Can the Christoffel connection be observed?

[pervect]

I was thinking over my answer, and maybe there's another question we should ask first. Can a choice of coordinates be "observed"? I.e. can we observe whether one uses polar coordinates, or rectangular? I would say that the answer to this is "no", it's not observable, it's part of our mental model, not the physics.

If we assume the answer to the above question "no", and we also know that the Chrsitoffel symbols depend on the choice of coordinates, then it seems they contain some elements that are observable, and some elements that are not.

I would argue that the 4-acceleration , being a tensor, is independent of coordinates. The Christoffel symbols contain information about the 4-acceleration, but also information about the coordinate choices.

 

[haushofer]

But is there really a sense in which something is intrinsically non-dynamic? I would think that you could always add some unobservable dynamics.

I'm not sure if I understand your "unobservable dynamics" meaning, but this reminds me of the Stückelberg trick, in which you add fields which can always be completely gauge-fixed such that certain symmetries become explicit. E.g. Newton-Cartan theory is a complicated and non-trivial version of that idea, although originally not constructed by the Stückelberg formalism.

 

[Demystifier]

That sounds harsh but I have to say I agree.

So you agree that sometimes non-covariant (non-tensorial) quantities, like energy-momentum pseudo-tensor, which depend on coordinates, may be physically more useful than covariant ones?

 

[Demystifier]

but this reminds me of the Stückelberg trick, in which you add fields which can always be completely gauge-fixed such that certain symmetries become explicit.

Where can I see more about that Stückelberg trick?

 

[martinbn]

So you agree that sometimes non-covariant (non-tensorial) quantities, like energy-momentum pseudo-tensor, which depend on coordinates, may be physically more useful than covariant ones?

I agree that what you propose in the paper is not useful. It seems to be just name giving. Of course I might be wrong and change my view if someone shows me why it may be useful.

 

[DrDu]

I know too few about GR, but this apparent usefullness of non-invariant quantities arises in several other fields. E.g. the spin is defined as the angular momentum in the center of energy reference frame of a particle. However, this does not mean that spin is not defined for other reference frames but only that it cannot be interpreted as an angular momentum in these frames. Another example are the London equations of a superconductor, which contain explicitly the electromagnetic vector potential, however, only if formulated in Coulomb gauge. A chain of more and more trivial examples can be developed.

 

[Demystifier]

I agree that what you propose in the paper is not useful. It seems to be just name giving. Of course I might be wrong and change my view if someone shows me why it may be useful.

OK, but what do you think about the gravitational energy-momentum pseudo-tensor? Do you think it's useful? Would you say that it is physical, given that it depends on the choice of coordinates?

 

[sweet springs]

Of course I might be wrong and change my view if someone shows me why it may be useful.

in high school physics gravity potential energy mgh is familiar but it is not of tensor.

 

[sweet springs]

So you agree that sometimes non-covariant (non-tensorial) quantities, like energy-momentum pseudo-tensor, which depend on coordinates, may be physically more useful than covariant ones?

bythe way not pseudo-tensor but nontensor , it should have been named.

 

[sweet springs]

In principle yes, but that's quite complicated and you cannot define their energy-momentum.

i have read in Dirac that in case of plane wave energy momentum is localized.

 

[martinbn]

I know too few about GR, but this apparent usefullness of non-invariant quantities arises in several other fields. E.g. the spin is defined as the angular momentum in the center of energy reference frame of a particle. However, this does not mean that spin is not defined for other reference frames but only that it cannot be interpreted as an angular momentum in these frames. Another example are the London equations of a superconductor, which contain explicitly the electromagnetic vector potential, however, only if formulated in Coulomb gauge. A chain of more and more trivial examples can be developed.

This is another example about observables depending on the reference frame. It is not about coordinates.

 

[martinbn]

OK, but what do you think about the gravitational energy-momentum pseudo-tensor? Do you think it's useful? Would you say that it is physical, given that it depends on the choice of coordinates?

Well, I don't know what it is, so i have no opinion. I'll take a look.

 

[haushofer]

Where can I see more about that Stückelberg trick?

See e.g.

https://arxiv.org/pdf/1105.3735

page 30 onwards.

 

[haushofer]

in high school physics gravity potential energy mgh is familiar but it is not of tensor.

So many things in classical mechanics are not tensors, and if they are, they are only under a specific group. E.g., the Newton potential is only a scalar under the Galilei group but not under accelerations; dito for Newton's second law being vectorial (from which you can derive the former). And kinetic energy is not even a scalar under the Galilei group.

 

[pervect]

OK, but what do you think about the gravitational energy-momentum pseudo-tensor? Do you think it's useful? Would you say that it is physical, given that it depends on the choice of coordinates?

I think the gravitational energy-momentum pseudo tensor is useful in the right circumstances, though I seem to recall that it's ambiguous. For instance, I recall seeing references to the Landau-Lifschitz pseudotensor in linearized gravity to distinguish it from other pseudotensors.

The lack of coordinate and gauge independence of the psuedotensors makes them not entirely satisfying as physical entities in my mind. But they're still useful in the right circumstances. One thing that bothers me is that the theoretical justification for their usefulness, asymptotically flatness, isn't a condition that's actually met in the FRW spacetimes used in cosmology.

 

[pervect]

On the general topic of tensors, one of the things that makes them so useful is that there is no need to specify a coordinate system. Sometimes coordinate based methods can be useful, and perhaps even easier. However, if one is going to use methods that depend on a particular choice of coordinates, I believe one has to communicate that choice clearly, so the reader knows when the results apply and when the results do not apply.

 

[atyy]

I think this is basically semantics, once we realize that some authorities (Wald) have defined coordinate-dependent tensors.

 

[martinbn]

I think this is basically semantics, once we realize that some authorities (Wald) have defined coordinate-dependent tensors.

You have to be more careful when you tell this joke. Some people may think you are serious.

 

[atyy]

Wald, p34: a Christoffel symbol is a tensor field associated ...

 

[cosmik debris]

Wald, p34: a Christoffel symbol is a tensor field associated ...

I thought the right hand side of the equation containing the Christoffel symbols was a Tensor field but not the individual symbols.

Cheers

 

[martinbn]

Wald, p34: a Christoffel symbol is a tensor field associated ...

I looked at that section the last time you mentioned it, but I thought you were joking. Apparently you are serious. Then I can only recommend that you read the whole section, and see how it is written in the context it is written. If you don't read the whole section but just search for the key words, then it is no surprise that you are confused.

 

[atyy]

It does not mean that if one has read the whole section, that one will not be confused.

 

[martinbn]

True, but you should at least try.