# ZPE in curved space-times

1. Apr 11, 2013

### friend

As I understand it, QFT predicts that there is a certain zero point energy (ZPE) in a given background of flat Minkowski space-time. And every curved space-time is locally a flat Minkowski space-time. So I'm wondering how the ZPE changes as the curvature of curved space-time changes from one flat space-time to a different flat space-time. For example, is there a higher ZPE closer to a gravitating body?

What motivates this question is if the ZPE and the Minkowski background space-time are unavoidably linked in QFT, then perhaps if there is a mechanism to change the ZPE, this might also change something in the Minkowski background which together may account for gravity. I have no idea what may change the ZPE, and I'm looking for thoughts on the subject. Thanks.

Last edited: Apr 11, 2013
2. Apr 12, 2013

### tom.stoer

ZPE is a rather strange artifact. In QFT you cannot calculate it b/c the result is always infinite. So you subtract this infinity in order to get zero, but that's not really a calculation, is it?

However there a spacetime curvature effects in QFT, the most famous one is Hawking radiation. I strongly recommend to read Hawking's original paper!

As a summary: yes, locally the manifold looks like flat space, therefore you want to apply standard QFT, but that does not work globally. Introducing a vacuum state requires to define positive and negative frequency solutions for classical e.o.m on which you then introduce quantum fields via creation and annihilation operators (which do no longer create plane wave states but distorted waves according to the e.o.m. on curved spacetime). The problem is that the definition of positive and negative frequencies is not possible globally, so an empty vacuum state w.r.t. to one observer is a non-vacuum state with physical particles w.r.t. a second observer (simply b/c they do not agree on the split for positive and negative frequencies). This effect works w/o any interacting fields, so we have free fields on curved spacetime (and that's why Hawking's explanation 'particle-antiparticle pair creation' is rather confusing b/c usually in QFT that requires an interaction term; but Hawking is brilliant both in his calculation, and in interesting but inappropriate popular explanations ;-)

Unfortunately I do have no idea how to make this story work for vacuum energy. The reason is that locally (for every local coordinate patch looking like flat space) you always introduce a normal ordering w.r.t. the local definition of vacuum, and therefore for every vacuum state you subtract the vacuum energy. So if you want to calculate something like <Ω|T00(x)|Ω> at some x for some (observer O, O', ... dependent) vacuum Ω, Ω', ... you always chose a normal ordering setting this to zero. The effect which survives this normal ordering is just Hawking radiation, so in some sense you could say that the energy density of the thermal radiation is the vacuum energy density of a state Ω defined by an observer O but measured by an observer O'.

Last edited: Apr 12, 2013
3. Apr 12, 2013

### friend

Thanks, that's a good start.

Whatever the cause of gravity, I suppose it would have to ultimately be the same effect as acceleration, if the principle of equivalence is to hold. Does acceleration in itself cause this shift in the split between positive and negative frequencies?

I wonder if interactions with matter slows down wave functions like water waves crashing onto shore. That might cause waves to bunch up and appear to be higher in frequency, raising the apparent zero point energy level.

4. Apr 12, 2013

### tom.stoer

Yes and no. The so-called Unruh effect shows that an accelerated observer in flat spacetime will see thermal radiation, too. Nevertheless QFT in curved spacetime is sensitive to the global structure of spacetime, and therefore I am not sure whether the principle of equivalence (which is a local principle) is sufficient.

5. Apr 13, 2013

### friend

The usual method of going from local to global is the process of integration. Would that apply here?