A basic question in general relativity

[David_cronin]

Dear all,

First, please forgive my english, I am French.

In most textbooks on general relativity (GR) the non trivial geometry is introduced in this way : the equivalence principle says that we can forget gravity as a force and consider an accelerated frame. As this frame is accelerated it is not inertial and therefore the interval (ds^2=dt^2-dx^2) is not conserved anymore (special relativity (SR) teaches us the the interval in conserved but only when going from an inertial frame to another). As the interval encodes the geometry, we see why a non-trivial (non-euclidean) geometry appears.
Sounds good to me.
BUT : in many course on GR, I see that people use the fact that the interval is in fact still conserved when changing frame in GR !
So, I am a bit confused...

Thanks for your help !

 

[haushofer]

Funny question, and I had to think a little about this :P But I think the confusion arises from the following.

In special relativity the symmetry group is the Poincare group ISO(d-1,1): translations and rotations in spacetime. This is a global symmetrygroup; the transformations don't depend on the coordinates, just on the velocity. Indeed, these transformations are between inertial observers.

In general relativity the symmetry group is GL(4): the group of general coordinate transformations (General Linear transformations in d=4). This is a local symmetrygroup; the transformations can depend on the coordinates.

So in special relativity, ds^2 is a scalar under Poincare transformations, so if you perform transformations which are NOT Poincare (for instance, you go to an accelerating observer) the interval ds^2 will change.

Does this answer your question?

 

[bcrowell]

I would suggest not using the word "conserved" for this. When something is conserved, we mean that it stays the same from one time to the next. The spacetime interval ds^2 isn't conserved, it's frame-invariant.

GR is locally the same as SR. Locally, you can apply a Lorentz boost, and ds^2 stays the same.

But notice that ds^2 is the square of an infinitesimally small number. What's different in GR is that GR isn't globally the same. In SR it makes sense to ask what is the interval between two widely separated events. In GR, this would only make sense if you specified a path from one event to the other.

 

[halcon]

You have to introduce the proper metric tensor $$g_{\mu\nu}$$ that conserves $$ds^2$$