How big is a black hole?

TL;DR

The paper defines a coordinate-free notion of interior volume for a spherically symmetric black hole as the maximal-volume, spacelike, spherically symmetric surface bounded by the horizon sphere. By recasting the problem as geodesics in an auxiliary 2D manifold with metric $ds^2_{M_{aux}} = r^4(-f(r)\,dv^2 + 2\,dv\,dr)$, it identifies the radius $r_V = \tfrac{3}{2}m$ that maximizes the relevant functional and derives the asymptotic volume growth $V(v) \approx 3\sqrt{3}\,\pi\, m^{2}\, v$ for large $v$. Numerically, the volume grows without bound with time due to a long steady phase at $r \approx r_V$, forming a cylinder-like interior; this holds across related spherically symmetric spacetimes, with appropriate adaptations (e.g., Kruskal or Reissner–Nordström). The results have implications for information-entropy considerations in black holes, highlighting a large interior space even as the exterior horizon area remains fixed.

Abstract

The 3d volume inside a spherical black hole can be defined by extending an intrinsic flat-spacetime characterization of the volume inside a 2-sphere. For a collapsed object, the volume grows with time since the collapse, reaching a simple asymptotic form, which has a compelling geometrical interpretation. Perhaps surprising, it is large. The result may have relevance for the discussion on the information paradox.

Figures (6)

• Figure 1: Conformal diagram of a collapsing object spacetime. The sphere $S_v$ is on the horizon, at time $v$. The spacelike surface $\Sigma_v$ whose volume we are computing is the one of maximum volume among those bounded by $S_v$.
• Figure 2: The area coordinate $r$, in $m=1$ units, as a function of the volume parameter $\lambda$, obtained integrating the equation \ref{['eq:rdoteq']}. As $A \rightarrow A_c$ , $\lambda_f=V / 4\pi \rightarrow \infty$.
• Figure 3: The integrand in \ref{['eq:rdotIntegral']} for different values of $A$ gradually approaching $A_c$. As $A\rightarrow A_c$ the volume contribution comes increasingly from $r_V=3m/2$.
• Figure 4: Black hole spacetime in Eddington-Finkelstein coordinates. The horizon is the vertical line $r=2m$. Dashed lines are the null geodesics. Maximum volume surfaces for different values of $A$, starting from the same sphere on the horizon are depicted.
• Figure 5: The maximal volume surfaces inside a black hole formed by a collapsing object. In red is an incoming spherical null shell that collapses and forms a singularity at $v=0$. (The region below the null shell is flat.) As (asymptotic) time passes, the interior grows. In green is the horizon.