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I read (extern link to a filehoster, page 163 section 4.8) that a variable celestial object like a pulsar or quasar must be smaller than the distance the light travels in its variation period.

Can someone give me a proof or a reference to a proof.

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  • $\begingroup$ Dear @user25575: It is best to supply title, author, etc, of link, so we can reconstruct the link in case of future link rot. $\endgroup$
    – Qmechanic
    Jun 9, 2013 at 10:06
  • $\begingroup$ Astronomie et Astrophysique: cinq grandes idees pour explorer et comprendre l'univers by seguin and villeneuve (2nd ed). this book is in french, you can download it from the 4shared link above $\endgroup$
    – user5402
    Jun 9, 2013 at 14:28
  • $\begingroup$ Note that sometimes it is important to distinguish between the object as a whole, and the emitting region. This can be important for things like pulsars - which can emit primarily from 'hot-spots $\endgroup$ Jun 9, 2013 at 23:16

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This is equivalent to the statement that the variations occur slower than the frequency produced by light that oscillates between the opposite points of the celestial object. But that's true because of special relativity simply because no signal can propagate faster than light.

The variations of the celestial object occur because of a signal – e.g. some pressure waves are moving from one side to another – and these waves are able to transmit information. If the variations of the objects were faster than light, in the sense discussed above, they would allow us to propagate the information faster than light. For a realistic model, we must really talk about waves that are similar to the standing waves, not waves that move indefinitely in one direction, but they're superpositions of two oppositely moving waves with a well-defined direction and their speed is therefore limited as well.

The precise numerical prefactor in the inequality may require some extra discussion that depends on the type of the signals that make the object variable. In most cases, the objects are vastly smaller than the limit because the waves causing the variability are a form of "sound" which is much slower than the light, at least in ordinary enough materials. With more exotic materials, one can get closer to the speed-of-light limit.

For example, the diameter of the Earth is 13,000 km or so, about 0.04 light seconds. Light only needs 0.04 seconds to travel this distance. All the inner cycles of the Earth take a much longer time.

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