This is the fourth part of the VERITAS series on General Relativity. Just to remind our readers: We are doing this series on Einstein’s Theory of Relativity to celebrate the 100th Anniversary of General Theory of Relativity. If you have missed any of the first three parts, you can read them here:
In the previous part we discussed the fundamental idea behind General Relativity- The Equivalence Principle. We used the equivalence principle to show that light is bent by gravity. And then we reasoned that gravity bends space-time and that objects moving along this distorted space-time seem to be moving under the influence of a force. If you have not read that part( or the previous ones), I suggest you read them before reading this one.
In this episode of this VERITAS series we will show how gravity causes time to slow down. We will also consider a practical application of this phenomenon.
If you have seen the movie Interstellar, you would know that time flows differently on Miller’s planet. Miller’s planet orbits the supermassive Black Hole Gargantua. Because Miller’s planet is under the influence of Gargantua’s gigantic gravitational forces, time flows very slowly there compared to Earth. We saw in the movie that one hour on Miller’s planet was equal to 7 years on Earth.! We also saw that tidal waves of more than 4000 feet raged across the planet. These massive tidal waves were also caused by Gargantua’s gravitational pull. If you have not seen Interstellar, I strongly recommend that you see it. It is a rare science-fiction movie in which science is not sacrificed. Almost every amazing thing shown in the movie is scientifically possible. This level of scientific accuracy was possible because Kip Thorne, one of the leading experts in General Relativity was the scientific consultant for the film. He solved Einstein’s equations for various situations described in the movie and then computer simulations were created for the solutions which were used in the movie. Kip Thorne has written a book, “The Science of Interstellar” which he describes more scientific aspects of the movie.
Now, let’s use equivalence principle to understand the effect of gravity on the flow of time.
See the above images that describe two situations. In the first case we have a rocket moving in empty space far away from any source of gravity. This rocket is accelerating at the rate of 9.8 m/s^2. In this rocket we have two clocks, one at the floor and one on the ceiling. In the middle we have an observer. The clocks are attached to a mechanism that sends a pulse of light at regular intervals( say 5 seconds). So the top clock and the bottom clock send a beam of light every 5 seconds. The beams reach the observer who records the times. Since the rocket in case 1 is accelerating at the rate of g( 9.8 m/s^2), the observer is moving towards the top clock and away from the lower clock. When the top clock’s light beam reaches the observer, the lower clock’s light beam has still not reached- it has a longer distance to cover because the observer has moved away from it. Thus the observer sees that the beams from the top clock come at a faster rate than the beams of the lower clock. The observer has the right to think that the bottom clock is running slower than the top clock. Now, according to equivalence principle( which we discussed in the last VERITAS of this series), an accelerating lift is equivalent to what happens in a gravitational field with the same g. So case 1 and case 2 are equivalent. What happens in case 1 also happens in case 2. Case 2 is the same arrangement but it is stationary on the surface of the earth. So even in case 2 the observer in the middle of the rocket will find that the beams of the upper clock come at a faster rate than the beams from the lower clock. So the observer will think that the clock at the floor of the rocket is slower than the clock at the top. Therefore, gravity slows down time! An clock in a stronger gravitational field slows down more compared to a clock in a weaker field. Note: as I said when we talked about time dilation in special relativity( episode 2 of this series), this is just one mechanism that shows that gravity slows down time. No matter what arrangement you come up with, you will find the exact same result- gravity does slow down time. This is called Gravitational time dilation.
The effect is so small that we do not notice it in our daily lives. Here is an example that would give you an idea of the magnitude of time dilation. Let’s take a 30 m high building( about 10 storeys). The clock on the ground floor would appear slower than the clock on the top floor by about a few microseconds in a 100 year period. So a person that resides on the top story( and never leaves home) would age faster than a person on the ground floor by a very small amount. This is because even the human heart is a kind of clock and this too should follow the principles of physics. Many experiments have experimentally verified Einstein’s calculations on gravitational time dilation. Physicists have observed the difference in time on an aeroplane vs the time on earth using atomic clocks.
Though the effect of gravitational time dilation is small, it is not without technological importance. The Global Positioning System which helps us navigate would not work properly if it did not take into account the time difference caused by gravitational time dilation. The GPS is a collection of 24 satellites with each satellite carrying an atomic clock. Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour. The satellites are so positioned that at any point on the earth, at least 4 satellites are always visible. Each satellite reports its position and time( from its atomic clock) at regular intervals. These signals( travelling at the speed of light) are intercepted by your GPS receiver, which calculates how far away each satellite is based on how long it took for the messages to arrive. The GPS receivers( like your smartphone) compare the time of reception of signals from 4 satellites to accurately determine your position on earth to an accuracy of about 5 meters. But since GPS calculations involve accurate knowledge of time, we have to take into account the time dilation caused by Gravitation and also special relativity. If time dilation is not accounted for, the GPS receivers will have a location error of about 10 Km per day! So general relativity is not just of interest to theoretical physicists studying black holes, even your smartphone’s GPS will not work without accounting for it.
In the next episode of this series, we will explore one of the most exotic objects in the universe: Black Holes.
Before we end, here is an Einstein quote:
“Whoever undertakes to set himself up as a judge of Truth and Knowledge is shipwrecked by the laughter of the gods.”
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Go wondrous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run,
correct old time, regulate the sun
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