Recent content by Jonsson

Distance over the speed: ##\Delta \tau = \frac{2 \pi r}{c} = 6 \pi GM ## I exected it to work if you replace ##\Delta t \to \Delta \tau##, but not this way around.

Hello there, We know that for lightlike paths, there are circular geodesics at ##r = 3GM## in Schwarzschild geometry. Suppose an observer flashes his flashlight at ##r=3GM## and after some time the light reappears from the other side of the black hole. The time he measures is ##6 \pi GM##. I...

Great, I don't know. I am trying to understand how this all works. Would you like to explain some more?

Hello there, Recently I encountered a type of covariant derivative problem that I never before encountered: $$ \nabla_\mu (k^\sigma \partial_\sigma l_\nu) $$ My goal: to evaluate this term According to Carroll, the covariant derivative statisfies ##\nabla_\mu ({T^\lambda}_{\lambda \rho}) =...

No, then I wouldn't have needed to ask. Thank you for your help :)

Carroll -- Spacetime and geometry. Page 333

This is in the context of deriving the Friedmann equations, so surely we are using comoving coordinates?

Inertial coordinates are free fall coordinates, the coordinates of a freely falling observer? This is in the context of deriblant the Friedmann equations, so surely means comoving coordinates?

$$ T^{\mu \nu} = g^{\mu \rho}g^{\nu \sigma} T_{\rho \sigma} = g^{\mu \rho}g^{\nu \sigma}\left[(\rho + p)U_\rho U_\sigma + pg_{\rho \sigma} \right] = (\rho + p)U^\mu U^\nu + p g^{\mu \nu} = \begin{bmatrix} \rho + p &0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{bmatrix}^{\mu \nu} + \begin{bmatrix} -p...

Hello there, I am learning GR and in the cosmology chapter, we are using the metric $$ ds^2 = - dt^2 + a^2(t) \left[ \frac{dr^2}{1 - \kappa r^2} + r^2 d \Omega \right]. $$ Suppose now that ##U^\mu = (1,0,0,0)## and the energy momentum tensor is $$ T_{\mu \nu} = (\rho + p)U_\mu U_\nu + p g_{\mu...

Thanks. I found the deflection angle is ##\frac{4GM}{b}##. How do you say I transform this quantity back?

That is a good thing, because we are not interested in all null paths. Yes, I agree these solve (*), but why is it important that you've found exactly one solution that don't interest us? Can you propose some strategy that will work?

I added a drawing to my previous post. There are three sets of coordinates. (1) ##(t,x,y,z)## as I have drawn in the picture above (2) ##(t,x^1,x^2,x^3)## where the mass is stationary (3) ## (t,\zeta^1,\zeta^2,\zeta^3) ## where the mass is moving. It may well be that ##(t,x,y,z) =...

Ok, I have been working away at this. I found the metric in the non moving frame. But how do I go from there. Suppose I write the up the geodesic equation or equations for conservation of momentum along a geodesic $$ 0 = g_{\mu \nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}, $$ We don't...

Alright, I take it then that $$ \frac{\partial x^\mu}{\partial x^{\alpha'}}\frac{\partial x^\nu}{\partial x^{\beta'}}g_{\mu \nu} = \Lambda^\mu_{\alpha'}\Lambda^\nu_{\beta'}(\eta_{\mu\nu}+ h_{\mu\nu}) \tag{1} $$ follows from using the Lorentz gauge then? Are we able to prove ##(1)##?