Precession and Lense-Thirring effect of hairy Kerr spacetimes

Einstein’s general relativity (GR) is a successful theory in modern physics and passes plenty of test in astrophysics as well as astronomy. The existence of black holes is a prediction of GR, and black holes provide natural laboratories to test gravity in the strong field regime. Recent observations on gravitational waves Abbott et al. (2016, 2019, 2020) and the supermassive black hole images Akiyama et al. (2019a, b, c, 2022a, 2022b) agree well with the predictions of Kerr black hole described by the Kerr metric from GR. More observations, such as those from Next Generation Very Large Array D. James et al. (2019) and Thirty Meter Telescope Skidmore et al. (2015), also provide significant properties in the regime of strong gravity of black hole spacetimes. In particular, these observations open a valuable window to explore, distinguish, or constrain physically viable black hole solutions that exhibit small deviations from the Kerr metric. On the other hand, due to the additional surrounding sources, the black holes in our Universe could obtain an extra global charge dubbed ‘hair’ and the spacetime may deviate from the Kerr metric. Recently, a rotating hairy Kerr black hole was constructed with the use of the gravitational decoupling (GD) approach Ovalle et al. (2021); Contreras et al. (2021), which is designed for describing deformations of known solutions of GR due to the inclusion of additional sources. The hairy Kerr black hole attracts lots of attentions. Plenty of theoretical and observational investigations have been done in this hairy Kerr black hole spacetime, for examples, thermodynamics Mahapatra and Banerjee (2022), quasinormal modes and (in)stability Cavalcanti et al. (2022); Yang et al. (2022); Li (2022), strong gravitational lensing and parameter constraint from Event Horizon Telescope observations Islam and Ghosh (2021); Afrin et al. (2021).

There are many theoretical scenarios in which a visible singularity exists, especially after it was found in Joshi et al. (2011) that a spacetime with a central naked singularity can be formed as an equilibrium end state of the gravitational collapse of general matter cloud. However, whether naked singularity (NS) really exists in nature and what is its physical signature distinguished from black hole (BH) are still open questions. In Mathematics, the Kerr spacetime metric in GR, as a solution of Einstein field equations, describes Kerr BH and the Kerr singularity is contained in the event horizon, otherwise, if the event horizon disappears, the metric describes a NS. In shadow scenario, the shadow cast by NS spacetime was found to have similar size to that an equally massive Schwarzschild black hole can cast Shaikh et al. (2019). More elaborate discussions on shadow from NS have been carried out in Joshi et al. (2020); Dey et al. (2020, 2021), which are important theoretical studies and match recent observations on the shadow of M87* and SgrA*. In addition, the orbital dynamics of particles or stars around a horizonless ultra-compact object could be another important physical signature which could be distinguishable from that around black holes, since the nature of timelike geodesics in a spacetime closely depends upon the geometrical essence of the spacetime. In this scenario, the timelike geodesics around different types of compact objects and the trajectory of massive particles or stars are widely studied Pugliese et al. (2011); Martínez et al. (2019); Hackmann et al. (2014); Potashov et al. (2019); Bhattacharya et al. (2020); Bambhaniya et al. (2019); Deng (2020); Lin and Deng (2021); Bambhaniya et al. (2021a); Ota et al. (2022) and reference therein. People expect that the nature of the precession of the timelike orbits can provide more information about the central compact object, as GRAVITY Bauböck et al. (2020) and SINFONI Eisenhauer et al. (2005) are continuously tracking the dynamics of ‘S’ stars orbiting around SgrA*.

For the timelike orbits around a rotating compact body, the geodetic precession de Sitter (1916) and Lense-Thirring (LT) precession Lense and Thirring (1918) are two extraordinary effects predicted by GR. The former effect is also known as de Sitter precessions and it is due to the spacetime curvature of the central body, while the latter effect is due to the rotation of the central body which causes the dragging of locally inertial frames. One can examine the dragging effects by considering a test gyro based on the fact that a gyroscope tends to keep its spinning axis rigidly pointed in a fixed direction relative to a reference star. The LT precession and geodetic effects have been measured in the Earth’s gravitational field by the Gravity Probe B experiment, in which the satellite consists of four gyroscopes and a telescope orbiting $642km$ above the Earth Everitt et al. (2011). The geodetic precession in Schwarzschild BH and the Kerr BH have been studied in Sakina and Chiba (1979). The LT precession is more complex and usually some approximations should be involved. In the weak field approximations, the magnitude of LT precession frequency is proportional to the spin parameter of the central body and decreases in the order of $r^{-3}$ with $r$ the distance between the test gyro from the central rotating body Hartle (2009). In the strong gravity limit, the LT precession frequency were studied in various rotating compact objects, for instances, in Kerr black hole Chakraborty and Majumdar (2014); Bini et al. (2016) and its generalizations Chakraborty and Pradhan (2013), in rotating traversable wormhole Chakraborty and Pradhan (2017) and in rotating neutron star Chakraborty et al. (2014). Those studies further show that the behavior of LT frequency in the strong gravity regime closely depends on the nature of the central rotating bodies. In particular, it was proposed in Chakraborty et al. (2017a) that the spin precession of a test gyro can be used to distinguish Kerr naked singularities from black holes, which was then extended in modified theories of gravity Rizwan et al. (2018, 2019); Pradhan (2020); Solanki et al. (2022).

Thus, one of the main aims of this paper is to investigate the spin precession frequency, including the LT frequency and geodetic frequency, of the test gyro in the hairy Kerr spacetime, and differentiate the hairy Kerr BH from hairy NS. We find a clear difference on the precession frequencies between hairy rotating BH and NS, and the effects of the hairy parameters are also systematically explored.

Additionally, we also investigate the accretion disk physics in the hairy Kerr spacetime, which is realized by studying the orbital precession of a test timelike particle around the hairy Kerr BH and NS. The motivation stems from the followings. Accretion in Low-Mass X-ray Binaries (LMXBs) occurs in the strong gravity regime around compact bodies, in which the quasiperiodic oscillations (QPOs) phenomena involves and their frequencies in the $Hz$ to $kHz$ range have been detected Torok et al. (2011); Bambi (2012); Tripathi et al. (2019). Plenty of efforts have been made to explain the QPOs phenomena, see van der Klisin (2006) for a review. It shows that the geodesic models of QPOs are related with three characteristic frequencies of the massive particles orbiting around the compact bodies, namely the orbital epicyclic frequency and the radial and vertical epicyclic frequencies. Therefore, since the QPOs provide a way to testify the strong gravity, these three frequencies could be used as a tool to study the crucial differences among the central compact objects Rizwan et al. (2019) or test/constrain alternative theories of gravity Aliev et al. (2013); Doneva et al. (2014); Azreg-Aïnou et al. (2020); Jusufi et al. (2021); Ghasemi-Nodehi et al. (2020). In particular, more recently it was proposed in Motta et al. (2018) that LT effect may have connection with QPOs phenomena and perhaps be used to explain the QPOs of accretion disks around rotating black holes. Thus, Along with using the spin precession frequency and LT precession frequency of the test gyro to indicate the differences between hairy Kerr BH and NS, we further study the three characteristic frequencies of massive particles orbiting around the corresponding central rotating objects. It is found that the LT precession and periastron precession of massive particle behave crucial differences between hairy Kerr BH and NS, as a support of the results from the test gyro in strong gravity regime of compact bodies.

Our paper is organized as follows. In section II, we briefly review the hairy Kerr spacetime derived from GD approach, and analyze the parameter spaces for the corresponding black hole and naked singularity. Then we derive the timelike geodesic equations in the spacetime. In section III, we derive the spin precession frequency of a test gyro in the hairy Kerr spacetime. Then we compare the LT frequency as well as the spin frequency between the hairy Kerr BH and NS cases. In section IV, we study the accretion disk physics by analyzing inner-most stable circular orbit (ISCO), and LT precession and periastron precession in terms of the three characteristic frequencies, of a timelike particle around the hairy Kerr BH and NS, respectively. The last section is our conclusion and discussion. Throughout the paper, we use $G_{\rm N}=c=\hbar=1$ and all quantities are rescaled to be dimensionless by the parameter $M$.

In this section, firstly, we will show a brief review on the idea of gravitational decoupling (GD) approach and the hairy spacetime constructed from GD approach by Ovalle Ovalle et al. (2021). Then we derive the timelike geodesic equations in the hairy Kerr spacetime.

The no-hair theorem in classical general relativity states that black holes are only described by mass, electric charge and spin Israel (1967); Hawking (1972); Mazur (1982). But it is possible that the interaction between black hole spacetime and matters brings in other charge, such that the black hole could carry hairs. The physical effect of these hairs can modify the spacetime of the background of black hole, namely hairy black holes may form. Recently, Ovalle et.al used the GD approach to obtain a spherically symmetric metric with hair Ovalle et al. (2021). The hairy black hole in this scenario has great generality because there is no certain matter fields in the GD approach, in which the corresponding Einstein equation is expressed by

$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi\tilde{T}_{\mu\nu}.$$

(1)

Here $\tilde{T}_{\mu\nu}$ is the total energy momentum tensor written as $\tilde{T}_{\mu\nu}=T_{\mu\nu}+\vartheta_{\mu\nu}$ where $T_{\mu\nu}$ and $\vartheta_{\mu\nu}$ are energy momentum tensor in GR and energy momentum tensor introduced by matter fields or others, respectively. $\nabla_{\mu}\tilde{T}_{\mu\nu}=0$ is satisfied because of the Bianchi indentity. It is direct to prove that when $\vartheta_{\mu\nu}=0$, the solution to (1) degenerates into Schwarzschild metric. The hairy solution with proper treatment (strong energy condition) of $\vartheta_{\mu\nu}$ was constructed and the detailed algebra calculations are shown in Ovalle et al. (2021); Contreras et al. (2021). Here we will not re-show their steps, but directly refer to the formula of the hairy metric

$$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})~{}~{}\mathrm{with}~{}~{}f(r)=1-\frac{2M}{r}+\alpha e^{-r/(M-l_{0}/2)}.$$

(2)

In this solution, $M$ is the black hole mass, $\alpha$ is the deformation parameter due to the introduction of surrounding matters and it describes the physics related with the strength of hairs, and $l_{0}=\alpha l$ with $l$ a parameter with length dimension corresponds to primary hair which should satisfy $l_{0}\leq 2M$ to guarantee the asymptotic flatness. The metric (2) reproduces the Schwarzchild spacetime in the absence of the matters, i.e., $\alpha=0$.

Later, considering that astrophysical black holes in our universe usually have rotation, the authors of Afrin et al. (2021) induced the rotating counterpart of the static solution (2), which is stationary and axisymmetric, and in Boyer-Lindquist coordinates it reads as

	$\displaystyle ds^{2}=$		$\displaystyle g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}+2g_{t\phi}dtd\phi$
	$\displaystyle=$		$\displaystyle-\left(\frac{\bigtriangleup-a^{2}\sin^{2}\theta}{\Sigma}\right)dt^{2}+\sin^{2}\theta\left(\Sigma+a^{2}\sin^{2}\theta\left(2-\frac{\bigtriangleup-a^{2}\sin^{2}\theta}{\Sigma}\right)\right)d\phi^{2}+\frac{\Sigma}{\bigtriangleup}dr^{2}+\Sigma d\theta^{2}$		(3)
			$\displaystyle-2a\sin^{2}\theta\left(1-\frac{\bigtriangleup-a^{2}\sin^{2}\theta}{\Sigma}\right)dtd\phi$

with

$\displaystyle\Sigma=r^{2}+a^{2}\cos^{2}\theta,~{}~{}~{}\bigtriangleup=r^{2}+a^{2}-2Mr+\alpha r^{2}e^{-r/(M-\frac{l_{0}}{2})}.$

(4)

It is noticed that the above metric is also proved to satisfy the equations of motion in the GD approach Ovalle et al. (2021). The metric describes certain deformation of the Kerr solution due to the introduction of additional material sources (such as dark energy or dark matter). In the metric, $a$ is the spin parameter and $M$ is the black hole mass parameter. Similar to those in static case, $\alpha$ is the deviation parameter from GR and $l_{0}$ is the primary hair which is required to be $l_{0}\leq 2M$ for asymptotic flatness. When $\alpha=0$, the metric reduces to standard Kerr metric in GR, namely no surrounding matters.

The metric (II) could describe non-extremal hairy Kerr black hole, extremal hairy Kerr black hole and hairy naked singularity which correspond to two distinct, two equal and no real positive roots of $g^{rr}=\bigtriangleup=0$ as well as $\Sigma\neq 0$, meaning they have two horizons, single horizon and no horizon, respectively. The parameter space $(l_{0}-a)$ of the related geometries for some discrete $\alpha$ has been show in Afrin et al. (2021). Here in FIG. 1, we show a 3D plot in the parameters space $(l_{0},a,\alpha)$ for various geometries. In the figure, for the parameters in the white region, the spacetime with the metric (II) is a non-extremal hairy Kerr black hole with two horizons, while it is a naked singularity without horizon for the parameters in the shaded region; and for the parameters on the orange surface, it is an extremal hairy Kerr black hole with single horizon. Besides, how the hairy parameters affect the static limit surface and the ergoregion of the hairy Kerr black hole has also been explored in Afrin et al. (2021). Moreover, as we aforementioned, some theoretical and observational properties of the hairy Kerr black hole have been carried out, such as the thermodynamics Mahapatra and Banerjee (2022), quasinormal modes and (in)stability Cavalcanti et al. (2022); Yang et al. (2022); Li (2022), strong gravitational lensing and parameter constraint from Event Horizon Telescope observations Islam and Ghosh (2021); Afrin et al. (2021).

In the following sections, we shall partly study the physical signatures in the strong field regime of those different central objects in the hairy Kerr spacetime (II). We will analyze the inertial frame dragging effects on a test gyro and on the accretion physics respectively, which is expected to show the difference between black hole and naked singularity. To proceed, we should analyze the timelike geodesic equations in the hairy Kerr spacetime. Since for the spacetime, we have two Killing vector fields $\partial_{t}$ and $\partial_{\phi}$, so we can define the conserved energy $\mathscr{E}$ and axial-component of the angular momentum $L$ of which the expressions are

$$\mathscr{E}=-g_{tt}\dot{t}-g_{t\phi}\dot{\phi},~{}~{}~{}~{}~{}~{}L=g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi},$$

(5)

where the dot represent the derivative with respect to the affine parameter $\lambda$. We employ the Hamilton-Jacobi method Carter (1968) and introduce the Hamilton-Jacobi equation for the particle with rest mass $\mu$

$$\mathscr{H}=-\frac{\partial S}{\partial\lambda}=\frac{1}{2}g_{\mu\nu}\frac{\partial S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu}}=\mu^{2}$$

(6)

where $\mathscr{H}$ and $S$ are the canonical Hamiltonian and Jacobi action, respectively, and $x^{\mu}$ denote the coordinates $t,r,\theta,\phi$ in the metric (II). Then after variation, we find that we can separate the Jacobi action as $S=\frac{1}{2}\mu^{2}\lambda-\mathscr{E}t+L\phi+S_{r}(r)+S_{\theta}(\theta)$ and define the constant $\mathscr{C}$ via

$$\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+\frac{(L-a\mathscr{E}\sin^{2}\theta)^{2}}{\sin^{2}\theta}=-\Delta\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{\left((r^{2}+a^{2})\mathscr{E}^{2}-aL\right)^{2}}{\Delta}=\mathscr{C}$$

(7)

to separate the outcome into four first-order differential equations describing the geodesic motion

	$\displaystyle\Sigma\dot{t}$	$\displaystyle=$	$\displaystyle a(L-a\mathscr{E}\sin^{2}\theta)+\frac{r^{2}+a^{2}}{\Delta}\left((r^{2}+a^{2})\mathscr{E}-aL\right),$		(8)
	$\displaystyle\Sigma\dot{\phi}$	$\displaystyle=$	$\displaystyle\frac{L}{\sin^{2}\vartheta}-a\mathscr{E}+\frac{a}{\Delta}\left((r^{2}+a^{2})\mathscr{E}-aL\right),$		(9)
	$\displaystyle\Sigma\dot{r}$	$\displaystyle=$	$\displaystyle\Delta\left(\frac{dS_{r}}{dr}\right)=\sqrt{\left(r^{2}+a^{2})\mathscr{E}-aL\right)^{2}-\Delta\left(Q+(L-a\mathscr{E})^{2}+\mu^{2}r^{2}\right)},$		(10)
	$\displaystyle\Sigma\dot{\theta}$	$\displaystyle=$	$\displaystyle\frac{dS_{\theta}}{d\theta}=\sqrt{Q+\cos^{2}\theta\biggl{(}a^{2}(\mathscr{E}^{2}-\mu^{2})-\frac{L^{2}}{\sin^{2}\theta}\biggr{)}},$		(11)

where $Q\equiv\mathscr{C}-(L-a\mathscr{E})^{2}$ with $\mathscr{C}$ defined in (7) is the Carter constant. Thus, the geodesics equations in the hairy Kerr spacetime (II) are separable as in Kerr case, and the modification from the hair appears in the metric function $\Delta$. Now we can proceed to investigate the influence of hair on the orbital precession. For convenience, we will assume that the gyros/stars/particles we will consider are minimally coupled to the metric, and also there is no direct coupling between them and the surrounding matters, i.e., the modified gravity effect or print of hair would only be reflected in the metric functions.

In this section, we will study the spin precession frequency of a test gyroscope attached to a stationary observer, dubbed stationary gyroscopes for brevity in the hairy Kerr spacetime, for whom the $r$ and $\theta$ coordinates are remaining fixed with respect to infinity. Such a stationary observer has a four-velocity $u_{stationary}^{\mu}=u_{stationary}^{t}(1,0,0,\Omega)$, where $t$ is the time coordinate and $\Omega=d\phi/dt$ is the angular velocity of the observer.

Let us consider stationary gyroscopes moving along Killing trajectory, whose spin undergoes Fermi-Walker transport along $u=(-K^{2})^{-1/2}K$ with $K$ the timelike Killing vector field. Since the hairy Kerr spacetime (II) has two Killing vectors which are the time translation Killing vector $\partial_{t}$ and the azimuthal Killing vector $\partial_{\phi}$, so a more general Killing vector is $K=\partial_{t}+\Omega\partial_{\phi}$. In this case, the general spin precession frequency of a test stationary gyroscope is the rescaled vorticity filed of the observer congruence, and its one-form is given by Straumann (2004)

$$\widetilde{\Omega}_{p}=\frac{1}{2K^{2}}*(\widetilde{K}\wedge d\widetilde{K})$$

(12)

where $\widetilde{K}$ is the covector of $K$, the denotations $*$ and $\wedge$ represent the Hodge dual and wedge product, respectively. According to the pedagogical steps of Chakraborty et al. (2017a), the corresponding vector of the covector $\widetilde{\Omega}_{p}$ is

$$\vec{\Omega}_{p}=\frac{\varepsilon_{ckl}{}}{2\sqrt{-g}\left(1+2\Omega\frac{g_{0c}}{g_{00}}+\Omega^{2}\frac{g_{cc}}{g_{00}}\right)}\left[\left(g_{0c,k}-\frac{g_{0c}}{g_{00}}g_{00,k}\right)+\Omega\left(g_{cc,k}-\frac{g_{cc}}{g_{00}}g_{00,k}\right)+\Omega^{2}\left(\frac{g_{0c}}{g_{00}}g_{cc,k}-\frac{g_{cc}}{g_{00}}g_{0c,k}\right)\right]\partial_{l},$$

(13)

where $g$ is the determinant of the metric $g_{\mu\nu}(\mu,\nu=0,1,2,3)$ and $\varepsilon_{ckl}(c,k,l=1,2,3)$ is the Levi-Civita symbol. Then in the hairy Kerr spacetime (II) which is stationary and axisymmetric spacetime, the above expression can be reduced as

$\displaystyle\vec{\Omega}_{p}=$

$\displaystyle\frac{\left(C_{1}\hat{r}+C_{2}\hat{\theta}\right)}{2\sqrt{-g}\left(1+2\Omega\frac{g_{t\phi}}{g_{tt}}+\Omega^{2}\frac{g_{\phi\phi}}{g_{tt}}\right)},$

(14)

with

	$\displaystyle C_{1}=$	$\displaystyle-\sqrt{g_{rr}}\left[\left(g_{t\phi,\theta}-\frac{g_{t\phi}}{g_{tt}}g_{tt,\theta}\right)+\Omega\left(g_{\phi\phi,\theta}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,\theta}\right)+\Omega^{2}\left(\frac{g_{t\phi}}{g_{tt}}g_{\phi\phi,\theta}-\frac{g_{\phi\phi}}{g_{tt}}g_{t\phi,\theta}\right)\right],$		(15)
	$\displaystyle C_{2}=$	$\displaystyle\sqrt{g_{\theta\theta}}\left[\left(g_{t\phi,r}-\frac{g_{t\phi}}{g_{tt}}g_{tt,r}\right)+\Omega\left(g_{\phi\phi,r}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,r}\right)+\Omega^{2}\left(\frac{g_{t\phi}}{g_{tt}}g_{\phi\phi,r}-\frac{g_{\phi\phi}}{g_{tt}}g_{t\phi,r}\right)\right].$

It is worth to emphasize that this expression is valid for observers both inside and outside of the ergosphere for a restricted range of $\Omega$

$$\Omega_{-}(r,\theta)<\Omega(r,\theta)<\Omega_{+}(r,\theta),$$

(16)

with $\Omega_{\pm}=(-g_{t\phi}\pm\sqrt{g^{2}_{t\phi}-g_{\phi\phi}g_{tt}})/g_{\phi\phi}$, which could ensure that the observer at fixed $r$ and $\theta$ is timelike.

Since the gyroscope has an angular velocity $\Omega$, the general precession frequency (14) stems from two aspects: spacetime rotation (LT precession) and curvature (geodetic precession). To disclose the properties of precession of gyroscope in hairy Kerr spacetime, we will study the influence of hairy parameters on the LT precession, geodetic precession and the general spin precession of the hairy Kerr black holes and naked singularities, respectively.

In this section, we will study the LT precession frequency and the periastron precession frequency of accretion disk physics around different geometries in the hairy Kerr spacetime, as the related physics could help to test the strong gravity of the central compact bodies. We expect that this study could further differentiate the hairy Kerr black hole and naked singularity as the central objects. To this end, we should study the geodesic motion of a test massive particle around the hairy Kerr spacetime. To study the accretion disk physic via the orbits of test massive particle, we should fix the stable circular orbit around the central bodies and then perturb it. One important stable circular orbit is the inner-most stable circular orbit (ISCO) which is the last or smallest stable circular orbit of the particle. Therefore, we will first recall the procedure of deriving an orbit equation to introduce the bound orbits, circular orbits and ISCO in the hairy Kerr spacetime. Then we explore the LT precession frequency and the periastron precession frequency by perturbing the circular orbit in equatorial plane. For convenience, we shall focus on the orbits in the equatorial plane of the spacetime.