The no-hair theorem and black hole shadows

In classical general relativity, black holes carry no charge other than mass, charge and spin, which is known in black hole physics as no-hair theorem(1967PhRv..164.1776I, ; 1968CMaPh…8..245I, ; 1971PhRvL..26..331C, ; 1972CMaPh..25..152H, ; 1975PhRvL..34..905R, ; 1982JPhA…15.3173M, ; 1995PhRvD..51.6608B, ; 2014ApJ…784….7B, ; 2019PhRvL.123k1102I, ; 2021arXiv211100953W, ; 2015PhRvL.114o1102G, ; 2015IJMPD..2442014H, ). However, it is possible that the interaction of black hole space-time with matter introduces other charges, such as internal norm symmetries and certain fields, which could make it possible for black holes to carry hairs. When these corresponding physical effects are introduced into the space-time of black hole, i.e., to make black hole hairy, the presence of these hairs will change the space-time background of black hole, thus bringing the quantum effects of black hole into the classical space-time geometry. Using extended gravitational decoupling (EGD) approach, J.Ovalle et al. obtained a spherically symmetric space-time metric with hair(2021PDU….3100744O, ), because in the EGD approach, there is no certain matter field, which makes the hairy black holes obtained by them have great generality. In EGD, the corresponding Einstein field equation can be expressed as follows:

where $R_{\mu\nu}$ is Ricci curvature tensor, $R$ is Ricci curvature scalar, $g_{\mu\nu}$ is the space-time metric, $\tilde{T}_{\mu\nu}$ is the total energy-momentum tensor, and can be written as

where $T_{\mu\nu}$ is the energy-momentum tensor in GR, $\theta_{\mu\nu}$ is the energy-momentum tensor caused by a quantum field or gravitational branch. According to general relativity, $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ needs to satisfy the Bianchi identity, namely $\bigtriangledown_{\mu}\tilde{T}^{\mu\nu}=0$. When $\theta_{\mu\nu}=0$, i.e., without considering hairs, it can be proved that the solution of field equation (1) degenerates into Schwarzschild black hole solution. By proper treatment of the tensor $\theta_{\mu\nu}$, the spherically symmetric solution of Einstein field equation can be obtained (2021PDU….3100744O, )

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$, $f(r)$ and $g(r)$ are the metric coefficients, and the expressions in this case are

Here, $\alpha_{0}$ describes the deformation parameters due to the consideration of hairs, so the size of $\alpha_{0}$ represents the physical meaning related to the strength of hairs. $L$ is a constant with the dimension of distance. In this work, we limit the values of $\alpha_{0}$ and $L$ as follows (2021PDU….3100744O, ; 2021PhRvD.103d4020C, ):

By adjusting the parameters $\alpha_{0}$ and $L$, we can analyse how hairs change the properties of the space-time metric of black holes. Meanwhile, the space-time metric provides a basis for testing the no-hair theorem of black holes from an astronomical perspective.

Using gravitational decoupling approach, E.Contreras et al. generalized the spherically symmetric black hole solution to the rotational case(2021PhRvD.103d4020C, ). By applying some restrictions on $S_{\mu\nu}$ in the energy-momentum tensor $\tilde{T}_{\mu\nu}=T_{\mu\nu}+S_{\mu\nu}$, such as meeting the strong energy condition (SEC), the space-time metric of the black hole with hair was obtained

where, the expressions of related symbols are

In these expressions, $a$ is the spin of the black hole, and other parameters have the same physical meaning as in the spherically symmetric case. Since rotational black holes are real in the universe, it is more feasible to test the no-hair theorem of black holes using observational data such as EHT and gravitational waves.

In addition, to generalize the rotational hairy black hole to the charged case(2021PDU….3100744O, ), we only need to replace the metric coefficient in the spherically symmetric case by

where $Q$ is the charge carried by the black hole. The form of Kerr-Newman black hole with hairs is similar to (7) $\sim$ (11).

When considering the coupling of gravity with some anisotropic fluids in general relativity, the black hole solution with hair can be obtained. These anisotropic fluids satisfy the following conditions, that is, in some conditions, the gravity will produce de Sitter and Reissner-Nordstron (RN) black holes, and in other conditions, the solution of the black hole with hair can be obtained(1997IJMPD…6..563D, ), whose linear element expression is

$Q_{m}$ is the strength parameter of the hair. When $k=1$, the space-time metric degrades into an RN black hole; when $k>1$, the space-time metric is a short-hair black hole, and $Q_{m}$ is the charged value of the short hair. In this work, we discuss the case of $k=\frac{3}{2}$ in order to understand the properties of short-hair black hole. For the black hole corresponding to the metric (13), its energy density and pressure are respectively

As discussed in the original literature, the black hole solution satisfies three classical conditions, namely the weak energy condition (WEC), the energy density decays faster than $r^{-4}$, and $T=T_{ab}g^{ab}\leqslant 0$. Therefore, this black hole does not violate the no-short-hair theorem.

Next, based on the metric of spherically symmetric short-hair black hole, we derive the solution of rotating short-hair black hole using Newman-Janis (NJ) method. In short, the NJ method is to generalize spherically symmetric space-time to rotational space-time by complex transformation. The key is to simplify Einstein field equation into a second-order partial differential equations. If there is an analytic solution to this equations, the rotation form of the corresponding space-time metric can be obtained. The content of the NJ method can be referred to the relevant literature (e.g.,(Newman:1965tw, ; 2014PhRvD..90f4041A, ; 2014PhLB..730…95A, )). Follow the idea of this method, we will show the key derivation.

Since the Schwarzschild coordinate system is used in the space-time form (III), if the space-time considered is a black hole, there is coordinate singularity in this kind of coordinate system. Therefore, in the NJ algorithm, this coordinate system needs to be transformed to the advanced null coordinates (ANC) $(u,r,\theta,\varphi)$, and the coordinate transformation is

In ANC, we can choose the advanced null basis vector $(e^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu})$ to expand the space-time metric, that is, the inverse form of the space-time metric is expressed as a linear combination of the basis vector, namely $g^{\mu\nu}=-e^{\mu}n^{\nu}-e^{\nu}n^{\mu}+m^{\mu}\bar{m}^{\nu}+m^{\nu}\bar{m}^{\mu}$. For the metric (III) considered here, its basis vector in ANC is

To generalize spherically symmetric space-time to the rotation case, we can perform the following operations in ANC. In complex space, the coordinate $(u,r)$ is rotated by an angle of $\theta$, that is, $u\rightarrow u-ia\cos\theta,r\rightarrow r+ia\cos\theta$; where $a$ is a constant, which can also be interpreted as the spin of the space-time, and $\theta$ is the rotation angle. At this time, the spherical symmetry metric coefficients $f(r)\rightarrow F(r,\theta,a),g(r)\rightarrow G(r,\theta,a)$ and $h(r)\rightarrow\psi(r,\theta,a)$. $h(r)$ is the coefficient of $d\Omega^{2}$ in the metric (III), and here $h(r)=r^{2}$. Through these operations, the basis vectors in ANC become functions expressed by $F(r,\theta,a)$, $G(r,\theta,a)$ and $\psi(r,\theta,a)$. After calculation, it is found that the components of the inverse metric $g^{\mu\nu}$ are as follows:

By processing these inverse metric, we can get the covariance metric tensor, and thus obtain the expression of its line element as

Next, using coordinate transformation, the space-time metric can be transformed from the Eddington-Finkelstin coordinates (EFC) to the Boyer-Lindquist coordinate (BLC)(???), and the final result is

Here, the relationship of the related symbols is

In this way, we get the general form of the metric of the rotational space-time in the NJ method. However, there is an unknown function $\psi$ in the metric expression, which needs to be solved by Einstein field equation. Since the space-time metric at this time satisfies rotation symmetry, the component of the Einstein tensor $G_{r\theta}=0$. Meanwhile, the metric (20) should also satisfy Einstein field equation(2014PhRvD..90f4041A, ; 2014PhLB..730…95A, ). Through a series of calculations, the equation of the gravitational field can be simplified into the following equations

For the short-hair black hole considered in this section, $f(r)=g(r)=1-\frac{2M}{r}+\frac{Q_{m}^{2k}}{r^{2k}},h(r)=r^{2}$, we can obtain $k(r)=h(r)\sqrt{\frac{f(r)}{g(r)}}=r^{2}$. Substituting $k(r)$ into equations (22) and (23), by solving the equations, we can get

So far, we have the expressions for all the unknown functions, thereby obtaining the rotation form of the short-hair black hole, which is

Where $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$, $\Sigma=(r^{2}+a^{2})^{2}-a^{2}\Delta(r)\sin^{2}\theta$, $\Delta=r^{2}-2Mr+\frac{Q_{m}^{2k}}{r^{2k-2}}+a^{2}$. If we consider a no-hair black hole, that is, $Q_{m}=0$, then the metric (2.25) degrades to a Kerr black hole, and $a$ can be interpreted as the black hole spin. When $Q_{m}\neq 0$ and $k>1$, it is a short-hair black hole in rotation situation.

When $k=\frac{3}{2}$, $\Delta=r^{2}-2Mr+\frac{Q_{m}^{3}}{r}+a^{2}=0$ will determine the structure of the black hole event horizon. By adjusting the range of values of $M$, $Q_{m}$ and $a$, the number of the black hole event horizon may be $1$, $2$ or $3$. This is caused by the introduction of the short hair. Then we analyse the specific properties of each case. From $\Delta=0$, it can be found that the event horizon of the black hole should satisfy the following equation

The discriminant of the root is

When $\Delta>0$, equation (26) has only one real root, so the black hole has only one event horizon, the radius of which is

When $\Delta=0$, equation (26) has three real roots, at least two of which are equal, so the event horizons of the black hole are

When $\Delta<0$, equation (26) has three unequal real roots, that is, the black hole has three different event horizons, and their values are

Through calculations, we find that the structure of the event horizon of the short-hair black hole is particularly complicated. $\Delta$ determines the number of black hole event horizons. Here, the mass of the black hole is taken as the unit, i.e., $M=1$. When the black hole spin $0\leqslant a\leqslant 1$, the critical value condition of $Q_{m}$ derived from $\Delta=0$ is

When $\Delta>0$, i.e., $Q_{m}^{3}<2(\frac{4}{9}-\frac{a^{2}}{3})^{\frac{3}{2}}+\frac{16}{27}-\frac{2a^{2}}{3}$, only $Q_{m}<0$ can meet this situation, which indicates that the short-hair black hole has no event horizon, so it is not the case we consider. When $\Delta=0$, the value range of $Q_{m}$ is $0\leqslant Q_{m}\leqslant\frac{2}{3}\times 4^{\frac{1}{3}}$. When $\Delta<0$, the value range is $Q_{m}>\frac{2}{3}\times 4^{\frac{1}{3}}$. In general, when $0\leqslant Q_{m}\leqslant\frac{2}{3}\times 4^{\frac{1}{3}}$, the short-hair black hole has three event horizons, two of which are equal. From a physical point of view, there are only two event horizons, which are modifications of Kerr black hole. When $Q_{m}>\frac{2}{3}\times 4^{\frac{1}{3}}$, the short-hair black hole has three different event horizons, which indicates that the appearance of the short-hair makes the black hole appear a new event horizon, and essentially changes the structure of the black hole event horizon.

As we all know, AdS-Kerr black hole has three event horizons $(r_{+},r_{-},r_{\Lambda})$. The radius of the event horizon generated by the introduction of cosmological constant $\Lambda$ is very large, namely $r_{\Lambda}>>r_{+}$, while the radius of the event horizon generated by the short hair is close to $r_{+}$. Since the physical effects of $r_{+}$ are relatively easy to measure, the corresponding effects caused by the short hair are also relatively easy to detect, which makes the third event horizon of the short-hair black hole interesting.

Once the geodesic of the photon is known, we can calculate the motion of the photon as measured by an observer anywhere. In general, we assume that the earth is infinitely far away from the black hole and the observer is a mass point, which makes the calculation very convenient. In the calculation of black hole shadows, physicists introduce the celestial coordinate system, which is a two-dimensional coordinate system(2009PhRvD..80b4042H, ), and its relationship with the BL coordinates of the black hole is

where $\theta_{0}$ is the angle between the line from the earth to the black hole and the axis of rotation of the black hole, and $r_{0}$ is the distance between the earth and the black hole. In a celestial coordinate system, photons emitted from the vicinity of the black hole correspond to the coordinates $(\alpha,\beta)$ one by one, so that the geometry of the black hole shadow to be understood by pushing back the edge of the geodesic motion of the photon by $(\alpha,\beta)$. Through calculation, we get the expressions of $\alpha$ and $\beta$ as

For Hairy black hole

For short-hair black hole

Using expressions (46) $\thicksim$ (49), the shape of the black hole shadow can be calculated, so as to make clear the influence of the hairs on the shape of black hole shadow, as shown in figures 1, 2, and 3. The main results are as follows:

We obtain the shadow shapes of two kinds of black holes with hairs. In model $1$ (corresponding to figures 1 and 2), the values of parameters $\alpha_{0}$ and $L$ have a significant influence on the shadow shapes (mainly referring to the size of the shadow boundary here). When the parameter $\alpha_{0}$, which represents the intensity of the scalar hair, increases gradually from $0$, the boundary of the black hole shadow will shrink continuously. When $\alpha_{0}$ increases to a certain critical value, the black hole shadow changes in reverse. On the other hand, the boundary of the black hole shadow will always increase with the increase of the characteristic length $L$. Therefore, the change of $\alpha_{0}$ and $L$ to the black hole shadow is complicated. In Model $2$ (corresponding to figure 3), when $k=1.5$, namely the short hair case, if the black hole spin $a=0$, the shape of the shadow boundary of the black hole is a standard circle. It can be found that when the charge $Q_{m}$ of the short hair keeps increasing, the shadow boundary of the black hole will decrease monotonically, but its shape is all circle. As the black hole spin $a$ increases, the shape of the black hole shadow is distorted. We find that the change of the hairy black hole in model $1$ to the shadow is basically consistent with the case $\varepsilon>0$ in reference (2021JCAP…09..028K, ), but the specific changes are different. The change of the short-hair black hole in model $2$ to the shadow corresponds to the case $\varepsilon<0$ in reference (2021JCAP…09..028K, ).

If the effects of various hairs carried by black holes on the metric of black holes are known, it is possible to test them by means of EHT observations. However, the parameter values of the real hairy black hole are smaller than those discussed by us (since the shape of the black hole is mainly contributed by the mass, and the quantum effect should not contribute much to it), which requires us to further improve the resolution of EHT before we can test the no-hair theorem of the black hole.

From the previous analysis, we can see that the boundary shape of the black hole shadow is a circle only when $a=0$. When the black hole spin $a\neq 0$, the boundary of the black hole shadow will be distorted in different degrees, which makes it necessary to introduce new parameters to accurately describe the shape of the black hole shadow. Generally speaking, scientists introduce the following two parameters to describe the shadow shape. The first parameter is the shadow radius $R_{s}$, which describes the scale of the shadow shape of the black hole. The second parameter is the degree of distortion of the shadow ($\delta_{s}$), which describes how far the boundary of the black hole shadow deviates from the circle. According to previous discussions, its definition is as follows:

The physical meanings of $\alpha_{p}$ and $\tilde{\alpha}_{p}$ are shown in reference (2009PhRvD..80b4042H, ). When the black hole spin $a=0$ (i.e. in the case of a spherically symmetric black hole), $\alpha_{p}=\tilde{\alpha}_{p}$, at this time, the black hole shadow is not distorted. When extreme black holes are considered (i.e. $a\rightarrow 1$), $\alpha_{p}-\tilde{\alpha}_{p}$ is at its maximum at this point. Through numerical calculation, we find the following results: (1) for model $1$, when Kerr black hole is considered, namely $\alpha_{0}=0$ (no-hair black hole), the change of the scale parameter $L$ hardly affects the value of $R_{s}$. When $\alpha_{0}=0.5$ (i.e. black hole carry hairs), $R_{s}$ increases slowly with the increase of parameter $L$. When $\alpha_{0}=0.8$, that is, more hairs are considered, $R_{s}$ rises rapidly and then changes gently with $L$ (see the left part of Figure 4). When $L=0$, $R_{s}$ decreases with the increase of $\alpha_{0}$. When $L=1$, $R_{s}$ decreases first and then increases with the increase of $\alpha_{0}$. As $L$ increases further, the size of $R_{s}$ varies with $\alpha_{0}$ to show some interesting phenomenon (see the left part of Figure 5). (2) For model $1$, when Kerr black hole (i.e. no-hair black hole) is considered, the distortion parameter $\delta_{s}$ is a constant function of the scale parameter $L$. When hair black holes are considered, i.e. $\alpha_{0}\neq 0$, $\delta_{s}(L)$ is a decreasing function and decays rapidly at the beginning, while $\delta_{s}$ changes slowly when $L$ is very large (see the right part of Figure 4). In the case that $L$ is constant, $\delta_{s}(\alpha_{0})$ is an increasing function when $L=0$, and the first half is an increasing function and the second half is a decreasing function when $L=1$. Its variation becomes more singular at larger spins of the black hole and $L$ (see the right part of Figure 5). (3) For model $2$, i.e., the short-hair black hole, $R_{s}(Q_{m})$ is a monotone decreasing function, $R_{s}(a)$, $\delta_{s}(Q_{m})$ and $\delta_{s}(a)$ are all increasing functions (see Figure 6). Since $R_{s}$ and $\delta_{s}$ vary significantly with scalar, quantum or short hairs, it is more likely to be tested in EHT observations.