Does gravity field really exist?

[harrylin]

What I said doesn't refer to how hard the theory is to understand. I was talking about the philosophical deviations from the thoughts that were the essentials of classical physics which of course are present in SR and GR no less than QM.

Thanks for the precision. :)

 

[stevendaryl]

Not so fast Kemosabe :p:p:p:p:p:p:p:p

I didn't want yo get into it because the book I pointed to by Ohanian examines the issue in detail - but it's not quite that simple.

You can model gravity in exactly the same way as EM and you get linearised gravity which actually is perfectly OK in explaining many phenomena.

The issue though is it contains its own destruction and inevitably leads to GR. First, it can be shown, particles do not move in flat space-time - but rather as if space-time had an infinitesimal curvature. Most importantly however it ignores the issue of gravity gravitating so it must be non-linear.

The interesting thing however is GR is rather strange, because, as Ohanion shows, the linear equations imply the non linear ones and you get full GR.

This is also related to the issue - is space-time curved or does the gravitational field simply make clocks and rulers behave as if it was curved. Experimentally there is no way to tell the difference.

BTW my bible on GR is Wald - do you can guess which side on that debate I come down on - but it's debatable 100% for sure.

Thanks
Bill

The route to GR via starting with a linear approximation and adding self-interaction consistently is interesting for the following reason:

Standard GR has the form:

G = \kappa T

where G is a tensor formed from the metric and its first and second derivatives, and T is the stress-energy tensor for nongravitational sources. So that's the sense in which gravity doesn't act as a source for gravity--there are no gravity terms on the right-hand-side.

If you start by treating the metric as a field whose source is stress-energy, and then you include self-interaction, you get something equivalent to standard GR, but not quite the same left-hand and right-hand sides. Instead, you get something like this: (This is intentionally hand-wavy, because I don't have a reference handy)

G_0 = \kappa (T_{non-grav} + T_{grav})
where G_0 is a different tensor (not G) involving the metric tensor and its derivatives, and T_{grav} is the stress-energy due to the metric, treating it as a field.

This is equivalent to GR because the combination of G_0 - \kappa T_{grav} happens to be equal to the tensor G.

So my very hand-wavy understanding is that there are terms that can be put on the left-hand side of the equation, and they become part of the Einstein curvature tensor. Or they can be put on the right-hand side, and they become the gravitational part of the stress-energy.

I was never able to follow the mathematics (described very briefly in Misner, Thorne and Wheeler) for how to bootstrap yourself from a linear spin-2 field theory to full GR, which is another reason this is hand-wavy. However, as I understand it, you can't directly figure out stress-energy from equations of motion. You have to concoct a Lagrangian that reproduces your equations of motion, then use that to compute the stress-energy. So there's conceptual an infinite iteration going on:

1. Start with a Lagrangian for non-gravitational fields and matter in flat spacetime.
2. Compute the stress-energy tensor from that.
3. Use that tensor as the source for the first approximation to an equation for the metric (as a spin-2 field).
4. Modify your lagrangian to give that field equation as the Euler-Lagrange equations of motion when you vary the metric.
5. Compute the stress-energy tensor for the modified Lagrangian.
6. Use that stress-energy as the second approximation to an equation for the metric.
7. Modify the Lagrangian again to give that field equation.
8. etc.

But it's possible to be clever and figure out (or guess) the "limit" theory.

 

[bhobba]

I was never able to follow the mathematics (described very briefly in Misner, Thorne and Wheeler) for how to bootstrap yourself from a linear spin-2 field theory to full GR, which is another reason this is hand-wavy.

I nutted it out from Feynmans Lectures many moons ago when I as really into GR:
https://www.amazon.com/dp/0813340381/?tag=pfamazon01-20

It took a bit of mucking around and tooing and frowing with Ohanions text.

Thanks
Bill

 

[PeterDonis]

there are terms that can be put on the left-hand side of the equation, and they become part of the Einstein curvature tensor. Or they can be put on the right-hand side, and they become the gravitational part of the stress-energy.

Yes, this is true. However, one key thing to note is that, with the terms on the LHS (i.e., part of the Einstein tensor), both sides of the equation have zero covariant divergence, which is an important conservation law. If you move the "gravitational field energy" terms to the RHS, the covariant divergence of the LHS and RHS is no longer zero.

 

[stevendaryl]

Yes, this is true. However, one key thing to note is that, with the terms on the LHS (i.e., part of the Einstein tensor), both sides of the equation have zero covariant divergence, which is an important conservation law. If you move the "gravitational field energy" terms to the RHS, the covariant divergence of the LHS and RHS is no longer zero.

That's a very good point, and a good argument for putting gravity on the LHS.

 

[ShayanJ]

The discussion remembers me something I read in a lecture note of general relativity so I want to mention it here both because it may be useful and also because to find out the connection I feel! But I can't give a reference because its actually a lecture note in my own language.
In that lecture note, at first it is argued that gravity should be a massless spin2 field. Then the free Fierz-Pauli Lagrangian is given:
<br /> L=\frac 1 2 (h_{\mu \ ,\lambda}^\lambda h^{\mu \nu}_{\ \ ,\nu}-h_{\mu \lambda}^{\ \ ,\mu}h^{,\lambda})+\frac 1 4(h_{,\mu}h^{,\mu}-h_{\mu\nu}^{\ \ ,\lambda} h^{\mu\nu}_{\ \ ,\lambda})<br />
(There may be some typing errors in the lecture notes so maybe this isn't right completely!)
Anyway, the EL equations coming from that Lagrangian are D^{\alpha \beta}_{\mu\nu} h_{\alpha \beta}=0, where D^{\alpha\beta}_{\mu \nu}=(\eta_{\mu}^{\alpha}\eta_{\nu}^{\beta}-\eta_{\mu\nu}\eta^{\alpha \beta})\partial^\rho \partial_\rho+\eta_{\mu\nu}\partial^{\alpha}\partial^\beta+\eta^{\alpha \beta}\partial_\mu \partial_\nu-\eta_{\mu}^{\beta}\partial^\alpha \partial_\nu-\eta_{\nu}^{\alpha}\partial^\beta \partial_\mu, which satisfies \partial^\mu D^{\alpha \beta}_{\mu\nu}=0.
But when we put the stress-energy tensor of matter in the RHS, we'll have D^{\alpha \beta}_{\mu\nu} h_{\alpha \beta}=T_{\mu\nu} which gives \partial^\mu T_{\mu\nu}=0. But this is an inconsistency because its not the stress-energy tensor of the matter alone that should be conserved but the total stress-energy tensor. So to solve this inconsistency, people started to add higher order terms to the LHS but it turns out that continuing to no finite order is enough and so an infinite series pops out. Now its Ogievetsky and Polubarinov that sum up this infinite series and show that it actually gives Einstein's Field Equations(1965,Ann.Phys. ,35, 107).
Then, as an easier solution, its proposed to vary the action S_0=\int [\phi^{\mu\nu}(\partial_\alpha\Gamma_{\mu\nu}^{\alpha}-\partial_\nu\Gamma_{\alpha\mu}^{\alpha})+\eta^{\mu\nu}(\Gamma_{\mu\nu}^{\alpha}\Gamma_{\rho\alpha}^{\rho}-\Gamma_{\beta\mu}^{\alpha}\Gamma_{\alpha\nu}^{\beta})] d^4x w.r.t. \Gamma_{\mu\nu}^{\alpha} and \phi^{\mu\nu} which gives an equivalent theory to the mentioned Fierz-Pauli action. Then the Minkowski metric is replaced by a dynamical metric and the variation of the new action w.r.t. the dynamical metric is taken as the stress-energy tensor of gravity. After modifying the action farther for getting a consistent theory, the action takes the form of the Einstein-Hilbert action and so we're led to GR.(This is mentioned as Deser's method!)
I said this because I failed to see the connection between this procedure and the discussion here, so maybe someone can clarify or give some references explaining this procedure farther.(For example in calculating that infinite series, I'm really interested in seeing it but I couldn't find the paper.)
(Here's a link to the lecture note, at least you can see the equations! The things I said start at page 20)