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For $g_{00} = 1 - 2GM/c^2r$ , the time interval $\Delta T$ measured in a stationary frame at a distance $r$ from the source and the time interval $\Delta t$ measured by a frame at $r= \infty$ are related via $$\Delta T = \sqrt{1 - \frac{2GM}{rc^2}} \Delta t$$ But, the invariant quantity $\Delta \tau^2$ in both frames should be the same: $$\Delta \tau^2 = \left( 1 - \frac{2GM}{c^2r} \right) (\Delta T)^2 = (\Delta t)^2$$ which gives the reverse relation between $\Delta T$ and $\Delta t$. What am I doing wrong?

Brain Stroke Patient
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1 Answers1

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The eigen time of a stationary frame at distance $r$ from a gravitational source is

$$(\Delta T)^2 =\frac{(\Delta s)^2}{c^2} = \left(1-\frac{2GM}{c^2 r}\right)(\Delta t)^2$$

i.e. the time measured in the stationary frame at distance $r$. The quantity $\Delta t$ is the coordinate time.

In order to know the coordinate time one can go to a frame in a region where there is no gravitational effect (i.e. an area with a Minkowski metric), formally achieved by putting $r\rightarrow \infty$. In such a frame we would have as eigen time (since there we have $(\Delta x)=(\Delta y)= (\Delta z)=0$.):

$$\frac{(\Delta s)^2_{r\rightarrow \infty}}{c^2} = (\Delta t)^2 -[(\Delta x)^2 -(\Delta y)^2 -(\Delta z)^2]/c^2 = (\Delta t)^2$$

In such a frame the eigen time corresponds to the coordinate time $\Delta t$.

So one can relate the eigen time measured at distance $r$ with the eigen time measured at $r\rightarrow \infty$ by

$$(\Delta T)^2_{r} = \left(1-\frac{2GM}{c^2 r}\right)(\Delta t)^2_{r\rightarrow \infty}$$

Therefore there is no contradiction at all.

Frederic Thomas
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  • What is an eigen time? Is it the same as proper time? – Brain Stroke Patient Jan 26 '23 at 19:02
  • Yes it is the same. – Frederic Thomas Jan 26 '23 at 21:29
  • I am still a bit confused. Normally when we use $d \tau^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}$, the $dt$ is the time difference between two events as seen from an inertial frame. Therefore, in this case in $d\tau^2 = g_{00}(r) dt^2$ , $dt$ should be the time difference between two clock ticks at distance $r$ from the source because it is being multiplied by $1 - 2 GM/r c^2$. But you're identifying it as the time difference between two clock ticks at distance $r= \infty$ – Brain Stroke Patient Jan 27 '23 at 06:44
  • @BrainStrokePatient May be the post https://physics.stackexchange.com/questions/582542/why-does-d-tau-instead-of-dt-denote-the-proper-time-is-it-a-definition can help you. – Frederic Thomas Jan 27 '23 at 14:21