Tensors and Relativity: Chapter 2

Relativistic Doppler shift

next up previous index
Next: Four- acceleration Up: Four- velocity, momentum and Previous: Four- velocity, momentum and

Consider a photon with energy tex2html_wrap_inline1384 moving at an angle tex2html_wrap_inline1238 with respect to the x- axis. Its three- velocity is tex2html_wrap_inline1390 , so its four- momentum is

equation997

where h is Planck's constant  and tex2html_wrap_inline1394 the frequency.

In a frame tex2html_wrap_inline1210 with three- velocity (v,0,0) relative to tex2html_wrap_inline1206 the frequency is tex2html_wrap_inline1402 . Using the Lorentz transformations we get

equation1004

therefore

equation1010

so we get the following result:

equation1013

If tex2html_wrap_inline1404 , so that the photon moves in the same direction as tex2html_wrap_inline1210 , we have

equation1017

For low velocities tex2html_wrap_inline1341 this reduces to

equation1020

This is the usual Doppler shift tex2html_wrap_inline1410 , modified at large v.

If tex2html_wrap_inline1414 , so the photon moves perpendicular to tex2html_wrap_inline1206 , we have

equation1024

This is the transverse Doppler shift  and is a consequence of time dilation.