# Effect of gravity on time

**Next:** Towards spacetime curvature **Up:** The principle of equivalence ** Previous:** Effect of gravity on

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*