I Proof of Lorentz Gauge Existence: Help Understanding Schutz 8.3

epovo
Messages
114
Reaction score
21
TL;DR Summary
In the derivation of the proof there is a step that I cannot make sense of
In Schutz 8.3, while proving that a Lorentz gauge exists, it is stated that
$$\bar h^{(new)}_{\mu\nu} = \bar h^{(old)}_{\mu\nu} - \xi_{\mu,\nu} - \xi_{\nu,\mu} + \eta_{\mu\nu}\xi^\alpha_{,\alpha}$$

where ##\bar h## is the trace reverse and ##\xi^\alpha## are the gauge functions. Then it follows with:
"Then the divergence is"
$$\bar h^{(new)\mu\nu}_{\,\,\,\,\,\,\,,\nu} = \bar h^{(old)\mu\nu}_{\,\,\,\,\,\,\,\,,\nu} - \xi^{\mu,\nu}_{\,\,\,,\nu}$$
I can't see why the divergence is that! I've tried and tried but I can't see it. Any help?
 
Physics news on Phys.org
epovo said:
I can't see why the divergence is that! I've tried and tried but I can't see it. Any help?
$$-\xi^{\nu,\mu}_{\,\,\,,\nu} + \eta^{\mu\nu} \xi^{\alpha}_{\,\,\,,\alpha \nu} = -\xi^{\nu,\mu}_{\,\,\,,\nu} + \xi^{\alpha,\mu}_{\,\,\,,\alpha} = 0,$$
since ##\alpha## and ##\nu## are both dummy summation indices.
 
  • Like
Likes epovo and vanhees71
True! Thank you
 
OK, so this has bugged me for a while about the equivalence principle and the black hole information paradox. If black holes "evaporate" via Hawking radiation, then they cannot exist forever. So, from my external perspective, watching the person fall in, they slow down, freeze, and redshift to "nothing," but never cross the event horizon. Does the equivalence principle say my perspective is valid? If it does, is it possible that that person really never crossed the event horizon? The...
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
Back
Top