Spherically symmetric solutions in modified Hořava-Lifshitz gravity

We find spherically symmetric solutions in the modified Hořava-Lifshitz gravity proposed recently by Blas, Pujolas and Sibiryakov. The nonlinear equations of the two-derivative action turn out to be similar to those stemming from the four-derivative action explored recently. We analyze the solutions and derive constraints on the relevant new coupling constant. We also analyze the case where the cosmological constant is nonzero. We derive the large-distance expansion of solutions and show that the power of the standard Newton’s law is modified in the presence of a cosmological constant.

Figure 1

Thin solid lines: $r/r_0$ (vertical axis) is plotted against $W$ (horizontal axis), for $a=-1$, $b=3$. There are four branches that reach asymptotic infinity at $W=-\frac{1}{|a|}$ and $W=\frac{1}{\sqrt{|a|b}}$. $r$ reaches zero when $W=-\frac{1}{\sqrt{|a|b}}$. Dashed line: $r/r_0$ (vertical axis) is plotted against $\dot{W}$. Thick solid line: The four-dimensional scalar curvature (in units of $r_0^{-2}$).

Figure 2

$N/N_0$ (vertical axis) is plotted against $W$ (horizontal axis), for $a=-1$, $b=3$.

Figure 3

Left panel: $r/r_0$ (vertical axis) is plotted against $g$ (horizontal axis), for $a=-1$, $b=3$ when $W>0$. The left branch has $g<0$ while the right branch has $g>\frac{2}{|a|}$. Asymptotic infinity $r\to \infty$ is at $g\to \pm \infty$. In the left branch $r_0<r<\infty$ while in the right branch $\sqrt{\frac{b}{|a|}}{r}_0<r<\infty$. The right branch corresponds to $\frac{1}{|a|b}<W<\infty$, while the left branch to $0<W<\frac{1}{|a|b}$. Right panel: $N/N_0$ (vertical axis) is plotted against $g$ (horizontal axis), for $a=-1$, $b=3$.

Figure 4

Left panel: $r/r_0$ (vertical axis) is plotted against $g$ (horizontal axis), for $a=-1$, $b=3$ when $W<0$. The left branch has $g<0$, the center branch has $\frac{b}{|a|}>g>\frac{2}{|a|}$ while the right branch has $g>\frac{b}{|a|}$. Asymptotic infinity $r\to \infty$ is at $g=\frac{b}{|a|}$. In the left branch $0<r<r_0$, in the center branch $\sqrt{\frac{b}{|a|}}{r}_0<r<\infty$ while in the right branch $0<r<\infty$. The left branch corresponds to $-\frac{1}{\sqrt{|a|b}}<W<0$, the right branch to $-\frac{1}{|a|}<W<-\frac{1}{\sqrt{|a|b}}$ and the center branch to $-\frac{1}{|a|}>W>-\infty$. Right panel: $N/N_0$ (vertical axis) is plotted against $g$ (horizontal axis), for $a=-1$, $b=3$, $W<0$.

Figure 5

Left panel: $a_{+}$ as a function of $|a|$. Right panel: $a_{-}$ as a function of $|a|$.

Figure 6

On the left panel, $r/r_0$ versus $W$ for $a=b=1$. On the right panel, $N/N_0$ versus $W$ for $a=b=1$.

Figure 7

On the left panel, $r/r_0$ versus $g$, with $W>0$ for $a=b=1$. On the right panel, $N$ as a function of $g$ normalized to one at $r\to \infty$.

Figure 8

On the left panel, $r/r_0$ versus $g$, with $W<0$ for $a=b=1$. There is no asymptotic infinity and $r$ varies from 0 to $r_0$. On the right panel, $N/N_0$ as a function of $g$.