Can Special Relativity Predict Gravitational Effects?

[DrGreg]

I've been thinking some more about what sort of congruence of curves that can define a quotient manifold in a useful way, and I think I get it. I just don't have an elegant way of saying it yet. I'm thinking that if we consider two small neighborhoods of two points on the same curve in the congruence, they should "look approximately the same" in their comoving inertial frames. The approximation must become exact in the limit where the size of the region goes to zero, and this must hold for any two points on any single curve in the congruence. To "look approximately the same" here means that any other curve that passes through the two neighborhoods must be at approximately the same spatial coordinates in the two comoving inertial frames.

As I understand it, and if I understand your point, the Killing vector condition ensures that your 2 neighbourhoods don't just look approximately the same, they are exactly the same (well, isomorphic).
In the original spiral example, the spiral has a symmetry: if you rotate about and translate along the time axis, all the spirals look the same as before. I'm not an expert with Killing vectors, but I believe the same thing happens there too.

 

[Fredrik]

As I understand it, and if I understand your point, the Killing vector condition ensures that your 2 neighbourhoods don't just look approximately the same, they are exactly the same (well, isomorphic).

I thought that might be the case. I'll have to think about this some more, but not today.

 

[DrGreg]

In another thread, a member Anamitra has defined concepts of "physical length", "physical time", "physical speed", all relative to a frame, which I have attempted to summarise by the following post in that thread:

I haven't come across Anamitra's technique before, but I think this is what is happening.

Given a metric of the form

ds^2 = g_{00}\,dt^2 + g_{ij}\,dx^i\,dx^j​

(where i, j take values 1,2,3 only) define two new metrics:

dT^2 = -g_{00}\,dt^2
dL^2 = g_{ij}\,dx^i\,dx^j​

T is being called "physical time" and L is being called "physical length". Both are evaluated by integrating along the same spacetime worldline that you would integrate ds along. And both are dependent on your choice of coordinate system.

It should be clear that if you were to evaluate T along a worldline of constant x¹,x²,x³ it would equal proper time. If you were to evaluate L along a curve of constant t it would equal proper length. But for an arbitrary worldline you evaluate both along the same worldline.

If you calculate |dL/dT| along a null worldline you get 1.

That seems to be closely related to what we're discussing here. In our case we have a more general metric of the form

ds^2 = g_{00}\,(dt - w_i\,dx^i)^2 + k_{ij}\,dx^i\,dx^j​

Rindler shows any stationary metric can be written in this form. So we can define metrics

dT^2 = -g_{00}\,(dt - w_i\,dx^i)^2
dL^2 = k_{ij}\,dx^i\,dx^j​

to define "physical time" and "physical length". We are effectively decomposing a 4-vector ds as sum of dT parallel to the Killing vector field and dL orthogonal to dT.