Tensors and Relativity: Chapter 5

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 tex2html_wrap_inline877 in the frame of the outside observer [ see Figure 5.6 ]. A second ray is emitted at tex2html_wrap_inline879 and received at tex2html_wrap_inline881 .

One can show [ Assignment 5 ] that

equation579

where we have assumed that tex2html_wrap_inline883 (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 tex2html_wrap_inline885 between two points B and A in a gravitational field. i.e.

equation581

If tex2html_wrap_inline891 where tex2html_wrap_inline893 [ no gravitational field ] and B is taken to be a general point with position vector tex2html_wrap_inline897 , we expect

equation585

Since tex2html_wrap_inline899 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:

equation591

Then the proper time measured by a clock at fixed (x,y,z) in a time tex2html_wrap_inline907 measured at infinity is

equation593

therefore

equation595

for tex2html_wrap_inline909 . This corresponds to spacetime curvature.

 

figure381
Figure 5.6: Space time diagram of rocket undergoing uniform acceleration g