E really does equal mc^2

[Frank]

Btw, Chalnot, while the math does fit, I couldn't find anything that could explain how gravitons could work at all.

Consider this: because it's just a cloud of particles, a photon can easily zip right next to a black hole without being affected, as there is a fair chance it wouldn't be hit by one. It's a rather small target, after all.

So, while gravitons might work for large masses, they cannot possibly for individual particles. Especially not when those are a fair distance away. And re-emitting a new cloud on being hit has it's own problems. Like where does the energy comes from (they are particles, after all), and how do they all travel in the right direction.

Then again, they might hit anything they would need, when you consider the whole volume of the shell (as by quantum mechanics) that holds the particle as the possible target. In that case, they cannot miss. But then again, they wouldn't be able to get anywhere, as space consists of nothing but those volumes (again, according to QM).

But on the other hand, if there are really that much and they hold even he most tiny bit of mass, they might make up the alleged dark mass that would be needed to make observations fit the theories.

And on and on.

But, while the math is easy, nobody has as yet shown any real-life proof that gravitons or garvity waves might actually exist. It's just math, so far.

And about the math: when you do things like:

1 + 1 = 2

but, we're not interested in 2, so let's eliminate that, and rewrite things as:

1 + 1 = 0

and see!

Or, for a more appropiate example:

E = mc2

But, let's use Planck units, so we can set c to 1, and we get:

E = m*1*1

or,

E = m

As we're not interested in mass, let's eliminate that:

E = 0

Which is about the same you did.



I know that things are done that way, and that you might be able to do so, as long as you're not re-introducing anything eliminated and all, but I don't know if that still reflects anything "real".

But then again, that kind of math is used more as a language to express ideas. And you have to look up the particular rules before you start.

Ok, I admit that very weird things like SR and QM turn out to be pretty accurate, but does that means that you can take any part of the math that lead to it, as a distinct whole, and still be able to talk sense? I'm not convinced about that.

Especially not, as there are plenty of things that only work in one of those theories, but make absolutely no sense in another one. Again, like with SR and QM.

I think it would be best to only apply those theories to which they work, and not try to make bits and pieces into universal truths, when they so easily clash with the theories that rule that particular case.

Common sense is needed, when discussing these things. And while I don't follow most of the math, I do think I have a reasonably good grasp of the issues.



Edit: If it was so easy to fit those pieces together, we would have had an Unified Theory of Everything a long time ago.

 

[KimB]

Btw, Chalnot, while the math does fit, I couldn't find anything that could explain how gravitons could work at all.

No, gravity waves don't have anything directly to do with gravitons. Rather we propose the idea of gravitons because it's the simplest way to consider gravity to be unified.

Consider this: because it's just a cloud of particles, a photon can easily zip right next to a black hole without being affected, as there is a fair chance it wouldn't be hit by one. It's a rather small target, after all.

Obviously the cloud needs to be dense enough that any object that has a measurable gravitational field is surrounded by a cloud of gravitons dense enough that this doesn't happen.

This really isn't a problem with the argument at all, because it's exactly the same issue with electromagnetism, but there's no problem with charged particles not feeling one another, and yet the theory is quite well-described by quantum mechanics.

But on the other hand, if there are really that much and they hold even he most tiny bit of mass, they might make up the alleged dark mass that would be needed to make observations fit the theories.

Such an effect, if it had to be calculated separately from what GR currently predicts, would be immediately-measurable in how gravity falls off with distance, and would have most effect where space is curved most (close to the mass). Gravity has been measured to be exactly what GR predicts to within about a micron.

And besides, dark matter is a smooth distribution that extends far beyond galaxies. It also is about 10 times as massive as normal matter. I don't think this could be described by the energy-momentum being carried by gravitons (which should be taken into account when one devises a solution to the Einstein equations anyway).

But, while the math is easy, nobody has as yet shown any real-life proof that gravitons or garvity waves might actually exist. It's just math, so far.

Gravity waves are on quite firm footing. Specifically, the decay of binary neutron star orbits is well-described by radiation of energy due to gravity waves.

As we're not interested in mass, let's eliminate that:

E = 0

Which is about the same you did.

No, what I did was to look at a region of space where there is no matter, and ask if the Einstein equations predict whether or not anything can happen in empty space. As it turns out, the Einstein equations predict that something can happen: gravity waves.

This is actually a standard mechanism in differential equations, as whenever you have a differential equation, there's always the particular solution (the one that depends upon the "souce" of the differential equation), and the homogeneous solution(s) (the solution(s) that are still there independent of the source). Whenever you are interested in solving a differential equation, you need to solve for both to get the full result.

 

[Frank]

No, gravity waves don't have anything directly to do with gravitons. Rather we propose the idea of gravitons because it's the simplest way to consider gravity to be unified.

Yes, I understand that. I even made a long post about that in this thread.

Obviously the cloud needs to be dense enough that any object that has a measurable gravitational field is surrounded by a cloud of gravitons dense enough that this doesn't happen.

This really isn't a problem with the argument at all, because it's exactly the same issue with electromagnetism, but there's no problem with charged particles not feeling one another, and yet the theory is quite well-described by quantum mechanics.


Such an effect, if it had to be calculated separately from what GR currently predicts, would be immediately-measurable in how gravity falls off with distance, and would have most effect where space is curved most (close to the mass). Gravity has been measured to be exactly what GR predicts to within about a micron.

And besides, dark matter is a smooth distribution that extends far beyond galaxies. It also is about 10 times as massive as normal matter. I don't think this could be described by the energy-momentum being carried by gravitons (which should be taken into account when one devises a solution to the Einstein equations anyway).

So, what do gravitons do when they miss their target? Decay?

Anyway, when gravitons worked exactly like photons, there would't be any need to unificate things in the first place. Which is just what all this is about.

Gravity waves are on quite firm footing. Specifically, the decay of binary neutron star orbits is well-described by radiation of energy due to gravity waves.

Yes, and Newton is pretty accurate as well. He was very good at describing our solar system. So good, that it took us centuries to spot the very small errors.

No, what I did was to look at a region of space where there is no matter, and ask if the Einstein equations predict whether or not anything can happen in empty space. As it turns out, the Einstein equations predict that something can happen: gravity waves.

Yes, I understand that. You even said so. Then again, if you just eliminate all matter from the equation, would it still work? There is no location in our galaxy where it could apply, for starters. We might not even know that it is wrong, as it seems to be in line with our current measurements.

This is actually a standard mechanism in differential equations, as whenever you have a differential equation, there's always the particular solution (the one that depends upon the "souce" of the differential equation), and the homogeneous solution(s) (the solution(s) that are still there independent of the source). Whenever you are interested in solving a differential equation, you need to solve for both to get the full result.

Yes, I said that as well. And you should keep the things apart, not re-introduce things you eliminate and all that. I know.

Then again, are you going to say that the rules for the math as used for SR are useable for any other field as well? They all have their own special rules.

 

[ninelven]

As it turns out, the Einstein equations predict that something can happen: gravity waves.

Would this not imply that the amount of energy in the universe is infinite? Perhaps I am being dense...

EDIT: Hmm... this gave me an idea, thanks.

 

[KimB]

So, what do gravitons do when they miss their target? Decay?

Gravitons don't have a "target" per se. To tell you the truth, I can't claim I can really understand this aspect of quantum interactions myself, though. But I can tell you that just because we do calculations in a form where we look at an interaction where two particles have interchanged a gauge boson (the graviton being one type: the photon, W, Z, and gluons being the others), that doesn't mean that's the only thing that can happen. It's just most useful to us to look at the case where two objects have exchanged a gauge boson (when the graviton has "hit" the object), because it's in this case that we can measure an interaction occuring.

Yes, I understand that. You even said so. Then again, if you just eliminate all matter from the equation, would it still work? There is no location in our galaxy where it could apply, for starters. We might not even know that it is wrong, as it seems to be in line with our current measurements.

One would expect so. We expect gravity to break down in the strong-field limit, not the weak-field limit. Of course, this is one possible area of study that physicists are considering in order to explain the observed accelerated expansion of the universe. But it's a really hard thing to do, as we have so little direction from current experiments as to how gravity might possibly be modified in the weak-field limit.

But that doesn't really have much bearing on this question, as you don't need zero energy-momentum in a region to have the gravity wave solutions.

Then again, are you going to say that the rules for the math as used for SR are useable for any other field as well? They all have their own special rules.

Well, obviously you need to be careful about what you're doing. But I'm more attempting to say that the math that works in quantum electrodynamics should work in a similar fashion with gravity. So it's more of an analogy than a rigorous analysis.

 

[KimB]

Would this not imply that the amount of energy in the universe is infinite? Perhaps I am being dense...

EDIT: Hmm... this gave me an idea, thanks.

Just because they can exist doesn't mean they're always there at all energies

Of course, by some measure, the amount of energy in the universe is infinite, but stating this typically requires we make some blanket statements of what we believe to be outside our observable universe, or that we draw unwarranted conclusions from quantum mechanics.

 

[Frank]

Well, obviously you need to be careful about what you're doing. But I'm more attempting to say that the math that works in quantum electrodynamics should work in a similar fashion with gravity. So it's more of an analogy than a rigorous analysis.

Agreed. But it turns out, that the math (or the effects it describes) in quantum electrodynamics don't exactly work as with gravity. Otherwise, we would very likely already have a Unified Theory of Everything by now. Some things are clearly in error. We just don't know which ones, or how to "fix" them. Which is the whole point.

And probably mostly, because both are completely consistent. You can explain pretty much everything covered by them. They just don't agree with one another. And the things that involve both are problematic. So, we have to look outside them to be able to solve the puzzle.

 

[Fred]

"A third method would be to attempt to quantize space, to somehow allow spacetime to be in a superposition of the two chosen states. The first problem that this has is that most obvious ways of quantizing space result in a breaking of Lorentz invariance (i.e. special relativity, which has been tested extensively, no longer holds...this is probably going to be an issue with your idea). The second problem is that if one attempts to create a particle description of gravity (the graviton), and looks at what properties it must have in quantum mechanics based upon its behavior on large scales, one finds that it is impossible to do any calculations: the theory is divergent."

Actually quantizing gravity isn't particularly difficult, at least in the path integral approach, and its not a big problem to have 'spacetime in superpositions'. Some of the popular accounts are a little bit vague on this.
The fundamental problem with quantum gravity is two fold.

1) how do you bypass the Coleman Mandula theorem?
2) What is a good observable?

We don't mind too much that Einstein gravity is nonrenormalizable, it makes good sense as an effective field theory. Moreover it could 'flow' to a nondivergent and perfectly sensible physical theory 'nonperturbatively' (this just means we might not be able to calculate things b/c we are stupid, not that the theory doesn't 'exist'). Again the fundamental problem is we are stuck with too simple a local description of our matter content and doesnt give us much wiggle room. String theory gives us the only currently known way around this by postulating an infinite tower of higher spin states.

The observable problem is more subtle otoh

 

[KimB]

We don't mind too much that Einstein gravity is nonrenormalizable, it makes good sense as an effective field theory.

Well, I'd say we do, as we can't do any calculations with the theory (at least, we don't yet know how).

String theory gives us the only currently known way around this by postulating an infinite tower of higher spin states.

Yeah, personally I think string theory is really fantastic. But it also seems to me that we are missing something significant that would place limits on string theory. On this subject, a couple of weeks ago I saw a talk given by Lisa Randall that was just excellent. Her idea was to consider a universe that is filled with all possible dimensions of d-branes, then allow that universe to evolve forward in time under the Friedman equation, then looking at which branes would come to dominate space (large dimensionality branes would fill too much space and annihilate with one another, low dimensionality branes would dilute). It turned out that 3-branes and 7-branes came to dominate. This is significant because one explaination is that our universe exists at a special intersection between 3-branes and 7-branes. Her next project is to explore the dynamics of collisions between 3-branes and 7-branes, which could prove to be extremely interesting for physics.

 

[Fred]

We can calculate just fine with Einstein gravity, just truncate the series after one loop. That will produce good answers down to where the coupling factor epsilon ~ E^2/Mpl^2 becomes comparable to 1 in fundamental units. After that the theory becomes strongly coupled, and all those infinite number of counterterms in principle becomes huge.

So while you won't learn much about the big bang, or parts of space where the curvature becomes extremely violent, we will know quite a lot about about the surrounding geometry where quantum mechanics *is* still important, but not too excessively violent. So its not quite right to say we dont know anything about how quantum mechanics and general relativity mix, we do, and in principle we can calculate to a pretty strong and controllable approximation a lot of things. Its just that there is a bit of a technical obstruction to solving it everywhere, (well there is a big conceptual difficulty too, but thats another story)

As far as Branes go and all that: Much love to Lisa, shes a friend, but I am not very satisfied with current D-Brane models for phenomonological reasons. Of course they are beautiful though.

 

[KimB]

We can calculate just fine with Einstein gravity, just truncate the series after one loop. That will produce good answers down to where the coupling factor epsilon ~ E^2/Mpl^2 becomes comparable to 1 in fundamental units. After that the theory becomes strongly coupled, and all those infinite number of counterterms in principle becomes huge.

With the divergences that appear in higher-order terms, can you really say this with certainty, though? I mean, we assume that the loop corrections, which appear to be divergent, will nearly cancel to a small number, but how do we know for certain? Granted, there are observational limits due to measurements of the strength of gravity, but I don't think you can really say with certainty that our current, divergent theory of quantum gravity can get anywhere close to the planck scale.

 

[Fred]

In the seventies, after the work of Wilson, where he founded the renormalization group, it became apparent that nonrenormalizable field theories weren't disastrous after all, they were merely only supposed to be taken to be valid within some energy range.

The analogue to think about is Fermis theory of the weak interaction. He basically had a four point interacting theory which is horribly divergent at the vertex, yet it produced perfectly sensible and experimentally accurate results long after theorists realized it couldnt be the last word on the matter. Weinberg, Salam, T'Hooft and others figured that it had to be mediated by W and Z particles to soften the divergence and renormalize the theory, and indeed lo and behold experimentally Fermis theory started to fail close to those resonances when experiment caught up.

But the point is, it wasnt necessarily a bad quantum theory, just incomplete, the 'real' theory merely limited too it.

In gravities case, we expect to go wrong when the energy scales are within say a few orders of magnitude from the Planck scale, but indeed we shoudl be able to see one loop effects (and be pretty darned accurate about them) at some intermediate scale.

Otoh, heres what I am ommitting in this whole discussion. We really need to know what Tuv is doing as well, in other words we *need* to know and have our matter spectrum pinned down as well (since we are an interacting field theory and gravity feels *everything*). The tacit assumption is that we know what our matter particles are doing all the way from the standard model, down to wherever it is we want to use our quantum gravity effective field theory. Thats a huge *if*, you very well could have new degrees of freedom enter somewhere down there that will ruin any chance of having predictive power barring some galaxy wide particle accelerator where we could measure them..

The other troubling thing is, we tacitly make the approximation Guv = guv + huv, namely the metric can be seperated into a fixed spacetime + saddle point perturbations away from it. Thats expected to no longer be true when spacetime foams too violently, ergo the whole field theoretic (and perhaps string theoretic) approach based on perturbation series might become untenable in certain violently curved spacetimes.

 

[KimB]

In gravities case, we expect to go wrong when the energy scales are within say a few orders of magnitude from the Planck scale, but indeed we shoudl be able to see one loop effects (and be pretty darned accurate about them) at some intermediate scale.

I'm not quite sure how we could be accurate about one-loop effects without renormalization.

Regardless, it is conceivable that gravity will break at a much smaller scale than the planck scale. Some have hypothesized that certain string theories will allow this, for instance. Of course, this is probably just wishful thinking, even though some of the arguments putting gravity unification at the TeV scale do seem to be fairly natural (or, at least, they attempt to be).

 

[Fred]

There are tons of people who work in this regime. Its called semiclassical gravity, or 'gravity in curved spacetime' and so forth. I believe Wald has a textbook on this. As far as the accuracy of one loop (it is renormalized) truncated gravity well thats a bit of an involved argument. You kinda sorta hope it behaves like a taylor series (each further contribution gets smaller and smaller), but this is not the case in quantum field theory as its generally believed to be asymptotic in nature only. This means smaller and smaller contributions, then poof turning points followed by heavily divergent contributions (read junk)

Say it has the form: (for n = number of loops)

sum_n (n!)^s alpha^n * c. Generally s is either 1 or 2 (depending on renormalization issues), so for s = 1, c = contants, the turning point ~ tends to be about 1/ alpha. Thats a very rough rule of thumb for the place where you want to cut off, b/c every term after that is going to start giving you nonsense. In Quantum electrodynamics, alpha is 1/127, so that means you are getting better and better accuracy up till about 127 loops.

Here in quantum gravity, s is going to be more like 2 (it is unrenormalizable), so its going to look more like alpha ^ (-1/2), but anyway.. we should expect pretty decent accuracy with 1 loop, so long as our coupling constant (which is scaling/running with energy) remains weakly coupled and small. We will certainly see some distinctly quantum effects over and above what general relativity will predict, and indeed this is how some of the original Hawking radiation calculations were done.

 

[KimB]

There are tons of people who work in this regime. Its called semiclassical gravity, or 'gravity in curved spacetime' and so forth. I believe Wald has a textbook on this.

I'm somewhat familiar with semiclassical gravity. But I don't think anything like it has been mentioned in the last few posts. From what I understand, with semiclassical gravity you're taking the approach that gravity is entirely classical, and it's our understanding of quantum mechanics that needs to be modified. So the approach that is typically taken is to state that the right side of the Einstein equation is <T>, which as I stated previously breaks down due to self interactions leading to runaway accelerations.

As far as the accuracy of one loop (it is renormalized) truncated gravity well thats a bit of an involved argument. You kinda sorta hope it behaves like a taylor series (each further contribution gets smaller and smaller), but this is not the case in quantum field theory as its generally believed to be asymptotic in nature only. This means smaller and smaller contributions, then poof turning points followed by heavily divergent contributions (read junk)

Say it has the form: (for n = number of loops)

sum_n (n!)^s alpha^n * c. Generally s is either 1 or 2 (depending on renormalization issues), so for s = 1, c = contants, the turning point ~ tends to be about 1/ alpha. Thats a very rough rule of thumb for the place where you want to cut off, b/c every term after that is going to start giving you nonsense. In Quantum electrodynamics, alpha is 1/127, so that means you are getting better and better accuracy up till about 127 loops.

Here in quantum gravity, s is going to be more like 2 (it is unrenormalizable), so its going to look more like alpha ^ (-1/2), but anyway.. we should expect pretty decent accuracy with 1 loop, so long as our coupling constant (which is scaling/running with energy) remains weakly coupled and small. We will certainly see some distinctly quantum effects over and above what general relativity will predict, and indeed this is how some of the original Hawking radiation calculations were done.

Well, right. But we only really have two "fundamental" field theories which are well-understood (QED and QCD). So I don't think we can really say that we have even this much understanding of quantum gravity without knowing the underlying quantum theory. For example, from what I understand, the Hawking radiation calculations have been debated rather intensely for some time now.

 

[Fred]

Yea actually really only QCD is believed to be fundamental, at least at an aesthetic eyeball level. QED suffers from the landau pole in the UV, and its generally believed that we can't get rid of that and makes the theory fatal. QCD on the other hand is asymptotically free and naively/manifestly valid through all energy scales. Most people don't think that will be the case though, stringy physics, or other stuff (technicolor, preons, etc) should appear as extra degrees of freedom at some point.

As far as the Hawking radiation calculation. Yea, its still kinda controversial, pretty much like the entire field of quantum gravity. But, you kinda throw your hands up in the air at some point though and just have to go with it to make progress.

 

[KimB]

Yea actually really only QCD is believed to be fundamental, at least at an aesthetic eyeball level. QED suffers from the landau pole in the UV, and its generally believed that we can't get rid of that and makes the theory fatal. QCD on the other hand is asymptotically free and naively/manifestly valid through all energy scales. Most people don't think that will be the case though, stringy physics, or other stuff (technicolor, preons, etc) should appear as extra degrees of freedom at some point.

Well, yeah. But, as you said, we expect QCD to break down at some point too, so in that respect QED is potentially a little bit better: the theory itself gives us a hint as to where it breaks down. Regardless, my main point was that unlike gravity, we can actually go in and test whether or not our assumptions are correct via experiment. We can't do that with quantum gravity. At least, not yet.

Pretty much the only tests that we have to date are that the theory must approach GR in the weak-field limit, and must match with close-range gravity experiments.

As far as the Hawking radiation calculation. Yea, its still kinda controversial, pretty much like the entire field of quantum gravity. But, you kinda throw your hands up in the air at some point though and just have to go with it to make progress.

Sure. And there has been a whole lot of useful knowledge that has been gained from the analysis of untestable theories. So even if your theory of quantum gravity may be wrong, it's possible that you will develop an excellent mathematical technique that will apply elsewhere. But personally, at least for the time being, I'm going to attempt to limit myself to doing examinations on that which is testable.

 

[Frank]

... I am not very satisfied with current D-Brane models for phenomonological reasons. Of course they are beautiful though.

That's my main problem with it as well (as far as I understand it): where is the locality? It might be really neat, but it's math only, as far as I'm concerned. There are way too few hooks into practical fields (none) to show us interactions and effects we can actually measure. Which is, of course, the main reason that it's really so very neat, but has as yet no practical applications whatsoever.

 

[Frank]

anyway.. we should expect pretty decent accuracy with 1 loop, so long as our coupling constant (which is scaling/running with energy) remains weakly coupled and small. We will certainly see some distinctly quantum effects over and above what general relativity will predict, and indeed this is how some of the original Hawking radiation calculations were done.

But that is exactly the point! It works, within it's own domain, and within it's own mathematical boundaries! Upper and lower limit, lowest and highest amplitude, diverging loops, it has it all.

Then again, that's also what should make it a quantified field theory. But on the other hand: quantum gravity still describes just one possible domain, just because you need to set up the specific boundaries and fields just right for that particuar application. Not a general solution, especially because you know it will break down in border situations, while we don't even know the boundary conditions. Which would be what you would need to describe to be able to quantify it in the first place.

Which is, again, what holds it back from becoming a Universal Theory in the first place.

 

[KimB]

I'd like to add an update to this topic on the finite transmission speed of gravity thingie. Sorry I didn't get up off my ass to walk down a floor to ask Steve Carlip about this myself back when we were having this conversation, but it turns out he's already written a very good paper on this exact subject:
http://xxx.lanl.gov/abs/gr-qc/9909087

(For those who don't recall, the originally-posted article that this is referring to can be found here)

As it turns out, one can obtain this apparent instantaneous transmission speed by taking a purely retarded form of the metric. The velocity terms in the Einstein equations cancel the abberation, giving the direction of the force of gravity as the extrapolated position of the object to second order.

A very similar affect happens in electricity and magnetism. How, then, do we see the Sun at a different position than its gravitational position if E&M works the same way? Well, it's simple, really: when we see the photons coming from the Sun, we're not seeing the direction of some hypothetical electromagnetic force (i.e. if the Sun and the Earth each had some overall charge). What we're seeing is the apparent direction from which the photons are arriving, which is something different.

The photons are still coming straight from the true location of the sun, but because we have some velocity with respect to this photon field, we see the photons coming from a slightly different direction (abberation). When you look at the force, though, the velocity terms in the equations conspire to exactly cancel this abberation (if there is no acceleration). Something nearly identical happens with gravity, though the cancellation is more substantial.