Undergrad Wilbert Zweistein's Objection to Equivalence Principle

[ergospherical]

I'd like to share the following competition problem:

My first reaction was that the thermometer will be able to measure heating due to the Unruh effect whilst his lab is accelerating.

 

[Dale]

Possibly, but I don’t think that the Unruh effect is local and I would also think that it would be very difficult to distinguish it from Hawking radiation.

 

[PeroK]

Does the equivalence principle extend to all theories of quantum gravity?

 

[ergospherical]

Possibly, but I don’t think that the Unruh effect is local and I would also think that it would be very difficult to distinguish it from Hawking radiation.

But something like a star or a planet doesn’t emit Hawking radiation, right?

 

[PeterDonis]

My first reaction was that the thermometer will be able to measure heating due to the Unruh effect whilst his lab is accelerating.

The Unruh effect is due to proper acceleration, and would be there, in principle, whether the lab was accelerating in free space or was sitting at rest in the gravitational field of a large mass. So it could not be used to distinguish those two cases.

OTOH, if by "gravitational attraction" the question means coordinate acceleration towards a large mass, the lab would be in free fall in this case, and although there would indeed be no Unruh effect in this case, there would also be a reading of zero on the lab's accelerometer (which could be as simple as a bathroom scale), which would be a much easier and more straightforward way of distinguishing the two cases.

 

[ergospherical]

Oh, I didn’t know that. Can we ascribe a physical origin to the heating effect in the case of hovering (by means of rockets, say) outside a star?

 

[PeterDonis]

Can we ascribe a physical origin to the heating effect in the case of hovering (by means of rockets, say) outside a star?

I should actually rephrase my previous statement. The Unruh effect is derived assuming flat spacetime and the quantum field being in the vacuum state according to inertial observers. (See below for why the qualifier "according to inertial observers" is necessary.) If a gravitating mass is present, spacetime is neither flat nor in the vacuum state according to inertial observers, so it's not clear whether the Unruh effect would even be predicted in this case. If we assume it would be predicted in the "hovering in the vacuum region above a gravitating mass" case based on something like the equivalence principle, then its physical origin would be what I describe below.

The qualifier I gave above about the vacuum state is necessary because the reason the Unruh effect is predicted at all is that which state of the quantum field is the "vacuum" is different for inertial observers and accelerated observers. More precisely, the concept of "vacuum state" for the quantum field requires a concept of "time translations" for its definition, and inertial observers and accelerated observers have different concepts of "time translations" (roughly speaking, because the inertial Killing vector field and the boost Killing vector field in Minkowski spacetime are different). The derivation of the Unruh effect amounts to showing that the state of the quantum field that is a vacuum state in flat spacetime for inertial observers is not a vacuum state for accelerated observers; instead, it turns out to look like a thermal state at the Unruh radiation temperature.

The reference from which I first learned all this is Wald's monograph, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Unfortunately I have never found it online and I don't know of a good online discussion of the above, although there probably is one.

 

[ergospherical]

Fascinating stuff. I’ll look for the book later this week (I recall it being referenced in H. Reall’s lecture notes); that’s probably more efficient than asking a bunch of groundwork questions here.

 

[PeterDonis]

I’ll look for the book later this week

It's on Amazon:

https://www.amazon.com/dp/0226870278/?tag=pfamazon01-20

When I said I have never found it online, I meant I have never found a digital version of it, or, for example, a preprint on arxiv.org or some other similar freely available online source that covers the same material.

 

[ergospherical]

Thanks, I can go borrow it but the next couple of days are slightly nuts so probably won’t stop by the lib until the weekend. :)

 

[PeterDonis]

Thanks, I can go borrow it but the next couple of days are slightly nuts so probably won’t stop by the lib until the weekend. :)

If you can get through the book in a weekend you are definitely quicker at it than I am.

 

[PAllen]

One thing or watch for in discussions of the equivalence principle is that it is local in time as well as space. Ohanian, for example, has published a number of alleged refutations of the principle that other physicists reject because they ignore the time element.

 

[Demystifier]

Does the equivalence principle extend to all theories of quantum gravity?

Does it extend to any theory of quantum gravity?

 

[vanhees71]

Good question. I'd say it the other way: A good theory of the gravitational interaction should obey (at least the weak) equivalence principle since it's empirically so well established.

As demonstrated by this discussion the problem with the equivalence principle or rather the equivalence principles is that it is usually discussed in the heuristic introductions to GR and then not further qualified given the fully exposed theory (a fate it shares with the discussions of the Michelson Morley experiment, which is usually also only discussed as a "crucial experiment" for the heuristical motivation of SR).

A clear way to state the principle is that the GR spacetime model is a Pseudo-Riemannian manifold with the fundamental form of the signature (1,3) or, equivalently (3,1). Physically that means that the notion of inertial frames is local an that at any point in spacetime there's always a local inertial frame of reference. That's of course not a good way to heuristically motivate GR but it should be the final clarifying statement about the content of the (then even strong!) equivalence principle.

For particle physicists another convincing semi-heuristic argument is that relativistic models of the gravitational interaction can be built by "gauging" Poincare invariance, which becomes then a local symmetry and thus you have a gauge theory with the general diffeomorphism invariance as the gauge group, which usually is called "general covariance". Together with the fact that there are particles with spin that leads to the statement that GR spacetime is described as a Einstein-Cartan manifold with torsion. In the usual "macroscopic" phenomenology, where gravitation plays a practical role, all you have is classical matter ("continuum mechanics") and the em. field as sources, and there the theory specializes to usual GR. If there is torsion, it's inside "polarized matter" and thus pretty difficult to observe.