General Coordinate transfos vs Lorentz transfos in GR

[kdv]

GR is invariant under general coordinate transformations. [it] If [/it] I understand correctly, this is basically devoid of any physical content. It just means that relabelling points does not change anything physical. So it's devoid fo physical content, right?

On the other hand, in special relativity, the equations are covariant under Lorentz transformations. Those have physical content. without imposing this symmetry the speed of light would not be the same in all frames.


My simple-minded question is: how is the Lorentz invariance ensured in GR? I used to think that it was somehow implemented as part of the general coordinate transformations but it seems now that I was completely in the left field.
we should be able to recover special relativity from GR in the limit of a flat spacetime so the Lorentz symmetry must be present at some level in GR. But how is it implemented?

 

[George Jones]

The collection of events in spacetime make up a differentiable manifold. At each event there is a tangent space that is isomorphic to Minkowski vector space. So, (local) Lorentz transformations act on stuff in a tangent space (think tangent plane to the surface of a sphere) while coordinate transformations are at the level of the manifold.

Sorry, I'm on my way out.

I wiil get back to the other thread, including more on this stuff, on spinors, and on the $\gamma$'s, but this might not happen for a few days.

 

[kdv]

The collection of events in spacetime make up a differentiable manifold. At each event there is a tangent space that is isomorphic to Minkowski vector space. So, (local) Lorentz transformations act on stuff in a tangent space (think tangent plane to the surface of a sphere) while coordinate transformations are at the level of the manifold.

Sorry, I'm on my way out.

I wiil get back to the other thread, including more on this stuff, on spinors, and on the $\gamma$'s, but this might not happen for a few days.

Ah! That is very helpful. Thank you. Of course it generates more questions
the first one being: what does a Lorentz transformation look like then, in GR?

Have nice evening and thanks a lot for your help!

 

[kdv]

The collection of events in spacetime make up a differentiable manifold. At each event there is a tangent space that is isomorphic to Minkowski vector space. So, (local) Lorentz transformations act on stuff in a tangent space (think tangent plane to the surface of a sphere) while coordinate transformations are at the level of the manifold.

Sorry, I'm on my way out.

I wiil get back to the other thread, including more on this stuff, on spinors, and on the $\gamma$'s, but this might not happen for a few days.

I have read a bit more and I see the basic idea (that one works on the tangent space to implement the Lorentz transformations) but I am still very perplex.

GR is invariant under arbitrary reparametrizations of the coordinates. If I understand correctly, this has no physical content in itself, is that right? If this is right, then my question is the foolowing: how does one show that GR is Lorentz invariant? Where is the Lorentz invariance included in GR?

( BY "GR" here, I mean Einstein's equation with Einstein's tensor (and the Ricci tensor and Ricci scalar and the Christoffel symbols and so on) defined the usual way.)

GR is formulated in an obvious reparametrization of the coordinates invariant way (using tensors ensures that) but is imposing Lorentz invariance then an extra requirement? I have never seen this mentioned. It always sounds as if Lorentz invariance is automatically built in in GR (when one reads GR books) which is why I thought the reparametrization invariance somehow included Lorentz invariance in the theory.

 

[Phrak]

kdv-
I understand the point of your question. I hope to see some answers as well.

Does 'general linear transform' mean any multilinear transform? But the Lorentz metric must underly both, so that the metric in local, othonormal coordinates is the Lorentz metric.

It seem, for instance, that a rotation in O(4) is a general linear transform, but that could invert the time coordinate.


How does the metric itself remap under a general linear transform?

So far, I've been happily developing field equations in manifestly covariant coordinates, in Riemann normal coordinates, satisfied as a matter of faith that they are equally valid in rotating or accelerating coordinate systems. But sooner of later I'll need know what I'm really doing.

 

[timur]

The fact that GR is invariant under arbitrary reparametrizations of the coordinates is not void of physical content, it has very much physical content. One that strikes me is that points on the mathematical manifold modeling the phyisical spacetime does not have physical meaning. Points can be physically distinguished only via the geometry (or equivalently via the matter content), at least when everything is smooth. So in particular, only the relative position of things matter, there is no 'absolute location', no 'absolute stationary point'.

In Relativity you cannot take O(4), but instead take the Lorentz transformation. Lorentz transformations are the natural replacement of rotations in pseudo-Euclidean (or Lorentzian) space.

 

[kdv]

The fact that GR is invariant under arbitrary reparametrizations of the coordinates is not void of physical content, it has very much physical content. One that strikes me is that points on the mathematical manifold modeling the phyisical spacetime does not have physical meaning. Points can be physically distinguished only via the geometry (or equivalently via the matter content), at least when everything is smooth. So in particular, only the relative position of things matter, there is no 'absolute location', no 'absolute stationary point'.

I am not sure I am understanding. What you seem to be describing is diffeomorphism invariance, not invariance under general coordinate transformation. In a GCT, the labels of the points are simply renamed (with the metric being changed accordingly) and the points of the manifold are left untouched.

At least that's the way I understand it. Let's say we have a 2D manifold 9so that we can visualize what's happening). There is the manifold itself (let's say a spehre) and then there is the coordinate system fixed on the sphere. A GCT is simply a distorsion of the coordinate grid we are placing on the sphere, the manifold itself is left completely unaltered. So the coordinates of the points changes and the form of the metric changes but the points are unaffected. In particular, the distance between any two points i sthe same as before.

Correct me if I am wrong.

 

[Phrak]

In Relativity you cannot take O(4), but instead take the Lorentz transformation. Lorentz transformations are the natural replacement of rotations in pseudo-Euclidean (or Lorentzian) space.

That's rather my point. Which means that either transforms in O(4) on a Riemann manifold are not general linear transforms, or that all general linear transforms are not physically meaningful.

 

[Ken G]

I am not an expert in GR, so I have the same kinds of questions, but I'll throw out what my answer would be in hopes that real experts will be able to correct any misconceptions. It seems to me classical mechanics proceeds in essentially (at least) three separate stages. The most elementary stage we might call "particle dynamics", which is essentially Newton's laws, which describe how the fundamental "atoms" (classically) of a system behave (i.e., accelerate) from the point of view of observers instantaneously and inertially moving with these fundamental dynamical elements. As different elements have different motions, it is thus necessary to "cobble together" the information obtained by all these hypothetical comoving observers into the "local dynamics" of a confined closed system, and the instructions for how to properly combine the information from all these comoving hypothetical observers into a view of one observer generates a "tangent space" where the local dynamics plays out from the perspective of anyone among this set of local observers, and the instructions are mediated soley by relative velocity (one can imagine noninertial observers too but I don't think that adds anything locally). However, tidal gravity effects warp the relative actions of neighboring tangent spaces, such that integration of nonlocal information into a more global whole requires additional instructions about how to treat gravity, and that's what general relativity supplies. So in summary, Newton's mechanics gives us the proper accelerations, special relativity allows us to track the local interrelations of these accelerations, and general relativity allows us to piece the local information into a global description including gravity.

Thus it doesn't matter how we coordinatize the tangent spaces, because GR will take that into account in the connecting instructions. But if we choose coordinates of each tangent space in a way that transforms under Lorentz transformations to different observers at the origin of the tangent space, then we can at least handle the "cobbling together" of each tangent space automatically by following simple covariant instructions for how to piece together Newton's laws, and will only need GR to handle tidal effects (second order effects) as we go between neighboring spaces.

So in other words, there are two different things going on here, the coordinatization, which is purely arbitrary and just involves giving labels to everything, and the physics, which takes into account the labels to determine how the information compiled by individual observers can be put together into a whole. That compilation proceeds in three stages, the particle dynamics, which are just Newton's laws for comoving inertial observers, the local dynamics, which require special relativity (which can be made transparent by choosing a covariant labeling system), and the global dynamics, which require general relativity because no globally covariant coordinatization exists to make the global dynamics transparent.

Thus my answer to "is there anything physical in a coordinate transformation" depends on whether one thinks of coordinates as being ways to achieve transparent dynamics, or if one is willing to do considerable work finding the dynamics after the coordinates have been chosen. It seems to me the point of a sensible coordinatization is to minimize that remaining work, so if that is the philosophy used, then there is something physical, not in the coordinates themselves, but in the way they get chosen. That holds both in the tangent space (Lorentz covariance) and in the global connections (which might, for example, respect the cosmological principle).