Is there an invariant tensor for metric under all 10 motions?

[jfy4]

Hi,

I was wondering, and I hope this isn't a ridiculous question, for the set of motions: 4 translations, 3 rotations, and 3 boosts; is there an invariant tensor for any metric under all 10 of these motions.

That is, preforming these 10 motions, is there a tensor which remains unchanged regardless of the metric/space-time one preforms these in?

 

[Mentz114]

Tensors transform covariantly and contractions are invariant. For a vector,

$$v^{\mu'}=P^{\mu'}_{\mu}v^\mu,\ \ v_{\mu'}=P_{\mu'}^{\mu}v_\mu,$$

$$v^{\mu'}v_{\mu'}=P^{\mu'}_{\mu}v^\mu P_{\mu'}^{\mu}v_\mu = (P^{\mu'}_{\mu}P_{\mu'}^{\mu})v^\mu v_\mu=v^\mu v_\mu$$

which can be done because $P^{\mu'}_{\mu}$ is a transformation matrix, not a tensor. This can be easily verified for any tensor contracted with any other tensor.

I would hazard that only scalars are always invariant.

 

[jfy4]

I wasn't clear enough, I know because this is kind of a weird question.

If I preform the 10 motions mentioned above, is there a geometric, or other tensor which is the same before and after preforming all 10 of those motions.

Example: If preform motions along Killing vector lines, my metric remains unchanged.

But the metric is not always invariant for translations and rotations for an arbitrary metric. I would like to know of a tensor, where If I preform those 10 motions, said tensor remains unchanged, whatever that tensor maybe.

 

[Mentz114]

I wasn't clear enough, I know because this is kind of a weird question.

If I preform the 10 motions mentioned above, is there a geometric, or other tensor which is the same before and after preforming all 10 of those motions.

Example: If perform motions along Killing vector lines, my metric remains unchanged.

But the metric is not always invariant for translations and rotations for an arbitrary metric. I would like to know of a tensor, where If I preform those 10 motions, said tensor remains unchanged, whatever that tensor maybe.

When you say 'unchanged' do you mean every component of the tensor is the same as before the 'motion' ? It seems to me that 'performing' the 'motion' is the same as a coordinate transformation, so your question is about tensors under coordinate transformations.

Killing vectors represent directions, so translations in those directions have an associated conserved scalar.

As I've said, components of tensors do not remain unchanged under non-trivial coordinate transformations.

If this is not true, I hope someone will point it out.

 

[LAHLH]

I don't believe there is any such tensor. For any maximally symmetric spacetime such as Minkowski the metric tensor itself is obviously such a tensor, that is indeed the definition of these ten 'motions', they are isometries. But for an arbitrary spacetime (arbitrary metric) I can't think of any tensor that would be invariant over a the orbits of all 10 KV.

 

[jfy4]

...I can't think of any tensor that would be invariant over a the orbits of all 10 KV.

Thank you, let me point out though that the motions need not preserve the metric tensor (killing vectors), just any tensor (if that wasn't clear).

 

[LAHLH]

Thank you, let me point out though that the motions need not preserve the metric tensor (killing vectors), just any tensor (if that wasn't clear).

Yeah, I'm aware, I was just stating that the metric tensor is such a tensor for max sym spaces, and the only one I can think of. I could be wrong however. I probably should have said motions instead of KV too.

How about the zero tensor? haha.

 

[jfy4]

Yeah, I'm aware, I was just stating that the metric tensor is such a tensor for max sym spaces, and the only one I can think of. I could be wrong however. I probably should have said motions instead of KV too.

How about the zero tensor? haha.

When you say 'unchanged' do you mean every component of the tensor is the same as before the 'motion' ? It seems to me that 'performing' the 'motion' is the same as a coordinate transformation, so your question is about tensors under coordinate transformations.

Killing vectors represent directions, so translations in those directions have an associated conserved scalar.

As I've said, components of tensors do not remain unchanged under non-trivial coordinate transformations.

If this is not true, I hope someone will point it out.

What about the Einstein curvature tensor $$G_{ab}$$ / the Ricci curvature $$R_{ab}$$ ? If I were to preform those 10 motions would I still get the same tensors?

 

[LAHLH]

No, not for a generic spacetime.

In flat spacetime the only tensors that remain invariant under Poincare I believe are the metric, Kronecker delta and Levi Cevita, and that is in flat space...If you leave the spacetime as absolutely arbitrary then I doubt there is anything.

I think Mentz is right, that coordinate change is probably the way to think about these things. The question would then be are there any *isotropic* tensors under the Poincare group in an arbitrary spacetime? and I think the answer is no, simply because as arbitrary spacetime will not have Poincare symmetry like Minkowski.

 

[jfy4]

Thank you,

That's sad...

Then, if I were to preform these motions/coordinate transformations, how would the metric change in an arbitrary case? For all 10 motions would it differ simply by a diffeomorphism? Would it differ for each case, 10 different diffeomorphisms?

EDIT: Of course coordinate changes are diffeomorphisms (thats me talking to me), What I'm wondering is, can these 10 motions for an arbitrary metric form a vector field that preserves something? So I'm trying to see if there is a connection between these 10 motions and some geometrical tensor.

 

[LAHLH]

I can only answer: not that I'm aware of. I should add the caveat that I've never seen reference to such things in any GR texts I'm familiar with, I'm just thinking about it off the cuff, so I guess it would be nice if someone else could answer more definitively for you, but I suspect the answer is no.

I do know however that a generic curved spacetime will not posses any symmetries at all (it is for this reason actually that things like particles are ill defined in curved space-times leading to the Unruh effect and so forth), and that I can say for definite. You only get a conserved quantity associated with KV for a spacetime that admits them, and of course in general a time translation say won't be the orbit of a KV, won't be a symmetry, won't lead to something conserved.

As for how the metric changes under an arbitrary diffeo the appendix A in Carroll is quite good.

As I say, take what I say with a pinch of salt, I'd be happy to be corrected by someone wiser on such matters...

 

[jfy4]

I realize this isn't a tensor, but what about the Christoffell symbols? would they be invariant under the coordinate transformations mentioned?EDIT: Nevermind.

EDIT2: This is a better way to ask. Would these coordinate transformations leave the connection of the tangent bundle invariant?