Allowed transformations in General Relativity

[kent davidge]

Does General Relativity allow for transformations which are not isometries of the metric?

 

[king vitamin]

Yes, the Einstein-Hilbert action is invariant under any diffeomorphism.

(Formally, in any sensible physical theory, any diffeomorphism is allowed. What is special about GR is that the action is also invariant under any diffeomorphism, which also makes the equations of motion covariant under these transformations.)

 

[haushofer]

What do you mean by "allowed"? Are you confused by gct's as gauge transformations keeping EOMs invariant and isometries keeping the metric invariant?

Isometries only make sense once you choose coordinates, i.e. once you chose a gauge. If in that coordinate system the form of the metric is preserved for specific transformations, you speak of isometries.

 

[haushofer]

Yes, the Einstein-Hilbert action is invariant under any diffeomorphism.

(Formally, in any sensible physical theory, any diffeomorphism is allowed. What is special about GR is that the action is also invariant under any diffeomorphism, which also makes the equations of motion covariant under these transformations.)

Well, you can also do that for Newtonian gravity. ;) General covariance is not the only issue, as Kretschmann already pointed out to Einstein.

 

[Orodruin]

An isometry is a map between two manifolds that preserves the metric. This has little to do with coordinate transformations and more to do with maps from a manifold (in the case of GR, spacetime) to itself.

Of course, given a coordinate system, any map from a manifold to itself is going to give you a new coordinate system by assigning the coordinates of the old point to the new point, which is a source of some general confusion.

Isometries only make sense once you choose coordinates

I would say they only make sense once you fix a metric (coordinates be damned!). Given a (pseudo-)Riemannian manifold ##(M,g)##, an isometry of that manifold would be a function ##f: M \to M## such that ##f^* g = g##. This is fundamental for example in the definition of Killing fields as generating fields of isometries, i.e., ##K## is a Killing field if its corresponding flow ##\gamma_K(s,p)## satisfies
$$
\mathcal L_K g = \lim_{\epsilon\to 0}\left[\frac{1}{\epsilon}(\gamma_K^* g - g)\right] = 0,
$$
which is true if ##\gamma_K^*g = g##.

 

[haushofer]

An isometry is a map between two manifolds that preserves the metric. This has little to do with coordinate transformations and more to do with maps from a manifold (in the case of GR, spacetime) to itself.

Of course, given a coordinate system, any map from a manifold to itself is going to give you a new coordinate system by assigning the coordinates of the old point to the new point, which is a source of some general confusion.I would say they only make sense once you fix a metric (coordinates be damned!). Given a (pseudo-)Riemannian manifold ##(M,g)##, an isometry of that manifold would be a function ##f: M \to M## such that ##f^* g = g##. This is fundamental for example in the definition of Killing fields as generating fields of isometries, i.e., ##K## is a Killing field if its corresponding flow ##\gamma_K(s,p)## satisfies
$$
\mathcal L_K g = \lim_{\epsilon\to 0}\left[\frac{1}{\epsilon}(\gamma_K^* g - g)\right] = 0,
$$
which is true if ##\gamma_K^*g = g##.

Yes, in the "tensors are coordinate independent entities" I totally agree. In actually calculating isometries of, say, the Schwarzschild metric, I personally have to choose coordinates.

I've had the same confusion (if I guessed TS's confusion right, that is) about the difference between general coordinate transformations as symmetries of GR, and isometries as symmetries of the metric. Sometimes it is said that a solution to the Einstein equations "breaks" the gct's to a subgroup of the gct's, which we then call isometries. But for me personally the best strategy is not all that formal stuff; for me it works best to take a concrete example and do an actual calculation in a certain coordinate basis and see what happens (i.e. transform the formal stuff into a component example which I think I understand).

That's also how in the end I really started to understand the "hole argument": by applying it to the Schwarzschild solution. And in my experience, people often can sketch the formal math behind it, but once you present an explicit example in a certain coordinate basis, people tend to get confused. I did a small experiment with this hole argument, and when I saw how many people were genuinely confused, I decides it was insightful enough to put it in my PhD-thesis (which was about gravity described as a gauge theory).

 

[Orodruin]

for me it works best to take a concrete example and do an actual calculation in a certain coordinate basis and see what happens (i.e. transform the formal stuff into a component example which I think I understand).

I agree that you usually need coordinates to do something more specific. The only beef I have with this is when people start using coordinate dependent statements as "truths" and don't consider that their interpretation may be coordinate dependent - such as considering cosmological redshift to be fundamentally different from a Doppler shift.

 

[kent davidge]

Yes, the Einstein-Hilbert action is invariant under any diffeomorphism

I guess what you are talking about is going from one coordinate system to another one, staying at the same point on the manifold, but what I'm asking is if you can vary the coordinates (of course in a given coordinate system), i.e., if you can go from one point to another point on the manifold, like @Orodruin describes in his post. Tensors will generally change if you do that. And I'm asking if these type of transformations where the metric changes are allowed in General Relativity.

What do you mean by "allowed"?

Please, see my reply to @king vitamin

Isometries only make sense once you choose coordinates, i.e. once you chose a gauge. If in that coordinate system the form of the metric is preserved for specific transformations, you speak of isometries.

Yes, I'm aware of that.

 

[stevendaryl]

I guess what you are talking about is going from one coordinate system to another one, staying at the same point on the manifold, but what I'm asking is if you can vary the coordinates (of course in a given coordinate system), i.e., if you can go from one point to another point on the manifold, like @Orodruin describes in his post. Tensors will generally change if you do that. And I'm asking if these type of transformations where the metric changes are allowed in General Relativity.

Please, see my reply to @king vitamin

Yes, I'm aware of that.

I don't see where you say what you mean by "allowed". Your reply to king vitamin seems to assume that he already knows what "allowed in General Relativity" means. A transformation is a mathematical object. It's either useful in General Relativity or it's not, but General Relativity can't allow or disallow you to use whatever mathematical techniques you want.

 

[haushofer]

Please, see my reply to @king vitamin

Why wouldn't they be? I'm missing context, I guess.

 

[haushofer]

Maybe it helps if you read page 26 onwards about the hole argument of my thesis. Maybe I'm misjudging your confusion, but the invariance of GR under diffeomorphisms is the fact that coordinates only make sense if you have introduced a metric. If you take e.g. the Schwarzschild solution, a diffeomorphism can not physically change the location of the event horizon, only the way you label the event horizon. However, after this coordinate system is chosen (or, as Orodruin remarks, "the form of the metric is fixed"), there are still diffeomorphisms which keep the form of the metric unchanged. This diffeomorphisms can be interpreted as active coordinate transformations, hence as physically moving through spacetime while keeping the geometry invariant. These are the isometries of your spacetime. If this is not your confusion, you have to add a little context to your question. For my thesis,

https://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_R_Andringa.pdf

 

[kent davidge]

For my thesis,

https://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_R_Andringa.pdf

very interesting, I am reading it

 

[haushofer]

Thanks, glad somebody reads it :P ;)