While I was reading Ta-Pei Cheng's book on relativity, I was unable to derive the correct relationship between coordinate time $dt$ (the book defined it as the time measured by a clock located at $r=\infty$ from the source of gravity) and proper time $d\tau$ from the definition of metric.
The book states that for a weak and static gravitational field, $g_{00}(r)=-\left(1+\frac{2\Phi(r)}{c^2}\right)$ (with the metric signature $(-1,1,1,1)$ and $\Phi(r)$ is the gravitational potential) and the proper time $d\tau=\sqrt{-g_{00}}\,dt$.
From the gravitational redshift result I know that the above result is correct (in a more unambiguous form $d\tau=\sqrt{-g_{00}(r_\tau)}\,dt$).
However, if I simply use the formula for spacetime interval $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ (assuming two clocks that measure proper time and coordinate time are at rest relative to each other), I have
$$ ds^2=g_{00}(r_\tau)c^2d\tau^2=g_{00}(r_t)c^2dt^2=-c^2dt^2\\ \implies \sqrt{-g_{00}(r_\tau)}\,d\tau=dt$$
This suggests that time flows faster with a lower gravitational potential which is incorrect.
I'm not sure why the above method lead to a wrong conclusion, did I misunderstood the the definition of proper time, coordinate time or spacetime interval?
Update:
- One mistake I've made is letting $ds^2=g_{00}(r_\tau)c^2d\tau^2$,
which should be $ds^2=-c^2d\tau^2$ by definition. However, I'm
confused about two definitions of $ds^2$ now.
$ds^2=-c^2d\tau^2=g_{\mu\nu}dx^\mu dx^\nu$, this suggests that
$g_{00}$ is always $-1$ for the frame that measures proper time, but
in my problem $g_{00}$ is a function of $r$ which is only equals to
$-1$ if $r=\infty$, how could two both be true at the same time? - Assuming $ds^2=-c^2d\tau^2$ is true, as all the answers pointed out that $d\tau=\sqrt{-g_{00}}\,dt$. But by the definition of $g_{00}$ and $ds^2$ the $g_{00}$ used here must be $-(1+2\Phi(r_t)/c^2)=-1$, but I want $g_{00}$ here to be $-(1+2\Phi(r_\tau)/c^2)$ so that
$$d\tau=\sqrt{-g_{00}}\,dt=\sqrt{1+2\Phi(r_\tau)/c^2}\,dt\approx (1+\Phi(r_\tau)/c^2)\,dt\\
\implies \frac{d\tau-dt}{dt}=\frac{\Phi(r_\tau)}{c^2}=\frac{\Phi(r_\tau)-\Phi(r_t)}{c^2}$$
Please correct me if I've made any mistakes!
Best Answer
Everytime I try and think of time dilation, length contraction or any other strange phenomenon predicted by this strangely beautiful theory I get confused!! Luckily we have a metric to do all the thinking for us. In coordinates $x^\mu=(ct,x,y,z)$ with spacetime signature $(+1,-1,-1,-1)$ the metric is given by \begin{equation} c^2d\tau^2 = ds^2 = c^2dt^2 - d\vec{r}^2 \end{equation} where $d\vec{r}^2=dx^2+dy^2+dz^2$. If the coordinates are parametrised by $\tau$ so that $t=t(\tau), x=x(\tau), y=y(\tau)$ and $z=z(\tau)$ then we may write the above equations as \begin{equation} d\tau = \sqrt{dt^2-d\vec{r}^2} \end{equation} which is equivalent to \begin{equation} d\tau = dt\sqrt{1-v^2} \end{equation} where we adopt a timescale for which $c=1$ and $d\vec{r}/d\tau$ is equvialent to the velocity and hence the relation between coordinate time and proper time between two events at $t_1$ and $t_2$ is \begin{equation} \tau = \int_{t_1}^{t_2}\sqrt{1-v^2}dt \end{equation}
To answer your question, a spacetime interval $ds^2=d\tau^2=-g_{tt}dt^2$, can be expressed as \begin{equation} d\tau=\sqrt{-g_{tt}}dt, \end{equation} by definition. Your definition of the spacetime interval $ds^2$ is slightly off, it should read $ds^2=d\tau^2 = -g_{tt}dt^2+...$