Effect of gravity on time

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Consider a rocket of height h undergoing acceleration g relative to an outside observer. Let a light ray be emitted from the top (B) at time t=0 and be received at the bottom (A) at time in the frame of the outside observer [ see Figure 5.6 ]. A second ray is emitted at and received at .

One can show [ Assignment 5 ] that

where we have assumed that (i.e non- relativistic motion).

Using the equivalence principle we know that the same relation must apply of there is a difference in gravitational potential between two points B and A in a gravitational field. i.e.

If where [ no gravitational field ] and B is taken to be a general point with position vector , we expect

Since is negative, the time measured on B's clock [ as seen by A at infinity ] is less than the time measured on A's clock, i.e. clocks run slow in a gravitational field.

One can interpret this by imagining that the spacetime metric has the non- Minkowski form:

Then the proper time measured by a clock at fixed (x,y,z) in a time measured at infinity is

therefore

for . This corresponds to spacetime curvature.

Figure 5.6: Space time diagram of rocket undergoing uniform acceleration g