Disproving the invariance of time…

Last time, I showed you a lot of stuff regarding reference frames, speed of light, observers, etc. on our quest to grasp the fraction of Einstein’s mind. I made you accept a lot of them without even giving you the courtesy of thinking what the heck all those math really meant. From today, I’ll clarify those one by one, so that you’ll finally be able to understand the mystery behind the speed of light. But, trust me on this. Pensiveness and intuition are something very difficult (or lousy) to speak of, in the theory of relativity which is simple but substantial…

Can we really “rely” on time?

Well, we saw Galilean relativity, when we convinced ourselves that space is something no one can rely upon (i.e) different observers disagree on the distance they measure. But time was considered absolute (i.e) an invariant quantity. This proposition was accepted by all scientists till the 19th century until Einstein came up and confused everyone*. The “variance of time” is something that’s really counter-intuitive. I agree with you. But, that’s the way nature has been constructed. Under such a circumstance, all we can do is find a carrot that makes this phenomena intuitive and more conceivable to our minds. And, we’ve found that. It’s a thought experiment.

[*]: Einstein was a great physicist. But, he’s not the only one behind relativity and the constant speed of light. Other contributors include Maxwell, Lorentz, Poincaré, and especially Michelson & Morley on their failure in detecting the luminiferous aether.

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This experiment is an usual one. Once you start reading special relativity in the internet, you’re more likely to find this example. It’s well explained in Wiki. I’ll go into that anyway…

Let’s imagine a clock. This isn’t an ordinary clock. It’s quite special (not because I stole it from Cassiopeia). We’re trying to disprove the invariance of time. Not the old pendulum, or other classic clocks, where we measure time by using motion of clock hands (which may bring back the issue again, euclidean distance measured depends on observers). We need something that we can rely upon. The only candidate here, is light which is a constant. Our clock is made of two perfectly reflective mirrors (in reality, all mirrors are reflective only to about 80%), just to ensure that the energy of light isn’t reduced after subsequent reflections. Alright, now a pulse of light is made to bounce between the mirrors (of course, in a straight line, or else it would fly out – severely hypothetical) which are separated by a distance of a meter. This bouncing is our measure of time. Light takes about 6.67 nanoseconds to bounce from both the mirrors. This corresponds to around 150 million reflections within one second (which is really an accurate clock, perfect for our experiment).

Slowing down the “time”…

Let the siege begin now. Okay, you and me are inertial observers (keyword – inertial, meaning there’s no acceleration). You’re at rest relative to me, while I’m moving relative to you. We’re inertial observers and both of us have this sophisticated wonderful light clock. Say I’m traveling at some velocity v. We don’t care how I accelerated from rest to achieve the velocity. We just begin our works by assuming that I’m going with some constant velocity. After all, it’s a thought experiment. Now, both of us measure the time taken by the clocks to tick from our frames of reference. Well, each of us have a rest frame. I mean, we carry a set of axes with us. With respect to that frame, our velocity is zero (and the vice versa). Only in that frame, we’re going to measure the motion of objects in the universe. For now, only two of us are in the universe.

It’s boring if we keep on measuring each others’ clocks. Because, we know it’s always gonna indicate the same 6.67 nanoseconds as one tick. So, you’re gonna sneak-a-peek into my clock. My rest frame (which has me & my clock) is moving relative to you. Now, you’re gonna find something interesting. My clock is moving slowly, relative to you.

Here, the GIF animations are perfectly synchronized (frames, frame rate, time step, etc.). Still, you’ll notice a time lag between both clocks. It’s not because they were loaded by your browser at different times. If that were the case, the clocks won’t meet the same point at the same time after a few seconds. But, that happens. Wait for a few moments and you’ll notice both the clocks reaching “1” at the same time and go out of sync again. Because, it represents the time dilation. I mean, (not really, but) that’s the time dilation you’d notice in my clock (top) relative to yours (bottom) if the thought experiment is carried out (with me going at 60% the speed of light, when time dilation is a factor of about 1.25).

How can light slow down?

You should pardon me for introducing these crude animations, because they have some flaws. Firstly, the usage of classic hand-clock. And next, the light pulse bounces only once and the clock indicates that as an hour. It may seem crazy. But, lemme define one unit of time (for the sake of bringing you this figure) not as one second, but the time taken for light to bounce off both the mirrors (which is hopefully 6.67 nanoseconds, but never mind, we don’t need those numbers) and that’s what my classic clock indicates. Now, you just ignore the minutes-hand (just like you ignore the seconds-hand in your everyday “time-checking”). Only the hour-hand shows the unit-of-time (defined above).

For our investigation today, we need only the moving clock. Because, that’s causing all these weird things. It should be noted that the mirrors would’ve moved a little bit when light starts bouncing because I’m moving with my clock. Well, you saw the visualization. Hence, the light pulse (as seen by you) traces an upside-down V-shaped path. Remember, that doesn’t happen in my frame. In my frame, the light moves just up & down. Recall the first postulate of relativity, that all physical laws are the same in inertial reference frames. In other words, you can’t do any experiment to determine who’s moving, and who’s not, in inertial frames. But, if I peek into your clock, I may see light tracing an inclined path. Because, you’re moving relative to me (Wiki has a nice GIF which illustrates how the light’s path inclines when viewed from different frames)…

Now, just by looking at the inclined path, one may easily declare that light has taken some extra time period to reach the other mirror, which may lead to the notion that light has slowed down as the clock moves. Well, how could that be? We’ve just declared that the speed of light is constant in all inertial reference frames. How would that just slow down?

It needs some not-so “grumpy” math…

Thumbs up for a lite math. Here, I’ve drawn an image, which doesn’t look bad. It’s analogous to light’s path. The three mirrors are drawn, just to represent that the mirrors have moved a little bit.

Okay, so the distance is simply one meter. The vT is the distance covered by the clock in my frame at a velocity v (which is also my velocity) within the time interval T as measured by you. And cT is the distance covered by light, as it traces the inclined path, which is also measured by you. Keep in mind that it’s you who see the inclined path. I see light doing the bouncing only vertically in my clock. In my frame, the time taken by light in the clock to cover the 1-meter separation is simply 1/c. Let this be t (time measured in my rest frame, by me). As it bounces two times, it becomes 2t=2/c

There you go, you get a right-angled triangle. Let Pythagoras work that out for you.

(vT)^2 + (1)^2=(cT)^2

\implies (1)=T^2(c^2-v^2)

\implies T= \frac{1}{\sqrt{c^2-v^2}}

That’s for the first bounce. Doubling the expression and rearranging…

2T= \frac{2}{c\sqrt{1-\frac{v^2}{c^2}}}

This large horror T is yours, while the simple one t is mine. Taking the ratios of both the time interval measurements…

\frac{T}{t}= \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\gamma

The thing \gamma which came out of the expressions is called the Lorentz factor, named after the guy who worked out the factor for the first time (like I said, special relativity has some nice history), but he thought it’s just math-play and not real. It’s Einstein who declared that the thing is real. The factor tells you how much a moving clock has slowed down as the velocity increases. The expression will now be reduced to T=\gamma t. The effect is shown to the left. Note the exponential curve. It shows that time dilation becomes significant only when the velocity nears very close to speed of light (probably greater than 0.5c)…

So, time dilation – Is it real?

Yep, it’s real. When I mean, “time is dilated”, I didn’t say that you’d see everything in slow motion (like those scifi movies). It’s not only the motion of clock or other moving objects that slow down, even the perception of your eye, the rate at which your brain functions, how fast your nerve carries information, the rate at which your heart beats, everything slows down. So, if you go fast, will you age less? Yep. But, you wouldn’t even know whether your time has slowed down. Like I said, if you experience that time slows down for you, then you’ll find out who’s moving and who’s not. The laws of physics are the same in every inertial frame of reference. Even the biological & atomic processes slow down, leaving you with no leftover evidence to conclude that time has slowed down for you…

Now, the real horrible counter-intuitive question jumps into the drama. I told you that I see the ticking of my clock (in my rest frame) moving only vertically. And, I’d see your clock slow down, since you’re moving relative to me. Again, that’s because the laws of physics are still the same for inertial motion. Just a food for thought…

Now, whose time really slows down? Mine or Yours?

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