Four-velocity in a static spacetime

by scottie_000
Tags: four vector, general relativity, geodesic, rest, spacetime
scottie_000 is offline
Feb9-10, 04:34 PM
P: 49
1. The problem statement, all variables and given/known data
I am given a static spacetime line element which has the property that the metric is time independent. I am asked to calculate some of the Christoffel symbols, which I have done.

The question asks to show that for an observer at rest, the four-velocity is given by [tex] V^a = (V^0,\textbf{0}) [/tex], where [tex] V^0 = V^0(\textbf{x}) [/tex] is a function of only spatial position

2. Relevant equations
Line element [tex] ds^2 = -e^{2\phi} dt^2 + h_{ij}dx^i dx^j [/tex]
Relevant Christoffel symbols (as calculated)
[tex] \Gamma^0_{00} = 0 [/tex]
[tex] \Gamma^0_{0i} = \frac{\partial \phi}{\partial x^i} [/tex]
[tex] \Gamma^0_{ij} = 0 [/tex]
Four-velocity [tex] V^a = \frac{dx^a}{d\tau} [/tex]
Geodesic equation:
[tex] \dot{V}^0 + 2\frac{\partial \phi}{\partial x^i}V^0 V^i = 0 [/tex]

3. The attempt at a solution
I am willing to believe that the spatial part of [tex] V^a [/tex] is 0, since I am told the observer is at rest. Is this correct?
Given this, I think the geodesic equation should become just
[tex] \dot{V}^0 =0[/tex]
but I don't see how this shows that [tex] V^0 [/tex] should be a function of just spatial variables, since the dot represents proper time, not coordinate time.
Is there any way to prove that it ought to be coordinate-time-independent?
Phys.Org News Partner Science news on
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
scottie_000 is offline
Feb10-10, 03:51 AM
P: 49
Anyone have any ideas? I'm sure it's very simple, but I can't think how to actually prove it

Register to reply

Related Discussions
Velocity from Static and Total pressures Aerospace Engineering 7
Use of pitot static tube for calculation of airplane velocity Aerospace Engineering 4
Banked Curve- find the Velocity when static MU=0 Introductory Physics Homework 1
Max. velocity with given coefficient of static friction (μ_s) Introductory Physics Homework 5