions are those inside the dashed blue lines. The dotted black curve separates spacetimes with a regular exterior (on the left of the curve) from those with naked singularities (on the right of the curve). From [179].

Figure 8

Continuum-fitting constraints on the CPR parameter ${\epsilon }_3^r$ from Cygnus X-1. The allowed region is between the two dashed blue lines. From [37].

Figure 9

Photon count (in black) and fit without the iron line (in red) of MGC-6-30-15. From [71].

Figure 10

Impact of the model parameters on the iron line profile: (top left panel) viewing angle $i$, (top right panel) emissivity index $q$, (bottom left panel) spin parameter $a_{*}$, and (bottom right panel) JP deformation parameter ${\epsilon }_3$. When not shown, the values of the parameters are $i=45°$, $q=3$, $a_{*}=0.7$, and ${\epsilon }_3=0$. The outer radius is $r_{\text{out}}=r_{\mathrm{ISCO}}+100$ $M$. $E_{\text{obs}}$ is the photon energy measured by the observer far from the BH. The vertical axis is the number flux of photons in arbitrary units. The iron line profiles are computed with the code described by [32].

Figure 11

$\mathrm{\Delta }{\chi }^2$ contours with (top panels) $N_{\text{line}}=10^3$ and (bottom panels) $N_{\text{line}}=10^4$ from simulations of measurements of iron line profiles. The background geometry is the CPR metric with ${\epsilon }_3^t$ and ${\epsilon }_3^r$ as the only possible nonvanishing deformation parameters. The reference model is a Kerr BH with spin parameter $a_{*}^{\prime }=0.95$ and inclination angle $i^{\prime }=70°$. In the left panels, we allow for a nonvanishing ${\epsilon }_3^t$ while we assume ${\epsilon }_3^r=0$. In the right panels, we illustrate the converse measurement, namely, ${\epsilon }_3^t=0$ while ${\epsilon }_3^r$ can vary. The ratio between the continuum and the iron line photon flux as well as the photon index of the continuum are also free parameters in the fit. The labels along the contour levels refer to the value of $\mathrm{\Delta }{\chi }^2$. See the text for more details. From [163].

Figure 12

Iron line profiles produced from the accretion disks around (left panel) a class of exotic compact stars and (right panel) a certain family of traversable wormholes. $E_{\text{obs}}$ is the photon energy measured by the observer far from the BH. The vertical axis is the number flux of photons in arbitrary units. See the text for more details. The iron line profiles are computed with the code described by [32], employing the metrics discussed in [48] and [33].

Figure 13

Examples of iron line profile from a Kerr BH with spin parameter $a_{*}=0.95$ and observed from an inclination angle $i=45°$. The emissivity is $I_{\mathrm{e}}\propto (r_{\text{break}}/r{)}^q$ for $r<r_{\text{break}}=5$ $M$ and $I_{\mathrm{e}}\propto (r_{\text{break}}/r{)}^3$ for $r>r_{\text{break}}$. See the text for more details. The iron line profiles are computed with the code described by [32].

Figure 14

Compare to Fig. 11. $\mathrm{\Delta }{\chi }^2$ contours with (top panels) $N_{\text{line}}=10^3$ and (bottom panels) $N_{\text{line}}=10^4$ from the analysis of the 2D transfer function. The reference model is a Kerr BH with spin parameter $a_{*}^{\prime }=0.95$ and inclination angle $i^{\prime }=70°$. In the left panels, we allow for a nonvanishing ${\epsilon }_3^t$ and assume ${\epsilon }_3^r=0$. In the right panels, we consider the converse case, namely, ${\epsilon }_3^t=0$ and ${\epsilon }_3^r$ can vary. The height of the source, the ratio between the continuum and the iron line photon flux, and the photon index of the continuum are also left as fit parameters. The labels along the contour levels refer to the value of $\mathrm{\Delta }{\chi }^2$. See the text for more details. From [163].

Figure 15

(Left panel) Polarization degree and (right panel) polarization angle as a function of the photon energy for a Schwarzschild BH (solid lines) and a Kerr BH with spin parameter $a/M=0.9$ (dashed lines) and viewing angles $i=45°$, 60°, and 75°. From [197].

Figure 16

Ray-tracing calculations of the direct image of a Kerr BH surrounded by an optically thin emitting medium. The dark area is the BH shadow and its boundary corresponds to the apparent photon capture sphere. From [119].

Figure 17

The function $R(\varphi )$ is defined as the distance between the center C and the boundary of the shadow at the angle $\varphi$ as shown in this picture. See the text for more details. From [135].

Figure 18

BH shadows and $R$ functions for CPR BHs with different values of the spin parameter $a_{*}$, the deformation parameters ${\epsilon }_3^t$ and ${\epsilon }_3^r$, and the inclination angle $i$. From [135].

Figure 19

Posterior probability density of spin and deformation parameters, marginalized over all other quantities, from the existing EHT data. The solid, dashed, and dotted white lines correspond, respectively, to the $1\sigma$, $2\sigma$, and $3\sigma$ boundaries. The dashed gray lines correspond (from top to bottom) to spacetimes with $r_{\mathrm{ISCO}}/M=6$, 5, and 4. The grayed region in the lower right corresponds to a metric with $r_{\mathrm{ISCO}}/M<4$ and it is neglected because the calculations may be affected by some pathological properties of this non-Kerr metric. Here $\epsilon$ is the deformation parameter of the non-Kerr metric of [142]. From [74].

Figure 20

Precession time scale from the mass ($M$), the spin ($S$), the quadrupole moment ($Q$), and stellar perturbation ($P$) for a pulsar orbiting ${\mathrm{SgrA}}^{*}$ as a function of orbital period $P_b$, assuming an orbital eccentricity of 0.5 and $10^3$ objects, all with mass $M=M_{\odot }$, within 1 mpc around ${\mathrm{SgrA}}^{*}$. From [200].

Figure 21

Light curve of a near-infrared flare of ${\mathrm{SgrA}}^{*}$ with the characteristic quasiperiodic substructure with a time scale of about 20 minutes. The arrows at the bottom indicate the peaks of the substructure. From [132] and [140].

Figure 22

Spectrogram of a hot spot orbiting a Schwarzschild BH at the ISCO radius from an inclination angle $i=60°$. The vertical axis is the ratio between the photon energy measured by a distant observer and the photon energy at the emission point. The color scale indicates the energy flux (in arbitrary units). We see two tracks because one is the spectrogram of the primary image, the other one is for the secondary image. The points labeled 1–5 refer to another figure in the original paper and can be ignored here. From [193].

Figure 23

Assuming that the hot spot is at the ISCO radius, the measurement of its light curve can select only the spacetimes with the same orbital frequency. In this plot, the reference model employs a Kerr BH with spin parameter $a_{*}=0.4$. The allowed regions are those between the two dotted blue lines ($\mathrm{\Delta }{\chi }^2<1$) and the two dashed red lines ($\mathrm{\Delta }{\chi }^2<10$). The possible measurement of the spin through a binary pulsar would provide the constraint given by the shaded yellow area, and the combination of the two observations could potentially break the degeneracy between $a_{*}$ and ${\epsilon }_3$. See the text for more details. From [193].

Figure 24

Left panel: Data reported by [121] to claim the absence of evidence for a correlation between the power of steady jets and BH spin measurements of the continuum-fitting method. Right panel: Data reported by [228] to claim the evidence for a correlation between the power of transient jets and BH spin measurements of the continuum-fitting method.

Figure 25

JP BHs with deformation parameter ${\epsilon }_3$. The solid black line corresponds to the equilibrium spin parameter $a_{*}^{\text{eq}}$ for a thin disk, as inferred from Eq. (31). The solid red, dashed green, and dotted blue curves are, respectively, the contour levels of the Novikov-Thorne radiative efficiency ${\eta }_{\mathrm{NT}}=0.15$, 0.20, and 0.25. Supermassive BH candidates in galactic nuclei must be on the left of the solid black line and observations show that their radiative efficiency can be high or even very high. The allowed region is thus on the left of the solid black line and on the right of the ${\eta }_{\mathrm{NT}}$ contour levels. The right panel is the enlargement of the region of intersection of the curves with higher spin. See the text for more details. From [21].

Figure 26

Evolutions of the (left panels) spin parameter $a_{*}$ and (right panels) the Novikov-Thorne radiative efficiency ${\eta }_{\mathrm{NT}}=1-E_{\mathrm{ISCO}}$ as a function of the amount of matter accreted onto an initially nonrotating BH for four different initial BH masses. The top panels are for Kerr BHs, and the bottom panels for JP BHs with ${\epsilon }_3=10a_{*}^2$. The horizontal solid red lines indicate the spin parameter $a_{*}=0.98$ in the top left panel, and the Novikov-Thorne radiative efficiency ${\eta }_{\mathrm{NT}}=0.234$ (which requires $a_{*}=0.98$ in the Kerr metric) in the right panels. From [40].