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Does Motion Affect The Speed At Which Time Passes? (Part 1)
Our immediate answer to this question might very well be, “HA!! Of course not, fool!! Only a bonehead would ask such a boneheaded question, because we all know that time is an independent and absolute quantity that ticks away the life of the entire universe at a constant, unchanging rate! Now beat it, punk!”. But even before we react irately, let’s first scrutinize special relativity’s two fundamental postulates, upon which the then twenty six year old Albert Einstein built his remarkable, revolutionary, and deeply elemental theory, to find out if this seemingly absurd question has any merit whatsoever.
The First Postulate – “Play Ball!”
In plain English, the first postulate, also known as the principle of relativity, asserts that observers in a system which is moving in a straight line and at a constant speed – that is, moving with uniform motion – cannot feel or sense this motion. Even more important is the fact that there is absolutely no physical experiment that they can perform that will detect or reveal this motion. In fact, the observers have every right to claim that they are the ones who are at rest and it is the outside world that is moving instead. A good example would be the passengers in a smoothly moving train cabin, who can consider themselves at rest while the station platform, trees, and the rest of the world whiz past them. In short, the first postulate says that one cannot distinguish between a state of uniform motion and a state of rest, by any means – the two situations are on equal terms. We can go even further and declare that all of the laws of physics operate in the same manner for those inside the uniformly moving cabin as they do for those on the stationary train platform. Hence, if the people in the moving cabin and the people on the stationary platform were to set up and perform identical experiments, then the results of these respective experiments, as measured by those in the moving cabin and by those on the platform, would likewise be identical. Let’s try to understand and remember the first postulate, as we will use it with astonishing results.
Indeed, this circumstance is a familiar experience. Whether we are riding the luxurious Orient Express (as just described), cruising down the highway in a spacious motor home or SUV, or cruising at altitude on a plane; so long as the motion is in a straight line and at a constant speed (i.e., the motion is non-accelerated), we won’t be aware of it. If we disregard the odd small bump or two (the ride can’t be perfectly smooth), our drinks will pour into our cups in the same way that they do at the dinner table. We can move about while consuming our drinks with the same ease as we do in our houses. Heck, if there is enough space, we can even pull out a baseball and some mitts, play a game of catch, and the ball would behave no differently than it would if we were playing catch at the park! We can consider ourselves at rest while the outside world is in motion instead. The laws of physics will operate in the same manner for us, in what we deem to be our stationary frame of reference, as they will for those in the outside world, who, from our point of view, are in uniform motion with respect to our stationary frame of reference. This principle of relativity was actually first theorized centuries ago by Galileo, one of the greatest physicists of all time, who expressed it quite eloquently in his book, Dialogue Concerning the Two Chief World Systems, published (way back) in 1632:
“…Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal… When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still…”
Special Relativity Is Absolutely Relative
One important aspect of Einstein’s special theory of relativity is its assertion that all uniform motion is relative. “What does this statement imply?”, we wonder. It implies that whenever we measure or observe the speed of a moving object, it is always relative to some reference point, or reference frame. For example, consider a lone car traveling at a constant speed down a straight country road. We are standing by the road, and we are stationary with respect to it. So, when we say, “the car is moving at a speed of 55 miles per hour”, what we are really saying is that “the car is moving at a speed of 55 miles per hour relative to us and the country road.”. We are using ourselves and the ground as the reference point (i.e., or reference frame), and the car is moving at 55 miles per hour with respect to it. And indeed we already know, from the principle of relativity and its examples, which we have examined, that the driver, who is stationary with respect to his car, has an equally valid point of view. He can say that, “those people, the country road and the rest of the outside world are moving past me at a speed of 55 miles per hour relative to me and my car.”. In this case, he is using himself and his car as the reference point, or reference frame, and the outside world is moving past at 55 miles per hour with respect to it. We also note the fact that this relative speed of 55 miles per hour is the same for each point of view, that is, it is the same for us, using ourselves and the ground as our reference frame, as it is for the driver, using himself and his car as his reference frame. This must be the case due to the symmetry of the principle of relativity. As a matter of fact, special relativity goes on to proclaim that the statement of a speed without the inclusion or implication of an associated reference point or frame is a meaningless statement. As an example, let’s examine the motion of the famed Apollo 11 spacecraft. It is moving from the earth to the moon. But the earth and the moon are both moving around the sun. And the sun and other planets in our solar system are all moving around the centre of the Milky Way Galaxy. Our galaxy, in turn, is moving away from the countless other galaxies in the universe at incredibly high relative speeds, as the universe continues to expand – and all of these galactic relative speeds are of different magnitudes! So it is only natural for us to wonder, “what, then, is the true or absolute speed of the Apollo 11 !?”. Special relativity states that we can never know, and moreover, it states that there is no such thing as an absolute speed, that is, a speed with a reference point or frame that is at rest with respect to everything in the universe itself. But what we can know and say is that, “the Apollo 11 spacecraft is traveling at the speed v (say, 25,000 miles per hour) relative to the earth.”.
Relative Speeds – Don’t Mess With Anyone Who Drives a Ferrari
Having said this, let’s examine some of our intuitive or “common sense” notions about calculating relative speeds, before proceeding any further. For the following situation, we will use the ground, as we did previously, as our reference frame for the speeds involved, unless we explicitly state otherwise. So… suppose we are driving in our race car at 300 mph (better have our seat belts on) and we pass our friend who is standing at the side of the road. Our friend then pulls out his paint ball gun and fires, sending a paint ball that chases us at 500 mph. As the ball overtakes us, we will measure the speed of the ball, this time relative to our car, to be only 200 mph (i.e., 500 mph – 300 mph = 200 mph). Annoyed at our friend’s little “practical joke”, we decide to play chicken with him. So we turn our car around and barrel towards him at our Ferrari’s top speed of 400 mph. Sure enough, he fires another paint ball headed our way at 500 mph. Now, we will measure the speed of the oncoming ball, again, relative to our car, to be a whopping 900 mph (i.e., 500 mph + 400 mph = 900 mph). No problems here, right? We have simply used our logical, common sense method of subtracting and adding the speeds of the paint balls and our car, which were all relative to the ground, to figure out the speeds of the paint balls relative to our car (fortunately, our friend missed us, and we steamrolled over him, flattening him into a pancake. But we drove him to a nearby hospital, where they unflattened him, and he’s okay now). Good; armed with this truly logical technique for calculating relative speeds, let’s get back to examining the postulates.
The Second Postulate – What’s Light Got To Do With It?
While the first postulate seems almost “instinctive”, special relativity’s second postulate is awfully hard to stomach (we might even hurl), because it goes against every grain of intuition and “common sense” that we have about the concept of relative speeds, which we have just demonstrated. Again, in plain English, the second postulate asserts that observers in uniform motion, no matter what the relative speeds between them may be, will always measure the speed of light to have the same value of: 186,000 miles per second (that’s 7.5 roundtrips around the world in one second!). Let’s think about what this means and let it stew in our brains for a while. Moreover, to really drive the point home, we’ll use a Galactic Space Cruiser instead of the Ferrari and give our friend a beam of light instead of the paint ball, so that we can dramatically increase the velocities involved. So, what does the second postulate really mean?
It means that when we redline the engines of our Space Cruiser and zip away from our friend on earth at a speed relative to him, of, say, 100,000 miles per second, and if our friend then shines a super powerful flashlight in the direction of our ship’s departure, we will measure this beam of light overtaking our ship NOT at the expected relative speed of 86,000 miles per second (i.e., 186,000 mps – 100,000 mps = 86,000 mps), but at the unvarying speed of exactly 186,000 miles per second instead! If we now order our helmsman to come hard about so that our ship is now moving towards our friend on earth at 100,000 miles per second, and if our friend now shines the flashlight in the direction of our ship’s approach, we will measure the oncoming light beam approaching our ship NOT at the expected relative speed of 286,000 miles per second (i.e., 186,000 mps + 100,000 mps = 286,000 mps), but again at the unvarying speed of exactly 186,000 miles per second! What in blazes is going on here!? Why isn’t the speed of light changing relative to our ship? Isn’t this scenario an exact analogy of the Ferrari and the paint balls? What has happened to our seemingly foolproof method for calculating relative speeds!? It is as though light itself is unaware of our or anyone else’s motion! Yet experiment after experiment carried out over the years, the most famous being that performed by Albert Michelson and Edward Morley in 1887, unquestionably confirmed this constancy and invariance of the speed of light, regardless of any uniform motion of the observers (the observers in this case not being the crew of a space cruiser, but instead, a bunch of horror stricken physicists panicking over why light coming from different directions and hitting the moving earth as it orbited the sun, was always measured to travel at the unwavering speed of exactly 186,000 miles per second!) – undeniably a shocking discovery, to say the very least.
Physics Rehashed – Not Again!
While most physicists of his time tried vigorously to invalidate this very result, primarily because it was in total and absolute contradiction to the intuitive or common sense notion of adding and subtracting relative speeds, it took the genius of Albert Einstein to theorize that the constancy of the speed of light is, in fact, a physical law of nature itself. He grasped that this speed is one of the universe’s very special ‘numbers’ (today, the speed of light in the vacuum of space is considered by physics to be one of the universal constants of nature, and is given the symbol c, i.e., c = 186,000 miles per second, or 300,000,000 meters per second). Einstein realized that light (and the laws of electromagnetism, since light itself is an electromagnetic wave) must also obey the principle of relativity, because, just as with all the other mechanical laws of physics, we cannot use the speed of light as a measuring stick in any experiment whatsoever, to determine whether we are in uniform motion or at rest. And why not? Because, as we have just discovered, we will always measure light to approach us or move away from us at the constant speed c, regardless of our own state of uniform motion or rest. And so, with Einstein’s skillful and inspired formulation of his special theory of relativity in 1905, in which he spelled out the new laws of relativistic mechanics for physics, came the beginning of the end for Galileo’s and Newton’s celebrated and long lived principles of classical, or absolute mechanics. Now, prepare to take a trip that needs no heavy narcotics, because we’ll soon discover, as Einstein did, that carrying just these two postulates to their logical limits reveals that ‘fact truly, truly is stranger than fiction’, and how!
A “Thought” Experiment – Does Motion Affect The Speed At Which Time Passes?
First, we need to build a very unique kind of clock. It consists of two square mirrors mounted onto a thin metal shaft so that they are facing each other, with the distance between the reflective surfaces of the mirrors being a height h. To keep time, we let a single particle of light, called a photon, bounce straight up and down between the lower and upper mirrors, with the photon moving at, of course, the speed of light c (186,000 miles per second, or 300,000,000 meters per second). Here is a diagram of our brand spanking new “photon clock”.
We define a single “tick” of our photon clock to be equal to one round trip journey made by the photon between the mirrors (as shown by the blue arrows), that is, the journey from the lower mirror to the upper mirror, and then back down to the lower mirror. The photon therefore travels a total distance of 2h in one tick of our clock, and since the photon travels at the speed of light c, the total time taken for one tick of our photon clock equals 2h / c seconds.
Just so there is no confusion about how this total journey time for the photon was calculated, consider this analogy: If we have to travel a distance d (say, 1000 miles) by car, and if we drive our car at a constant, unchanging speed v (say, 50 miles per hour), then the total time for our journey will equal d / v hours (i.e., 1000 miles / 50 miles per hour = 20 hours, or 72,000 seconds). That’s great mileage – gas ain’t cheap.
Now, if we construct our photon clock so that the distance, h, between the reflective surfaces of the lower and upper mirrors has a value of 5 feet (or 1.524 meters), then the total time taken for one tick will be equal to 2(1.524 m) / (300,000,000 m/s) = 0.00000001 seconds, or one–one hundred millionth of a second! Definitely a minuscule time interval, to say the least. Luckily for us, there is no time interval too small that can’t be measured by our brand new, state of the art Tyrex watch, a finely tuned marvel of timekeeping. We now synchronize our photon clock and Tyrex watch so that the secondhand of our Tyrex moves through exactly 0.00000001 seconds (also expressed as 10-8 seconds) for every tick of our photon clock. Next, we ask our friendly engineer, Scotty, to assist us. We give him a photon clock and Tyrex watch that are exact duplicates of the ones we have. He then synchronizes his new clock and watch in the same manner as we did ours, and hence both his clock and watch are now ticking at exactly the same rate as our clock and watch.
The experiment is just about ready to begin. “What experiment?”, we might ask, in case we have forgotten. And the answer is, the experiment that will prove conclusively whether or not motion affects the passage of time. But before we start, we provide Scotty with one extra timepiece that he synchronizes with his photon clock and Tyrex watch – an atomic clock (atomic clocks use the oscillations of individual atoms to keep time, much like grandfather clocks use the swinging of their pendulums. Atomic clocks are the most exact timekeepers ever conceived, accurate to within one second over 30 million years!).
At last, we can commence. We part ways with Scotty, and he boards a train, taking with him his photon clock, Tyrex watch and atomic clock, all of which are ticking in synchronization. Scotty’s sleek diesel hums down a smooth and straight set of tracks at a constant speed, so that Scotty and his clocks are moving with uniform motion. As for us, we are standing at the train crossing, with our photon clock and Tyrex watch also ticking in sync. When the warning bells sound, the safety poles come down, and the T.O. Special roars through the crossing at a speed equal to v relative to us. As Scotty’s passenger car rolls by, we see his photon clock inside, and we carefully observe the path that the photon takes as his clock ticks. What do we observe? Unlike our photon clock, in which the photon moves straight up and then straight down between the mirrors for each tick, the photon in Scotty’s clock must make two diagonal paths for every tick. “Why?”, we might ask. Well, because the mirrors of his photon clock are moving with the train (say, from left to right), we therefore see the photon in Scotty’s clock first travel diagonally to the right and up, in order for it to hit the top mirror, and then diagonally to the right and down, in order for it to hit the bottom mirror and register a tick. Now, we might ask, “Uuuh, but why a diagonal path? Why don’t we just see his photon travel straight up (or straight down), so that it misses the moving mirrors completely and flies off into oblivion?”. For the answer to this question, let’s put ourselves in Scotty’s shoes. To Scotty, he is the one at rest, along with his photon clock (and his Tyrex watch, and his atomic clock, and everything else that is in his passenger car…). Hence he sees his photon move only straight up and then straight down between the mirrors of his clock, for each tick. From his perspective, therefore, the photon must hit the mirrors of his clock. And from our perspective, the only way for his photon to hit the moving mirrors is to follow the diagonal paths described. Here is a diagram of the paths that each photon (i.e., ours and Scotty’s) takes, as seen from our point of view.
A Thought Experiment – The Million Dollar Question
It’s now time to ask ourselves the one million dollar question. Recall that before Scotty boarded his train, our and his clocks were all happily ticking in synchronization. So, from our vantage point standing at the train crossing, will we see Scotty’s now moving photon clock still tick at the same rate as our stationary photon clock? We remember that a “tick” equals one roundtrip journey that each photon makes from the bottom mirror to the top mirror, and then back down to the bottom mirror. We analyze the previous diagram for clues to the answer, and at first, we might reply, “Even though Scotty’s photon has a longer distance to travel than ours does to complete a tick due to its diagonal paths, his photon will receive a ‘forward speed boost’ from the moving train equal to the train’s speed v, which will be just enough to offset the longer distance that it has to travel, and therefore, Scotty’s photon clock will still tick at the same rate as ours. Now show us the dough, Gerrard!”.
If we study our previous diagram, then the first part of our answer is indeed correct. Scotty’s photon does have a longer distance to travel than ours for each tick, because unlike ours, which travels a total vertical distance equal to 2h straight up and down, his travels this same vertical distance and also the extra horizontal distance from left to right, which results in a total diagonal distance equal to 2d that is therefore greater than 2h. Stating it another way, if we notice that L is the total left to right distance that Scotty’s photon – and train – move during one tick of his photon clock, then d is the length of the longest side (i.e., the diagonal, or hypotenuse) of the right angled triangle whose other two (shorter) sides measure h and L/2. Hence d must be greater than h, so Scotty’s photon travels a longer distance than ours does, since 2d must then also be greater than 2h. Moreover, the relationship between d and h, as per the great mathematician Pythagoras (remember him?), is d2 = h2 + (L/2)2, and therefore, the total distance that Scotty’s photon travels to complete one tick, as observed from our point of view, is given by: Distance = 2d = 2 x sqrt [ h2 + (L/2)2 ].
Then what about the ‘forward speed boost’ given by the train? If his photon does in fact receive an extra amount of horizontal speed equal to v, which is the speed of the moving train, then we can show, using some vector algebra, that this extra speed does just offset the photon’s longer travel distance so that Scotty’s photon clock will still tick at the same rate as ours. But we need not carry out this further calculation nor even ponder it, because we know, from the second postulate, that light travels at one and only one speed, and that speed is c (186,000 mps). Light cannot be sped up (or slowed down) even one iota; light cannot be made to acquire or shed any amount of velocity via any object in (uniform) motion. So, Scotty’s photon cannot be given any sort of speed boost by the train whatsoever, and thus, contrary to our initial response, the answer to the million dollar question must be this: From our perspective, Scotty’s photon has to travel a longer distance to complete a tick than our photon does to do the same, but since his photon and our photon both travel at the same speed of light c, it will thus take Scotty’s photon a greater amount of time than it takes our photon, as observed from our point of view, to complete a tick, and therefore we will observe Scotty’s moving photon clock ticking at a slower rate than our stationary photon clock!
Our first deduction: Provided that they are constructed with identical dimensions, a moving photon clock will always tick at a slower rate than its stationary counterpart. Our second deduction: The greater the speed of a moving photon clock relative to its stationary counterpart, the longer the diagonal paths that the photon must travel to complete one tick, and therefore the slower the moving clock will tick. Maybe Mr. Gerrard Magueer can show us half of the dough instead…
A Thought Experiment – The Gazillion Dollar Question
And now, we ask ourselves the one gazillion dollar question: From our vantage point standing at the train crossing, we see Scotty’s moving photon clock ticking at a slower rate than ours. But will we see the hands of his Tyrex watch also slow down, so that it remains in synchronization with his photon clock? Uneasy about this question, we scrutinize the first postulate for clues to the answer. At first, we will reply, “Absolutely and positively not!! Get real, fellow! There is no mechanical, electromagnetic, chemical, nuclear, or any other kind of connection between his photon clock and his Tyrex watch. The slowing down of Scotty’s photon clock is just a consequence of the relative motion of his train and of the constancy of the speed of light, and hence his photon clock cannot be keeping the real time – it’s just the geometry of the situation! The real time is being kept by his Tyrex watch and his atomic clock, which must therefore remain ticking at their original rate, that is, the same rate that our stationary Tyrex watch has always been ticking at. Hell no, for the reasons cited, Scotty’s Tyrex will definitely not slow down as his photon clock does, and will thus lose its synchronization with his photon clock. Now show us our retirement greenbacks, fool!”.
A windbag of an answer, but one that is well argued. However, there is a very deep connection between his photon clock, Tyrex watch, and atomic clock. In fact, this connection also profoundly involves Scotty himself, and every other person and object on the train. But for now, we’ll stick with Scotty and his three timepieces. To grasp and appreciate this profound link, we must once more put ourselves in Scotty’s shoes. Again, Scotty has every right to claim that he is at rest, and it is we and everything else outside his train that are moving instead. Moreover, consider this scenario: Scotty decides to pull down all of the blinds of his passenger car’s windows because of the glaring sun, and as a consequence, he can no longer see the outside world. But then, being the immense klutz that he is, he stumbles while walking back to his seat, slamming his head into the floor and giving himself amnesia so that he does not remember boarding the train in the first place. When he finally comes to his senses, because his train is traveling flawlessly straight and smooth, and at a constant speed, he will not be able to tell whether his passenger car is in motion or at rest whatsoever. Furthermore, there is no experiment that Scotty can perform, in the whole of physics, that will detect or reveal whether he is in uniform motion or at rest. And why is this so? Because as the first postulate decreed from the very beginning of our discussion, all of the laws of physics behave in exactly the same manner for those in a state of uniform motion as they do for those in a state of rest. Mother Nature does not conduct herself differently depending on one’s state of uniform motion or one’s state of rest – the two situations are equivalent. Now then, let us lay down the law and answer this question correctly.
We have argued at the outset that Scotty’s Tyrex watch will keep ticking at its original rate and will not slow down, unlike his photon clock which has slowed down, as we have seen firsthand from our vantage point at the train crossing. This means that Scotty will also clearly see his Tyrex and photon clock fall out of their original synchronization, apparently for no reason at all! Since we have essentially attributed this breakdown in synchronization to the motion of his train, Scotty, being a very astute engineer, will also come to realize that the failure of his watch and clock to keep their synchronization can only be due to one thing – the fact that his train is definitely moving. And so, even with his window blinds down, his perfectly smooth and straight ride at constant speed, and his damned amnesia, Scotty will be able to deduce that he is unquestionably the one who is truly in motion, a deduction that is in utter and absolute contradiction with the first postulate, that is, the principle of relativity!
Let’s calm down and rewind our thoughts, however, because the first postulate must hold firmly with zero contradictions, or else all of physics is in seriously deep manure. Hence, contrary to all of our arguments, the only correct answer to the gazillion dollar question must be this: From our vantage point, we see Scotty’s moving photon clock ticking at a slower rate than ours. If his Tyrex watch does not also slow down, the resulting loss of synchronization would enable Scotty to conclude that he is unquestionably the one who is truly moving, violating the principle of relativity outright. Therefore, we must and will see his Tyrex watch also slow down so that it remains in synchronization with his photon clock, thus upholding the principle of relativity! Sorry, not able to cash any retirement cheques today…
A Thought Experiment – The Deep End Of The Pool
But what is Scotty’s Tyrex watch other than a collection of gears, springs, and dials? What is the ‘deep link’ that it shares with his photon clock? Since his watch has slowed down (as observed from our point of view), and clearly there is nothing special or magical about his Tyrex watch, then we are forced to conclude that any and all timekeeping devices on board Scotty’s moving train must also slow down by the same amount that his photon clock (and Tyrex watch) have slowed down, so that they all still remain in synchronization. From our point of view, then, we see that the rate at which the pendulum of a grandfather clock swings slows down, we see that the frequency at which a metronome ticks slows down, and, among other things, we see that the rate at which Scotty’s atomic clock ticks also slows down. But since his atomic clock uses the natural vibrations of atoms to keep time, then this means that these atoms themselves have slowed down their vibrations. And again, since there is nothing unique or magical about this particular set of atoms, we are forced to conclude that all of the atoms on the moving train – the atoms in Scotty’s body, the atoms in the other passengers’ bodies, the atoms in other organic materials such as food and plants, the atoms in inanimate objects such as the chairs and tables, the atoms that make up the train itself – have slowed down their natural vibrations. Moreover, these atoms have slowed down their vibrations so that they will still remain in synchronization with Scotty’s slowed down photon clock, his slowed down Tyrex watch, his slowed down atomic clock, and all of the other (slowed down) timekeeping devices on board. And therefore, as we observe all of these things happening on the train, we must conclude that the very deep and profound link that we have been pondering from the outset is the fact that: As seen from our point of view standing at the train crossing, we will observe that the passage of time itself has slowed down for Scotty and everyone and everything else on board the moving train! Let’s take a moment to thoroughly inhale this end result. Take a monstrous, deep breath in, then let it out, very slowly, and read through the experiment again, if we deem it to be necessary!!
Our first conclusion: Time itself slows down for objects that are in relative uniform motion. Our second conclusion: The greater the relative speed of a moving object, the slower time passes for the object, because the distance the photon must travel to complete 1 tick will increase as relative speed increases, and since light always travels at the speed c, then the photon clock (and all other clocks) will tick at a slower rate. Moreover, we also take note of the fact that the rate at which Scotty’s photon clock (and, of course, all of his other clocks) tick, as observed by us, is obviously in direct proportion to the rate at which time itself is passing for Scotty on board his train. For instance, if we observe Scotty’s moving clocks ticking at a rate that is 3 times slower than the ‘normal’ rate of our stationary clocks, then we must obviously conclude that time itself is passing 3 times slower for Scotty as compared to the ‘normal’ rate of time passage that we, as stationary observers, are experiencing.
And so, as we see Scotty’s photon clock complete each tick, we must still see the secondhand of his Tyrex watch move through its corresponding synchronized time intervals of 0.00000001 (i.e., 10-8) seconds, but to us, the secondhand will be moving through these intervals (literally) in slow motion, since time itself is passing at a slower rate on Scotty’s train, as observed from our point of view. Therefore, as we see the 10-8 seconds elapse on his moving Tyrex watch, we will see a corresponding time interval that is greater than 10-8 seconds elapse on our stationary Tyrex watch, because Scotty’s watch is ticking at a slowed down rate compared to ours, which is ticking at the ‘normal’ rate of time passage. The situation is pretty much analogous to watching a slow motion replay of, say, a few seconds of a pro basketball game. If the replay slows down the normal game speed by a factor of 3, then the players and the ball will move 3 times slower, and the game clock will tick at a rate 3 times slower than normal. It will still tick down by 1 second intervals, but for every 1 second interval that elapses on the slowed down game clock, 3 seconds of ‘normal’ time will have elapsed on our clocks and watches. Hence, 1 second of replay time has been “stretched out”, or dilated, over the course of 3 seconds of normal time.
Of course, what is happening on Scotty’s moving train is no visual effect – time has actually slowed down. But now, we might ask, “Then why don’t we observe this phenomenon for moving objects such as planes, trains, or automobiles?”. A logical question, with the answer being that time does in fact slow down aboard these vehicles, but since their speeds, relative to us observers, are so small compared to the speed of light (e.g., an airplane traveling at 500 mph vs. light traveling at 186,000 mps, or 669,600,000 mph!), the amount by which time is slowed is infinitesimally minuscule, and therefore virtually unnoticeable. Even the space shuttle, at its fastest, travels at a relative speed that is still way too slow for us to notice anything. As we will soon discover mathematically (okay, let’s stop hyperventilating, because the derivation isn’t as violent as we think), this phenomenon of time dilation, as it is formally called in special relativity, becomes perceivable only at relative speeds that are significant fractions of the speed of light.
A Thought Experiment – “Mr. Scott, You Are The Ultimate Cheapskate!”
Alright, then, with that said, let’s imagine that Scotty’s train is heading for the train crossing at a speed, again, relative to us, of 0.866c. At this phenomenal speed, the relativistic time dilation factor has a value exactly equal to 2. This means that for every second that we observe to elapse on Scotty’s clocks, 2 seconds of normal time will elapse on our clocks. Stating it another way, as observed from our point of view, for every second of normal time that elapses on our clocks, we will observe only ½ of a second to elapse on Scotty’s clocks. Hence, 1 second of Scotty’s time will be stretched out, or dilated, over the course of 2 seconds of our normal time. From our point of view, then, time on Scotty’s train will be passing at a rate that is 2 times slower than our normal rate. And so, if we could “peer” into his passenger car as it moves through the vicinity of the train crossing, we would observe that not only are his clocks ticking in slow motion, but in fact, everything is happening in slow motion – specifically, 2 times slower than normal!
Therefore, when Scotty decides to do a little relaxing and lights up a Cuban cigar, he will do so in slow motion – specifically, 2 times slower than normal (i.e., ‘normal’ meaning had his train not been in motion, or, stated more correctly, had his train been at rest relative to us, so that time on his train was also passing at the normal rate). The smoke from his cigar will rise to his cabin’s ceiling 2 times slower than it normally would. Also, it will take Scotty 2 times longer, as measured by our clocks, to finish off his cigar, because he will be puffing on it 2 times slower. The maitre d’ will open a bottle of the finest red wine for Scotty, all in ‘2x slow motion’. The wine itself will flow from the bottle and into Scotty’s glass 2 times slower, and the wine’s aroma will travel to Scotty’s nose 2 times slower than it normally would. Scotty will reach into his pocket, pull out his wallet, leave a measly tip for the maitre d’, and sip on his wine, again, all in ‘2x slow motion’. Scotty will read 2 times slower than his normal rate, he will write 2 times slower, he will think 2 times slower, and, along with all of his fellow passengers and crew, Scotty will age 2 times slower than we do! And all of this time dilation is due to SR’s very potent fundamental postulates, and the fact that his train is in uniform motion, relative to us.
Continue To Part 2
- Does Motion Affect The Speed At Which Time Passes? (Part 2)
The 1st in a series of articles on Albert Einstein’s Special Theory of Relativity