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I have a question about the Unruh effect and energy conservation; I'd originally asked part of this in another forum, but I thought it might be more appropriate to post here.

I understand that, as per the Unruh effect, a uniformly accelerated observer and an inertial observer will see different vacua, i.e., the notion of 'particles' end up being observer-dependent. A uniformly-accelerating observer will see a thermal bath at some temperature T depending on the magnitude of the acceleration. The bath will contain all types of particles with some probability, even though that probability may be vanishingly small is certain cases.

Now my question: consider, say, proton decay (to a neutron, positron and neutrino). Once he proton is uniformly accelerated (by, e.g., a linear accelerator), according to the Unruh effect there should be a non-zero probability that the proton will interact with a Rindler electron in thermal bath and 'decay' to produce a neutron.

From the inertial observer's perspective, the energy required for the decay comes from whatever is powering the acceleration.

Here's where I become confused: if the magnitude of the acceleration is small, and/or the duration is short, the energy imparted to the proton *may not be sufficient* for the decay process to occur.

So in the accelerated frame there is a non-zero transition probability, and in the inertial frame the probability is zero because insufficient energy was supplied. And energy overall must be conserved.

Clearly, this can't be right - I'm wondering what I'm missing?

If the duration of the acceleration is finite, does the alter the description of the process in the accelerated frame (i.e., in Rindler coordinates)?

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# Unruh effect and conservation of energy

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