Numerical thermodynamic studies of classical gravitational collapse in $3\mathbf{+}1$ and $4\mathbf{+}1$ dimensions

Recent numerical work on the gravitational collapse of a five-dimemsional (5D) [$4+1$] Yang-Mills instanton has provided numerical evidence that the free energy $F=E-TS=E/3$ of a 5D Schwarzschild black hole of mass $E$ can be obtained classically via the Lagrangian. Although there is no Hawking radiation, these numerical results suggest that the quantity $TS$ has a classical meaning. We investigate this association for the physically relevant case of $3+1$ dimensional collapse. We track numerically the negative of the total Lagrangian $-L$ during the gravitational collapse of a massless scalar field to a Schwarzschild black hole in isotropic coordinates. We show that $-L$ approaches the free energy $F=E-TS=E/2$ of a four-dimensional Schwarzschild black hole to within 5%. We also show that the matter contribution to the free energy tends towards zero so that the entropy at late stages of the collapse is gravitational in origin. The entropy $S$ makes a negative contribution to the free energy and this feature is observed in our numerical simulation. There is a pronounced dip (negative contribution) in a thin slice just inside the event horizon precisely where the metric field is nonstationary. This is in accord with recent work suggesting black hole entropy is connected with the nonstationary phase space hidden behind the event horizon. We also obtain thermodynamic results for the 5D collapse of a massless scalar field which confirms that previous 5D results are universal and independent of the type of matter undergoing the collapse.

Figure 1

Space profile of the metric field $\psi$ for different times. At late times ($t=20$), the exterior region matches almost exactly the Schwarzschild form.

Figure 2

Space profile of the lapse function $N$ for different times. The event horizon ($r=0.334$) is the radius where $N$ crosses zero at late times. After crossing zero at $r=0.334$, $N$ is negative in the interior and approaches zero from below.

Figure 3

Space profile of the matter field $\chi$ for different times. The initial matter distribution is a shell. It first expands into the interior ($t=4$) before collapsing to a thin shell near the event horizon at late times. Note the propagation of an outgoing matter wave in the exterior region.

Figure 4

Accumulation plot for the gravitational Lagrangian at different times. At late times, there is a negative contribution (the large dip) stemming from a thin slice just inside the event horizon where the metric field $\psi$ is nonstationary.

Figure 5

Accumulation plots for the Klein-Gordon Lagrangian for different times. It is close to zero at late times.

Figure 6

Time dependence of the metric field $\psi$. $\dot{\psi }$ peaks just inside the event horizon, is decreasing in magnitude in the rest of the interior and is zero outside.

Figure 7

Time profile of the free energy and the horizon radius. The total free energy $F_{\mathrm{tot}}$ and the horizon radius $r_0=E/2$ converge towards the same value at late times.

Figure 8

Space profile of the lapse function $N$ at different times. $N$ crosses zero at the location of the apparent horizon. This occurs for the first time at $t=2.6$. At late times, $N$ crosses zero at $r=0.49$ and this is identified as the radius of the event horizon. After $N$ crosses zero at $r=0.49$, it is negative (albeit very small) throughout the interior ($r<0.49$) and never crosses zero inside.

Figure 9

Space profile of the matter field $\chi$ at different times. At late times, the matter has collapsed to a thin shell near the event horizon. Note the propagation of an outgoing matter wave in the exterior region.

Figure 10

Space profile of the metric field $\psi$ at different times. At late times ($t=8$), the exterior region matches almost exactly the Schwarzschild form. Note that $\psi \to 0$ in the interior.

Figure 11

Time dependence of the metric field $\psi$. It is static except for a thin slice just inside the event horizon.

Figure 12

Accumulation plot for the gravitational Lagrangian at different times. There is a negative contribution (the large dip) coming from the region just inside the event horizon.

Figure 13

Accumulation plot for the Klein-Gordon Lagrangian at different times. It approaches zero at late times except for the small contribution of the outgoing matter wave.

Figure 14

Time profile of the free energy and the mass of the black hole. The total free energy approaches a value of a third of the mass $E$ at late times.