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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.

Lagrangian
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