Circular geodesics of naked singularities in the Kehagias-Sfetsos metric of Hořava’s gravity

We discuss photon and test-particle orbits in the Kehagias-Sfetsos (KS) metric of Hořava’s gravity. For any value of the Hořava parameter $\omega$, there are values of the gravitational mass $M$ for which the metric describes a naked singularity, and this is always accompanied by a vacuum “antigravity sphere” on whose surface a test particle can remain at rest (in a zero angular momentum geodesic), and inside which no circular geodesics exist. The observational appearance of an accreting KS naked singularity in a binary system would be that of a quasistatic spherical fluid shell surrounded by an accretion disk, whose properties depend on the value of $M$, but are always very different from accretion disks familiar from the Kerr-metric solutions. The properties of the corresponding circular orbits are qualitatively similar to those of the Reissner-Nordström naked singularities. When event horizons are present, the orbits outside the Kehagias-Sfetsos black hole are qualitatively similar to those of the Schwarzschild metric.

Figure 1

The $g_{tt}$ coefficient of the KS metric. For $L=0$, this is also the effective potential $V_{\text{eff}}=f$, given by Eq. (23). For the black curves (positive everywhere), corresponding to naked singularities, $\omega M^2$ varies from 0.05 to 0.45 in steps of 0.05, from the higher to the lower curve; for the gray curves (having zeros), corresponding to black holes, $\omega M^2$ has the values 0.5, 0.6, 0.8, and 2.0, increasing in the same sense. The function $f\le 1$ satisfies $f\to 1$ as $x\to 0^{+}$ and $x\to \infty$, and is continuous everywhere. The zeros of $f$ correspond to the radial location of the event horizons.

Figure 2

Effective potential for photons at the equatorial plane, Eq. (22). Different curves correspond to different values of $\omega M^2$. The upper, gray curves have $\omega M^2=0.30$, 0.32, 0.34, 0.36, from top to bottom. They do not allow circular photon orbits. The black short-dashed curve corresponds to ${(\omega M^2)}_{\mathrm{ms}.\mathrm{ph}}$, Eq. (28). The positive black curves have $\omega M^2$ varying from 0.40 to 0.48 in steps of 0.02 and correspond to the naked singularity case. The dashed black curve corresponds to $\omega M^2=0.5$. The black curves below it have $\omega M^2=0.6$, 0.9, 2.0 and 100. Black dots correspond to stable photon orbits, whereas gray dots correspond to unstable photon orbits. The black dot on the short-dashed curve represents the marginally stable photon orbit. The vertical dashed line at $x=3$ marks the Schwarzschild value of the photon orbit radius.

Figure 3

Regions of (in)stability of timelike circular geodesics and relevant radii in the KS spacetime. Black: Region between horizons, where the vector ${\partial }_t$ is spacelike. Dark gray: Stability region for timelike circular geodesics. Light gray: Instability region for timelike circular geodesics. White: Region of no circular timelike geodesics. Solid line: $x_{0\mathrm{G}}$. Dotted lines: $x_{\mathrm{ms}}$. Dot-dashed lines: $x_{\mathrm{ph}}$. We see that the $x_{\mathrm{ms}}$ line meets the photon orbit lines, at ($x_{\mathrm{ms}.\mathrm{ph}},(\omega M^2{)}_{\mathrm{ms}.\mathrm{ph}}$); see Eq. (28).

Figure 4

Angular momentum (squared) of test particles in circular orbits as a function of $x$. Different curves are for different values of $\omega M^2$, varying from 0.08 (bottom) to 0.44 (top) in steps of 0.04. Solid black lines correspond to stable motion, whereas dotted gray lines correspond to unstable motion. The lower six curves correspond to the case of no marginally stable orbits, $\omega M^2<(\omega M^2{)}_{\mathrm{ms}}$. The next two (middle) curves, for ${(\omega M^2)}_{\mathrm{ms}}<\omega M^2<{(\omega M^2)}_{\mathrm{ms}.\mathrm{ph}}$, exhibit two marginally stable orbits, an ISCO and an OSCO. For the upper two curves, ${(\omega M^2)}_{\mathrm{ms}.\mathrm{ph}}<\omega M^2$, the angular momentum in the inner stable branch goes to infinity at the stable photon orbit, and the angular momentum in the unstable branch goes to infinity at the unstable photon orbit. There are no circular orbits between the photon orbits. See Sec. 4 and Fig. 3.

Figure 5

Effective potential $V_{\text{eff}}$ as a function of $x$ for different values of $L^2$ (decreasing from top to bottom). Depicted in the figure is the KS black hole case, $\omega M^2=1>1/2$, qualitatively similar to the Schwarzschild solution.

Figure 6

Same as Fig. 5, but for $(\omega M^2{)}_{\mathrm{ms}.\mathrm{ph}}<\omega M^2=0.4<0.5$, the regime with two photon orbits and one marginally stable orbit.

Figure 7

Same as Fig. 5, but for $\omega M^2=0.2<(\omega M^2{)}_{\mathrm{ms}}$, the regime of stable orbits alone, qualitatively similar to Newtonian gravity for $x>x_{0\mathrm{G}}$; cf. Eq. (8).

Figure 8

Same as Fig. 5, but for $(\omega M^2{)}_{\mathrm{ms}}<\omega M^2=0.32<(\omega M^2{)}_{\mathrm{ms}.\mathrm{ph}}$, the regime with two marginally stable orbits.

Figure 9

Angular frequency of the circular orbits as a function of $x$. Different curves are for different values of $\omega M^2$ in the naked-singularity regime, varying from 0.08 (bottom) to 0.44 (top) in steps of 0.04. Solid black lines correspond to stable motion, whereas dotted gray lines correspond to unstable motion in the region between the OSCO and the ISCO (or the photon orbit and the ISCO, for the top two curves).

Figure 10

Frequency shift at the surface of the antigravity sphere, at $x=x_{0\mathrm{G}}(\omega M^2)$, Eq. (8). The Doppler frequency shift is not present here, since $\mathrm{\Omega }(x_{0\mathrm{G}})=0$. Accordingly, the shape of the line profile is unaffected, but the line is broadened and its center is shifted by the factor $g$, owing to gravitational redshift, Eq. (31).

Figure 11

Direct images of a Keplerian disk in the Schwarzchild (left) and KS metric with $\omega M^2=0.1$ (right). The inner radius of the disk is at $r_{\text{in}}=r_{\mathrm{ms}}$ in the Schwarzchild case and $r_{\text{in}}=r_{\mathrm{\Omega }\max }$ in the KS case. The outer radius is $r_{\text{out}}=20M$. Images are generated for three representative values of observer inclination ${\theta }_o=30^{\circ }$, 60° and 85° (from top to bottom). The color scale is given by normalized frequency shift $g*=(g-g_{\min })/(g_{\max }-g_{\min })$.

Figure 12

First indirect images of a Keplerian disk in the Schwarzchild (left) and KS metric with $\omega M^2=0.1$ (right). The inner radius of the disk is at $r_{\text{in}}=r_{\mathrm{ms}}$ in the Schwarzchild case and $r_{\text{in}}=r_{\mathrm{\Omega }\max }$ in the KS case. The outer radius is $r_{\text{out}}=20M$. Images are generated for three representative values of observer inclination ${\theta }_o=30°$, 60° and 85° (from top to bottom). The color scale is given by normalized frequency shift $g*=(g-g_{\min })/(g_{\max }-g_{\min })$.

Figure 13

Line profiles generated in Keplerian orbits at radius $x_{\mathrm{e}}=1.1x_{0\mathrm{G}}$ for four representative values of parameter $\omega M^2=0.1$, 0.2, 0.3, 0.4, and three inclination angles (top to bottom) ${\theta }_{\mathrm{o}}=30°,60°,85°$. The original emissivity function is $\epsilon (\nu )={\epsilon }_0\exp [-10^{-5}(\nu /{\nu }_0-1{)}^2]$.

Figure 14

Line profiles generated in Keplerian orbits at $r=10M$ for four representative values of parameter $\omega M^2=0.1$, 0.2, 0.3, and 0.4 and three inclination angles (top to bottom) ${\theta }_{\mathrm{o}}=30°$, 60°, and 85°. The original emissivity function is $\epsilon (\nu )={\epsilon }_0\exp [-10^{-5}{(\nu /{\nu }_0-1)}^2]$.

Figure 15

Line profiles generated at ISCO for two values of parameter $\omega M^2=0.3$, and 0.4, and three inclination angles ${\theta }_{\mathrm{o}}=30°$, 60°, and 85°. The original emissivity function is $\epsilon (\nu )={\epsilon }_0\exp [-10^{-5}(\nu /{\nu }_0-1{)}^2]$.

Figure 16

Line profiles generated at radius $r_{\mathrm{e}}=r_{\mathrm{\Omega }\max }$ for three representative values of parameter $\omega M^2=0.1$, 0.2, 0.28, and three inclination angles ${\theta }_{\mathrm{o}}=30°,60°,85°$. The original emissivity function is $\epsilon (\nu )={\epsilon }_0\exp [-10^{-5}(\nu /{\nu }_0-1{)}^2]$.

Figure 17

Comparison of spectral line profiles generated from a Keplerian ring at the ISCO of a Schwarzchild black hole (solid), the ISCO of a KS naked singularity with parameter $\omega M^2=0.3$ (dashed) and at the orbit of maximum angular velocity (at $r=r_{\mathrm{\Omega }\max }$) of a KS naked singularity with parameter $\omega M^2=0.2$ (dotted). Observer inclination is ${\theta }_{\mathrm{o}}=85°$.