- #1
unscientific
- 1,734
- 13
Homework Statement
(a)Find Christoffel symbols
(b) Show the particles are at rest, hence ##t= \tau##. Find the Ricci tensors
(c) Find zeroth component of Einstein Tensor
Homework Equations
The Attempt at a Solution
Part (a)[/B]
Let lagrangian be:
[tex] -c^2 \left( \frac{dt}{d\tau}\right)^2 + a^2 \left[ \left(\frac{dx}{d\tau}\right)^2 + \left(\frac{dy}{d\tau}\right)^2 + \left(\frac{dz}{d\tau}\right)^2 \right] [/tex]
Euler-Lagrangian is given by:
[tex] \frac{d}{d\tau} \left( \frac{\partial L}{\partial (\frac{\partial x^{\gamma}}{\partial \tau})} \right) = \frac{\partial L}{\partial x^{\gamma}} [/tex]
Applying Euler-Lagrangian to temporal part:
[tex]\frac{d}{d\tau} \left( -2c^2 \frac{dt}{d\tau} \right) = 2 a \dot a \left[ \left(\frac{dx}{d\tau}\right)^2 + \left(\frac{dy}{d\tau}\right)^2 + \left(\frac{dz}{d\tau}\right)^2 \right] [/tex]
[tex] \frac{d^2 (ct)}{d\tau^2} + \frac{a \dot a}{c} \left[ \left(\frac{dx}{d\tau}\right)^2 + \left(\frac{dy}{d\tau}\right)^2 + \left(\frac{dz}{d\tau}\right)^2 \right] = 0 [/tex]
This implies that ## \Gamma_{11}^0 = \Gamma_{22}^0 = \Gamma_{33}^0 = \frac{a \dot a}{c} ##.
Applying Euler-Lagrangian to spatial part:
[tex]\frac{d}{d\tau} \left( 2a^2 \frac{dx^{\gamma}}{d\tau} \right) = 0 [/tex]
[tex]2 a \dot a \frac{dt}{d\tau} \frac{dx^{\gamma}}{d\tau} + a^2 \frac{d^2 x^{\gamma}}{d\tau^2} = 0 [/tex]
[tex] \frac{d^2 x^{\gamma}}{d\tau^2} + \frac{2 \dot a}{a c} \frac{dx^{\gamma}}{d\tau} \frac{d (ct) }{d\tau} = 0 [/tex]
This implies that ## \Gamma_{10}^1 = \Gamma_{20}^2 = \Gamma_{30}^3 = \frac{2 \dot a}{ac}##. Am I missing a factor of ##\frac{1}{2}## somewhere or is the question wrong?
Part (b)
I can't see why the particle is at rest in this frame. From the transport equation, the change in vector ##V^{\mu}## when transported through length ##\delta x^{\beta}## is:
[tex] \delta V^{\mu} = -\Gamma_{\alpha \beta}^{\mu} V^{\alpha} \delta x^{\beta}[/tex]
Letting ##\delta x^{\beta}## represent the time component and ##\alpha## represent the spatial component:
[tex] \frac{\delta V^{\mu}}{\delta x^{\beta}} = - \Gamma_{\alpha \beta}^{\mu} V^{\alpha} [/tex]
The LHS represents 'velocity' while the right hand side represents ##\int \hat r dt##. The Christoffel symbols are clearly non-zero: ## \Gamma_{10}^1 = \Gamma_{20}^2 = \Gamma_{30}^3 = \frac{2 \dot a}{ac}##.
Last edited: