quantum mechanics - Einstein's box - unclear about Bohr's retort

I was reading a book on the history of Quantum Mechanics and I got intrigued by the gendankenexperiment proposed by Einstein to Bohr at the 6th Solvay conference in 1930.

For context, the thought experiment is a failed attempt by Einstein to disprove Heisenberg's Uncertainty Principle.

Einstein considers a box (called Einstein's box; see figure) containing electromagnetic radiation and a clock which controls the opening of a shutter which covers a hole made in one of the walls of the box. The shutter uncovers the hole for a time Δt which can be chosen arbitrarily. During the opening, we are to suppose that a photon, from among those inside the box, escapes through the hole. In this way a wave of limited spatial extension has been created, following the explanation given above. In order to challenge the indeterminacy relation between time and energy, it is necessary to find a way to determine with adequate precision the energy that the photon has brought with it. At this point, Einstein turns to his celebrated relation between mass and energy of special relativity: $E = mc^2$. From this it follows that knowledge of the mass of an object provides a precise indication about its energy.
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Bohr's response was quite surprising: there was uncertainty in the time because the clock changed position in a gravitational field and thus its rate could not be measured precisely.

Bohr showed that [...] the box would have to be suspended on a spring in the middle of a gravitational field. [...] After the release of a photon, weights could be added to the box to restore it to its original position and this would allow us to determine the weight. [...] The inevitable uncertainty of the position of the box translates into an uncertainty in the position of the pointer and of the determination of weight and therefore of energy. On the other hand, since the system is immersed in a gravitational field which varies with the position, according to the principle of equivalence the uncertainty in the position of the clock implies an uncertainty with respect to its measurement of time and therefore of the value of the interval Δt.

Question: How can Bohr invoke a General Relativity concept when Quantum Mechanics is notoriously incompatible with it? Shouldn't HUP hold up with only the support of (relativistic) quantum mechanics?

Clarifying a bit what my doubt is/was: I thought that HUP was intrinsic to QM, a derived principle from operator non-commutability. QM shouldn't need GR concepts to be self consistent. In other words - if GR did not exist, relativistic QM would be a perfectly happy theory. I was surprised it's not the case.

Answer

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.