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How is it that time runs faster at regions further out from the center of a gravitational field if:

A clock's arm is made to make one cycle for a complete wavelength of a particular light beam. Consider the clock moving upwards from the surface of the earth as the light also moves up. Since the wavelength of the light increases as it loses energy, then the frequency must decrease:

$$c = λν$$

c is a constant, therefore

$$ λ ∝ \frac{1}{ν}$$

But the frequency is the number of cycles the light beam makes in a second: this number reduces. Since the period — the duration of time for once wavelength of the light (one cycle of the clock's arm) — is

$$ T = \frac{1}{f}$$

it increases for a decrease in frequency. Therefore, the duration of time for one cycle increases; the clock runs slower. Thus the clock runs slower with increased de-energisation of the light as it moves further away from the center of the gravitational field.

I get that experimental evidence differs, but I'm having a hard time finding the fault in the above claim.

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  • $\begingroup$ There are several flaws in your question. The main one is that you don't have a defined frame of reference, in which you would evaluate your equations: "Consider the clock moving upwards from the surface of the earth as the light also moves up" - No frame can be defined at the speed of light. Secondly: "c is a constant" - No, the speed of light is variable in General Relativity. It is constant only locally, but you don't have a frame, in which c would be local. Finally: "light [...] loses energy" - No, it doesn't. This thinking is a result of you not having a defined frame of reference. $\endgroup$ – safesphere Sep 29 at 18:43
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Therefore, the duration of time for one cycle increases; the clock runs slower.

Yes. The clock at a lower altitude runs slower as observed by a clock at a higher altitude. As the light from the lower clock goes higher it loses energy and has a longer period. Since the higher observer sees that the lower clock ticks at the same rate as the light, the gravitational redshift of the light implies that lower clock is slower as measured by the higher clock.

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  • $\begingroup$ So is it that the longer period measured by the higher observer is not that which he perceives in his time, but that which he measures for the lower observer? $\endgroup$ – allan e Sep 29 at 17:43
  • $\begingroup$ Yes! That is correct $\endgroup$ – Dale Sep 29 at 17:53
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    $\begingroup$ Great! Thank you! $\endgroup$ – allan e Sep 29 at 17:55
  • $\begingroup$ "As the light from the lower clock goes higher it loses energy" - It doesn't. In the frame of the emitter, the ascending photons do not redshift. In the frame of the receiver, these photons are emitted already "redshifted" and do not lose energy in flight. In either frame of reference, light does not lose energy in a gravitational field. $\endgroup$ – safesphere Sep 29 at 18:34
  • $\begingroup$ In the Schwarzschild coordinates it does lose energy. It doesn’t really make sense to talk about the frame of either the emitter or the receiver since they are non inertial and the frame of a non inertial observer is not uniquely defined. You could use Fermi normal coordinates along the light’s worldline to make your argument. But since the OP seemed to be using Schwarzschild coordinates in the question I chose to do so in the answer also. $\endgroup$ – Dale Sep 29 at 18:57
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I think the scenario you're envisioning amounts to a light signal being broadcast from some location on earth, and a clock at a different (perhaps higher) location picking up the signal. Such clocks will stay in sync with each other at any altitude (once initially synchronized, and if they never miss a pulse), while clocks that don't have an external reference and rely on some local physics like the vibration of a quartz crystal will tick faster at higher altitudes. So your analysis is correct: your clock will run slower than a quartz clock. It's backwards from the usual statement about clocks simply because the ratio is normally stated as the speed of the local clock relative to the global reference.

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