Recent content by chartery

Durr... Got fixated on second term of ## \eta^{\rho\lambda} {\partial_{ \mu}}\epsilon h_{\nu\lambda} - \epsilon h^{\rho\lambda} {\partial_{ \mu}}\epsilon h_{\nu\lambda}## (just in case someone of similar density looking up). Many thanks.

Sorry for gap. I can see Vanhees understands, though it seems to me if ##\partial h## is order ##1/\epsilon## then ##\epsilon h^{\rho\lambda} {\partial_{ \mu}}\epsilon h_{\nu\lambda}## is only order ##\epsilon## but needs to be order ##\epsilon^2## to be ignored in OP equation?

Sorry @vanhees71 I can't get the multiple quote insert to work!Yes, my problem was being sure that ##h^{\rho\lambda}{\partial_{ \mu}}h_{\nu\lambda}## terms were order ##h^2## It makes sense that ##\epsilon h^{\rho\lambda} {\partial_{ \mu}}\epsilon h_{\nu\lambda}## would be order ##\epsilon^2##...

Sorry, you've lost me. Were you referring to ##g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}##? My problem was how to know that the partial derivative (i.e. variation) of a small item was necessarily also small. if ##\frac{1}{1+x}## is how I should think of ##\partial_{\mu}## here, I'm afraid I need...

Carroll linearising by perturbation ##g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}## has: (Notes 6.4, Book 7.4) ##\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\lambda}\left( {\partial_{ \mu}}g_{\nu\lambda}+{\partial_{ \nu}}g_{\lambda\mu}-{\partial_{...

@ergospherical, again many thanks. Self-taught, I realise I don't have enough knowledge of variational (and tensor) manipulations, but there does seem to be a lot of casually presupposed ability packed suddenly into that one bald equality in his notes !

@ergospherical, thanks very much. I wasn't doubting that logic, just wondering whether my application of index rules was shaky. It seemed to me that the combination of symmetry ## \nabla _{(\mu}V_{\nu)} ## with the dual contraction meant there was no need to rely on symmetry of the...

Queries on Carroll's derivation of matter action ## S_M ## under a diffeomorphism: (Book B.23+4 Notes 5.35+6) ##\frac{\delta S_{M}}{\delta g_{\mu\nu}} \delta g_{\mu\nu} = \frac{\delta S_{M}} {\delta g_{\mu\nu}} \left( 2 \nabla _{(\mu}V_{\nu)} \right) =\left( 2 \right) \frac{\delta...

@PeterDonis, thanks very much again for your help and patience

Contracted but no sum?These look exactly what I think I need to understand to relate tensors and operators, and 'pictorialise': ##R_{\sigma \nu} = \Sigma_X \langle \mathscr{R}(X, e_\sigma) e_\nu, X \rangle## ##\text{Ricci}(A, B) = \Sigma_\alpha \langle \mathscr{R}(e_\alpha, A) B, e_\alpha...

@PeterDonis Thanks again. Torsion-free: yes, my misapprehension, I got misled by Eq 3.112 in Carroll's book $$\left[ \bigtriangledown _{\mu},\bigtriangledown _{\nu} \right]V^{\rho}=R^{\rho}\text{ }_{\sigma\mu\nu}V^{\sigma}-T_{\mu\nu}\text{ }^{\lambda}\bigtriangledown _{\lambda}V^{\rho}$$I'm...

@PeterDonis, many thanks for reply and apologies for the lax terminology. References are unavailable, as the end result is my attempted distillate from a blizzard of websites. I realise ##R^{\rho}\text{ }_{\sigma\mu\nu}\text{ }X^{\mu}Y^{\nu}V^{\sigma}\partial_{\rho}## is a contraction, but...

I'm having trouble with notations and visualisations regarding Ricci curvature. For Riemann tensor there is variously: ##R^{\rho}\text{ }_{\sigma\mu\nu}\text{ }X^{\mu}Y^{\nu}V^{\sigma}\partial_{\rho}## ##[\nabla _{X},\nabla _{Y}]V## ##R(XY)V\mapsto Z## ##\left\langle R(XY)V,Z...

Thanks very much for the spoon-feeding. My knowledge is patchy, and being 'woolly' on the flurry of notations, it helps me avoid overlooking subtleties and errors applying generalisations, which of course are obvious in hindsight. (Sorry @PeroK, was deficiency in grounding rather than...

@strangerep, sorry, I missed you were separate, so forgot to thank you. Now I am working through your reply, I have a question. In the first equality just quoted, what happened to the ##~ \left( \frac{1}{ \sin\theta} \right)\partial_\theta~## term from the commutator? (Assuming the missing...