### A satellite orbits the earth. Due to it’s speed, special relativity says time should be going slower than if it was on earth for it, but in fact, it goes quicker. Why?

This is the third of three posts on time. We have already discussed The Special Theory of Relativity in the first post, and the concept of space-time in the second one. I would recommend reading them first, for this most may not make much sense if you have not. In true “TheCompBlog” style (a great tech blogger/friend-of-mine), this post is being written whilst on a airplane flight, and just how topical it is! We learnt in the aforementioned posts, that the faster an object move, the slower time runs for it. So up here, in this plane, time for me is moving quicker than for someone standing below me right? Well, there is more to it than that. You see, another of Einstein’s discoveries was that not only is time affected by speed, it is also affected by gravity. How does this come about? well really, it is just applying Special Relativity:

The best way to explain Gravitational Time Dilation, this process of time being slowed by gravity, is through imagining a perfect vacuum in the middle of space. You are in a rocket in this vacuum. No external forces, including gravity, are acting on the rocket. At the front end of rocket is a lamp, emitting regular pulses of light every second. At the other end is a target.

The pulses arrive at the target at the same rate as when they left the lamp, namely 1 per second. The time at the front of the rocket would be moving at exactly the same rate as the time at the back. However, what would happen if the rocket started accelerating? By the time the pulse reached the other end of rocket, the rocket would have aquired a speed. We know that velocity equals acceleration times time taken, or v=at, and the time it takes for the light pulse to travel the distance is distance divided by the speed of light, or t=xc. Therefore the speed gained whilst the pulse is traveling is v=a xc. Well firstly we have to look at something called the Doppler shift. We have all experienced the effect when an ambulance passes you. The siren sound seems to drastically change as it is approaching, passing and moving away. This is because the frequency changes. This change can be measured with this formula:

 f’=f⁄(1± v ) c

The bottom of that fraction is similar to that from our other equations from Special Relativity, though this is only because, just like in time in special relativity, the frequency is affected by velocity. For very large velocities, time dilation has to be taken into account, but in this situation, the speeds are not large enough for it to matter. The equation can be rearranged to be

(f’-f) ≈ -fvc

and from our previous equation for v, we get

(f’-f) ≈ -fahc2

The pulses of light reach the target at a different frequency as when they started:

Now imagine that the rocket is not in space but instead on a earth. The lamp is on the floor of the rocket now, and the target is on the top:

Gravity is, in essence, just an acceleration. If a feather and hammer are dropped in a near vacuum, they will fall at exactly the same rate. On earth, that rate is 9.81 m/s2. So the rocket, in this situation, is in fact feeling an acceleration downwards, just as we feel pushed towards the ground. The person standing at the top would conclude that the pulses are being released at longer intervals a person standing next to the lamp would measure the frequency to be, this is called the gravitational redshift. Now imagine that the pulses are some sort of clock, each one emitted a second after the last. Seeing that the frequency of this would be less for the person at the target, he would, correctly, conclude that the time closer in the gravitational field was moving slower.

Many experiments have been made to prove this theory. In 1960, Robert Pound and Glen Rebka verified it by firing gamma radiation up and down a 22.5m tower. The shift would only have been about -2.5 x 10-15 but it was verified with a margin for error of only 1%. When the first GPS satellites were launched they did not take into account time dilation from gravity or from speed and were therefore very far off when calculating someones position after only a day or two.

I really hope you like this post and this series of posts. I may revisit it one more time, just to run over the main points, sort of as a conclusion, but I hope that this has interested you in the mean time. Please email me at theaftermatter@gmail.com, comment or get in touch on Twitter if you have any suggestions,  complaints or just want to have a chat about physics/maths/engineering, I love hearing from you.

Check out my last two posts:
What is Energy? – The word energy is used so loosely these days, so what does it actually mean?
“We don’t serve neutrinos in here!”…A neutrino walks into a bar – What are neutrinos and what is all the fuss about?

Main source for this and the other articles based around relativity is the book about relativity here. It goes into more detail on all these subjects and the other books they do are brilliant too, I recommend them a lot.