What is the definition of general covariance ?

1. Apr 20, 2007

pervect

Staff Emeritus
What is the definition of "general covariance"?

What is the definition of "general covariance"?

Is it sufficient for a theory to be written only in terms of tensors for it to be generally covariant? Is it necessary for a theory to be able to be written solely in terms of tensors for it to qualify as generally covariant?

Last edited by a moderator: Aug 10, 2013
2. Apr 20, 2007

jimbobjames

From Einstein: "The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant)"

So the definition would appear to be: "laws of nature are generally covariant if the corresponding equations hold good for all systems of co-ordinates".

Your questions:

1) Is it sufficient for a theory to be written only in terms of tensors for it to be generally covariant?

Don't know if this helps but: Can you think of a theory which can be written only in terms of tensors which does not hold good for all systems of coordinates? If not then the answer to your first question is yes, otherwise no.

2) Is it necessary for a theory to be able to be written solely in terms of tensors for it to qualify as generally covariant?

Can you think of a theory which can be written without the use of tensors and nonetheless holds in all systems of coordinates?

Maybe this from Dirac will be useful in answering your question: "Even if one is working with flat space (which means neglecting the gravitational field) and one is using curvilinear coordinates, one must write one's equations in terms of covariant derivatives if one wants them to hold in all systems of coordinates".

So covariant differentiation appears to be a necessary requirement and, as far as I know, you cant have covariant differentiation without introducing tensors.

Dirac elsewhere writes: "The laws of Physics must be valid in all systems of coordinates. They must thus be expressible as tensor equations."

So tensors appear to be a necessary, but perhaps not sufficient, condition for a generally covariant theory.

I hope that was somewhat helpful.

3. Apr 21, 2007

Old Smuggler

No, the theory cannot have any "prior geometry" either; i.e., a theory that has absolute
geometric elements independent of matter sources does not qualify as general covariant.
See MTW §17.6.

4. Apr 22, 2007

pmb_phy

From Gravitation and Spacetime by Ohanian and Ruffini. page 371 - footnote, which I'm sure you'd agree with it already, but for others
On general covariance for equations. From page 373
I hope that was of so use pervect.

Best regards

Pete

5. Apr 22, 2007

pervect

Staff Emeritus
Yes, that is helpful, thank you.

6. Apr 23, 2007

pmb_phy

You're welcome.

Pete

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