Recent content by GR191511

Thank you! ##df=\frac {\partial f} {\partial r}dr+\frac{1}{r}\frac {\partial f} {\partial \theta}rd\theta## and ##(dr,rd\theta ) \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix} dr \\ rd\theta \\ \end{pmatrix}\Rightarrow...

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...

##df=\frac {\partial f}{\partial r} dr+\frac {\partial f}{\partial \theta}d\theta\quad \nabla f=\frac{\partial f}{\partial r}\vec{e_r} +\frac{1}{r}\frac{\partial f}{\partial \theta }\vec{e_\theta }## On the other hand ## g_{rr}=1\:g_{r\theta}=0\:g_{\theta r}=0\;g_{\theta\theta}=r^2\;##So...

"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...

Why"all the the other terms independent of ##\phi## and ##\theta##"indicates spherical symmetry?

But the ##d\theta^2+\sin^{2}\theta d\phi^2## vary when ##\theta## are varied (because of ##sin^{2}\theta##)

What about ##dx^2+dy^2+dz^2##?Does the line element vary when x and y and z are varied?How?

"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...

o_O...OK!##X_{[a}\partial_bX_{c]}=\frac 1 6(X_{a}\partial_bX_{c}+X_{c}\partial_aX_{b}+X_{b}\partial_cX_{a}-X_{a}\partial_cX_{b}-X_{b}\partial_aX_{c}-X_{c}\partial_bX_{a})##?

any of the three indices?Doesn't the notation ##X_{[a}\partial_b X_{c]}##means that "a" exchanges with "c" only?

Is the" totaly anti-symmetric part"of that equation##X_{[a}\partial_{b}X_{c]}+X_{a[}\partial_{b}X_{c]}+X_{[a}\partial_{b]}X_{c}##?

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...

Thanks！

"the christoffel symbols are all zero at the origin of a local inertial frame" Why must it be at the origin? If it is not?Thanks!