What is the definition of general covariance ?

In summary, the definition of "general covariance" is that the laws of nature must be expressible by equations that hold true for all systems of co-ordinates, and are co-variant with respect to any substitutions. It is necessary for a theory to be written solely in terms of tensors in order to qualify as generally covariant, as tensors are necessary for covariant differentiation. However, this may not be a sufficient condition, as the theory must also be invariant, meaning that the form and content of the equations must remain unchanged under coordinate transformations. This principle of general covariance is important in ensuring that the laws of physics are valid in all systems of coordinates.
  • #1
pervect
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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?
 
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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 can't 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
pervect said:
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?

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
pervect said:
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?

From Gravitation and Spacetime by Ohanian and Ruffini. page 371 - footnote, which I'm sure you'd agree with it already, but for others
* We have "covariant" vectors, "covariant" derivatives and now "covariant" equations. The word "covariant" is unfortunately much overworked, and which the meaning of the three meanings is intended must be guessed from context.

On general covariance for equations. From page 373
Principle of general covariance: All laws of physics shall bve stated as equations which are covariant with respect to general coordinate transformations.
...
Invariance goes beyond covariance in that it demands that not only the form, but also the content of the equations be left unchanged by the coordinate transformation. Roughly, invariance is achieved by imposing the extra condition that all "constants" in the equation remain exactly the same.

I hope that was of so use pervect.

Best regards

Pete
 
  • #5
pmb_phy said:
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

Yes, that is helpful, thank you.
 
  • #6
pervect said:
Yes, that is helpful, thank you.
You're welcome. :smile:

Pete
 

What is the definition of general covariance?

General covariance is a principle in general relativity that states that the laws of physics should be the same for all observers, regardless of their position or motion. This means that the laws of physics should not depend on a specific coordinate system, but rather should be described in a way that is independent of the observer's frame of reference.

How is general covariance related to the theory of relativity?

General covariance is a fundamental principle in the theory of relativity, specifically in general relativity. It is one of the key principles that Einstein used to develop his theory of gravity, which describes the relationship between space, time, and matter.

Why is general covariance important in physics?

General covariance is important because it allows for the development of a theory of gravity that is consistent with the principles of relativity. It also allows for the description of physical phenomena in a way that is independent of the observer's frame of reference, making it a more universal and accurate representation of reality.

How does general covariance differ from other principles in physics?

General covariance differs from other principles in physics, such as Galilean invariance or Lorentz invariance, as it is a more general and comprehensive principle that applies to all physical laws, not just those related to motion. It also takes into account the effects of gravity and the curvature of space-time.

Can general covariance be tested or proven?

General covariance is a fundamental principle in physics and cannot be directly tested or proven. However, the predictions and equations derived from general relativity have been extensively tested and have been shown to accurately describe a wide range of physical phenomena, providing strong evidence for the validity of general covariance.

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