# How can curved spacetime creat particles ?

## Main Question or Discussion Point

How can curved spacetime creat particles ??

I have heard that in curved spacetime the usual QFT theory does not hold and that the 'metric' (curvature) of spacetime produces particles how can it be ? for example given the Metric

$$-N(t)dt^{2}+ g_{ij}(X) dx^{i}dx^{j}$$ (Einstein summation is assumed on i and j)

what kind of particles are created by it ? , then if 2 observers move on a different metric would they observe a different "sea" of particles ??

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I have to wonder if it isn't fast changing spacetime curvature that isn't creating real particles to begin with. We know that there is a zero point energy that is creating the sea of virtual particles in flat spacetime. These virtual particles pop into existence and then quickly recombined so that the net result over time is no real particles. We also know that real particles are created near black-hole event horizons where spacetime is curved. The more curved the space near the horizon, the more particle radiation is created. And we also know that any curved spacetime is locally flat so that virtual particles should exist locally. So I'm thinking that any spacetime curvature creates changes in the virtual particle content from one place to another or from one time to another, which is equivalent to creating real particles. What do you think? Could this account for dark matter?

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This is probably related to the Unruh effect. http://en.wikipedia.org/wiki/Unruh_effect Unruh showed that accelerated observers in flat space time would observe the vaccuum to be a warm gas of particles. Since acceleration is locally equivalent to first order gravitational effects, then there would likely be the same effect for an observer in curved spacetime.

This is probably related to the Unruh effect. http://en.wikipedia.org/wiki/Unruh_effect Unruh showed that accelerated observers in flat space time would observe the vaccuum to be a warm gas of particles. Since acceleration is locally equivalent to first order gravitational effects, then there would likely be the same effect for an observer in curved spacetime.
Agreed, but isn't the simplest way to see it black-hole evaporation ?

nrqed
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Gold Member
Agreed, but isn't the simplest way to see it black-hole evaporation ?
But that's a quite different situation than the Unruh effect. In the black hole case there is the event horizon where timelike vectors become spacelike and vice-versa. This is what is at the origin of Hawking radiation. The way I see it, this is quite different than the Unruh effect.

But that's a quite different situation than the Unruh effect. In the black hole case there is the event horizon where timelike vectors become spacelike and vice-versa. This is what is at the origin of Hawking radiation. The way I see it, this is quite different than the Unruh effect.
I don't know that those situations are so different, for sure at the level I understand them, but even also at the level they are understood at all. The analogy is quite simple : the Unruh and Hawking temperatures are the same T = a/2pi in natural units, and I have read a few times (and it seems reasonable to me) that the Hawking radiation is in fact a direct consequence of the equivalence principle on the horizon (together with the Unruh effect). I need to check Wald's book, which talks about both.

nrqed
Homework Helper
Gold Member
I don't know that those situations are so different, for sure at the level I understand them, but even also at the level they are understood at all. The analogy is quite simple : the Unruh and Hawking temperatures are the same T = a/2pi in natural units, and I have read a few times (and it seems reasonable to me) that the Hawking radiation is in fact a direct consequence of the equivalence principle on the horizon (together with the Unruh effect). I need to check Wald's book, which talks about both.

I don't know that those situations are so different, for sure at the level I understand them, but even also at the level they are understood at all. The analogy is quite simple : the Unruh and Hawking temperatures are the same T = a/2pi in natural units, and I have read a few times (and it seems reasonable to me) that the Hawking radiation is in fact a direct consequence of the equivalence principle on the horizon (together with the Unruh effect). I need to check Wald's book, which talks about both.
And would the Unruh effect have any application to the "accelerating" expansion of space, to inflation, and the big rip? These are also situation where separate reference frames are accelerating with respect to each other, right?

Haelfix
A local 'inertial' pointlike observer will see particle creation that is just like flat space. That follows by the definition of a manifold. Now, if he/she is accelerating, they will see a rindler horizon (or a generalization thereof) and the vacuum will no longer be the simple one.

Now in curved spacetime, the problem is two inertial observers at spacelike seperation can no longer agree on what the real vacuum is in general. They will of course agree about the results of an experiment, but they will not be able to agree upon the interpretation thereof. For instance was it radiative corrections that induced such an such a transition, or was it inductance or thermal contamination.

Mathematically, this is simply a consequence that the holonomy group they use to define their vacuum has no appropriate automorphisms that are continously connected.

Is it agreed that more tightly curved spacetime is caused by a greater energy density? Mass/energy curves space, right? So then I offer this intuitive explanation for how real particles can be created by changing curvature due to changing gravitational fields. Start at some point in curved spacetime that is locally flat. It will have a zero point energy of some value creating and anihilating virtual particles. These particles are created, move apart a bit, then come back together when they anihilate. Now if a small distance away the curvature is slightly more, then the energy density is a little greater, and the zero point energy is creating slightly more virtual particles than the first place I mentioned. My explanation is that this slightly more energetic zero point energy will not be in equalibrium with the first place. But the greater number of virtual particles will occassionally interact with the lesser number of virtual particles at the first place, preventing some of the virtual pairs from recombining, thus creating real particles due to changes in curvature in adjacent points in space. Does this sound right?

reilly
Let me pose a simple minded question -- which may expose my limited knowledge of the topics covered herein.. If gravity can be approximated by spin-2 particles, then why can't a mass of some sort emit a graviton.which in turn creates a pair? Similarly, a gas of gravitons could create quite a few pairs, as well as a curved space-time.

Yes, I'm aware that the spin-2 approach has some difficulties with renormalization.Nonetheless, my sense is that as a phenomenological approach -- cf. Bethe's initial Lamb shift calculation -- such pair production makes a certain amount of sense.

Regards,
Reilly Atkinson

I recommend Sean Carroll - Spacetime and Geometry, Chapters 9.5 and 9.6.

Carroll tries to argue that the Unruh effect implies the Hawking effect (under reasonable assumptions).

Haelfix
Let me pose a simple minded question -- which may expose my limited knowledge of the topics covered herein.. If gravity can be approximated by spin-2 particles, then why can't a mass of some sort emit a graviton.which in turn creates a pair? Similarly, a gas of gravitons could create quite a few pairs, as well as a curved space-time.

Reilly Atkinson
It does. Actually gravity couples to everything in sight, so any matter, itself (like QCD) and so forth. Typically you want to truncate the series relatively early (usually one loop) to get a reasonable effective theory (valid when E>>Mpl) or you will end up with nonsense. This is called 'semiclassical gravity' (stupid name, b/c we have a quantum of gravity involved, but there it is)

Mathematically, particles are representations of the Poincare group. However, in a curved spacetime, there is no canonical identification of the Poincare group at different spacetime points. The metric (or connection, depending on how you take your poison) induces a path-dependent mapping, which manifests in the fact that the identity of particles depend on the motion of the observer.

Mathematically, particles are representations of the Poincare group. However, in a curved spacetime, there is no canonical identification of the Poincare group at different spacetime points. The metric (or connection, depending on how you take your poison) induces a path-dependent mapping, which manifests in the fact that the identity of particles depend on the motion of the observer.
If that were true, then why is this an issue even in flat spacetime? (the Unruh effect doesn't need curved spacetime as far as I know)

Also, while I may be misunderstanding, this seems to imply a mixing of types of particles but still agreement on number of particles?

To help me understand this better, can someone explain an example: What does Hawking radiation look like to a freefall observer? He sees the vaccuum randomly pop out not just virtual particles but REAL particles?

Let me guess....the freefall observer doesn't see any effect at all. The accelleration of the observer is correlated to the appearance of matter particles. You should consider that the difference between real particles and virtual particles is part of your frame of reference. Every change in the motion of the observer results in a change in the set of observable particles. The full set of virtual and 'real' particles should be nearly perfectly dense.

Just a hunch.

R.

reilly
It does. Actually gravity couples to everything in sight, so any matter, itself (like QCD) and so forth. Typically you want to truncate the series relatively early (usually one loop) to get a reasonable effective theory (valid when E>>Mpl) or you will end up with nonsense. This is called 'semiclassical gravity' (stupid name, b/c we have a quantum of gravity involved, but there it is)
Thanks, Haelfix.

As an older guy, I have no problem with the notion of a semiclassical approach, at least for E&M. Jackson does a nice job with this topic. However in the E&M version, many photons are "sorta'" involved -- for monochromatic fields, with the number of photons= (E**2 + B**2)/2(hV/2pi) where v is the frequency. This was big in the days of low energy particle and nuclear physics, brehmsstrahlung, stopping power and so on. But I agree that what we have briefly discussed is not semiclassical by normal standards.

Regards,
Reilly

There are two modes of acceleration that might produce particles during the Big Bang. One is the accelerated expansion of the universe. And the second is the curvature of a small sized universe. When the universe was "small", its curvature was tightly curled, and the acceleration of expansion does not compare with the curvature. The acceleration was mostly determined by the curvature which was the same everywhere. So expansion acceleration did not cause much particle creation. But when the universe unfolded so that the curvature was much less, then the expansion acceleration caused particle creation. And these particles slowed the expansion acceleration. Could this theory be testable, and would it confirm acceleration, whether by expansion or curvature, as the cause of particle creation?

But when the universe unfolded so that the curvature was much less, then the expansion acceleration caused particle creation. And these particles slowed the expansion acceleration. Could this theory be testable, and would it confirm acceleration, whether by expansion or curvature, as the cause of particle creation?
Hi friend, it seems to me that before your theory can be testable, someone needs to provide a demonstration that the concept of (nonflat) spatial curvature actually exists as a physical phenomenon. At the moment it seems to be little more than a mathematical construct which fits neatly into the Einstein equations and solutions to them. It is widely accepted because of its mathematical ubiquity and usefulness.

And as I understand it, the physical existence of spatial curvature itself depends on the physical existence of 4 spatial dimensions. So we also need a physical, not merely mathematical, demonstration of the reality of a 4th spatial dimension.

Personally I don't project any of this being demonstrated anytime soon, and probably not ever. I don't believe that spatial curvature and a 4th spatial dimension are valid physical phenomena.

Of course I am distinguishing spatial curvature from spacetime curvature. The latter is a useful mathematical description and coordinate system which was never intended to embody a physically real geometry.

Jon

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jal
jonmtkisco
Of course I am distinguishing spatial curvature from spacetime curvature. The latter is a useful mathematical description and coordinate system which was never intended to embody a physically real geometry.
Of course…. Symmetry …. It is just a “cosmic coincidence”
Would you say the same thing about Planck Scale?
jal

Hi jal,

I'm having trouble interpreting the specific point you're making, and the question you ask. Sorry about that. Perhaps you could explain a bit more?

I don't think I'm being heretical when I say that the spacetime coordinate construct was intended to be a mathematical convenience rather than a tangible description of physical geometry.

I didn't mention the concept of symmetry. However, it strikes me that there are an infinite number of examples of symmetry in mathematics which don't necessarily manifest themselves in the behavior of "real" physical particles. The fact that a particular mathematical construct is elegant, beautiful and symmetrical tells us in itself very little about whether it models a real physical mechanism.

I don't have a lot to say about sub-Planck scale physics. I can say that, as with all quantum mechanics theories, the burden must lie on the proponent of a theory to demonstrate that a mathematical construct not only accurately describes observable physical behaviors, but also is able, without sacrificing logical consistency, to limit or exclude predictions of physical behaviors that are currently undemonstrable and are likely to remain so in the near future. Like the existence of more than 3 spatial dimensions.

I also think that any sub-Planck scale theory needs to logically describe how physical phenomenon which exist only at the sub-Planck scale would aggregate into observable behaviors at the above-Planck scale without introducing anomolies.

Jon

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I don't know that those situations are so different, for sure at the level I understand them, but even also at the level they are understood at all. The analogy is quite simple : the Unruh and Hawking temperatures are the same T = a/2pi in natural units, and I have read a few times (and it seems reasonable to me) that the Hawking radiation is in fact a direct consequence of the equivalence principle on the horizon (together with the Unruh effect). I need to check Wald's book, which talks about both.
theyre very different (although I know nothing about the Unruh effect). The cause of Hawking radiation is the event horizon (a change from spacelike to timelike regions). Without an event horizon it can't occur. The Unruh effect, as Ive gathered from the other posts, takes place in purely spacelike regions (possibly purely timelike regions too).

strangerep
Mathematically, particles are representations of the Poincare group. However, in a curved spacetime, there is no canonical identification of the Poincare group at different spacetime points. The metric (or connection, depending on how you take your poison) induces a path-dependent mapping, which manifests in the fact that the identity of particles depend on the motion of the observer.
If that were true, then why is this an issue even in flat spacetime? (the Unruh effect doesn't
need curved spacetime as far as I know)
The Unruh effect involves accelerations (mostly the texts talk about uniform accelerations).
The group that includes these is larger than the Poincare group, and has different
Casimir operators. E.g: the Poincare Casimir P^2 (i.e., mass^2) is no longer a Casimir
in the larger group. Therefore, one cannot reasonably expect particles classified under
the Poincare group to make sense under the larger group. The Unruh effect is one
result of attempts to do so regardless.

What really matters is the underlying abstract group and its unitary
irreducible representations (unirreps). You can't reasonably take the unirreps
from one group, transplant them naively into a situation founded on a
different group, and expect to get sensible results.

Also, while I may be misunderstanding, this seems to imply a mixing of types
of particles but still agreement on number of particles?
Even if there's only one type of particle, one observer's "empty vacuum" is another's
thermal bath of particles.

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