Einstein's box - unclear about Bohr's retort

Bohr realized that the weight of the device is made by the displacement of a scale in spacetime. The clock’s new position in the gravity field of the Earth, or any other mass, will change the clock rate by gravitational time dilation as measured from some distant point the experimenter is located. The temporal metric term for a spherical gravity field is $1~-~2GM/rc^2$, where a displacement by some $\delta r$ means the change in the metric term is $\simeq~(GM/c^2r^2)\delta r$. Hence the clock’s time intervals $T$ is measured to change by a factor $$ T~\rightarrow~T\sqrt{(1~-~2GM/c^2)\delta r/r^2}~\simeq~T(1~-~GM\delta r/r^2c^2), $$ so the clock appears to tick slower. This changes the time span the clock keeps the door on the box open to release a photon. Assume that the uncertainty in the momentum is given by the $\Delta p~\simeq~\hbar/\Delta r~<~Tg\Delta m$, where $g~=~GM/r^2$. Similarly the uncertainty in time is found as $\Delta T~=~(Tg/c^2)\delta r$. From this $\Delta T~>~\hbar/\Delta mc^2$ is obtained and the Heisenberg uncertainty relation $\Delta T\Delta E~>~\hbar$. This demands a Fourier transformation between position and momentum, as well as time and energy.

This argument by Bohr is one of those things which I find myself re-reading. This argument by Bohr is in my opinion on of these spectacular brilliant events in physics.

This holds in some part to the quantum level with gravity, even if we do not fully understand quantum gravity. Consider the clock in Einstein’s box as a blackhole with mass $m$. The quantum periodicity of this blackhole is given by some multiple of Planck masses. For a blackhole of integer number $n$ of Planck masses the time it takes a photon to travel across the event horizon is $t~\sim~Gm/c^3$ $=~nT_p$, which are considered as the time intervals of the clock. The uncertainty in time the door to the box remains open is $$ \Delta T~\simeq~Tg/c(\delta r~-~GM/c^2), $$ as measured by a distant observer. Similary the change in the energy is given by $E_2/E_1~=$ $\sqrt{(1~-~2M/r_1)/(1~-~2M/r_2)}$, which gives an energy uncertainty of $$ \Delta E~\simeq~(\hbar/T_1)g/c^2(\delta r~-~GM/c^2)^{-1}. $$ Consequently the Heisenberg uncertainty principle still holds $\Delta E\Delta T~\simeq~\hbar$. Thus general relativity beyond the Newtonian limit preserves the Heisenberg uncertainty principle. It is interesting to note in the Newtonian limit this leads to a spread of frequencies $\Delta\omega~\simeq~\sqrt{c^5/G\hbar}$, which is the Planck frequency.

The uncertainty in the $\Delta E~\simeq~\hbar/\Delta t$ does have a funny situation, where if the energy is $\Delta E$ is larger than the Planck mass there is the occurrence of an event horizon. The horizon has a radius $R~\simeq~2G\Delta E/c^4$, which is the uncertainty in the radial position $R~=~\Delta r$ associated with the energy fluctuation. Putting this together with the Planckian uncertainty in the Einstein box we then have $$ \Delta r\Delta t~\simeq~\frac{2G\hbar}{c^4}~=~{\ell}^2_{Planck}/c. $$ So this argument can be pushed to understand the nature of noncommutative coordinates in quantum gravity.

How can Bohr invoke a General Relativity concept when Quantum Mechanics is notoriously incompatible with it?

You may have misheard, Sklivvz. General relativity is perfectly compatible with quantum mechanics. If you want the full and completely accurate theory that answers questions that depend on both GR and QM, in any regime, it is called string theory.

But obviously, you don't need the sophisticated cannon of string theory to answer these Bohr-Einstein questions. String theory is only needed when the distances are as short as the Planck length, $10^{-35}$ meters, or energies are huge, and so on. Whenever you deal with ordinary distance scales, semiclassical GR is enough - a simple quantization of general relativity where one simply neglects all loops and other effects that are insanely small. And indeed, string theory does confirm (and any other hypothetical consistent theory would confirm) that those effects are small, suppressed by extra powers of $G$, $h$, or $1/c$.

And in this Bohr-Einstein case, you don't even need semiclassical general relativity. You don't really need to quantize GR at all. This is just about simple quantum mechanics in a pre-existing spacetime, and Bohr's correct answer to Einstein is just a simple comment about the spacetime geometry. The extreme phenomena that make it hard to unify QM and GR surely play no detectable role in this experiment. They don't even play much role in "quantum relativistic" phenomena such as the Hawking radiation: all of their macroscopic properties may be calculated with a huge accuracy.

Shouldn't HUP hold up with only the support of (relativistic) quantum mechanics?

Nope. The Heisenberg uncertainty principle is a principle that holds for all phenomena in the Universe. Moreover, it's a bit confusing why you wrote "only" in the context of relativistic quantum mechanics - relativistic quantum mechanics is the most universally valid framework to describe the reality because it includes both the quantum and relativistic "refinements" of physics (assuming that we do the relativistic quantum physics right - with quantum field theory and/or string theory).

Einstein, in his claim that he could violate the uncertainty principle, used gravity, so it's not surprising that the error in Einstein's argument - one that Bohr has pointed out - has something to do with gravity, too. Because we talk about the uncertainty principle, you surely didn't want to say that we should be able to describe it purely in non-quantum language. If you wanted to say that non-relativistic quantum mechanics should be enough to prove Einstein wrong, then it's not true because photons used in the experiment are "quantum relativistic" particles.

In particular, the mass of a photon that he wants to measure is $m=E/c^2 = hf/c^2$. Because photons and electromagnetic waves in any description are produced at a finite frequency $f$, we cannot let $c$ go to infininity because the change of the mass $m=hf/c^2$ that Einstein proposed to measure (by a scale) would vanish so he couldn't determine the change of the mass - and he wanted to calculate the change of the energy from the mass so he wouldn't be able to determine the energy, either.

So Einstein's strategy to show that $E,t$ may be determined simultaneously uses effects that depend on the finiteness of both $h$ and $c$, so he is using relativistic quantum phenomena. To get the right answer or the right predictions what will happen and what accuracy can be achieved, he should do so consistently and take into account all other relevant phenomena "of the same order" that also depend both on relativity and the quanta. The time dilation, as pointed out by Bohr, is one such effect that Einstein neglected, and if it is included, not surprisingly, HUP gets confirmed again.

Such proofs are somewhat redundant in theories that we know. Whenever we construct a quantum theory, whether the gravity is described relativistically or not, with fields or not, the uncertainty principle is automatically incorporated into the theory - by the canonical commutators - so it is never possible to find a measurement for which the theory predicts that HUP fails. This conclusion is safer than details of Bohr's particular "loophole" - I am not going to claim that Bohr's observation is the only (or the main) effect that Einstein neglected. There are probably many more.