Quantum gravity effects in the Kerr spacetime

We analyze the impact of the leading quantum gravity effects on the properties of black holes with nonzero angular momentum by performing a suitable renormalization group improvement of the classical Kerr metric within quantum Einstein gravity. In particular, we explore the structure of the horizons, the ergosphere, and the static limit surfaces as well as the phase space available for the Penrose process. The positivity properties of the effective vacuum energy-momentum tensor are also discussed and the “dressing” of the black hole’s mass and angular momentum are investigated by computing the corresponding Komar integrals. The pertinent Smarr formula turns out to retain its classical form. As for their thermodynamical properties, a modified first law of black-hole thermodynamics is found to be satisfied by the improved black holes (to second order in the angular momentum); the corresponding Bekenstein-Hawking temperature is not proportional to the surface gravity.

Figure 1

The radial distance $d(r)$ in the equatorial plane for $m=10$ and various values of $a$. All quantities are expressed in Planck units. The gray scale runs from black to gray for increasing $a$.

Figure 2

The function $q(\Omega )$ in the 4 cases discussed in the text.

Figure 3

The figures show examples of the 3 possible configurations the function $Q_{\tilde{b}}^{\overline{w}}$ of Eq. (4.6) can assume, with two, one, and no zero on the positive real axis.

Figure 4

The figures show the $m$ dependence of the radii $r_{b\pm }$ (thick lines) and $r_{b\pm }^I$ (thin lines) for $a=5$ and several values of $\theta$. The continuous lines represent $r_{\pm }$ and $r_{\pm }^I$. The dashed lines represent $r_{S\pm }$ and $r_{S\pm }^I$.

Figure 5

The figures show a cross section through the event horizons (continuous lines) and static limit surfaces (dashed lines) in the $xz$ plane for a quantum black hole with $\tilde{m}=6$ and $\tilde{a}=5$. To facilitate the comparison with the classical case, in (b) the corresponding classical surfaces are superimposed (thick lines).

Figure 6

The solution $\tilde{m}(\tilde{a})$ of the quantum extremality condition for the improved Kerr black hole [with $d(r)=r$ and $\overline{w}=1$]. The dashed line represents the $\tilde{m}(\tilde{a})=\tilde{a}$ dependence of the classical Kerr spacetime. For $\tilde{a}\to 0$, $\tilde{m}$ assumes its minimum value at $\sqrt{\overline{w}}$, while it approaches the classical behavior for $\tilde{a}\to \infty$.

Figure 7

The figures show the $m$ dependence of the improved radii $r_{b\pm }^I$ obtained from the exact $d(r)$ given in (2.5) at $\theta =\frac{\pi }{2}$, for $\overline{w}=5$ and several values of $a$. The outer curves are the improved static limits $r_{S\pm }^I$ and the inner ones the improved event horizons $r_{\pm }^I$. The structure of the curves is essentially the same as in the $d(r)=r$ approximation.

Figure 8

The figure shows schematically the $r$ dependence of $v_{\pm }^{\mathrm{light}}$, $v_0^{\mathrm{eq}}$, and $v_{\mathrm{dragging}}$ at the equatorial plane. The hatched region corresponds to pairs $(r,v)$ for which the test particle has negative energy.

Figure 9

The plots on the LHS of this figure display the $r$ dependence of $v_{\pm }^{\mathrm{light}}$ and $v_0^{\mathrm{eq}}$ and are analogous to Fig. 8. They refer to classical Kerr black holes with $M=15m_{\mathrm{pl}}$, $a=13.5m_{\mathrm{pl}}$ and $M=15m_{\mathrm{pl}}$, $a=12.6m_{\mathrm{pl}}$, respectively. They have the same ratio $a/m=0.9$. On the RHS the radius of the critical surfaces is displayed for all masses up to $20m_{\mathrm{pl}}$, for the constant ratio $a/m=0.9$ as in the corresponding plots on the LHS. The dashed vertical line in the plots on the RHS symbolizes the mass values used on the LHS.

Figure 10

The same type of plots as in Fig. 9, but now for the improved Kerr black hole, with masses ranging from $M=5m_{\mathrm{pl}}$ down to $M=0.5m_{\mathrm{pl}}$. All examples considered have an identical ratio $a/m=0.9$.