Does GR have Background?

Passionflower

Who says so?

It is unusual if it does not but in GR matter does not necessarily have to be attractive.

waterfall

How do you model negative curvature in GR such that an object with gravity shielding can just float in air without propulsion?

atyy

I'm unsure of this point, but I think that harmonic coordinates include the event horizon of a black hole, as well as the expanding universe. Some references that discuss this are:
Cook, Initial Data for Numerical Relativity, section 3.3.2
Weinberg, Gravitation and Cosmology, section 8.1 - 8.3

waterfall

I tried to read the reference above. But not completely understood it.
What events do the harmonic coordinates exclude?

Is harmonic coordinate related to linearized gravity?

Lastly, are you saying that if gravity is a real physical field (versus mere geometry), it can't explain all gravity dynamics but only those belonging to a subclass compatible with harmonic coordinate? I wonder what dynamics it excludes.

bahamagreen

I'm not up on any of this, but I wonder if answering a primitive question might give me or others a grasp on the implications brought out in this thread:

Does background or no background have anything to do with whether an inertial frame mass is subject to any acceleration/momentum (or however the influence/interaction would be properly characterized?) due only to the general universal expansion of space geometry itself?

atyy

I don't know what harmonic coordinates exclude, my guess is that they fail close to black hole and cosmological singularities, or when the spacetime topology gets weird. Harmonic coordinates mean that the nonlinear curved spacetime is exactly equivalent to a spin 2 field on flat spacetime, so they don't imply linearization about a flat spacetime. They are related to linearization on flat spacetime, since using them makes it easy to see gravity as a nonlinear field on flat spacetime. Then, when the nonlinear gravitational field is weak, it can be successfully approximated as a linear field on flat spacetime. (There may be other sorts of linearizations about curved spacetimes.)

As long as harmonic coordinates are possible, you can view gravity both as having a background, and as having no background. In the no background view, an inertial frame is only local, due to the curvature of spacetime, which in our universe can be approximately described as an expansion of space.

waterfall

We know that gravitons or gravitational waves self-interact and distort the spacetime as they travel. Except the case when these gravitational waves are very weak, where they can be seen as tiny ripples disturbing a fixed geometry. Is this what you were describing? the latter directly related and what is the true meaning of harmonic coordinates where spacetime is not distorted significant by strong gravitons? But wherever you are, gravity is still strong such that if you go outside the windows of your 2nd floor, you'd fall to the ground. Does this approximate flat spacetime?

atyy

No. Harmonic coordinates don't assume weak field, so the weak field condition is not the "true meaning of harmonic coordinates". For example, the proof of local existence of solutions to the nonlinear Einstein equation relies on harmonic coordinates. How about the glocal existence? Well, maybe there is no global existence, if the spacetime contains singularities.

waterfall

In the Weinberg Gravitation book you mentioned above. I saw inside the line "Another related advantage of the harmonic coordinate condition is that, as shown in Chapters 9 and 10, its use produces a very great simplication in the weak-field equations, similar to the simplication brought to Maxwell's equations by the use of the Lorentz gauge."

And you said "Harmonic coordinates mean that the nonlinear curved spacetime is exactly equivalent to a spin 2 field on flat spacetime, so they don't imply linearization about a flat spacetime."

And Lee Smolin said in "Trouble with Physics": "But Heisenberg and Pauli thought it would be simpler to first study cases in which the gravitational waves are extremely weak and can be seen as tiny ripples on a fixed background. This allowed them to apply the same methods they had developed to study quantum electromagnetic fields moving on a fixed background of spacetime. And in fact it was not difficult to apply quantum mechanics to very weak gravitational waves moving freely. The result was that each gravitational wave could be seen quantum mechanically, as a particle called the graviton - analogous to the photon, which is the quantum of the electromagnetic field. But at the next step, they faced a big problem, because gravitational waves interact with each other. They interact with anything that has energy, and they themselves carry energy."

I'm still a bit confused about this connection with Harmonic coordinate. Both use the same concept of spin-2 over flat spacetime. But like the Smolin quote, this can only occur when the gravity is weak. And you mentioned how the harmonic coordinate is related to the concept of spin-2 over flat spacetime. So one thing in common for them is weak field. Yet you deny this. Why, can you model spin-2 over flat spacetime in strong field? This isn't possible because in strong gravity, it is highly self-interacting and there are no solutions!

atyy

By strong field, Smolin means when the curvature is Planck scale. I'm talking about cases where the field isn't as strong, but still strong enough to be in the nonlinear regime.

waterfall

No. He was not referring to planck scale, the context he was referring was this:

"Describing the self-interaction of gravitons consistently turned out to be a tough nut to crack. We now understand that the failure to solve this problem is a consequence of not taking Einstein's principle of background independence seriously. Once the gravitational
waves interact with one another, they can no longer be seen as moving on a fixed background. They change the background as they travel.

So when you say "strong", it means the planck scale. So you mean outside it when you said "Harmonic coordinates mean that the nonlinear curved spacetime is exactly equivalent to a spin 2 field on flat spacetime"

Since this is outside the planck scale. So all gravity outside it can be described as spin 2 field on flat spacetime. Now why don't physicists accept this as primary instead. That is, that gravity is a spin 2 field on flat spacetime? Then one can apply gauge theory on it. In other words. Why assume the geometry dual context is the primary and the spin 2 field as secondary. Why not assume the spin 2 field as primary and the geometry just secondary? Please address this issue as it's the source of all confusion. Thanks.

atyy

That's what I like to think. You can find this view in reviews

by Hinterbichler: The real underlying principle of GR has nothing to do with coordinate invariance or equivalence principles or geometry, rather it is the statement: general relativity is the theory of a non-trivially interacting massless helicity 2 particle. The other properties are consequences of this statement, and the implication cannot be reversed."

and by Carlip: "Note that even though the perturbation theory described here does not provide an ultimate quantum theory of gravity, it can still provide a good effective theory for the low energy behavior of quantum gravity. Whatever the final theory, gravity at low energies is at least approximately described by a massless spin two field, whose action must look like the Einstein-Hilbert action plus possible higher order terms. If we restrict our attention to processes in which all external particles have energies of order E ≪ M_Planck, we can write an “effective action” that includes all local terms allowed by dffeomorphism invariance."

waterfall

So Hobba was right in the other thread we were discussing when he said ""Up to about the plank scale the assumption it is flat is fine, with gravitons making it behave like it had curvature or actually giving it curvature (we can't determine which) works quite well."

In reply to the above, remember Marcus wrote the following:

Marcus thought it was my idea when it was not even mine nor Hobba's but from others which you also saw. Hence Marcus was wrong here or not aware of the source you also have, agree? Try to agree and case closed.. it was what confused me a week ago because I thought he was right that I and Hobba were wrong.

atyy

Yes, I think Hobba was right.

waterfall

Ok. But there was something you said later in the thread that perplexed me. You said:

"BTW, although massless spin 2 can be equivalent to Einstein gravity in spacetimes that can be covered by harmonic coordinates (or similar), I don't think the reverse is true that the existence of a spin 2 field is sufficient to produce Einstein gravity.

Zhang and Hu, A Four Dimensional Generalization of the Quantum Hall Effect
Elvang and Polchinski, The Quantum Hall Effect on R^4

Bekaert et al, How higher-spin gravity surpasses the spin two barrier"

How could that be. You said massless spin 2 in harmonic coordintes can produce Einstein gravity, then you followed it immediately with the conflicting passage " I don't think the reverse is true that the existence of a spin 2 field is sufficient to produce Einstein gravity." But you just mentioned in the first sentence that it can! This has been perflexing me for a week so hope you can explain the context of what mean in your conflicting paragraph. Thanks.

atyy

A chair can be made of wood, but not everything made of wood is a chair.

waterfall

Ok. So you mean full GR includes black holes dynamics *near* singularity which spin-2 field over flat spacetime doesn't cover. Good. Thanks for the clarification.