General covariance and tensors

[Goldbeetle]

The fact that physics laws must have the same form in any reference frame (general covariance) is guaranteed by expressing them in tensor notation (, if possible). Considering also non linear coordinate transformations the tensor transformation rules are defined by means of the partial derivatives of the coordinates transformations. This means that we are linearising the coordinate transformations. In other words, the requirements of general covariance requires more precisely that the law of physics have the same form with respect to the linear approximation of a general transformations among reference systems.

Am I correct?
Thanks,
Goldbeetle

 

[Matterwave]

In what way are we "linearising" the coordinate transformation? We have that tensors must obey, in general:

$$T'^{ij}=\frac{\partial x'^i}{\partial x^k}\frac{\partial x'^j}{\partial x^l}T^{kl}$$

In general then, even if the x primes are not linear functions of the x's, the partial differentials will still capture that. Although, for SR, the relevant transformations are the 3 boosts, 3 rotations, and origin displacement, which happen to be linear.

 

[Goldbeetle]

If you take x=x(x') or the inverse x'=x'(x) and compute their jacobians matrixes. These define the action of the transformations locally, up to first order. In other words the general coordinates transformation is considered locally approximated when defining the tensors.

 

[bcrowell]

The fact that physics laws must have the same form in any reference frame (general covariance) is guaranteed by expressing them in tensor notation (, if possible). Considering also non linear coordinate transformations the tensor transformation rules are defined by means of the partial derivatives of the coordinates transformations. This means that we are linearising the coordinate transformations. In other words, the requirements of general covariance requires more precisely that the law of physics have the same form with respect to the linear approximation of a general transformations among reference systems.

This sounds like a reasonable formulation to me. One thing to watch out for is that a frame of reference is not the same thing as a coordinate system in GR. GR doesn't have global frames of reference.

 

[TrickyDicky]

If you take x=x(x') or the inverse x'=x'(x) and compute their jacobians matrixes. These define the action of the transformations locally, up to first order. In other words the general coordinates transformation is considered locally approximated when defining the tensors.

I agree , Goldbeetle, at least when I read in general relativity textbooks (like Ryder for instance) that general covariance is a mathematical statemement of the Equivalence principle, what I understand is that the way this principe is inserted in the GR theory is precisely by imposing that the tensors are locally(infinitesimally) approximated by the SR linearity. Or how else could we make the local frames of GR equivalent to SR inertial frames?
So you are right when you say that general covariance in GR implies that we are linearising the coordinate transformations and that the laws of physics have the same form with respect to the (SR) linear approximation of general coordinate transformations.

However, and even though your definition of general covariance has been approved by others here, you must be aware that is not the mainstream interpretation, where general covariance is literally taken as absolute coordinate transformation freedom and also to mean that physics laws must have the same form in any coordinates.

 

[atyy]

Tensors (tensor fields) are locally (multi)linear objects. For example, a vector field, which is a particular case of a tensor field, is the choice of a vector from the tangent space at each point in spacetime. The tangent space at any point in spacetime is a vector space, which is what one means by linear.

This local linearity is what is preserved under arbitrary nonlinear changes of coordinates by the tensor transformation rule.

 

[TrickyDicky]

Tensors (tensor fields) are locally (multi)linear objects. For example, a vector field, which is a particular case of a tensor field, is the choice of a vector from the tangent space at each point in spacetime. The tangent space at any point in spacetime is a vector space, which is what one means by linear.

This local linearity is what is preserved under arbitrary nonlinear changes of coordinates by the tensor transformation rule.

Sure, that's the mainstream doctrine as I said in my post, but this has been criticized in the past as a trivial statement about differential geometry and Riemannian manifolds. In the sense that it empties general covariance of any specific meaning in GR, in other words any physical theory, like say electrodynamics ,can be formulated in tensorial form with general covariance.
I, however, tend to think that general covariance in GR has a more specific function which is to realize the Equivalence principle.

 

[atyy]

The principle of equivalence was heuristic.

Within GR, it is formalized as minimal coupling between non-gravitational fields and the gravitational field.

 

[TrickyDicky]

The principle of equivalence was heuristic.
Within GR, it is formalized as minimal coupling between non-gravitational fields and the gravitational field.

This feels like a bit of a deja vu, but yeah, that minimal coupling thing looks heuristic enough, too.

 

[atyy]

This feels like a bit of a deja vu, but yeah, that minimal coupling thing looks heuristic enough, too.

Yes. I'm sure you'll find this funny. http://arxiv.org/abs/0707.2748

 

[TrickyDicky]

Yes. I'm sure you'll find this funny. http://arxiv.org/abs/0707.2748

Thanks, it is interestingly funny. And captures part of my point.
Let me quote the points that better reflect my thoughts about this:

"What does “non-gravitational fields” mean?
There is no precise definition of “gravitational” and “non-gravitational” field. One could say that a field non-minimally coupled to the metric is gravitational whereas the rest are matter fields. This definition does not appear to be rigorous or sufficient and it is shown in the following that it strongly depends on the perspective and the terminology one chooses."

"As a conclusion, the concept of vacuum versus non-vacuum, or of “matter field” versus “gravitational field” is representation-dependent. One might be prepared to accept a priori and without any real physical justification that one representation should be chosen in which the fields are to be characterized as gravitational or non-gravitational and might be willing to carry this extra “baggage” in any other representation in the way described above. Even so, a solution to the problem which would be as tidy as one would like, is still not provided."

"many misconceptions arise when a theory is identified with one of its representations and other representations are implicitly treated as different theories."

"it is not only the mathematical formalism associated with a theory that is important, but the theory must also include a set of rules to interpret physically the mathematical laws"
END of quotes.

I'll just add that it has become a habit to say that the math in a theory like GR is self-evident and actually has no possibility of different interpretations (representations in the article language), those that say it of course give you their "interpretation" as the only one possible, even conceiveable given the equations of the theory. One is never sure if they really believe that there is only one interpretation of the math or they are cinically trying to impose you their interpretation with ulterior motives.
To me ultimately the guide to the right representation should be the observable and predictive consequences of a certain physical interpretation of the equations.

To come back to our specific issue, the minimal coupling formalism between gravitational and non-gravitationa fields is clearly an empty formalism if we are not specifying how we define their distinction and based on what representation of the theory we do it.
The alternative way to insert the requirement of considering local frames in GR as SR inertial frames that I bring up (triggered by the distinction about general covariance made in the OP) is of course simply a different possible interpretation that just might be worth checking.