Particle motion in Hořava-Lifshitz black hole space-times

We study the particle motion in the space-time of a Kehagias-Sfetsos black hole which is a static spherically symmetric solution of a Hořava-Lifshitz gravity model. This model reduces to general relativity in the infrared limit and deviates slightly from detailed balance. Taking the viewpoint that the model is essentially a ($3+1$)-dimensional modification of general relativity we use the geodesic equation to determine the motion of massive and massless particles. We solve the geodesic equation exactly by using numerical techniques. We find that neither massless nor massive particles with nonvanishing angular momentum can reach the singularity at $r=0$. Next to bound and escape orbits that are also present in the Schwarzschild space-time we find that new types of orbits exist: manyworld bound orbits as well as two-world escape orbits. We also discuss observables such as the perihelion shift and the light deflection.

Figure 1

The effective potential $V_{\mathrm{eff}}(r)$ for a massive ((a)-(c)) and a massless ((d)-(f)) test particle, respectively, for different values of $\omega$ and $L_z$.

Figure 2

The values of $E^2-\epsilon$ and $1/L_z^2$ corresponding to the absolute minimum of the effective potential $V_{\mathrm{eff}}(r)$ at small $r$ for different values of $\omega$, $m=1$ and $\epsilon =1$. Note that while here we treat $E^2-\epsilon$ as a parameter that can take arbitrary values, we should have $E^2-\epsilon \ge -1$ when looking for zeros of $\mathcal{E}-V_{\mathrm{eff}}(r)$.

Figure 3

The two regions of the potential for which BOs of massive test particles exist. In region 1, we have MBOs, while in region 2 there exist BOs. Here $L_z=7.0$, $E^2=0.9787$, $m=1.0$, while $\omega =0.51$ and $\omega ={10}^4$, respectively. The red dotted-dashed line represents the total energy ($E^2-\epsilon$) of the test particle.

Figure 4

The values of $E^2-\epsilon$ and $1/L_z^2$ corresponding to the maximum (thick upper line) and relative minimum at large $r$ (thin lower line) of the effective potential $V_{\mathrm{eff}}(r)$ are given for $m=1$, $\epsilon =1$, $\omega =5.1$ (left) and $\omega =5.1\times {10}^4$ (right), respectively. In the dark shaded region (region I) there exist MBOs and BOs, while in the light shaded region (region II) MBOs as well as EOs exist. In region III there are MBOs, while there are TEOs in region IV.

Figure 5

The effective potential $V_{\mathrm{eff}}(r)$ for radial trajectories ($L_z=0$) of massive particles ($\epsilon =1$) in the space-time of a KS black hole with $m=1$ and different values of $\omega$.

Figure 6

Examples of MBOs and BOs of a massive test particle ($\epsilon =1$) with $L_z=7.00$, $E^2=0.9787$ in the space-time of a KS black hole with $m=1.00$ as well as $\omega =0.51$ (left) and $\omega ={10}^4$ (right). We show MBOs (region 1, top) and BOs (region 2, bottom), respectively. The red dashed circles in the plot represent the horizons of the KS black hole. Note that we are plotting two radial periods during which the particle moves from $r_{\min }$ to $r_{\max }$ and back again.

Figure 7

Examples of TEOs of a massive test particle ($\epsilon =1$) with $E=1.8$, $L_z=4$ in the space-time of a KS black hole with $m=1$ and different values of $\omega$. The dashed circles correspond to the two horizons of the KS space-time.

Figure 8

Examples of MBOs of a massless test particle ($\epsilon =0$) with $E=0.3$, $L_z=1.75$ in the space-time of a KS black hole with $m=1$ and different values of $\omega$. The dashed circles correspond to the two horizons of the KS space-time. Note that we are plotting one radial period during which the particle moves twice from $r_{\min }$ to $r_{\max }$ and back again.

Figure 9

Examples of TEOs of a massless test particle ($\epsilon =0$) with $E=1.8$, $L_z=4$ in the space-time of a KS black hole with $m=1$ and different values of $\omega$. The dashed circles correspond to the two event horizons of the KS space-time.

Figure 10

Examples of EOs of a massless test particle ($\epsilon =0$) with $E=0.47$, $L_z=2.2$ that passes very close by a KS black hole with $m=1$ and $\omega =0.51$ [Fig. 10b]. For much larger values of $\omega$ (here: $\omega ={10}^4$) the particle crosses the horizon on a TEO [Fig. 10c]. We also show the corresponding effective potential [Fig. 10a]. The red dashed line in Fig. 10a corresponds to the value of $E^2$, while the dashed circles in Fig. 10b, 10c correspond to the two event horizons of the KS space-time.

Figure 11

The value of the perihelion shift per period $T$ as a function of $\omega$ for a massive test particle ($\epsilon =1$) with energy $E^2=0.9787$ and angular momentum $L_z=7.00$ in the space-time of a KS black hole with $m=1$. We show the perihelion shift per period $T$ for an MBO (left) and for a BO (right), respectively.

Figure 12

The absolute value of the difference $\delta r_{\min }=r_{\min ,\mathrm{KS}}-r_{\min ,\mathrm{S}}$ ($\delta r_{\max }=r_{\max ,\mathrm{KS}}-r_{\max ,\mathrm{S}}$) between the minimal (maximal) radius of the BO of a massive particle in the KS space-time $r_{\min ,\mathrm{KS}}$ ($r_{\max ,\mathrm{KS}}$) and the minimal (maximal) radius of a BO in the Schwarzschild space-time $r_{\min ,\mathrm{S}}$ ($r_{\max ,\mathrm{S}}$) is shown in dependence on $\omega$ for two different values of $m$ (left). We also give the difference $\delta r_{\min }$ for an EO of a massless test particle (right). In both cases the energy of the particle is $E=0.99$ and the angular momentum $L_z=7.00$.

Figure 13

The value of the light deflection of a massive test particle in the space-time of the KS black hole with $m=1$ as a function of the impact parameter $b=L_z/E$. We give the light deflection for the TEO (left) and for the escape (EO) (right) for three different values of $\omega$. Note that the light deflection diverges at $b=b_{\mathrm{crit}}$ with $b_{\mathrm{crit}}=4.6937$ for $\omega =0.50$, $b_{\mathrm{crit}}=5.0950$ for $\omega =2.00$ and $b_{\mathrm{crit}}=5.1961\approx \sqrt{27}$ for $\omega ={10}^4$, respectively.