# Light frequency and time relation

I am reading Stephen Hawking "A Brief History of Time" recently, and I cannot understand perfectly this little fragment:

(...) Another prediction of general relativity is that time should appear to slower near a massive body like the earth. This is because there is a relation between the energy of light and its frequency (that is, the number of waves of light per second): the greater the energy, the higher frequency (...)

How is that light influence time? He writes also:

As light travels upward in the earth’s gravitational field, it loses energy, and so its frequency goes down. (This means that the length of time between one wave crest and the next goes up.) To someone high up, it would appear that everything down below was making longer to happen.

And that I can understand, it just a matter of perspective. But what's most confusing for me is this:

This prediction was tested in 1962, using a pair of very accurate clocks mounted at the top and bottom of a water tower. The clock at the bottom, which was nearer the earth, was found to run slower, in exact agreement with general relativity

How is that related to light? How light affects situation when clock at the bottom was found to run slower? I know basics of general relativity, I am just confused why did he mentioned "relation between the energy of light and the frequency" in that. I'am probably missing something obvious here, that's why I am waiting for someone to help me.

## 1 Answer

We can think of time as a way to measure intervals that separate two events. In this case, light is introduced to help you visualize how time dilation happens.

Imagine there's a machine composed of two plates separated by one meter, and from the bottom light is emitted. Since the light is a wave, we can measure our time as the interval it takes for two consecutive crests to arrive (i.e. a time unit). In the absence of external influences, we should measure the same time no matter our position with respect to the clock.

Now let's put our clock near an object with an intense gravitational field and you stay near it while I position myself away from it. As Hawking says, as the light is traveling upwards it loses energy (remember energy, frequency, and wavelength are related by $$E=h\nu=hc/\lambda$$). Since the speed of light $$c$$ must be conserved, that means that the frequency $$\nu$$ decreases and thus the wavelength $$\lambda$$ increases.

Now if you measure the time between two consecutive crests, you will measure one-unit of time (say, one second).. However, I see the clock from far away and I notice that, due to the gravitational field, the wavelength of the light now has increased. This means it takes longer for two consecutive crests to arrive to the top plate, so I measure a longer time (say 2 seconds). That means that time has been affected by the presence of a gravitational field (this is actually called gravitational redshift).

Now this may seem like a particular case for light, but remember that all matter is essentially made up of elementary particles with their own interactions, and time is measured as the interval between each interaction. So their "time" is also affected by the gravitational field in the same was that light was affected by the gravitational field.

Since atomic clocks are basically clocks that measure time according to specific atomic transitions, they're affected by the gravitational effect I just mentioned before. This is why we observed a small but non-zero difference on the intervals of time measured by the clocks on the ground and the clocks on an airplane.

Hope this helps make it more clear.