Why exactly do we need black holes to satisfy the 2nd law of thermodynamics in finite time?

“No, let’s not assume that.”

Penrose diagram of flat Minkowski space where we conduct versions of the Bekenstein thought experiment in his 1972 paper. Black holes (cyan) form and exist (we are not assuming evaporation because black hole entropy is a prior) until future timelike infinity (𝒾₊ as a white dot). We (worldline A) send blackbody radiation into one of the black holes (Bekenstein 1972; left side of diagram). Purportedly it is a problem that the entropy of that blackbody radiation is “lost”. However, we could ostensibly send those blackbody photons into the abyss of space where they land on future null infinity ℐ ⁺ (colloquially pronounced scri-plus) and are also “lost” to us.

In light of my previous post (and because I have no social life¹) I decided to embark on a mission to rehash the past 50+ years of black hole physics starting with Bekenstein 1972. I’d actually not read the original; I have to say the paper is oddly constructed. Equation 1 (!) is the black hole entropy S = η k A/ℓₚ² with η as an unknown constant — it is justified by the dimensional analysis and scaling arguments that follow. First comes the handwaving:

By choosing [the entropy of a black hole] proportional to A … we ensure the total black hole entropy … depends only on the total horizon area … which is of fundamental importance in Hawking’s work. … The introduction of … a length squared … is necessitated by dimensional considerations. We choose Planck’s length because it is the only truly universal constant with units of length. … Although a black hole is a thoroughly classical entity, black hole entropy contains ℏ because it relates to the interaction of a black hole with material systems. It is well known that the expression of entropy for any material system always contains ℏ; thus, the appearance of ℏ in [the black hole entropy] is understandable.

Then comes the thought experiment — we consider a beam² of blackbody radiation of energy E that we shine on a black hole (see the middle left side of the diagram above). That beam has an entropy; that entropy could be “lost” to the black hole if we could not observe a change in its only observable quantities: mass, charge, or angular momentum (these all contribute to the size of a Kerr black hole, and therefore the area of the event horizon). If we assume the area law, we have on two sides of the entropy ledger (in units where G = c = 1 where ℏ = ℓₚ²):

\(\displaystyle \Delta S_{beam} \sim k \frac{M E}{\hbar} \qquad \Delta S_{BH} \sim \eta \; k \frac{M E}{\hbar}\)

But! I yell from the back of the classroom, preparing to dodge an eraser — the energy of the beam of N photons you sent to their doom is E ~ N ℏ ω. The ℏ drops out of both sides of the ledger. All we learned is

\(\displaystyle \Delta S_{beam} \sim N \qquad \Delta S_{BH} \sim N\)

We sent in N photons and the “black hole entropy” increased by N. But we can keep track of that by knowing we sent N photons in. We don’t exactly need to observe³ the result via properties of the black hole to keep the ledger balanced. Some observer might be like “wut?” when the black hole area increases, but we can get together latter to discuss. I’m not confused — I sent the beam in!

This is not to say there is something erroneous here. Hawking and Gibbons later re-derived the area law using a (Euclidean⁴) path integral — giving it some amount of credence. However, the “engineering dimension” considerations lead to the same place. If I say the entropy I send in is at a temperature ~ ℏ and on the other side I only have G, c, and area A then that A is going to get measured in units of ℓₚ² = Gℏ/c³. I just need the entropy of the black hole to go up by N and A/ℓₚ² is one way to do it⁵.

The question I have is: why exactly do we have to be able to observe the “lost” entropy by some finite time in the future? Impatience? Sure, the 2nd is violated because we’re missing information that is causally disconnected from us (i.e. inside a black hole) — but why should we have access to information that is causally disconnected from us? If something happens in the “elsewhere” portion of a spacetime diagram (i.e. an event spacelike separated from us) we don’t feel entitled to that information. We are comfortable waiting until enough time has elapsed for a signal to reach us. I don’t think the 2nd law is violated during the time it takes for a photon that tells me I’ve done it to reach me after I drop some entropy elsewhere. That that time is the infinite future (timelike infinity 𝒾₊) for matter in a black hole is not an insurmountable problem given we’ve already assumed we can manage to send a blob of entropy into a black hole and measure what happens. And if I send a blob of blackbody photons into the abyss of space⁶ (see the right side of the diagram above), I won’t be able to catch up with them until I make it to the tip of the census taker’s hat and look back from 𝒾₊ on ℐ ⁺. In any practical sense, that entropy is lost to me. But somehow the same blob terminating on the singularity of a black hole results in entropy that isn’t. Or so black hole thermodynamics believes is necessary.

Don’t get me wrong — I think this is a perfectly cromulent assumption guiding a line of inquiry. I also think this, and the entire industry it has spawned, is a consistent approach⁷. But as one would say in order to try to win an argument with Milton Friedman — no, let’s not assume that.

1. I pretended to be social by doing most of this at a coffee shop.

2. There is an assumption here that the blackbody radiation we shine on the black hole has a characteristic wavelength (e.g. from the Wien displacement law) is small compared to the size of the black hole so that we can just think of geometric optics (kinda sorta a blackbody radiation “laser”). This gives us the mass of the black hole (i.e. its size / radius via the Schwarzschild radius) on the “beam” side of the ledger.

3. I have some serious philosophical issues with the idea that there’s a problem with an as yet completely inaccessible physical process that we can “solve” with a thought experiment that assumes we must be able to access completely inaccessible information. It’s an abstract mathematical ledger. Why do we care it is balanced using elements of the system (i.e. properties of the black hole allow us to extract N) rather than by using our knowledge of elements of the system (i.e. we know we sent in N units of entropy and the ledger balances)?

4. There are some interpretational issues here that I’m still mulling as to their significance of my argument. I think not? It’s fine if we’ve already given up the assumption that we are allowed to know the entropy of stuff that is causally disconnected from us.

5. *cough* Lorentz invariance *cough*

6. For like thirty seconds, I believed this thought experiment was novel. But of course it wasn’t. Still, am somewhat proud that after being out of academia for twenty years (plus this wasn’t even my field), I’ve finally made it to the state of the art in 2003.

7. *cough* Lorentz invariance *cough*