Can GR be formulated as background dependent?

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In summary, there are different formulations of general relativity that use various approaches such as acting over a spacetime metric or using geometric algebra. While these formulations may differ in their treatment of causality and background dependence, they are all consistent and can reproduce the key results of general relativity. However, there are caveats and technical issues that must be considered, such as global topologies and the use of coordinates. Ultimately, there are valid formulations of GR that may break the diffeomorphism group and rely on background structures such as frame fields.
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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?
 
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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
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".
 
  • #5
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
 
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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)
 

1. Can General Relativity (GR) be formulated as background dependent?

Yes, there are formulations of GR that treat spacetime as a dynamical entity and do not rely on a fixed background structure. These formulations include loop quantum gravity and causal dynamical triangulations.

2. What is the significance of formulating GR as background dependent?

Formulating GR as background dependent allows for a more complete and consistent understanding of the theory, especially in relation to quantum mechanics. It also opens up the possibility of resolving issues such as singularities and the unification of gravity with other fundamental forces.

3. How does a background dependent formulation of GR differ from the traditional formulation?

The traditional formulation of GR, known as the Einstein-Hilbert action, treats spacetime as a fixed background on which matter and energy interact. In contrast, a background dependent formulation treats spacetime as a dynamical entity that is influenced by matter and energy.

4. Are there any experimental tests that can distinguish between a background dependent and a background independent formulation of GR?

At the current time, there are no experimental tests that can definitively distinguish between the two formulations. However, ongoing and future experiments in quantum gravity could potentially provide evidence for one over the other.

5. What are the limitations of a background dependent formulation of GR?

One limitation is that the mathematical complexity of these formulations makes it difficult to make concrete predictions and perform calculations. Another limitation is that they do not yet have a complete and consistent framework for incorporating quantum mechanics, which is necessary for a complete understanding of gravity.

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