When you are finding the length of a space-time interval between two events in special relativity, you can find it is either positive or negative; (c dt)^2 - dx^2 - dy^2 - dz^2
If that turns out to be positive, you know that its a time-like interval, but if it turns out to be negative, you know it's a space-like interval.
Let me go ahead and post what I was going to post before I realized I had that question, keeping in mind that I probably made a BLUNDER in confusing space-like and time-like intervals:
I have been wondering a great deal lately about the question of physical interpretation of line elements ds and or dτ vs.
dr, in the Schwarzschild metric. I notice, you are mostly focusing on the length of ds. However, I was able to make a
bit of progress thinking about the length of dτ, i.e. the timelike component between two events. Perhaps some insight
into the nature of the time element could help resolve some confusion about the line element.
Consider the Schwarzschild metric:
c^2 {d \tau}^{2} = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1-\frac{r_s}{r}\right)^{-1} dr^2 - r^2 \left(d<br />
<br />
\theta^2 + \sin^2\theta \, d\varphi^2\right)
where
r_s= \frac{2 G M}{c^2}
There is a fairly straightforward but lengthy process for calculating the coefficient of g_{00} = \left ( \frac<br />
<br />
{\partial \tau}{\partial t} \right )^2 = \left ( 1 - {\frac{2 G M}{c^2 r}} \right ) which I explored in some detail
https://www.physicsforums.com/showpost.php?p=3415913&postcount=18".
we were comparing frequencies, which is essentially the same as comparing Δτ to Δt. Or more specifically, the length of
the four-vector from (0,r,θ,Φ) to (Δt,r,θ,Φ). Δt is the time as measured from the external coordinate system. But from
the point-of-view of an internal coordinate system, that four-vector appears to go from (0,r,θ,Φ) to (Δτ,r,θ,Φ). In
other words, the clock that is located at that point is going to go slower, but that is the only difference.
Since then, I've been trying to figure out how to get the second coefficient,g_{11}=\left(1-\frac{2 G M}{r c^2}\right)^{-1}
, which has been difficult, because I have not even really been able to establish a common-sense interpretation to what that coefficient even means. However, this morning, I think I was able to grasp something.
The key is that instead of considering a time-like interval in (t,r,θ,Φ), we will be considering a space-like interval in the r-component alone. Specifically, where dt=dθ=dΦ, but r is allowed to change. Hence, two events which appear to happen simultaneously, but at different locations outside the gravitational field, [STRIKE]but will appear to happen at different times from within the gravitational field.
This is a significant qualitative (big) difference from the Rindler coordinates, so [/STRIKE]I think it would be enlightening to try to make a distinctive comparison of the Schwarzschild metric to the Rindler metric at this point.[STRIKE]
Why do I say this is a big difference?
Because [/STRIKE] in the Rindler coordinates, events that happen at t=0 also happen at τ=0.
If events seem to happen simultaneously in the "rocket" frame, then they also seem to happen simultaneously in the momentarily comoving rest-frame. Here is a diagram:
In this diagram, the future events, C and D, are on a line of constant τ. However, using a line of constant t, it appears that event C will happen before event D. As time passes, the line of constant τ will rotate (technically a Lorentz skew Transform, with eigenvectors of slope ±c) down and coincide with the line of constant t. At the point in time where event C and D actually HAPPEN, they are simultaneous.
Similarly, though events A and B appear to HAVE happened at different times, at the time that they actually HAPPENED, (when they crossed the t=0 line) they were simultaneous.
In the Rindler coordinates, events that appear to be simultaneous in the rocket also appear to be simultaneous in the momentarily comoving reference frame. But this is the point that surprised me: In the schwarzschild coordinates, events which appear to be simultaneous from the external reference frame DO NOT appear to be simultaneous locally.
[STRIKE]That means that the t=0 line and the τ=0 line will only intersect at one point, and do not represent the same set of events. I'm not sure what the implications of this are. [/STRIKE]
But now that this has occurred to me, I wonder whether anyone has a graph of Schwarzschild coordinates just showing curves of constant τ and curves of constant r' in the Schwarzschild metric, analogous to the image I posted above for the Rindler coordinates, but perhaps on a space-time scale near a body of high gravity?
