Thanks! Does ##g_{\theta\theta}=r^2## apply to any transition?from the holonomic basis to the normalised covector basis? I have already checked,it doesn't.
Thank you! I tried and I found ##g_{\hat{\theta}\hat{\theta}}=1## apply to a transition from the normalised basis ##(\frac{\partial}{\partial r}, \frac{1}{r} \frac{\partial}{\partial \theta})## to the holonomic covertor basis ##(dr, d\theta)## ...But when can I use...
"A space-time is said to be stationary if and only if it admits a timelike Killing vector field"
"...given a a timelike Killing vector field,then there always exists a coordinate system which is adapted to the Killing vector field##X^a##,that is,in which##X^a=\delta^a_0##holds..."
How to...
"Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##"
I wonder why:
"the line element does not vary when##\theta## and##\phi## are...
I'm reading "Introducing Einstein's Relativity_A Deeper Understanding Ed 2"on page 271:
##X_{a}\partial_{b}X_{c}=\lambda f_{,a}\lambda_{,b}f_{,c}+\lambda^2f_{,a}f_{,cb}##
Taking the totally anti-symmetric part of this equation and noting that the first term on the right is symmetric in a and c...