GR & QM: When Do They Disagree?

[H_A_Landman]

The exact formula for gravitational time dilation is $$T_d = exp(m\phi / mc^2)$$. (We usually cancel out the ms, but I want to show this as a ratio of energies.) Since $$m\phi$$ is the potential energy, this says that the rates or frequencies of things in GR depend exponentially on their energy. On the other hand, $$E = h\nu$$ implies that the frequencies of things in QM are linearly proportional to their energy. It's quite obvious that these things cannot both be universally true. So, which one is wrong, and when?

 

[mjc123]

I'm no expert on relativity, but I think you're confusing things. Time dilation isn't a frequency. A rate isn't a frequency. QM doesn't say that the "frequencies of things" are linearly proportional to their energy. E = hν relates the energy and frequency of a photon, specifically.

Consider for example a linear rotor, such as a diatomic molecule. The energy levels are given by E = BJ(J+1) and the angular momentum is ħ√{J(J+1)}. They are related by E = p²/2I. Now the angular momentum is directly proportional to the frequency of rotation, p = 2πIv. So the energy is proportional to the square of the frequency. A photon emitted or absorbed in a transition between two levels will have a frequency given by ΔE = hv_photon, but that is a different thing.

Or consider a harmonic oscillator. Its energy levels are E = (n+1/2)hv. A transition between adjacent levels has the energy hv, so the photon will have the same frequency as the oscillator. But the frequency of the oscillator doesn't change with energy. In a higher energy state, it is oscillating with greater amplitude and speed, but at the same frequency.

 

[Vanadium 50]

"frequencies of things" is vague, and assumes exactly the same scaling laws in all cases, which we know is wrong. A spring feels a force proportional to r. A charged body feels a force proportional to 1/r².

 

[Dale]

I am with @Vanadium 50 on this. In QM the frequency of what scales linearly with which energy? In GR the frequency of what scales exponentially with which energy? Unless both the "what" and the "which" match between the two cases, there is no contradiction and no need to claim one must be wrong.

 

[jartsa]

Summary: In QM frequencies scale linearly with energy, but in GR they scale exponentially. These can't both be true.

The exact formula for gravitational time dilation is $$T_d = exp(m\phi / mc^2)$$. (We usually cancel out the ms, but I want to show this as a ratio of energies.) Since $$m\phi$$ is the potential energy, this says that the rates or frequencies of things in GR depend exponentially on their energy. On the other hand, $$E = h\nu$$ implies that the frequencies of things in QM are linearly proportional to their energy. It's quite obvious that these things cannot both be universally true. So, which one is wrong, and when?

GR: Time dilation is infinite at the event horizon of a black hole.

QM: Well, in that case everything has zero frequency and zero energy at the event horizon of a black hole. $exp(m\phi / mc^2)$ should be infinite at the event horizon of a black hole.
So how does that work? Does the numerator go to infinity, or does the denominator go to zero, or does something else happen?

 

[H_A_Landman]

Since people seem to be having trouble seeing a concrete example, let's take the Colella-Overhauser-Werner experiment as our starting point (R. Colella, A. W. Overhauser, and S. A. Werner, “Observation of Gravitationally Induced Quantum Interference,” Phys. Rev. Lett. 34 (23), 1472–1475 (1975)). In this, a neutron beam is split into a part that goes vertical then horizontal, and a part that goes horizontal then vertical (similar to a Mach-Zender interferometer). The vertical parts are assumed identical and cancel out. But the horizontal parts are at different altitudes and have different gravitational time dilations, causing a relative phase shift that is visible when the two parts interfere. This phase shift is a function of the gravitational potential energy difference between a neutron on the upper path and one on the lower path. QM predicts that this dependence will be linear; GR predicts that there are higher-order terms (see e.g. https://aapt.scitation.org/doi/abs/10.1119/1.19454 for a recent survey and analysis). They agree to first order, but for large deltas they disagree and can't both be right.

 

[Vanadium 50]

You are surprised that non-relativistic quantum mechanics doesn't work in the relativistic regime?

 

[H_A_Landman]

This isn't necessarily relativistic. The mismatch is still there in the low-speed limit, or even at zero speed. Also, are you sure that relativistic QM solves the problem? If it did (say, if the Dirac equation matched GR even though the Schrödinger equation doesn't) then that would be a satisfactory answer to my question.