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Gravitational Redshifts and Twisted Time

In addition to the ``ordinary'' redshifts of distant stars caused by the relativistic Doppler shift due to the fact that they are actually receding from the observer on Earth, there is a graviational redshift of the light from near a large mass M when observed from a position far from the source, even if the source and observer are at rest relative to one another. This is not too surprising if we recall that a gravitational field has to be indistinguishable from an accelerated reference frame, and an accelerated object cannot be at rest for long! But an easier way to see the result is to remember that a massless particle like a photon still has an effective mass where (if I may borrow a hitherto undemonstrated result from quantum mechanics) for a photon. Here is the frequency of the light and J-s is Planck's constant. Anyway, if the energy of a photon far from M is (at ) then its effective mass there is and as the photon ``falls'' toward M it should pick up kinetic energy until at a finite distance r its energy is where the new effective mass is . Thus and if we collect the terms proportional to E we get where . Dividing through by gives the formula for the gravitational redshift,

(I have fudged in that extra factor of 2 that turns into the correct Schwarzschild radius ). This derivation is completely bogus, of course, but it does indicate why there is a gravitational redshift.

Given that any mechanism for generating electromagnetic waves constitutes a ``clock'' of sorts, the waves emitted by such a device constitute a signal from it telling distant observers about the passage of time at the origin. (Think of each wave crest as a ``tick'' of the clock.) The very existence of a gravitational redshift therefore implies that time passes slower for the clock that is closer to the mass - a result that was referred to earlier without proof.



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Next: About this document Up: Gravity Previous: Mutability