# Question about gravitational time dilatation

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?

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.

• What is an eigen time? Is it the same as proper time? Commented Jan 26, 2023 at 19:02
• Yes it is the same. Commented Jan 26, 2023 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 $r= \infty$ Commented Jan 27, 2023 at 6:44
• @BrainStrokePatient May be the post physics.stackexchange.com/questions/582542/… can help you. Commented Jan 27, 2023 at 14:21