# Does the interior volume of a black hole grow forever?

Recently, I was reading about a article which tells about something known as "Susskind Complexity". The article states that the interior volume of a black hole grows forever.

How/why does the internal volume of a black hole grow forever? What is the mechanism/reason behind this?

• For questions like this, always specify context. People generally have very different pictures for the interior of the black hole, especially people like Susskind who're working on new ideas. These pictures are usually drastically different from the old, well-established picture that is given by GR. In the GR picture, inside the black hole time collapses onto the central singularity (or wraps around the ringularity for Kerr black holes). The term "forever" doesn't really make much sense in this context. Commented Jun 11, 2023 at 9:52
• How do you define "volume"? What do you use to measure it? Commented Jun 11, 2023 at 12:29
• @AgnibhoDutta If you're using an exotic, unmeasurable definition of volume, real-world physics cannot answer your question. Commented Jun 11, 2023 at 15:41
• If you are Susskind you can conjecture anything and ask other people to prove its correctness. In the article you mention there is no even one mathematical equation. In my view such theories are not physics but juggling with words.
– JanG
Commented Jun 11, 2023 at 16:19
• @JanG: You are confusing theory with popular science article reporting on that theory. Original papers by Susskind do have equations. And while quantum gravity implications are mostly conjectural (note, well motivated conjectures) his definition of volume is standard GR. Commented Jun 11, 2023 at 17:38

Volume is purely spatial, i.e. three dimensional, concept, while the natural arena of general relativity is a 4-dimensional spacetime, so whenever we want to calculate a volume, we need to define “spatial slice” over which to perform integration to obtain that volume.

For some situations this choice of slice is fairly obvious. For example if we have a static gravitating body (e.g. a star or a planet) we can choose static time coordinate $$t$$, such that metric is invariant under both shifts $$t\to t+c$$ and reflections $$t\to -t$$ and then choose a spatial slice as a hypersurface $$t=\mathrm{const}$$. Four-dimensional spacetime metric would induce three-dimensional Riemannian metric on this slice, so we can integrate the volume form of this 3D metric over the interior of a body and obtain the unambiguous quantity: the volume of a static body.

However, this procedure would fail if we try to apply it to the black hole interior (in what follows we limit ourselves to static black holes only). The reason is that the “static time coordinate” becomes spacelike inside the event horizon, whereas radial coordinate $$r$$ becomes timelike, so that for an observer inside the black hole the singularity $$r=0$$ is the inevitable future. There is thus no “natural” and unambiguously defined spatial slice over which we can integrate.

There are many choices of time coordinates and of possible spatial slices inside the black hole besides “static time” that can be more appropriate for calculating black hole volume, but they give varying results: for some choices we have corresponding volume of black hole constant in time, for some the volume varies, with complicated time dependence, while for some the volume can even be zero (see here for a few pedagogical examples, aimed at students of GR).

Susskind proposes to eliminate this ambiguity by defining black hole volume as a volume inside horizon on a “maximal slice” (that is a slice with maximal volume from a set of all “nice slices” asymptoting to a given moment of time $$t$$ of distant static observer). The main purpose of this definition is the conjecture relating this volume to complexity of black hole quantum state.

On a Penrose diagram such maximal slices for different times look like this:

And here is the embedding diagram for those maximal slices at different moments in black hole evolution:

First frame of embedding diagram is before the singularity is formed, second frame is soon after singularity formation, last frame is much later. Note, that this is schematic representation of spherically symmetric 3-space as a 2-surface, circles represent spheres.

The most important part of the maximal slice geometry is the “tube” of almost constant radius corresponding to $$r\approx\mathrm{const}$$ part of the slice geometry. Since inside the black hole $$r$$ is the timelike coordinate, this tube would be spacelike. The length of the tube grows linearly with time corresponding to (approximately) linear with time growth of black hole volume. The rest of the maximal slice geometry is the “cap” at one end of the tube (at the initial stage of black hole evolution before the singularity formation) and the throat at the other end of the tube connecting it to outside of event horizon at specific moment of outside time.

More details could be found in lecture by Susskind:

• Susskind, Leonard. Entanglement is not enough. Fortschritte der Physik 64.1 (2016): 49-71, arXiv:1411.0690.

The concept of a volume should be accessible to anyone with basic knowledge of GR while the rest of the paper requires at least some familiarity with area of quantum gravity.

The volume proposal of Susskind is not the only attempt at defining coordinate/slicing independent notion of black hole volume. Closely related is the proposal of Christodoulou & Rovelli that differs in boundary conditions for maximal slices. This black volume also grows approximately linearly with time.

But there are also definitions of black hole volume that do not change with time (as long as black hole mass remains constant). One is the notion of thermodynamic volume (see here) where volume is defined as a quantity thermodynamically conjugate to pressure (with pressure being related to cosmological constant).

Another constant in time definition is the geometric volume of black hole due to Parikh where volume is defined as a rate of change with asymptotic time of 4D volume swept by any time invariant slice through black hole interior.

Does the interior volume of a black hole grow forever?

In classical general relativity where black hole cannot disappear, yes (for Susskind's definition of volume). If we take into account quantum effects, then probably not forever, but for a very long time. First, black holes can evaporate, but even if we prevent it (by feeding back into the black hole the mass it looses due to evaporation) there should be a maximal possible volume corresponding to a maximal complexity of black hole quantum state. Once the black hole reaches it, its volume would stop growing, and over a very long time due to Poincare recurrence it could even shrink back to small values, “rejuvenating” the black hole.

• Inside a black hole, the $r$ coordinate becomes timelike and also "grows" like time, so the volume is also growing. Is my understanding correct now?
– user355398
Commented Jun 12, 2023 at 4:22

Black holes (of constant mass) grow only in a certain rather artificial technical sense, not in the ordinary sense of the word.

I wrote an answer about this to a question that isn't a duplicate of this one, but it's close enough that I'll copy parts of the answer here.

Suppose you're sitting outside the event horizon of a black hole (never mind how you avoid falling in; perhaps you're in a rocket ship, or perhaps there's a Dyson sphere supporting you or something). You toss an object of negligible mass into the hole, then you wait a short time, then you toss a second object in (an exact duplicate of the first toss, but at a later time).

If you plot the objects' worldlines (paths in spacetime), they have the same shape, because you threw the objects identically, but they are offset from each other in time.

That's true before they cross the event horizon. The worldlines after they cross the horizon are still the same shape, but they are spacelike separated, meaning they are "side by side" with neither one objectively earlier or later in time than the other.

Extend this to a large number of people located all around the hole (all at the same height) continually tossing objects into it. The events of the tosses now form a cylinder in spacetime, or rather a "hypercylinder" whose cross section is a sphere instead of a circle. The lengthwise direction on this hypercylinder is the time direction. The spacetime locations of the tossed objects shortly after they cross the horizon also form a cylinder, but now the lengthwise direction is a spatial direction: all of the tossed objects are "side by side" on this cylinder and they all head for the singularity "in parallel" in some sense.

(That's the cylinder shown in A.V.S.'s answer to this question.)

If you start tossing the objects shortly after the hole initially forms from collapsing matter, then the longer you keep doing it, the longer this internal cylinder ends up being. That's essentially why they say that the "interior volume" increases with time.

This doesn't mean that objects tossed in later are farther away from the singularity. All of the objects are almost exactly the same distance from the singularity. If they were stopwatches that started ticking at 0 when they were tossed, they'd all show almost exactly the same elapsed time just before hitting the singularity. The cylinder is not a space through which they have to pass to get to the singularity. It isn't even a space they could in principle explore, if they were equipped with thrusters; they can only reach a portion of it in their limited time before hitting the singularity, and the size of that portion is almost exactly independent of when they were tossed in.

The expanding cylinder appears to have no physical significance. In every measurable sense, black holes are extremely stable objects. They don't get larger in any measurable sense when they aren't eating matter. This includes even measurements performed inside the event horizon.