Time like geodesics in three-dimensional
rotating Hořava AdS black hole

The three-dimensional models of gravity are of interest because it is possible to investigate efficiently some of their properties which are shared by their higher dimensional analogs, and also exhibit interesting solutions such as particle-like solutions and black holes. In this context, three-dimensional general relativity (GR), which has no local gravitational degrees of freedom, and is Lorentz invariant presents the well know Bañados–Teitelboim–Zanelli (BTZ) black hole solution with a negative cosmological constant Banados:1992wn . Also, it presents interesting properties at both classical and quantum levels and the BTZ solution shares several features of the Kerr black hole Carlip:1995qv . An important issue in gravitational physics is to know the kind of orbits that test particles follow outside the event horizon of a black hole. This information can be provided by studying the geodesics around these black holes, in this context for the BTZ background it was shown that while massive particles always fall into the event horizon and no stable orbits are possible Farina:1993xw , massless particles can escape or plunge to the horizon Cruz:1994ir .

The three-dimensional Hořava gravity gravity Sotiriou:2011dr admits a Lorentz-violating version of the BTZ black hole, i.e. a black hole solution with AdS asymptotics, only in the sector of the theory in which the scalar degree of freedom propagates infinitely fast Sotiriou:2014gna . Remarkably, in contrast to GR, the three-dimensional Hořava gravity also admits black holes with positive and vanishing cosmological constant. Nowadays, one could think that Lorentz invariance may not be fundamental or exact, but is merely an emergent symmetry on sufficiently large distances or low energies. It has been suggested in Ref. Horava:2009uw that giving up Lorentz invariance by introducing a preferred foliation and terms that contain higher-order spatial derivatives can lead to significantly improved UV behavior, the corresponding gravity theory is dubbed Hořava gravity. It was shown that the propagation of massive scalar fields is stable in the background of rotating three-dimensional Hořava AdS black holes and by employing the holographic principle the different relaxation times of the perturbed system to reach thermal equilibrium were found for the various branches of solutions Becar:2019hwk . Also, concerning to the collision of particles, it was shown that the particles can collide on the inner horizon with arbitrarily high CM energy if one of the particles has a critical angular momentum being possible the BSW process, for the non-extremal rotating Hořava AdS black hole. Also, while that for the extremal BTZ black hole the particles with critical angular momentum only can exist on the degenerate horizon, for the Lorentz-violating version of the BTZ black hole the particle with critical angular momentum can exist in a region from the degenerate horizon Becar:2020hix . Concerning to the null geodesics, it was shown that for the motion of photons new kinds of orbits are allowed, such as unstable circular orbits and trajectories of first kind. Also, it was shown that an external observer will see that photons arrive at spatial infinity in a finite coordinate time Gonzalez:2019xfr .

In this work we study the motion of particles in the background of a three-dimensional rotating Hořava AdS black hole Sotiriou:2014gna , with the aim of analyzing the effect of breaking the Lorentz symmetry by calculating the time like geodesic structure. We will show that Lorentz-violating version of the BTZ black hole posses a more rich geodesic structure, where the planetary orbits are allowed, which does not occur in the BTZ background. Also, we will have a complete knowledge of the geodesic structure for the rotating three-dimensional Hořava AdS black hole allowing us to understand in depth the Lorentz-violating effects on the BTZ black hole. For other studies about geodesics in three-dimensional spacetimes, see Fernando:2003gg ; Cruz:2013ufa ; Panotopoulos:2020zvi ; Kazempour:2017gho .

The work is organized as follows. In Sec. II we give a brief review of three-dimensional rotating Hořava AdS black hole. In Sec. III we find the motion equations for particles and we present the time like geodesic structure in Sec. IV. Finally, our conclusions are in Sec. V.

The three-dimensional Hořava gravity is described in a preferred foliation by the action Sotiriou:2011dr

$$S_{H}=\frac{1}{16\pi G_{H}}\int dTd^{2}xN\sqrt{g}\left[L_{2}+L_{4}\right]\,,$$

(1)

being the line element in the preferred foliation

$$ds^{2}=N^{2}dT^{2}-g_{ij}(dx^{i}+N^{i}dT)(dx^{j}+N^{j}dT)\,,$$

(2)

where $g_{ij}$ is the induced metric on the constant-$T$ hypersurfaces. $G_{H}$ is a coupling constant with dimensions of a length squared, $g$ is the determinant of $g_{ij}$ and the Lagrangian $L_{2}$ has the following form

$$L_{2}=K_{ij}K^{ij}-\lambda K^{2}+\xi\left({}^{(2)}R-2\Lambda\right)+\eta a_{i}a^{i}\,,$$

(3)

where $K_{ij}$, $K$, and ${}^{(2)}R$ correspond to extrinsic, mean, and scalar curvature, respectively, and $a_{i}$ is a parameter related to the lapse function $N$ via $a_{i}=-\partial_{i}\ln{N}$. $L_{4}$ corresponds to the set of all the terms with four spatial derivatives that are invariant under diffeomorphisms. For $\lambda=\xi=1$ and $\eta=0$, the action reduces to that of general relativity. In the infrared limit of the theory the higher order terms $L_{4}$ (UV regime) can be neglected, and the theory is equivalent to a restricted version of the Einstein-aether theory, the equivalence can be showed by restricting the aether to be hypersurface-orthogonal and the following relation is obtained

$$u_{\alpha}=\frac{\partial_{\alpha}T}{\sqrt{g^{\mu\nu}\partial_{\mu}T\partial\nu T}}\,,$$

(4)

where $u_{\alpha}$ is a unit-norm vector field called the aether, see Ref. Jacobson:2010mx for details. Another important characteristic of this theory is that only in the sector $\eta=0$, Hořava gravity admits asymptotically AdS solutions Sotiriou:2014gna . Therefore, assuming stationary and circular symmetry the most general metric is given by

$$ds^{2}=Z(r)^{2}dt^{2}-\frac{1}{F(r)^{2}}dr^{2}-r^{2}(d\phi+\Omega(r)dt)^{2}~{},$$

(5)

and by assuming the aether to be hypersurface-orthogonal, it results

$$u_{t}=\pm\sqrt{Z(r)^{2}(1+F^{2}(r)U^{2}(r))}\,,u_{r}=U(r)\,.$$

(6)

The theory admits the BTZ “analogue” to the three-dimensional rotating Hořava black holes described by the solution

$$F(r)^{2}=Z(r)^{2}=-M+\frac{\bar{J}^{2}}{4r^{2}}-\bar{\Lambda}r^{2}~{},~{}\Omega(r)=-\frac{J}{2r^{2}}\,,~{}U(r)=\frac{1}{F(r)}\left(\frac{a}{r}+br\right)\,,$$

(7)

where

$$\bar{J}^{2}=\frac{J^{2}+4a^{2}(1-\xi)}{\xi}~{},~{}\bar{\Lambda}=\Lambda-\frac{b^{2}(2\lambda-\xi-1)}{\xi}~{}.$$

(8)

The sign of the effective cosmological constant $\bar{\Lambda}$ determines the asymptotic behavior (flat, dS, or AdS) of the metric. Also, $\bar{J}^{2}$ can be negative, this occurs when either $\xi<0$ or $\xi>1$, $a^{2}>J^{2}/(4(\xi-1))$. The aether configuration for this metric is given explicitly by

$$u_{t}=\sqrt{F^{2}+\left(\frac{a}{r}+br\right)^{2}}\,,~{}u_{r}=\frac{1}{F^{2}}\left(\frac{a}{r}+br\right)\,,~{}u_{\phi}=0\,,$$

(9)

where $a$ and $b$ are constants that can be regarded as measures of aether misalignment, with $b$ as a measure of asymptotically misalignment, such that for $b\neq 0$ the aether does not align with the timelike Killing vector asymptotically. Note that for $\xi=1$ and $\lambda=1$, the solution becomes the BTZ black hole, and for $\xi=1$ and $\lambda\neq 1$, the solution becomes the BTZ black hole with a shifted cosmological constant $\bar{\Lambda}=\Lambda-2b^{2}(\lambda-1)$. However, there is still a preferred direction represented by the aether vector field which breaks Lorentz invariance for $\lambda\neq 1$ and $b\neq 0$. The locations of the inner and outer horizons $r=r_{\pm}$, are given by

$$\ r_{\pm}^{2}=-\frac{M}{2\bar{\Lambda}}\left(1\pm\sqrt{1+\frac{\bar{J}^{2}\bar{\Lambda}}{M^{2}}}\right)~{}.$$

(10)

Considering $M>0$, a negative cosmological constant $\bar{\Lambda}<0$, and $\bar{J}^{2}>0$ the condition $-\bar{J}^{2}\bar{\Lambda}\leq M^{2}$ must be fulfilled for the solution represents a black hole. For $0<-\bar{J}^{2}\bar{\Lambda}<M^{2}$ the black holes have inner and outer horizons $r_{-}$ and $r_{+}$, the extremal case corresponds to $-\bar{J}^{2}\bar{\Lambda}=M^{2}$, while that for $\bar{J}^{2}<0$ the black holes have outer horizon $r_{+}$, but no inner horizon $r_{-}$.

Besides the existence of inner and outer horizons, also there are universal horizons, which are given by Sotiriou:2014gna

$$\ (r_{u}^{\pm})^{2}=\frac{M-2ab}{2(b^{2}-\bar{\Lambda})}\pm\frac{1}{2(b^{2}-\bar{\Lambda})}\left[(M-2ab)^{2}-(4a^{2}+\bar{J}^{2})(b^{2}-\bar{\Lambda})\right]^{\frac{1}{2}}\,.$$

(11)

On the other hand, the existence of a well-defined spacelike foliation is essential in Hořava gravity. As, it was shown in Ref. Sotiriou:2014gna , this can be achieved by imposing the condition $F^{2}+(a/r+br)^{2}>0$ or

$$\frac{1}{r^{2}}\left((b^{2}-\bar{\Lambda})r^{4}+(2ab-M)r^{2}+\left(\frac{\bar{J}}{4}+a^{2}\right)\right)>0.$$

(12)

The Fig. (1), show the behavior of the horizons as a function of the parameter $b$, and as a function of $a$ in Fig. (2), for a choice of parameters. There are different zones, one of them is limited by $r_{-}$ and $r_{+}$; and it is described by the existence of the aether, where the roots $r_{u}^{\pm}$ are imaginary and therefore there are no universal horizons. Other zones are characterized by two real and distinct universal horizons inside the region between $r_{-}$ and $r_{+}$, outside $r_{-}$, and inside $r_{+}$; and an especial zone where both universal horizons coincide and is given by

$$r_{u}^{2}=\frac{M-2a_{\pm}(M,\bar{J},b)b}{2(b^{2}-\bar{\Lambda}(b))}\,,$$

(13)

where $a_{\pm}$ are the roots of

$$\frac{(4a^{2}+\bar{J}^{2})(b^{2}-\bar{\Lambda}(b))}{\xi(M-2ab)^{2}}=1\,.$$

(14)

In the region between $r_{u}^{-}$ and $r_{u}^{+}$, the aether turns imaginary and the foliation cannot be extended until the singularity.

In order to analysis the roots of the lapse function the case $J=0$, we will consider a set of values for the parameters that satisfied the existence of two horizons $r_{-}$ and $r_{+}$, so the condition $0<-\bar{J}^{2}\bar{\Lambda}<M^{2}$ must be satisfied, and also there are not universal horizons, thereby the roots of $r_{u}^{\pm}$ are imaginary, as in our previous analysis. So, in order to satisfy the above condition the parameter $\xi$ must satisfied $\xi_{e}<\xi<\xi_{c}$, where $\xi_{e}$ corresponds to the value of $\xi$ for which the black hole is extremal

$$\xi_{e}=\frac{2a^{2}\left(\sqrt{a^{2}\left(\Lambda-2b^{2}(\lambda-1)\right)^{2}+b^{2}(2\lambda-1)M^{2}}-2b^{2}\lambda-\Lambda\right)}{M^{2}-4a^{2}\left(b^{2}+\Lambda\right)}\,,$$

(15)

and $\xi_{c}$ is the value of $\xi$ for which the black hole passes from having two horizons to having one horizon and it is $\xi_{c}=1$ for non-rotating black holes. For $\xi>\xi_{c}$ the black holes are described by one horizon Becar:2019hwk . In Fig. 3, we plot the behaviour of the horizons for $\xi=0.9$ for non-rotating black holes.

In the following we will focus mainly on a set of values for the parameters that satisfied the existence of two horizons $r_{-}$ and $r_{+}$, with $a>0$, and $b>0$, where there are not universal horizons, thereby the roots of $r_{u}^{\pm}$ are imaginary, see Figs. 1, 2, and 3. Note that under the conditions $M>0$, $\bar{J}^{2}>0$, and $\bar{\Lambda}<0$, the roots of $r_{u}^{\pm}$ are imaginary when $M<2ab$ from Eq. (11).

In this section, we find the motion equations of probe massive particles around the three-dimensional Hořava AdS black hole. It is important to emphasize Cropp:2013sea that in a Lorentz-violating scenario, particles will be generically coupled to the aether field generating UV modifications of the matter dispersion relations, it is more, one can also expect radiative corrections in the infrared sector but these contributions are suppressed by knows mechanisms. In our analysis we are interested on the infrared limit of the theory; so the presence of higher-order terms ($L_{4}$) related to the UV behaviour of the theory are ignored and in this case the theory can be formulated in a covariant fashion, and it then becomes equivalent to a restricted version of Einstein-aether theory Sotiriou:2014gna . Since our analysis is focused in the low energy part of the theory, the interaction between the massive particle and the aether field is ignored, thus the presence of aether field only affect the background spacetime geometry. It is worth mentioning that a similar analysis was performed in Zhu:2019ura where the authors analyzed the evolution of the photon around the static neutral and charged aether black holes using the Hamilton-Jacobi equation. Therefore, the massive particles follow the typical geodesics in such given black holes spacetime, that can be derived from the Lagrangian of a test particle, which is given by chandra

$$\mathcal{L}=\frac{1}{2}\left(g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\right)~{}.$$

(16)

So, for the three-dimensional rotating Hořava AdS black hole described by the metric (5), the Lagrangian associated with the motion of the test particles is given by

$$2\mathcal{L}=-[\mathcal{F}(r)-r^{2}\Omega^{2}(r)]\dot{t}^{2}+2r^{2}\Omega(r)\,\dot{t}\,\dot{\phi}+{\dot{r}^{2}\over\mathcal{F}(r)}+r^{2}\,\dot{\phi}^{2}~{},$$

(17)

where $\dot{q}=dq/d\tau$, and $\tau$ is an affine parameter along the geodesic. Here, we have defined $F(r)^{2}=Z(r)^{2}=\mathcal{F}(r)$. Since the Lagrangian (17) is independent of the cyclic coordinates ($t,\phi$), then their conjugate momenta ($\Pi_{t},\Pi_{\phi}$) are conserved. Then, the equations of motion are obtained from $\dot{\Pi}_{q}-\frac{\partial\mathcal{L}}{\partial q}=0$, and yield

$$\dot{\Pi}_{t}=0\,,\quad\dot{\Pi}_{r}=-[\mathcal{F}^{\prime}(r)/2-r\Omega^{2}(r)-r^{2}\Omega^{\prime}(r)]\dot{t}^{2}-{\mathcal{F}^{\prime}(r)\,\dot{r}^{2}\over 2\mathcal{F}^{2}(r)}+r\,\dot{\phi}^{2}\,,\quad\textrm{and}\quad\dot{\Pi}_{\phi}=0~{},$$

(18)

where $\Pi_{q}=\partial\mathcal{L}/\partial\dot{q}$ are the conjugate momenta to the coordinate $q$, and are given by

$$\Pi_{t}=-[\mathcal{F}(r)-r^{2}\Omega^{2}(r)]\dot{t}+r^{2}\Omega(r)\,\dot{\phi}\equiv-E~{},\quad\Pi_{r}={\dot{r}\over\mathcal{F}(r)}~{}\textrm{and}\quad\Pi_{\phi}=r^{2}\Omega(r)\dot{t}+r^{2}\,\dot{\phi}\equiv L~{},$$

(19)

where $E$ and $L$ are integration constants associated to each of them. Therefore, the Hamiltonian is given by

$$\mathcal{H}=\Pi_{t}\dot{t}+\Pi_{\phi}\dot{\phi}+\Pi_{r}\dot{r}-\mathcal{L}\,.$$

(20)

Thus,

$$2\mathcal{H}=-E\,\dot{t}+L\,\dot{\phi}+{\dot{r}\over\mathcal{F}(r)}\equiv-m^{2}~{}.$$

(21)

where $m=1$ for timelike geodesics or $m=0$ for null geodesics. Therefore, we obtain

	$\displaystyle\dot{\phi}=-{1\over(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})\bar{\Lambda}}\left[{EJ\over 2}+L\left({-\bar{\Lambda}r^{2}}-M-{J^{2}-\bar{J}^{2}\over 4r^{2}}\right)\right]~{},$		(22)
	$\displaystyle\dot{t}=-{[Er^{2}-JL/2]\over(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})\bar{\Lambda}}~{},$		(23)
	$\displaystyle\dot{r}^{2}=\left(E-\frac{JL}{2\,r^{2}}\right)^{2}-\left(-M+\frac{\bar{J}^{2}}{4\,r^{2}}-\bar{\Lambda}r^{2}\right)\left(m^{2}+\frac{L^{2}}{r^{2}}\right)=\left(E-V_{-}\right)\left(E-V_{+}\right)~{},$		(24)

where $V_{\pm}(r)$ is the effective potential and it is given by

$$V_{\pm}(r)=\frac{JL}{2\,r^{2}}\pm\sqrt{\left(-M+\frac{\bar{J}^{2}}{4\,r^{2}}-\bar{\Lambda}r^{2}\right)\left(m^{2}+\frac{L^{2}}{r^{2}}\right)}~{}.$$

(25)

Since the negative branches have no classical interpretation, they are associated with antiparticles in the framework of quantum field theory Deruelle:1974zy , We choose the positive branch of the effective potential $V=V_{+}$. In the next section we will perform a general analysis of the equations of motion. Note that if $\dot{t}>0$ for all $r>r_{+}$ the motion is forward in time outside the horizon, so from Eq. (23) for $\bar{\Lambda}<0$ the following condition must be fulfilled

$$Er^{2}-JL/2>0\,.$$

(26)

On the other hand, Eq. (22) can be rewritten as

$$\dot{\phi}=-\frac{1}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})\bar{\Lambda}}\left[\frac{J}{2r^{2}}\left(Er^{2}-\frac{JL}{2}\right)+LF(r)^{2}\right]\,,$$

(27)

due to the term $Er^{2}-JL/2>0$ and $F(r)^{2}>0$ for $r>r_{+}$, when $L$, and $J$ have the same sign ($JL>0$), the term in square-brackets have not zeros outside the event horizon. However, when $L$ and $J$ have different signs ($JL<0$), the zeros can be in the relevant domain, and it would not necessarily indicate a turning point. In fact, the positive root ($R$) of the term in square-brackets in Eq. (22) is

$$R^{2}=\frac{JE}{4\bar{\Lambda}L}-\frac{M}{2\bar{\Lambda}}-\frac{\sqrt{(4LM-2JE)^{2}-16\bar{\Lambda}L\left(J^{2}L-\bar{J}^{2}L\right)}}{8\bar{\Lambda}L}\,,$$

(28)

and it corresponds to the point where the angular velocity of the test particle $\omega(r)$

$$\dot{\phi}/\dot{t}={d\phi\over dt}={{EJ\over 2}+L\left({-\bar{\Lambda}r^{2}}-M-{J^{2}-\bar{J}^{2}\over 4r^{2}}\right)\over Er^{2}-JL/2}\equiv\omega(r)\,,$$

(29)

is null. Also, note that for a motion with $L=0$, $\omega(r)={J\over 2\,r^{2}}$. Thus, a particle dropped ‘straight in’ ($L=0$) from a finite distance is ‘dragged’ just by the influence of gravity so that it acquires an angular velocity ($\omega$) in the same sense as that of the source of the metric ($J$), effect called as ”dragging of inertial frames”.

In this section we analyze the motion of particles, $m^{2}=1$, so the effective potential is given by

$$V(r)=\frac{JL}{2\,r^{2}}+\sqrt{\left(-M+\frac{\bar{J}^{2}}{4\,r^{2}}-\bar{\Lambda}r^{2}\right)\left(1+\frac{L^{2}}{r^{2}}\right)}~{},$$

(30)

whose behaviour is showed in Fig. 4 for $J>0$, and positive values of the angular momentum of the particle (direct geodesics), and for $J<0$, and positive values of the angular momentum of the particle (retrograde geodesics). We can observe for direct geodesics (left panel), there is a critical value of the angular momentum ($L_{LSCO}$) where the last stable circular orbit (LSCO) is present at $r=r_{LSCO}$; and for $L>L_{LSCO}$ we can distinguish four orbits, the planetary, the second kind, the circular unstable at $r=r_{U}$, the circular stable at $r=r_{S}$, and critical orbits are allowed. We will study the existence of circular orbit in detail in subsection B. Note that, in the BTZ background is allowed orbits of second kind. So, the spacetime analyzed present a richer geodesic structure. Also, for retrograde geodesics (right panel) we can observe that, for $r>r_{+}$, circular and planetary orbits are not allowed. Here, the trajectory always have a turning point, from which the particle plunge in the event horizon, knowing as trajectory of second kind.

The orbit in polar coordinates is given by ¹¹1In this work, we have considered a choosing of values for the parameters such that the zeros of term in square-brackets are not located in the relevant domain they are inside the event horizon.

$$-{r^{2}\over(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})\bar{\Lambda}}\left[{EJ\over 2}+L\left({-\bar{\Lambda}r^{2}}-M-{J^{2}-\bar{J}^{2}\over 4r^{2}}\right)\right]\left(\pm\frac{dr}{d\phi}\right)=\sqrt{P(r)}\,.$$

(31)

where we have used Eq. (22) and Eq. (24), $P(r)$ corresponds to the characteristic polynomial, and it is given by

	$\displaystyle P(r)=r^{6}\,\bar{\Lambda}+r^{4}\left(E^{2}+{L^{2}\bar{\Lambda}}+M\right)+r^{2}\left(L^{2}M-JEL-\bar{J}^{2}/4\right)+(J^{2}-\bar{J}^{2})L^{2}/4$
	$\displaystyle P(r)\equiv-\bar{\Lambda}\left(-r^{6}+\alpha\,r^{4}+\beta\,r^{2}+\gamma\right)\,.$		(32)

where

	$\displaystyle\alpha=-{E^{2}+M+\bar{\Lambda}L^{2}\over\bar{\Lambda}},$		(33)
	$\displaystyle\beta={JEL+\bar{J}^{2}/4-ML^{2}\over\bar{\Lambda}},$		(34)
	$\displaystyle\gamma=-{(J^{2}-\bar{J}^{2})L^{2}\over 4\bar{\Lambda}},$		(35)

Therefore, we can see that depending on the nature of its roots, we can obtain the allowed motions for this spacetime.