How do you know this for sure? Maybe there are regions that are Euclidean and small and haven't been discovered. Certainly Euclidean spacetime is used as a tool for path integration in QFT. The point of the question is to see why is the necessity for timelike Killing vectors in either the Komar...
So what happens if the norm is positive? What if there were no singularities but the space were of Euclidean signature (with no time dependence in metric, in some coordinate system)? I don't understand why the norm has to be negative. It's not about the ability to calculate the norm.
It means...
OK, though I still have to take the timelike (in the sense of negative squared norm, not in the sense of in the direction of time in some coordinate system) part on faith. Maybe I need to get Wald. Which hypersurface are you referring to?
No disagreement on that. We were quibbling about the...
I can't tell you because PF is not for original research. So far we have only been talking about known physics.... If you want to know why, maybe we should take this offline?
Right, we agree. So I am interested in the time component of this 4-vector, not its norm. It's different than the Komar mass (or equivalently the mass defined by 23.19) because of the \Sqrt[-det[g]] instead of just the Sqrt[gtt] factor, though it looks the same for a S. metric as eqn 23.19, but...
vanhees71, thanks for that reply. I admit I don't understand it, so I hope you will be patient in continuing to try to explain it to me.
1. Why must the (3-surface) integration of T^{ij} d\Sigma_j be over all of space, for fixed time, where \Sigma is the 3-surface? Why can't it be over a region...
Peter Donnis: After reading the article you suggested, I am even more confused than before, because it claims that equation 4.19 (or (1) in box 1) of MTW for the mass energy inside a radius r IS the same as the Komar mass, so my original hunch was right? But this can't be, because m(r), at least...
OK, sorry I reposted that. Something is not working in my browser. Yes I agree. The T^00 integral is the time component of a 4 vector, not invariant. See my other reply
Having trouble replying, it quotes the wrong message, so I'll do it by hand
"You do if you want the mass you calculate to be an invariant, which you should."
You're right. I'm interested in the mass energy inside a volume, which is more than just the rest mass/energy. So NOT invariant, but...
As far as Komar mass, I'm not convinced it's something different from the T^{00} (not T_00) integral, except it does have hope for a non-zero finite value for a black hole where T00 can go to 0 at r=0, whereas Sqrt[-g_{00}] goes to infinity, so their product could be m.
We don't need an invariant quantity, but the time component of a 4-vector, which that integral would be if we have the correct differential 1-form integrated over a 3-volume (with an appropriate Sqrt[-det[g]] in any coordinate system), and in a frame where there is no kinetic energy (all the...
Why doesn't this work if the field is strong? Or does it work as long as there are no singularities?
Mentor's Note: Original thread title was, "Calculating rest mass by integrating T_{00} over a 3 volume for static metric"
Ultimately, the only thing that matters for symmetry is whether the Lagrangian (or more accurately the action) remains invariant under some transformation. Whether the invariance comes from an active or passive transformation doesn't matter. The Noether currents are the same.