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The usual explanation for photon's higher frequency in lower altitudes (higher gravity), when the photon is going downward towards a massive body, is that gravitational potential energy is converted into photon's energy which, by its turn, through $E=hf$ implies higher frequency. Because, since $c$ is constant, it doesn't make sense to turn that potential energy into kinetic energy.

That, I understand.

My question is, why is it affirmed that this implies that time runs slower in the lower altitude? I mean why is it such that this higher frequency is equivalent to slower running clocks?

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"Because, since c is constant, it doesn't make sense to turn that potential energy into kinetic energy." Why not? The energy of a photon is as you said: $E=hf$, which does not depend on its speed. If you are thinking of the classical formula $E=\frac{1}{2}mv^2$, clearly it doesn't apply to photons, since photons are massless and their speed is constant. The correct relativistic formula is $E^2 = (mc^2)^2 + (pc)^2$. Setting $m=0$ gives the correct energy for photons. – Michael Brown Jan 8 '13 at 1:51
I corrected attidude to altitude - if that's not what you meant, please roll back. – Nathaniel Jan 8 '13 at 2:09

Imagine two identical sources of monochromatic light, emitting at a fixed frequency. Then they are also clocks, as one can think of the oscillations in the light wave as 'ticks' of the clock (or perhaps count photons, if one wishes to make this explicitly non-classical).

The clock rates match when they are together or in Minkowski spacetime, but in a gravitational field, gravitational redshift (or blueshift) means they don't. The rate of any other physical process in their vicinity can is can be measured by those clocks, so it is natural to interpret this as gravitational time dilation.

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