Mass inflation in Brans-Dicke gravity

A detailed nonlinear analysis of the internal structure of spherical, charged black holes that are accreting scalar matter is performed in the framework of the Brans-Dicke theory of gravity. We choose the lowest value of the Brans-Dicke parameter that is compatible with observational constraints. First, the homogeneous approximation is used. It indicates that mass inflation occurs and that the variations of the Brans-Dicke scalar inside the black hole, which could in principle be large in the absence of mass inflation, become small when mass inflation does occur. Then, a full nonlinear numerical study of the black hole interior perturbed by a self-gravitating massless uncharged scalar field is performed. We use an algorithm with adaptive mesh refinement capabilities. In this way, the changes in the internal structure of the black hole caused by mass inflation are determined, as well as the induced variations of the Brans-Dicke scalar, confirming, qualitatively, the indications given by the homogeneous approximation.

Figure 1

Initial conditions in the numerical integration box, defined by $u_0=0\le u<30$, $v_0=5\le v<20$, for the case where the black hole is fed by a compact influx of scalar field $\phi$. The unperturbed black hole has mass and charge $m_0=1$, $q=0.95$. The influx along the null segment $u=u_0$ starts at $v_0=5$, has amplitude $A=0.05$ or $A=0.1$, and endures for $\Delta v=1$. The influx, depicted by a curve, is nonzero only for $v_0<v<v_0+\Delta v$. The scale in the $u$ and $v$ axes is different to make closer contact with the Carter-Penrose diagrams shown in Figs. 3, 9.

Figure 2

The radial coordinate $r$ as a function of $u$ and $v$ close to the inner ingoing apparent horizon (top panel) and outer ingoing apparent horizon (bottom panel), for the unperturbed RN black hole with $m=1$ and $q=0.95$. Horizons are at $r_{-}\approx 0.6878$ and $r_{+}\approx 1.31$, in agreement with analytical expectation. Contours are plotted only for radii in the range of the color bar, $r\in [0.6876,0.6879]$ in the top panel, $r\in [1.2,1.4]$ in the bottom panel.

Figure 3

Carter-Penrose diagram for the eternal RN black hole. Some constant $r$ lines are depicted. The dotted line encloses a rectangular region in the $u$, $v$ plane, where the integration takes place (in the chosen coordinates $u_0=0\le u<30$, $v_0=5\le v<20$). The dashed lines enclose two regions corresponding to the regions enclosed by dashed lines seen in Fig. 2. The same patterns of $r=\mathrm{\text{constant}}$ lines are seen.

Figure 4

The evolution of $|g_{rr}|$, $|g_{tt}|$ (top panel, solid and dashed lines, respectively) and $\Phi -1$ (bottom panel) with $r$, in the case $w=0$ with ${\rho }_i={10}^{-4}$. This behavior of $|g_{rr}|$, $|g_{tt}|$ indicates the presence of a horizon around $r\simeq 0.69$. The evolution of $\Phi -1$ indicates that the BD scalar diverges as the horizon is approached.

Figure 5

The evolution of $|g_{rr}|$, $|g_{tt}|$ (top panel, solid and dashed lines, respectively) and $\Phi -1$ (bottom panel) in the $w=1$ case for various choices of ${\rho }_i$ (from right to left in top figure and from dotted to solid line in bottom figure, ${\rho }_i={10}^{-4}$, $2\times {10}^{-4}$, $4\times {10}^{-4}$, $8\times {10}^{-4}$). This behavior of $|g_{rr}|$, $|g_{tt}|$ indicates the existence of mass inflation. A noticeable jump in $\Phi$ occurs at the beginning of mass inflation, but it is followed by a gentle variation of $\Phi$.

Figure 6

The evolution of $|g_{tt}|/|g_{rr}|$ for $w=0$ (dark solid line) and for $w=1$ (same cases as exhibited in Fig. 5).

Figure 7

The radial coordinate $r$ as a function of the null coordinates $u$ and $v$ close to the inner apparent horizon when a single compact pulse with amplitude $A=0.05$ (top panel) or $A=0.1$ (bottom panel) and during $\Delta v=1$ is fed on to an initially unperturbed RN black hole with $m_0=1$ and $q=0.95$. The radius $r$ is no longer a monotonic function of $u$ as in the unperturbed case (Fig. 2, top panel). The locus of the inner apparent horizon is a curve that interpolates between the maxima $\partial r/\partial v=0$ of the $r=\mathrm{\text{constant}}$ curves. For the pulse with larger amplitude, the inner apparent horizon changes more abruptly with $v$.

Figure 8

This plot shows a detail of the radius $r$ in the vicinity of the outer horizon for the pulse with amplitude $A=0.05$ (top panel) or $A=0.1$ (bottom panel). The behavior is similar to the case without accretion $A=0$ (Fig. 2, bottom panel), which can be attributed to the fact that the accreted mass is small. The value of $u$ at large $v$ (in the future) differs noticeably from the zero accretion case.

Figure 9

Heuristic Carter-Penrose diagram for the perturbed RN black hole. The initial perturbation is the curve depicted at the beginning of the integration box. The diagram is drawn to include the features observed in Figs. 7, 8, namely, that $u_a$ increases/decreases with $v_a$ for the inner apparent horizon (IAH)/outer apparent horizon (OAH). One expects the curvature to blow up at the Cauchy horizon (CH), and to become Planckian on a spacelike surface before the CH.

Figure 10

${log}_{10}(m)$ as a function of $u$ and $v$ for $A=0.05$ (top panel) and $A=0.1$ (bottom panel), showing that mass inflation occurs as $v$ increases, approaching the Cauchy horizon of the black hole. The mass function is quite sensitive to $A$.

Figure 11

Lines of constant Ricci scalar (actually ${log}_{10}|R|$) for $A=0.05$.

Figure 12

$\Phi -1$ as a function of $u$ and $v$ for $A=0.05$. The variations of $\Phi$ are small slowly fading away after the initial collapse of the scalar field pulse.

Figure 13

Noncompact influx: ${log}_{10}(m)$ as a function of $u$ and $v$ for the perturbation (41), with $A=0.1$. The plot shows that mass inflation occurs close to the Cauchy horizon of the black hole.

Figure 14

Noncompact influx: ${log}_{10}|R|$ as a function of $u$ and $v$ for the perturbation (41), with $A=0.1$. The behavior of the curvature is qualitatively different from the compact pulse case (cf. Fig. 12).

Figure 15

Noncompact influx: $\Phi -1$ for the perturbation (41), with $A=0.1$. The Brans-Dicke scalar variations are qualitatively different from the compact pulse case (cf. Fig. 11).