Written By: Shummas Humayun
NOTE: I wrote this essay based on the assumption that the reader has no prior knowledge of the basic implications of special relativity, which for our concern is Time Dilation. In my text, I tried to be as simple as I could while covering the underlying concept. For that reason, I divided the essay into 5 sections.
-[The relativity principle]-
At the heart of special relativity lies a simple fact, that the laws of nature are invariable. The principle of relativity, which is often wrongly credited to Einstein and actually dates back to the time of Galileo under the title ‘Galilean relativity’, states that the laws of physics will work the same way both in the state of rest or of uniform motion. This forms the backbone of special relativity and this simple sounding fact has very far reaching and counter intuitive consequences. It also suggests that any experiment conducted by us in our frame that is either at rest or in uniform motion, will give us the same experimental result in both cases. Another way to put up the relativity principle is to say that the states of rest and of uniform motion are actually indistinguishable from each other. Both of these states are equivalent in all aspects and because you will get the same experimental results in both of them, there is actually no way to test whether you are absolutely at rest or absolutely in uniform motion. The concept is totally relative. If I say that I’m driving at 40 m/s it would mean that I’m moving at this speed in relevance to the trees, buildings and roads which are themselves not at rest but in a state of uniform motion all the time due to the earth’s orbit. (Earth’s motion is not fully constant but it has a negligible angular acceleration due to it’s slightly elliptical orbit around the sun). Hence motion is relative and absolute state of uniform motion or rest does not exist. Only change in motion can be absolutely measured which in scientific terms, is called acceleration. Furthermore, If I’m moving at a speed of 40 m/s to the right from your perspective, it is equivalent to say that you are moving at negative 40 m/s from my perspective (opposite direction). This is simply because none of us can claim to be in a state of absolute rest in the the entire universe. In the above case we both happen to be in relative motion. Similarly, if we both move at the same speed in the same direction, it is logically equivalent to say that we both are at rest with respect to each other but at the same time, we both are in motion with respect to the earth. Thus in order to understand the basic implications of special relativity, one has to unlearn the false concept of absolute motion and absolute rest. As you will later see that even though one will get the same results for experiments conducted in his own state of rest or of uniform motion, he will not agree with the experimental results of someone who is moving with a relative velocity. One of these disagreements will be regarding the passage of time. Time may seem to pass slowly for one person who is in uniform motion with respect to the other person. Because the relativity principle asserts that none of them can claim to be in a state of absolute rest or absolute uniform motion, time itself is not absolute for them under these circumstances.
What is time? No one has a definite answer and many physicists seem to have varying opinions. Perhaps this question comes under the domain of the philosophical side of physics but what we do know is that we as physical beings, are bound by it. Ever heard the phrase, ‘Time is relative’? You must have if you are used to watching science fiction movies but could it really happen? A non-scientist tends to think, “How can that even be possible? it’s science fiction anyway”. Well, when it comes to the understanding of reality, truth can actually be stranger than the strangest science fiction. The fact is that not only time travel is theoretically possible but it happens all the time, all around us. Furthermore, the type of relativistic time travel which special relativity allows is just one of the several theoretically possible ways to travel time. But, apparently this is comparatively an easier way. You won’t really leap frog into the future to meet your old self, instead you will live slowly through time as compared to others. Also, this type of time-travel is totally one directional. There is really no way to reverse time and go back into the past.
In the 1960′s, a group of physicists conducted an experiment in the vicinity of Mount Washington. This experiment is most famously known as the Muon rain experiment and is one of the simplest experiments done for the verification of time dilation and length contraction. Due to the cosmic rays, a steady rain of subatomic particles called Muons falls on Earth. These particles are created high in the atmosphere and they race down to earth at speeds as high as 99% of the speed of light. In normal conditions, these particles destruct themselves almost as soon as they are formed. 6000 ft high on the top of Mount Washington, experimenters observed approximately 550 Muons being detected by the Muon detector in one hour. When they repeated the experiment on the sea level they found roughly the same amount of Muons landing on their detector. The question that arises here is that how did the muons survive such a long time period of their flight even though in normal conditions they are supposed to immediately destruct themselves just as soon as they are formed? The answer is ‘Time dilation’. Time in frames moving at a high relative velocity slows down. The muons were actually experiencing a slower time with respect to us (the observers). Hence they managed to live through time far longer than their expected age. ‘Moving clocks run slow’ a phrase often used in special relativity, is practically correct. Even though in a strictly scientific language this phrase is slightly incorrect and quite misleading, but for a non scientist it is enough to assert that time is not absolute. The correct sentence would be, “Clocks that are in relative motion are observed to run slow”. This phenomenon of time dilation occurs due to the fact that the laws of Nature work in the same way at both rest and in uniform motion (the relativity principle). The constancy of the speed of light (and all other electromagnetic waves), is one of the laws of nature as recognized by Einstein and it happens to be one of the two postulates of special relativity. Einstein’s other postulate states that nothing can catch up to the speed of light. Laws of nature are really stubborn. It appears that in order keep the speed of light constant for you, and hence in order to maintain the law, nature can significantly change other quantities that include your time, length, and mass. Consider this. You are in a super fast train moving at 99% of the speed of light while I’m standing on the ground observing you. Now, if we both conduct an experiment involving the use of a stopwatch to measure a time interval, I will observe your time as moving slower than mine by a certain factor. However, you in the train will feel your time as running in the normal way because according to the relativity principle, you are also in a perfectly valid state to carry out the experiments of physics in your own frame. I’ve provided the theoretical proof of time dilation directly due to the constancy of the speed of light below. You may skip it if you want to and scroll directly on to the following section.
-[Theoretical proof for Time Dilation]-
Picture, a light pulse clock. It consists of two mirrors at each end. A light ray leaves one mirror let’s say, mirror F and strikes mirror G, then reflects and comes back at mirror F. This is one cycle of the light ray. The length of the light clock is l, so in this case the light ray covers a distance of two times l. Then the speed of light is given by c = [(2l)/t] where t is the proper time. This is the time as measured in a frame of reference where events F and G occur at the same point in space. Suppose this light clock is placed vertically in a frame of reference, call it S*. The light clock remains at rest with respect to this frame. A person in a different frame S which is in uniform motion with respect to S* will see the vertical path of the light beam’s complete cycle as elongated and tilted due to it’s relative motion. So the length that the light beam covers from this observer’s perspective is no longer 2l but greater than that. Applying Pythagorean theorem to calculate this elongated length L, we get
L² = (uT/2)² + l² where u is the relative speed of the frame S*
L = [(uT/2)² + l²]^½ Putting l = ct/2 we get,
L =[(uT/2)² + (ct/2)²]^½ .
We know that light’s speed has to remain invariable. So L is also = (ct/2). Hence, (ct/2) = [(uT/2)² + (ct/2)²]^½ .
In the above equation the lengths have been substituted and now what we do have is the time T which is the elongated time of events F and G in S*, as observed from S and, the time t, which is the proper time of events F and G in S*. After simplifying the above expression for T in terms of t, we finally get
T = [t²/(1-u²/c²)^½] which is the time dilation equation.
This equation suggests that as u approaches c, the denominator becomes 0 and then, the limit T = t²/0 = approaches infinity. Hence if we could move at the speed of light, time dilation will be infinite. But that’s quite impossible. This result sounds contradictory so we can conclude that nothing can ever catch up to the speed of light. But for sufficiently closer values to c it is obvious that T (Time dilation) will give larger and larger values. This T is the stretched time (slowed down time) as seen by the observers in relative motion. Also, for extremely small values of u, the denominator remains sufficiently close to 1 which asserts that T ≈ t (T is approximately equal to t) and this is where the Newtonian domain emerges from within special relativity. For slow speeds, the relativistic effects are negligible and under such conditions, Newtonian mechanics serves the purpose well.
-[A contradiction emerging out from the Newtonian domain]-
Truly, time dilation seems go against common sense, but common sense is simply based on everyday experiences in everyday life, we never get to deal with such high speed motion. Furthermore, Time Dilation in special relativity is not only theoretically convincing but has also been proved experimentally. In normal circumstances, Newtonian physics works quite well which involves the assumption that time in both reference frames is the same throughout. This is what Newton actually thought. According to him, there was an absolute universal time, running in the universe free from any intervention. To picture how Newtonian concepts contradict the experimented special relativity, think of a spaceship that is moving towards the right with speed v. It fires a light beam which has speed c, towards right. Now Newtonian mechanics predicts that due to the combined effect of their velocities, the resultant speed of the light beam must become (c + v) which asserts that the cosmic speed limit c can be trespassed. But we know that electromagnetic waves move at a constant speed c no matter how we look at them. So qualitatively, this is what actually happens. In this case, special relativity tells us that the spaceship’s time will just slow down by the right amount so as to decrease it’s speed in a way that due to the combined effect of spaceship’s velocity, the speed of light will still remain equal to c (remains constant). The time in the spaceship’s frame gets stretched just by the right amount as needed in order to keep the speed of light constant with respect to us and with respect to the spaceship itself. The clocks inside the spaceship in this case, run slow. The spaceship moving at such a high relative velocity will hence work just like a time machine. Being very precise, time dilation also occurs on slower speeds but in this case, the time difference turns out to be just approximately a millionth of a second which is totally negligible. So in that case T ≈ t (This is also theoretically shown above, in the paragraph regarding the mathematical proof of time dilation).
-[Further experimental evidences of Time Dilation]-
Not so long ago, another famous experiment was conducted for the verification of time dilation. Two identical atomic clocks were synchronized and one was kept on earth, while the other was flown in a jet plane at a constant velocity for several hours. Later, on comparison it was found that the one which was kept on earth was ahead in time as compared to the one flown in the jet plane, by a difference of just a millionth of a second. The clock in motion experienced a slowed down time. A negligible difference but yes at a relativistic speed (a speed comparable to that of light), this time difference can be stretched up to a hundred years (This can also be verified using the same Time Dilation equation derived above). Furthermore, the Large Hadron Collider (a particle accelerator) located in Geneva, Switzerland, which is one of the most expensive experiments ever designed in the entire history of scientific research, is used to study the constituents of atomic particles. By applying electric fields, it accelerates the charged particles up to 99% of the speed of light in underground constructed tunnels as long as 25 kilometers. When these particles collide, due to the sufficiently high energy of motion, they get broken down into their constituent subatomic particles. The subatomic particles formed in normal conditions, must destroy themselves nearly the moment they are created. But it’s due to time dilation that these particles live up for a long enough time relative to us, so that their trajectories are observed and data is derived from those trajectories through calculations. This is again analogous to the Muon rain experiment as discussed above.
I really hope that at some point in our technological evolution, we will achieve such relativistic speeds by maximizing power efficiency and by minimizing the resistive effects, in technologically and structurally advanced vehicles. Since we have come so far from steam engines to bullet trains, there is no reason to assume that further improvement is impossible. Just in the last century, humans have advanced at a faster rate than they ever did in the entire history of their existence. It is obviously very likely that in the next 400 years, the face of Earth will be unexpectedly different with human capabilities far beyond our current approximations.