# Can GR be formulated as background dependent?

1. Aug 9, 2009

### ensabah6

as a field acting over a spacetime metric (rather than encoding it in the metric) but still reproduce key GR results including time-varying phenomena?

2. Aug 9, 2009

### marcus

No, because events would have a different causal ordering in the two cases.
The causal order is determined by the metric---which defines the lightcones and therefore which events can influence which---who is in the causal past of whom.

In the real GR there is the one dynamically determined metric g, and that decides causality.

In the fake case you have a fixed metric h and on top of that a small field k so that the combined result is supposed to reproduce the real GR case and give g = h + k.

The difference is that in the fake case causality is determined by the fixed h.

3. Aug 10, 2009

### Fredrik

Staff Emeritus
Kip Thorne's book "Black holes and time warps: Einstein's outrageous legacy" mentions a formulation of GR in which spacetime is flat, and instead of curving spacetime, matter deforms "measurement devices" to get the same predictions about the results of experiments as standard GR. That sounds like a formulation that we might want to call "background dependent".

4. Aug 10, 2009

### Ich

Last edited by a moderator: Apr 24, 2017
5. Aug 10, 2009

### atyy

I tried to find the formulation that Thorne refers to, but the closest I could find is Section 4.3 "Einstein's equations in relaxed form", which requires that one can set up harmonic coordinates "Equation 62 is exact, and depends only on the assumption that spacetime can be covered by harmonic coordinates." http://relativity.livingreviews.org/Articles/lrr-2006-3/ [Broken]

There is apparently a different approach to GR using geometric algebra in which spacetime is flat: http://arxiv.org/abs/gr-qc/0405033

Last edited by a moderator: May 4, 2017
6. Aug 10, 2009

### Haelfix

My answer would be yes it is completely consistent, modulo the caveats found in MTW chapter 18 or in Weinbergs book. Starting from linearized gravity, you have to bootstrap your way up to the full nonlinear equations, but that works fine and there is no problem with causality.

There is a small technical issue for global nontrivial topologies which means in practise that you have to be careful and glue together several coordinate charts, but again that has been done and there isn't much of a problem.

The bigger problem is what if you want the local chart to be something other than a connected Euclidean topology. We know of no physical example where that would not be the case, but if you insist on being completely general (perhaps more general than nature herself) then you would have a case.

Anyway, more generally there are other perfectly valid formulations of GR other than 'pure' linearized gravity, where you do need to break the diffeomorphism group and specify coordinates from the onset. For instance, everytime you use a vielbien you are explicitly providing a background dependant picture of GR (in terms of frame fields)