Black Hole Interior Volume

• Black hole interior volume is defined as the maximal proper volume inside the event horizon, computed via variational methods over spacelike hypersurfaces.
• In general relativity, the interior volume grows linearly with advanced time despite a fixed horizon area, offering insights into quantum information storage.
• Modified gravity models with hollowgraphy reveal vanishing interior volume while retaining finite surface area, reinforcing the holographic area law.

A black hole's interior volume refers to a geometric construct: the maximal proper volume contained within a bounded spacelike hypersurface inside the event horizon. Unlike the naive expectation from flat-space geometry, where volume is simply a function of the radius, in a black hole spacetime the notion of "the volume inside" is highly nontrivial and depends crucially on the underlying gravitational dynamics, quantum structure, and even the theory of gravity itself. The research literature reveals two fundamentally different paradigms: (1) the standard general relativistic treatment, which typically finds a large and growing interior volume, and (2) exotic modifications exhibiting "hollowgraphy," wherein the invariant spatial volume can vanish despite finite horizon area.

1. Black Hole Interior Volume in General Relativity

The coordinate-invariant prescription for the interior volume of a black hole was formalized by Christodoulou and Rovelli. Given a 2-sphere $S$ on the event horizon, the interior volume is defined as the proper volume of the largest (i.e., maximal) spherically symmetric spacelike 3-hypersurface $\Sigma$ that is bounded by $S$ (Christodoulou et al., 2014). In Schwarzschild geometry, writing the line element in ingoing Eddington–Finkelstein coordinates,

$$ds^2 = -f(r) dv^2 + 2 dv dr + r^2 d\Omega^2,\;\; f(r) = 1 - \frac{2M}{r},$$

and parameterizing spacelike hypersurfaces as $\Sigma = \gamma \times S^2$, with $\gamma: \lambda \to (v(\lambda), r(\lambda))$, the volume functional reduces to

$$V_{\Sigma}[\gamma] = 4\pi \int d\lambda\, r^2 \sqrt{-f(r)\left(\frac{dv}{d\lambda}\right)^2 + 2\frac{dv}{d\lambda}\frac{dr}{d\lambda}}$$

Maximizing this functional yields an interior slice with $r = (3/2)M$ over a long segment, resulting in

$V_{\mathrm{CR}} = 3\sqrt{3} \pi M^2 v$

where $v$ is the advanced Eddington-Finkelstein time coordinate. Thus, the maximal interior volume grows linearly and unboundedly with advanced time, even when the horizon area remains fixed (Christodoulou et al., 2014). This growth is also found for other black hole solutions (charged, rotating, higher-dimensional, AdS, and BTZ), with the details of the maximal hypersurface and growth rate being solution-specific (Ong, 2015, Christodoulou et al., 2016, Zhang, 2019, Maurya et al., 23 Apr 2024).

Notably, the relationship between interior volume and horizon area is not generally monotonic or even direct beyond Schwarzschild. For instance, in AdS spacetimes with toral or lens-space horizon topology, two black holes of identical mass and different horizon areas may have identical interior volumes (Ong, 2015), and the maximal volume can exhibit more complex dependence on black hole parameters (Chew et al., 2020).

2. Interior Volume and The Information Paradox

The unbounded growth of the interior volume under classical and semiclassical evolution (including Hawking evaporation) implies that, at least geometrically, black holes admit vast interior "storage capacity" for quantum information (Christodoulou et al., 2016, Ong, 2015). Quantum stat-mech arguments suggest that an effective number of field-theoretic degrees of freedom can increase as the volume grows, leading to speculation that the black hole interior could act as an information reservoir. This bulk storage scenario is sometimes invoked as a potential clue toward resolving the black hole information paradox, particularly since the Bekenstein–Hawking entropy—proportional to the horizon area—remains finite and even decreases as the black hole shrinks, while the volume continues to grow (Ong, 2015, Zhang, 2015).

Explicit calculations of quantum field entropy (e.g., massless scalar modes) inside the maximal interior volume show that the entropy is proportional to, but always strictly less than, the Bekenstein–Hawking entropy (Zhang, 2015, Majhi et al., 2017, Zhang, 2017). The precise result for Schwarzschild is

$S_{\mathrm{in}} = \alpha\, S_{BH},$

where $S_{BH} = A/(4\hbar)$ and $\alpha < 1$ is a solution-dependent constant (e.g., $5/(192\pi)$). Thus, the horizon still carries the maximal entropy, consistent with holographic bounds and reinforcing that the main thermodynamic bookkeeping resides at the surface, not in the volume, despite the latter’s potential immensity (Majhi et al., 2017, Zhang, 2017). This bulk-surface entropy dichotomy is intertwined with quantum gravity considerations regarding the fate of information and the statistical mechanical origin of black hole thermodynamics.

3. Hollowgraphy: Vanishing Volume Interiors and Area Law

A particularly striking alternative construction arises in certain scalar-tensor gravity theories with "spontaneously induced general relativity," in which a phase transition occurs precisely at the would-be Schwarzschild radius (Davidson et al., 2010). For such theories, the interior solution is "hollow": the invariant spatial volume inside any sphere of finite surface area vanishes in the $\delta \to 0$ limit (where $\delta$ parameterizes the deviation from GR through a small scalar charge). The metric inside is characterized by

$$ds^2 \simeq -\frac{\delta}{6} \left( \frac{r}{2GM} \right)^{2(3/\delta-2)} dt^2 + \frac{6}{\delta} \left( \frac{r}{2GM} \right)^{2(3/\delta-3)} dr^2 + r^2 d\Omega^2,$$

and the invariant spatial volume (for $r \leq 2GM$) is

$$V(r) \simeq 4\pi \sqrt{ \frac{2\delta}{3} (2GM)^3 \left( \frac{r}{2GM} \right)^{3/\delta} }.$$

As $\delta \to 0$, $V(r) \to 0$ for any $r < 2GM$. Thus, while the area of any interior sphere remains finite, the corresponding enclosed volume vanishes. This starkly contrasts with GR and provides a geometric mechanism for the "holographic" area law: since no bulk volume exists, there is literally no room for additional quantum degrees of freedom in the interior. All entropy must be associated with the boundary, matching the Bekenstein–Hawking area law: $S_{BH} = \frac{c^3 A_{BH}}{4 G \hbar}.$ In this sense, "hollowgraphy" enforces the area law not by microscopic counting, but as a macroscopic geometric consequence of the vanishing volume (Davidson et al., 2010).

An additional, remarkable feature is that the effective Newton constant $G_{in}(r)$ becomes radius-dependent, reflecting the nontrivial gravitational dynamics in the hollow interior: $$G_{in}(r) \sim G \left( \frac{r}{2GM} \right)^{2-\delta}.$$

4. Methodologies for Maximal Volume Construction

The maximal volume prescription is universal across most constructions (Christodoulou et al., 2014, Ong, 2015, Christodoulou et al., 2016, Susi et al., 2016, Ali et al., 1 Aug 2024). The essential steps are:

1. Express the black hole metric in an appropriate coordinate system (e.g., Eddington–Finkelstein).
2. Consider spherically symmetric spacelike hypersurfaces of the form $\Sigma = \gamma \times S^{D-1}$ for a $D$-dimensional black hole.
3. Derive the volume functional for $\Sigma$:

$$V[\Sigma] = 4\pi \int d\lambda\, r^{D-1} \sqrt{ -f(r)\left(\frac{dv}{d\lambda}\right)^2 + 2\frac{dv}{d\lambda}\frac{dr}{d\lambda}}$$

or, equivalently, recast this as the proper length of a geodesic in an auxiliary 2-dimensional space with metric $\tilde{g}_{AB} = r^{2(D-1)} g_{AB}$.

1. Use symmetry or variational principles to identify a "steady" segment, typically at constant $r = r_c$, that dominates the volume.
2. Maximize the integrand with respect to $r$ to obtain $r_c$ for the particular black hole solution.

For rotating or charged spacetimes, the maximal hypersurface generally becomes angle-dependent (e.g., $r = r_R(\theta)$ in Kerr spacetimes (Maurya et al., 23 Apr 2024)), complicating analytic treatments but preserving the conceptual structure.

In 2+1 dimensions (BTZ black holes), similar variational procedures are employed, leading to characteristic results such as $V_{BTZ} \sim (\pi/2) m v$ using an optimal slice at $r = \sqrt{m/2}$ (Zhang, 2019, Ali et al., 2020).

5. Generalizations, Extensions, and Limitations

The interior volume paradigm has been extended to:

• Higher-dimensional Schwarzschild black holes, where the rate of growth $dV/dv$ decelerates with increasing dimension $D$ and becomes negligible for very large $D$ (Bhaumik et al., 2016).
• Black holes with nontrivial cosmological constant and horizon topology, such as AdS black holes with toral or lens space horizons, which display intricate relationships between the horizon area and the maximal interior volume (Ong, 2015, Chew et al., 2020).
• Rotating and charged solutions in four and lower dimensions, where the maximal hypersurface is not at constant $r$ and requires careful treatment (Wang et al., 2018, Maurya et al., 23 Apr 2024, Maurya et al., 2022).
• Modified gravity theories exhibiting "hollowgraphy," which geometrically enforce the area law in a way fundamentally distinct from standard GR (Davidson et al., 2010).

A fundamental limitation is that, in the semiclassical regime, all results rely on mass $M \gg M_\text{Pl}$, so that full quantum gravity effects, including uncertainty relations involving position and energy, remain negligible. Near the Planck scale, these approaches fail and new physical principles must enter (Ali, 3 Sep 2025, Zhang, 2015).

6. Summary Table: Interior Volume Growth in Selected Models

Black Hole Type	Maximal Volume $V$	Volume Growth with Time	Volume–Area Relationship
Schwarzschild (4d)	$3\sqrt{3}\pi M^2 v$	Linear in $v$	$V \propto M^2 v$; $A \propto M^2$
Reissner–Nordström (4d)	Solution-dependent $r_c(Q, M)$	Linear in $v$	Nonmonotonic with $A$
Kerr (slow rotation)	$3\sqrt{3}\pi M^2 v - O(a^2 v)$	Linear in $v$	Corrections depend on $a$
BTZ (3d, AdS)	$(\pi/2) m v$	Linear in $v$	$V$ independent of $A$ (for toral)
Hollowgraphy driven	$V_{\text{in}}(r)\to0$ as $\delta\to0$	N/A (vanishing)	$S_{BH} \propto A$ only (area law)

For all cases in GR—save for vanishing volume interiors—the maximal interior volume grows linearly with advanced time, while the Bekenstein–Hawking entropy is dictated by the horizon area.

7. Conceptual and Physical Implications

The research strongly differentiates between spacetime construction leading to a "filled" black hole interior (possibly capable of storing vast information) and those in which the interior is geometrically hollow. In standard GR, the enormous and ever-growing volume may suggest a role in the storage and retrieval of information in the context of the information paradox (Christodoulou et al., 2014, Ong, 2015, Christodoulou et al., 2016, Maurya et al., 23 Apr 2024). Nonetheless, the entropy carried by interior quantum fields is strictly less than the Bekenstein–Hawking bound, confirming that, despite the large spatial volume, the horizon remains the ultimate ledger for black hole information.

In exotic modifications ("hollowgraphy" (Davidson et al., 2010)), the lack of bulk volume provides a natural geometric rationale for the area law of black hole entropy—quantum degrees of freedom simply cannot reside in a vanishing-volume interior. Here, the entire information content is localized at the boundary, providing sharp support for the holographic principle and aligning with the interpretation of the black hole entropy as a boundary phenomenon.

The persistence of these geometric, thermodynamic, and quantum relationships under various black hole dynamical processes—including Hawking evaporation—points toward potential universal "volume laws," but ultimately confirms the uniqueness of the area law in black hole thermodynamics.