# Gravitational Frequency Shift

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1. May 4, 2015

### epsilon

I have been studying the "fallen" photon experiment, in which the frequency of a photon changes as it falls through a height H.

$f'=f_0(1+\frac{gH}{c^2})$

It is often stated that this is a huge factor in the application of GPS. However, I do not understand why.

I understand that the photon will arrive at Earth with a different frequency, however I do not see why the time taken should vary - it will simply arrive in a different part of the electromagnetic spectrum.

The distance is not changing and the photon is travelling at the speed of light too. Surely this also rules out the effects of special relativity, as the time dilation, length contraction and mass increase equations all incorporate the Lorentz factor:

$\gamma=\sqrt[]{1+\frac{v^2}{c^2}}$

However as $v=c, \gamma=1$ for all 3 of the relativistic effects: $t'=t_0, l'=l_0, m'=m_0$.

So why is the gravitational frequency shift important for GPS?

2. May 4, 2015

### Staff: Mentor

The important thing for GPS is not the frequency shift of the photons in the GPS signals; it's the fact that the clocks on board the GPS satellites run faster than clocks on Earth. (Note that there are actually two effects--the gravitational effect due to altitude, and the kinematic effect due to the motion of the satellites. At the altitude of the GPS satellites, the first effect is stronger, so the satellite clocks run faster than clocks on Earth. But for a satellite in low Earth orbit, such as the ISS, the kinematic effect is stronger and clocks actually run slower than clocks on Earth.) The rates of the clocks on the GPS satellites have to be adjusted so that they effectively run at the same rate as Earth clocks.

3. May 4, 2015

Staff Emeritus
A GPS signal is essentially a clock ticking "one two three...". That frequency shifts, and that's the frequency that matters - as you say, the radio frequency is unimportant.

4. May 4, 2015

### epsilon

Now that you have said that, I can't believe I didn't realise it. Thank you.

5. May 4, 2015

Staff Emeritus
Glad to have been of help.

6. May 4, 2015

### pervect

Staff Emeritus
You need to combine the observed gravitational frequency shift with another principle of GR, the principleof equivalence. The principle of equivalence says, very loosely speaking, that if you have a cesium atom, the frequency of it's emission doesn't depend on height (or anthing else, for that matter, but the height is what's important in this application).

Reconciling this, with the fact that the frequency does shift, you come to the conclusion that the clocks must change frequency too.

Underlying this point of view is the idea that sort of time clocks (such as the cesium standard that defines the SI second) is a sort of time called "proper time", which is in general different from the notion of time a a label that we assign to events that happen "at the same time", which is called coordinate time. It's also implied that the proper time is more fundamental than coordinate time, because physics is most simply expressed in terms of proper time.

This is a rather quick and over-simplifed description, but hopefully it will point you in the right direction.

To recap, with perhaps a bit of philsophy added. What GR says is that the proper time that clocks keep iis more fundamental, because it's simpler, while while the "coordinate time" that we introduce when we compare clocks at different locations is a less fundamental human invention.

7. May 4, 2015

### Janus

Staff Emeritus
It is because the distance is not changing that the frequency change is important. Imagine I have a clock that counts the oscillation of an atom for keeping time. Moving charges give off radiation, one oscillation per wave cycle.

The atom oscillates at 5e14 Hertz which means it gives off light in the visible range at that same frequency. IOW, the frequency of light the atom emits is tied to its oscillation rate. Thus for this clock 5e14 oscillations equal 1 second.

Now imagine that this clock is high above me in a gravity field. As per gravitational frequency shift the light reaching me from that clock will be higher than 5e14 Hertz. I am also counting the oscillations of the atom as I watch it. Now here's the thing, I can no more see a disconnection between the oscillations of the atom and the light it gives off than someone next to the clock does. The one wave cycle per oscillation rule must still hold. In other words, the frequency of the light I see and the number of oscillations I see the atom make in one sec must match. Anything else would be be a violation of the laws of physics. I cannot see the atom emit yellow light but vibrate at a frequency equal to that of Red light.

Thus I will count 5e14 oscillations of the atom in less than 1sec by my clock. I also cannot alter the 5e14 oscillation per sec that the clock at the higher elevation measures. Since by my clock 5e14 oscillations of the atom takes less than 1sec. and 5e14 oscillation takes 1 sec according to the higher clock, this means when I compare my clock to the higher clock, the higher clock is running faster.

8. May 8, 2015