# Geodesic equation and spatial variation of time

I am trying understand the interpretation of geodesic equations. For simplicity, let us take a metric $$ds^2 = g_{00}(x)dt^2 + a(x,y,z)(dx^2 + dy^2 + dz^2).$$

I interpret the metric to be a spacetime, where time, $$t$$, varies with $$x$$. Now how does $$t$$ vary with $$x$$? We know that $$dt = \frac{d\tau}{\sqrt{g_{00}}}$$, where $$\tau$$ is proper time. Since $$\tau$$ is an invariant quantity, is it right to say that time $$t$$ varies with $$x$$ as $$t = \frac{t_0}{\sqrt{g_{00}}}$$, where $$t_0$$ is some constant? or should one use the geodesic equation to determine the spatial variation of $$t$$?

The geodesic equation would obviously depend on $$a(x,y,z)$$, and hence the result should be very different from giving the time variation as $$t \approx \frac{t_0}{\sqrt{g_{00}}}$$.

If the geodesic equation does not provide the spatial variation of $$t$$, what would it signify here?

• Why do you say that $\tau$ is an invariant quantity? Invariant with respect to what? – Avantgarde Sep 5 at 13:57
• invariant w.r.t the location of the observer. Observer at infinity and observer inside the gravitational field should measure the same $ds$. In one case $ds^2 = d\tau^2$, and in the other $ds^2 = g_{00}dt^2$. Here we are measuring pure timelike distances. – Angela Sep 5 at 16:16

Your metric is indipendent on time. So, there is a Killing vector $$\xi_\mu=(1,0,0,0)$$ and you have the conserved quantity ($$c=1$$) $$\xi_\mu\frac{dx^\mu}{d\tau}=g_{00}(x)\frac{dt}{d\tau}=\frac{E}{m}.$$ Also, from the metric you get $$1=g_{00}(x)\left(\frac{dt}{d\tau}\right)^2+a(x,y,z)\left[\left(\frac{dx}{d\tau}\right)^2+\left(\frac{dy}{d\tau}\right)^2+\left(\frac{dz}{d\tau}\right)^2\right]$$ and using the conserved quantity it yields $$1=g_{00}^{-1}(x)\left(\frac{E}{m}\right)^2+a(x,y,z)\left[\left(\frac{dx}{d\tau}\right)^2+\left(\frac{dy}{d\tau}\right)^2+\left(\frac{dz}{d\tau}\right)^2\right].$$ Therefore, you also need the geodesic equation to solve for $$t=t(x,y,z)$$ and $$\tau=\tau(x,y,z)$$ as independent variables.