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Axiomatic approach to GR

by PAllen
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PAllen
#1
May5-11, 01:46 PM
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Mostly I have little interest in this topic, but the discussion came up in a recent thread about what is the axiomatic basis of GR? I threw out what I would consider a 'physicist view': the action or field equation plus rules for relating mathematical object to natural objects are all you need; nothing else need be assumed. It was pointed out that this is not much like axioms in the mathematical sense of fundamental postulates from wich (hopefully) something substantive can be derived.

I bumped into a couple recent papers on axiomatic basis of GR. I haven't looked at them in great detail, but they might be interesting to those concerned with 'foundational issues'. One thing that comes out is that there are no axioms resembling any common statement of equivalence principle, general covariance, or that objects follow geodesics under any particular circumstances (except that an inertial observer is defined one following a timelike geodesic).

So, of possible interest:

http://arxiv.org/abs/1005.0960
http://arxiv.org/abs/1105.0885

Their references for related points seem mostly available in arxiv.

[EDIT: Actually, there is an analog of general covariance, accomplishing what Einstein intended, rather than what he wrote in his GR papers. It is not a stated axiom but is contained in the process of changing the SR axioms into the GR axioms].

[EDIT2]: I see at the very end that the view is not so different from mine. The Einstein field equations are added at the very end as either a definition(of the stress energy tensor) or another axiom, your choice.
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TrickyDicky
#2
May6-11, 08:09 AM
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Quote Quote by PAllen View Post
Mostly I have little interest in this topic, but the discussion came up in a recent thread about what is the axiomatic basis of GR? I threw out what I would consider a 'physicist view': the action or field equation plus rules for relating mathematical object to natural objects are all you need; nothing else need be assumed. It was pointed out that this is not much like axioms in the mathematical sense of fundamental postulates from wich (hopefully) something substantive can be derived.
EFE+ the right symmetries (cosmological and Weyl principles, etc) and conditions (vacuum, etc) work fine for me as an axiomatic basis. The only snag I had in the other thread was that I thought you meant just the EFE, but that was already clarified.
dextercioby
#3
May6-11, 08:41 AM
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What's wrong in assuming the Lagrangian action and the associated variational principle for the grav. field coupled to matter as the only axiom ? The background on which the dynamics takes place is assumed to be a 4-dimensional paracompact manifold without boundary.

TrickyDicky
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May6-11, 12:26 PM
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Axiomatic approach to GR

Quote Quote by dextercioby View Post
What's wrong in assuming the Lagrangian action and the associated variational principle for the grav. field coupled to matter as the only axiom ? The background on which the dynamics takes place is assumed to be a 4-dimensional paracompact manifold without boundary.
You have answered yourself,you are adding the symmetries of the manifold assumed.
dextercioby
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May6-11, 02:32 PM
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I don't understand that. What symmetries am I bringing in ? The manifold has very few restrictions.
TrickyDicky
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May6-11, 03:03 PM
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Quote Quote by dextercioby View Post
I don't understand that. What symmetries am I bringing in ? The manifold has very few restrictions.
All I talked about in my post #2 are the EFE without anything else, now of course the EFE can be derived from the least action principle and the appropriate Lagrangian but if you are including those my snag is no longer valid since with them symmetries and restrictions are introduced that have axiomatic value.
dextercioby
#7
May6-11, 05:17 PM
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Yes, but even giving the EFE as you say <without anything else> is actually unaccomplishable, because you still have to define the terms (tensors or spinor tensors) entering the equation, hence specifying the manifold's geometrical and topological structure.

So why not simply postulate the manifold, the matter fields and the variational principle ?
TrickyDicky
#8
May6-11, 06:03 PM
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Quote Quote by dextercioby View Post
Yes, but even giving the EFE as you say <without anything else> is actually unaccomplishable, because you still have to define the terms (tensors or spinor tensors) entering the equation, hence specifying the manifold's geometrical and topological structure.

So why not simply postulate the manifold, the matter fields and the variational principle ?
That is precisely my point.When one tries to solve the field equations, one builds into their paramaterization the symmetries and assumptions most compatible with the observable universe or with the problem at hand (i.e. spherical symmetry for the vacuum solution).
But the same EFE with other assumptions can lead to absurd, unphysical solutions. And in this probably trivial sense is in which I pointed out that the EFE by themselves can lead to different axioms.


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