- #1

- 67

- 9

## Main Question or Discussion Point

Consider the following metric for a 2D spacetime:

##g_{tt} = -x ##

##g_{tx} = g_{xt} = 3##

##g_{xx} = 0##

i.e.

[tex]

g_{\mu \nu} = \left(

\begin{array}{cc}

-x & 3\\

3 & 0

\end{array}

\right)

[/tex]

Now, since the metric is independent of time (t), there is supposedly a conservation law containing ##\frac{dx(\tau)}{d\tau}## and ##\frac{dt(\tau)}{d\tau}##. What is this conservation law, and why does the time independence of the metric imply it?

##g_{tt} = -x ##

##g_{tx} = g_{xt} = 3##

##g_{xx} = 0##

i.e.

[tex]

g_{\mu \nu} = \left(

\begin{array}{cc}

-x & 3\\

3 & 0

\end{array}

\right)

[/tex]

Now, since the metric is independent of time (t), there is supposedly a conservation law containing ##\frac{dx(\tau)}{d\tau}## and ##\frac{dt(\tau)}{d\tau}##. What is this conservation law, and why does the time independence of the metric imply it?