One hundred years ago this month, twenty-six-year-old Albert Einstein published a paper entitled "Zur Elektrodynamik bewegter Körper*"* or "On the Electrodynamics of Moving Bodies". As we all know by now, 1905 was Einstein's *annus mirabilis*, the miraculous year in which he published four papers in the *Annalen der Physik. *The first was a paper on the photoelectric effect; the second on Brownian motion; and the third, which we have already mentioned, spelled out the ideas which would come to be known as special relativity. In case you are less than sure why Einstein's name has become a metonym for extreme intelligence, consider that there is broad consensus among physicists that any one of these three papers *by itself* would have been more than enough to win Einstein a Nobel Prize. The fourth paper, by the way, used the axioms of the third to derive a nice little result equating energy and mass: E = mc^{2}, probably the most famous equation of all time. Not only that, he would certainly have won another Nobel for general relativity, which he published a decade later. In other words, you can safely think of Einstein as someone who, in a fairer world, would have been *at least* a four-time Nobel winner. As it is, the Nobel committee cited only the photoelectric effect when they awarded him the prize in 1921.

Einstein's results were disseminated and understood so slowly (especially in the English-speaking world) that when Sir Arthur Eddington lead an expedition to prove general relativity correct by showing that the light from stars near the Sun in the sky would be bent by its gravity (during a solar eclipse in 1919 when you could actually see stars close to the Sun), and a journalist asked him: "Is it true that only three people in the world understand relativity?" Eddington reportedly responded, "Who's the third?"

This year has been declared the World Year of Physics in commemoration of the centennial of Einstein's *annus mirabilis*, and in that same spirit, I would like today to attempt to give you a sense of what the theory of special relativity (SR) is, whose 100th birthday we celebrate this month. Most of you have at least some vague idea of what SR implies: you have heard that time slows down when you start traveling very fast (near the speed of light); that lengths contract; and almost everyone knows the Twin Paradox, where one twin travels out into space at high speed, then returns, say 6 years later to find that her twin on earth has aged 39 years while she was gone. You have probably also heard that Einstein is responsible for inextricably entwining space and time into spacetime. I will explicate all these aspects of SR, and I will not do it by using crude analogies, which tend to confuse more than they illuminate; instead, I will use the actual math so that you fully understand the beauty of this theory. Wait! Don't stop reading just yet. The math required is no more than simple high school algebra, so if you remember how to do that, stay with me. If the sight of even the simplest equation makes you tremulous, then I can only say: learn some math! Einstein himself famously stated that "The presentation of science to the public must be made as simple as possible, but not more so," and we cannot but follow his dictum here, where our very aim is to celebrate his science. (The beauty of SR lies in the incredible *conceptual* leap which Einstein made. The mathematics is relatively (!) straightforward, and this is what makes my elucidation of it possible. The mathematics of general relativity is much more advanced, indeed far beyond the abilities of most people who, like me, are neither mathematicians nor physicists. Even Einstein needed some help from mathematicians to work it all out.)

THE BACKGROUND

A century ago, this was the situation: Galilean and Newtonian physics said that any descriptions of motion by any two inertial observers (for such observers, bodies acted on by no forces move in straight lines) in uniform (not accelerating) relative motion are equally valid, and the laws of physics must be exactly the same for both of them. Bear with me here: what this means is, for example, if you see me coming toward you at a speed of 100 mph, then we could both be moving toward each other at 50 mph, or I could be still and you could be moving toward me at 100 mph, or I could be moving toward you at 30 mph while you are coming at me at 70 mph, and so on. All these descriptions are equivalent, and it is always *impossible* to tell whether one of us is "really" moving or not; all we can speak about is our motion relative to each other. In other words, all motion is relative to something else (which is then the inertial frame of reference). So for convenience, we can always just insist that any one observer is still (she is then the "frame of reference") and all others are in motion relative to her. This is known as the Principle of Relativity. Another way to think about this is to imagine that there are only two objects in the universe, and they are moving relative to one another: in this situation it is more clearly impossible to say which object is moving. (Think about this paragraph, reread it, until you are pretty sure you get it. Just stay with me, it gets easier from here.)

At the same time, James Clerk Maxwell's equations of electricity and magnetism implied that the speed of light in a vacuum, *c*, is absolute. The only way that this could be true is if Maxwell's equations refer to a special frame (see previous paragraph) of reference (that in which the speed of light is *c*) which can *truly* be said to be at rest. If this is the case, then an observer moving relative to that special frame would measure a different value for *c*. But in 1887, Michelson and Morley proved that there is *no* such special frame. Another way of saying this (and this is the way Einstein put it in 1905) is that the speed of light is fixed, and is independent of the speed of the body emitting it. (The details of the Michelson-Morley experiment are beyond the scope of this essay, so you'll have to take my word for this.)

Now we have a problem. We have two irreconcilable laws: 1) The Principle of Relativity, and 2) The absoluteness of the speed of light for all observers. They cannot both be true. It would be another eighteen years before a young clerk in the Swiss patent office would pose and then resolve this problem. Here's how he did it: he asked what would happen if they *were* both true.

Next, I will show how the various aspects of SR fall straight out of the assumption that both of these laws are true. I will focus in greater detail on the slowing down (dilation) of time, and then speak more briefly about length contraction, and the intertwining of space and time.

TIME DILATION

As I have mentioned, Einstein began by assuming that the following two postulates always hold true:

1) The Principle of Relativity, and

2) The speed of light will always be measured as

cby all observers

Now, keeping these in mind, let us consider a simple mechanism that we will call a light clock (shown in Fig. 1). The way it works is this: the top and bottom surfaces are perfect mirrors. The distance between the top and bottom mirror is known exactly. The light clock's period is the time that it takes light to go from the bottom to the top, and then to come reflected back. Since the mirrors are perfect, light will keep on bouncing back and forth like this forever. All observers can build identical clocks with exactly the same distance between the two mirrors, ensuring the same period. And since the speed of light is always *c*, and the distance between the two mirrors can be measured precisely, we know exactly how long one "tick" or period of the clock is in seconds.

Let us now say that there are two observers, each of whom has such a clock. If one of them is moving past the other at a velocity *v*, something close to the speed of light, then the first observer, F, will see the second observer S's clock as something like what is shown in Fig. 2. Of course, by symmetry, and the principle of relativity, S will see F's clock the same way. Take a little time to look at Fig. 2 and convince yourself of this. (This is basically like thinking about a man moving a flashlight vertically up and down on board a train; if the train is stationary relative to you, you will see what is shown in Fig. 1; if the train is moving by you, you will see what is shown in Fig. 2. It should be quite obvious once you try to imagine it.)

Since S's clock seems to be moving to F, it will seem to F that the light travels a longer path than just the vertical distance between the two mirrors, because after the light leaves the bottom mirror, the top mirror keeps moving to the right, and the light beam travels a diagonal path up to where the top mirror has moved to. (Imagine the whole apparatus moving to the right as the light beam goes up from the bottom mirror, or look at Fig. 2.) Since the speed of the light must still be measured as *c* by *both* observers, and according to F, the light beam is traveling a greater distance, it must be taking longer to make the trip to the top mirror and back. Therefore, according to F, S's clock is ticking more slowly, and vice versa!

Someone might object that this is a special kind of clock, and maybe we could construct a different type of clock that would not slow down when seen speeding along relative to some other observer. This cannot be true. The reason is that if we were able to construct such a clock, it would violate our first postulate, the Principle of Relativity. Remember that in saying that only relative motion is physically significant, we are insisting that nothing done by S can tell her whether it is she who is moving past F, or vice versa. Suppose that two different types of clock were synchronized (one of them a light clock of the type we have been describing), then both of them are sent off with S at high speed, if they do not behave exactly the same way and were to fall out of synchronization, this would tell S that it is she who is really moving, and this contradicts the first postulate. All clocks must therefore slow down in the same way when they are observed in relative motion close to the speed of light. In other words, this time dilation is not a property of any particular type of clock, *but of time itself*.

So how much exactly is S's time seen to be slowing down by F? To answer this question, consider the situation in Fig. 3. As shown, the light clock is moving to the right at a velocity *v*. At rest, the light clock has a period *T. *In half that time, when stationary, the light beam travels the distance *A*, but when moving, it has a slower period *T' *because it travels the distance *C* at the same speed.* *This means that the ratio of the distances *A/C* is the same as the ratio of the times taken to traverse them, *T/T'. *(At the same speed, if you travel twice the distance, it will take you twice as long.) So,

(1) A/C = T/T'

Now while the light beam travels the distance *C*, the mirrors have moved a distance *B* to the right. The ratio of these two distances *B/C*, traveled in the same amount of time, is the same as the ratio of the speeds with which the distances are covered. (Moving for a given time at double the speed just doubles the distance covered.) So,

(2)

B/C=v/c

Look at Fig. 3 again, and notice that the sides *A*, *B*, and *C* form a right triangle, so by the Pythagorean Theorem:

(3)

A+^{2}B=^{2}C^{2}

Dividing both sides by *C ^{2}* we get:

(A/C)+^{2}(B/C)= 1^{2}

Subtracting *(B/C) ^{2}* from both sides we get:

(A/C)= 1 -^{2}(B/C)^{2}

Taking the square root of each side we get:

A/C= sqrt ( 1 -(B/C)^{2 })

Now if we substitute for A/C and B/C from equations (1) and (2) above, we get:

T/T' = sqrt ( 1 -

(v/c)^{2 })

And finally, inverting both sides, we get:

(4) T'/T = 1 / sqrt ( 1 -

(v/c)) = γ (^{2}gamma-- the relativistic time dilation factor)

Since the speed of light is so high (186,000 miles per second or 300,000 kilometers per second), *gamma* is not significant at speeds that are common to our experience. For example, even at the speed which the space shuttles must attain to escape Earth's gravity (11 km/sec), *gamma* is 1.000000001. At fifty percent of the speed of light (0.5 *c*), *gamma* is 1.155. You can confirm these values by plugging in the speeds into the time dilation equation above. One way in which we know that Einstein was correct about time dilation is that particles with known half-lives decay much more slowly when they are accelerated to near the speed of light in particle accelerators. For example, muons, which have a half-life of 1.5 microseconds, are observed to decay in 44 microseconds on average in a CERN experiment which accelerated them to 0.9994 *c, *at which speed gamma can be calculated using the equation above to be 28.87. In perfect agreement with the theory, 1.5 microseconds multiplied by 28.87 comes out to 44 microseconds, exactly what is seen in the experiment. There are countless other very exact confirmations of relativistic time dilation effects.

LENGTH CONTRACTION

This time, imagine the light clock lying on its side (in other words, Fig. 1 rotated by 90 degrees counter-clockwise). Now the motion of the light pulse is back and forth in the same direction that the whole clock is moving. What happens this time? Well, as the light pulse leaves one mirror and heads toward the other, that mirror advances forward to meet it. This trip is shorter than when the clock is stationary. On the way back, though, the light pulse is chasing a retreating mirror, and the trip takes longer than it would in a stationary situation. This round trip period, T'', is longer than T' by the factor 1/γ. (This is similar to the case where an airplane traveling across the Atlantic with a steady headwind against it, and then returning with the same wind at its back, will take a longer time for the round trip than if there were no wind at all. I leave the simple math here as an exercise for the reader.)

Now if this were all there is to the story, the amount of time dilation would depend on the orientation of the clock relative to the direction of motion, but then this would violate the Principle of Relativity. What prevents this violation is a shortening of lengths along the direction of motion. The distance between the two mirrors would thus contract by the factor 1/γ, reducing T'' to the correct value T' as it should be. So, lengths are observed to contract along the direction of motion by a factor of 1/γ. Again, this only becomes noticeable at very high speeds, approaching *c*.

SPACETIME

Newton's notion of absolute space and absolute time are no longer valid for us. We have seen that measures of time are relative to the observer, as are measures of space. The good news is that different observers of the same reality *can* agree on something. And this is what it is: we know that

T/T' = 1/γ

Substituting for 1/γ from equation (4) and squaring both sides we get:

(T/T')= 1 -^{2}(v/c)^{2}

Multiplying both sides by *(c*T'*) ^{2}* we get:

(cT)=^{2}(cT')T'^{2}- (v)^{2}

Here, *v*T' is just the distance *L'* that the moving clock travels in time T'. Meanwhile the stationary clock doesn't go anywhere in time *T*, so *L* = 0, and by substituting that *L ^{2}* = 0 and

*L'*=

^{2}*(v*T'

*)*into the equation above, we get:

^{2}

(cT)-^{2}L=^{2}(cT')-^{2}L'^{2}

Here, finally, is a quantity that is the same for both observers. It is not a measure of time or a measure of space; instead, it is a *spacetime* measure. So we find that in the end, though observers cannot agree about measures of space or time by themselves, it is possible to weave them together into a spacetime measure that everyone does agree on. This is what is meant when it is said that space and time have become interwoven after Einstein.

The account I have followed in explaining special relativity is essentially that used by Richard Feynman (who invented light clocks as a way of explaining SR) and Julian Schwinger. The two of them shared the 1965 Nobel in physics with Sin-Itiro Tomonaga.

Thanks to Margit Oberrauch for all the light clock illustrations.

Have a good week!

My other recent *Monday Musings*:

Vladimir Nabokov, Lepidopterist

Stevinus, Galileo, and Thought Experiments

Cake Theory and Sri Lanka's President

One of the reasons it is so difficult for most of us to visualize relativistic effects is because the velocity light is so large, 186,000 miles/sec or 3 x 10(10) cm/sec. Also, wave effects inside an atom are also difficult to visualize because of the smallness of Planck's constant. There is a neat book which illustrates what our everyday experience would be like if these constants were about unity: "Mr Tompkins in Wonderland" by George Gamov. Also, all physicists did not share the kudos for Einstein. For example, read the the introduction in the Russian physicist's book "Relativity" by Fock.

Posted by: Winfield J. Abbe | Monday, June 06, 2005 at 06:08 AM

An excellent explanation of SR. Thanks for this brief but very clear elucidation. The last time I read about this was in Amir Aczel's book: The God's Equation, a few years ago. You have sumarized it in a short essay and it is simpler. What a nice way to celebrate the 100th year of this discovery!

Posted by: Tasnim | Monday, June 06, 2005 at 08:00 AM

Very nice Monday Musing. I'm saving up for a light clock.

Posted by: Levi | Monday, June 06, 2005 at 10:19 AM

Brilliant as this is, I still can't math my way out of a paper bag. But then again, I'm a math dummy and actually hopeless.

Posted by: J. M. Tyree | Monday, June 06, 2005 at 11:28 AM

As I imagined the light and movement of light and steadiness of clock, the concept of relativity got much more clearer and tooo very easily.

Its rightly said that Imagination is Power.

Posted by: AMOL KINGAONKAR | Wednesday, July 13, 2005 at 09:46 AM

I don't buy it. Once you start a clock you can't move it. There is absolutely no law of physics that would compel the beam of light to move with the mirrors. So the idea of starting two clocks and them moving them away is simply impossible because the bouncing beam of light will stay put and spill off the edge of the mirror.

What if an observer knows he's in motion? He's on a rail car and he sees the car moving along some tracks. How does he start his clock? He must shoot the beam of light at an angle such that the forward progression of the beam bouncing between the mirrors matches the forward progression of the rail car. At this point the observer on the rail car still sees a clock with what appears to be a beam of light bouncing up and down. However, the time taken to bounce between mirrors is longer than the distance between mirrors would dictate. This tells the observer that he must be moving, and he can use trigonometry to calculate his forward speed.

The observer standing by tracks will see the light bouncing between the mirrors at the same rate as the observer on the rail car. But he can clearly see the angle that the light is moving at. The observer on the rail car doesn’t see the light moving at an angle, but knows it is because that’s how he started it.

This must be so because a beam of light moves with absolute independence from its environment. Nothing can cause a beam of light that is bouncing straight up and down to move sideways (except, of course, for a very large gravitational force.) So if you want to bounce a beam of light around with mirrors you have to make sure that YOU go to where the beam of light dictates...not the other way around.

There are other explanations for the muons that don’t require the notions of relativity.

Posted by: Willy L | Friday, July 29, 2005 at 11:04 PM

I loved the explanation. I hope you don't wait until the anniversary of general relativity to write a short essay that will plainly explain that theory.

Posted by: Andrew | Monday, August 08, 2005 at 05:16 PM

I have to agree with willy l. Make the mirrors on the order of the width of the beam. Then move the mirrors to a distance of y=1lightsecond. Now begin moving the emitter and reflectors at a velocity of 1m/s. Assuming your light beam is much less than a meter wide, why would the light emitted at t=0 hit the mirror at t=1second. The mirror would not be where the emitter was initially pointed. Instead you would have to point the emitter ahead to hit the mirror where it will be 1 second in the future. Therefore your clocks timing is being set different than the clock in the "non-moving" frame of reference. The only way the light would not have to be pointed ahead is if there was an aether that traveled with the mirrors which was the anti-result of the M-M experiment.

Posted by: cwes | Friday, September 23, 2005 at 01:45 PM

Okay, Willy and Cwes, I'm going to explain this one last time: you are getting unnecessarily confused by something very simple.

Imagine the following: A man stands on a train going by you while holding a ball in his right hand in front of and above his head, while he holds his left hand in front of his stomach, palm up, directly below the other hand. Now he let's go of the ball. The ball will drop vertically straight down (as he sees it) into his left hand. But this is obviously not how you will see it. You will see the ball follow a parabolic arc (because it is being accelerated by gravity), moving sideways as well as down into his left hand. It is a matter of who's perspective the event is being described from. It is exactly the same if you replace his hands with mirrors, and the ball with a ray of light. To him, the light just bounces back and forth, up and down vertically between the mirrors. What you will see is the light traveling diagonally (since it is not accelerating but traveling with constant speed) to where the mirror has moved to. Do you get it? What you are asking is, in the earlier case, since his left hand moved while the ball traveled down, how come it didn't miss his hand? Think about it. Hope this helps.

Posted by: Abbas Raza | Friday, September 23, 2005 at 02:37 PM

I understand the physics of a ball falling in a moving object. The problem is that light does not require a medium to move in. If we could slow down time (as with a super high speed video camera) we would see the light begin leaving the source vertically, and continue vertically, of course the source is moving, so we will see the origin of the light moving with the source, basically leaving a \ not a / . If we were moving as fast as the speed of light then the light would make an angle with respect to the horizontal motion of the source of 45 degrees.

Why is this too hard for you? The light from our sun leaves 8 min. before it reaches earth. When we look at the sun, we are looking at where it existed 8 min before, not where it actually is located in the sky. The same principle applies to this light clock that has been proposed.

Posted by: cwes | Thursday, September 29, 2005 at 02:47 PM

You can think of it as the time lapse footage of the car passing by in the night. The camera captures the exact trail of the car where it was and where it ends up, not where it is going (though those speeds are hardly relativistic, so without traveling faster I guess the difference would be like a few nanometers difference between the two.

Posted by: cwes | Friday, September 30, 2005 at 05:35 PM

No, cwes, I am afraid you do not understand what you call the "the physics of a ball falling in a moving object." And what you do not understand is the Newtonian principle of relativity. The fact that you refer to "a ball falling in a moving object" is what betrays your misunderstanding: there simply is no sense in taking the train to be a moving object. The whole point of the principle of relativity is that you can choose a frame of reference such that the train is NOT moving, instead the Earth is speeding by it in the opposite direction. The laws of physics must stay the same in both cases, and they do. The ball just falls straight down. It is only if you consider the Earth still and the train moving, that you see the ball drop in a parabolic arc, moving to the right at the same time as down.

As for light not requiring a medium to move in, nor does a ball, for your information. What does that have to do with anything? We are obviously assuming a vacuum in all cases, for simplicity.

Look, I'll try to say it once again: if you use the camera that you propose, you must specify where the camera is. If it is on the train itself. It will record the ball (or beam of light) traveling vertically. If the camera is on the ground outside, it will record the ball falling in a parabola, and the beam of light traveling in a diagonal (because we are ignoring gravity in the case of the light beam, which is corrected in general relativity). By the time the light reaches the other mirror, the mirror will have moved over horizontally just enough so that the light beam will strike it dead center.

If you don't get it, I accept my failure to explain this to you. Maybe you could try some of the other explanations I have linked to. You seem to have some very fundamental intuition about how the light ray will behave, and I am sorry to say, it is wrong.

A last desperate attempt to get you to see what is happening: imagine shooting a gun while lying on the floor of the train at a small target directly above you on the ceiling of the train. To you, the bullet just leaves the gun and travels straight up and hits the target. But to someone standing outside (or a camera there) it seems like after the bullet exits the gun, then the target on the ceiling moves to the right, so how can the bullet hit it? It hits the target, because the outside observer will see the bullet travel diagonally upwards to the right and hit the target. This is just a matter of perspective, and there is no "reality" to the matter. Light does exactly the same thing in going from the lower to the upper mirror. All this is Newtonian/Galilean physics, and doesn't even have anything to do with Einstein's relativity. Please do slowly try to imagine what I am urging you to. I am pulling my hair out at this point, and don't know what else to do to try and make you understand. Sorry.

Posted by: Abbas Raza | Friday, September 30, 2005 at 06:46 PM

I think you are subscribing directly to the theory that has dominated for the last century, which we hold to be flawed because of a paradox. I want you to observe from a different point of view, that of the person on the train car. The problem is that time dilation and length contraction has to be true from both standpoints. The problem is that it can't be true from both standpoints, only one can have a clock moving slower than the other persons clock, unless the explanation is that they do not exist at all in the same space-time, which would mean that nothing in the entire universe exists in the same space time, so what prevents us from passing right through one another?

Hopefully you see the flaw.

I understand the theories of how the tennis ball will move, however I also know that should you drop the tennis ball out of a window of a moving car that the tennis ball will not drop straight down and travel straight back up to the hand that dropped it. Why? Because it moves through whatever medium is acting on it, in this case air. Light does not move through air, it travels through the space between molecules (effectively a vacuum). If you were to say that it moves through the molecules, then you would be talking about light scattering which does not follow a straight line path.

Think outside the box and free your mind from the books you learned relativity from and you too will see the problems.

Now, you can say it has to be true because experiments have used the theories to explain the actions of muons, and other particles, but obviously there could be some other answer. After all for centuries, the brightest and smartest scientists in the world explained the motion of the stars and the sun as revolving around the earth. Galileo and Newton et al corrected them. Now Einstein is correcting them both and saying that both are true and both are wrong. Well he can't be right.

Posted by: cwes | Monday, October 03, 2005 at 12:17 PM

I guess all of this really stems from the idea that you can't have it both ways. Either light can or can't gain momentum from it's source. If it can then light must be traveling faster than c, if it can't then the light clock can't work as proposed.

Posted by: cwes | Wednesday, October 05, 2005 at 09:33 AM

I agreed with Abbas Raza.

Everyone should not be confused by the fact that: how could the photon go diagonally between 2 mirrors?---(1)

From stationary frame (of the first observer), the photon does go straight between the 2 mirrors.

To answer (1), can I say that: for the second observer, the photon goes diagonally because his frame of reference is moving with the velocity v?

Thanks

Trieu

Posted by: Nguyen Thu Trieu | Friday, October 28, 2005 at 01:33 PM

I agreed with Abbas Raza.

Everyone should not be confused by the fact that: how could the photon go diagonally between 2 mirrors?---(1)

From stationary frame (of the first observer), the photon does go straight between the 2 mirrors.

To answer (1), can I say that: for the second observer, the photon goes diagonally because his frame of reference is moving with the velocity v?

Thanks

Trieu

Posted by: Nguyen Thu Trieu | Friday, October 28, 2005 at 01:34 PM

Perhaps I'm overlooking somethings, but I do not feel comfortable with the explanation. The computation of the time dilation is indeed straight-forward as it is based on simple geometry, but it is based on the fact that the photon's direction of movement is perpendicular to the direction of the movement of the object - this is how the light clock is positioned. Were we to place the light clock in such a way so that the photon's path is on the same axis as the object's (light clock's) movement, then we will find that there's no time dilation at all because the sum of the forward and the backwards photon movements between the two mirrors will be same as in a static situation. Furthermore, placing the light clock at an arbitrary angle with respect to the direction of motion will yield other slightly more complex formulas for the time dilation. Is this a paradox or I am missing something?

Posted by: evgueni | Thursday, November 17, 2005 at 07:17 PM

Evgueni,

The answer to your question is explained in the next section on Length Contraction. To recapitulate: if the time dilation were different depending on the orientation of the clock, we would violate the Principle of Relativity (we would know that we were moving). Since that cannot be (by our own two axioms), lengths must get shorter in the direction of motion. Do you get it? Thanks for writing.

Posted by: Abbas Raza | Thursday, November 17, 2005 at 07:44 PM

Abbas,

I am trying to get my head around the twin paradox, with spectacular lack of success, and am wondering if you could help. My problem is the changes of inertial frame, which are said to make the travelling twin arrive back on Earth younger than the remaining one. But imagine two spaceships A and B, stationary and back-end-to-back-end, with an explosive charge between them, and a very long rubber bungee attached to their noses. The charge is detonated, each goes off at speed v, A to the left and B to the right, till the bungee tightens, and they then both do a U-turn, returning back to become stationary at their point of departure, thanks to good shock absorbers. Since no direction in space is privileged, the acceleration effects will be the same for each, and will cancel out, meaning that thanks to time-dilation, each will be younger than the other. How come?

My other question is if there was a third spaceship C between the original two, and now two explosive charges, one between A and C, and the other between B and C, so A and B zoom of as before, but C remains stationary. If C sees both going off with velocity v, A to the left and B to the right, how do I calculate the velocity of A as seen by B (or vice versa)?

If you can help, thanks in advance.

Jeremy

Posted by: Jeremy | Thursday, December 08, 2005 at 02:36 PM

Maybe Einstein didn't win the Nobel prize for SR because the committee also couldn't see how the light clock could possibly work. I think the reason so many people can't see the failure of the light clock is because they do not realize that light travels without respect to the motion of its source--even the perpendicular motion of its source.

I thought I must be missing something, but now that I see the only way this writer can support the light clock is by turning it into a bouncing ball or a flashlight--objects the movement of which IS affected by the motion of their source--I am shocked. Einstein chose light for the very reason that it's motion is independent. But he didn't get that?? Unthinkable! If anyone can explain this very eloquently-posed paradox that this writer has clearly avoided like the plague, please post now!

Another correction this article seems to need is a mention that the suggestions that it is "impossible to say which one is moving" and that there is no frame "which can truly be said to be at rest" are wrong. By using the same quality of light's constancy, can't we say what is at rest? Something is at rest that light emanating undisturbed from reflects back to through the same distance to the same exact spot upon.

No?

Is this light clock thing the only proof that time is different from different viewpoints??

Thanks!

Posted by: motormanmark | Thursday, January 05, 2006 at 03:26 PM

no, no, no... (as to that last idea) because the light would prove only an inertia relative to the mirror you are reflecting off of.

Posted by: motormanmark | Thursday, January 05, 2006 at 03:34 PM

For all those who do not believe in or do not conceptualize the notions/postulates of SR, there is a simple scenario to consider.

Imagine you are observing a laser device, sitting at rest in the reference frame of the laboratory, and emmiting a red beam of light that travels through a vacuum tube. According to what have been said before, that is, the motion of the light beam is independent of the motion of the source, one would definitely observe a deflection of the light beam in the vacuum tube. The explanation for this senario is most intuitive, since there are three superimposing motions that the vacuum tube, together with the entire lab and the observer him/herself participate in. There is the rotation of the Earth around its axis, around the Sun, and its comotion with the entire solar system around the center of galaxy. Therefore, the light beam would definitely deflect in some crazy unpredictable direction in the tube, since its motion is supposedly independent of the source, in our case the laser device. However, this is not what actually happens in reality. The light beam ironically seems to be moving in a straight line in the tube, whatever the motion of the reference frame might be! The same thing happens with the light clock and the mirrors that move with respect to the observer.

There is no absolute frame of reference, at least in the universe we live in. In other words, there is no absolute rest or no absolute motion. The Earth is at rest when seen from the observer on its surface, but it is at motion when seen from an observer on Mars. Who is right about the state of motion of the Earth? Apparently, there is no physical evidence to prove that one observer is right and the other is wrong. The terms "rest" and "motion" can only be used when they occur with respect to some reference frame. Space itself cannot be a reference frame, no matter how counterintuitive that might sound to some people. A final comment: How could someone perceive one's state of motion is there are no reference points around? Obviously, in that case the terms "rest" and "motion" would make absolutely no sense.

Posted by: odysseus | Friday, January 27, 2006 at 03:10 AM

Thank you, Odysseus! I am eager to get this straightened out. I still do not understand, though. You wouldn't need a vacuum tube, as light is only minimally deflected by air particles. In attempting to debunk the idea that the motion of the light beam is independent of the motion of the source... dude -- light beams don't move. they go--that's all. you point--they go. once they're on their way, they just go straight. picture a lighthouse that you turn off just before it points at a ship. i don't care how fast that light is turning, if you turn it off just before it is directly pointed at the ship, the ship will never ever get a direct beam of light.

As for your example --light moves at such a rapid speed, it is impossible to detect its directional "tail" or "trail" (think "nude descending a staircase" to think what it might look like.) a laser in a lab is moving at, let's see.. about 900 miles an hour with earth's rotation? depending on the time of day, this might be boosted by its orbit another 66000 mph. as for its galaxial "comotion," i haven't a clue, but i'm sure we're still talking miles per hour.

now -- light is in a different world entirely at 186,000 miles per SECOND. picture an overhead freeze-frame of a gangster spraying a machine gun in an italian restaurant. if you could see the bullets in mid air, even though the guy was moving his torso, the bullets would still appear to go in a straight line, because they move so much faster than the guy.

As for your last point, we could construct an "rest" beacon just like the failed "light clock." we would have three sets of mirrors, each light laser going in a different of the 3 dimensional directions, all attached. we would run a fog machine beside it so we could observe the light beams and it would have to be thousands of miles long, so we could notice tiny changes. as long as the 3 beams of light reflected back constantly and perfectly, we could say the beacon is at rest.

of course, you could argue that the pathways light travels upon may be moving, but that would be a whole new concept, wouldn't it?

Posted by: motormanmark | Saturday, January 28, 2006 at 03:08 AM

I agree with Odysseus and in fact the comments relate directly to the Michelson-Morley experiment, the purpose of which was to deduce the motion of the earth through an absolute reference frame (comprised of the ether). It was precisely the failure of this and similar experiments that served as a catalyst to the Lorenz contraction and subsequently Special Relativity. The light clock demonstration is therefore coherent.

Posted by: Mark Lutton | Friday, September 29, 2006 at 10:04 AM

It would be just great if a supporter of this light clock notion could explain directly to the points. Why say you agree with something if you can't explain it? This is science, not religion. Why not just say, I agree because I don't like to hear that things beyond my scope of understanding are in disarray? Anybody with a coherent explanation?

Posted by: motormanmark | Wednesday, December 20, 2006 at 11:23 PM

Lovely article! Thanks for bumping it up again...

I'll point out that Einstein did not refer to the Michelson-Morley experiment in his work kbecause he did not have to. Being a theorist, he understood that if there were no preferred inertial frames of reference (as Galileo had proposed 300 years before in his famous Parable of the Ship), Maxwell's Equations would have to appear invariant under the Lorentz transformation and "c" would be the same in all inertial frames. As a result, all the time dilation and length effects come automatically.

Einstein would, of course, go on to show that even accelerated frames of reference could be shown to be equivalent in his General Theory of Relativity.

Posted by: Bill | Friday, March 15, 2013 at 02:42 AM

As Bill says, it's great to read it again, even every year.

I like the fact that Einstein would rather his theory be called a Theory of Invariance. This fits more with a central theme in his work: The Laws of Physics (Nature) are exactly the same for each observer, regardless of the position, momentum, and time relative to any other observer.

I started reading the popular books on Einstein and his theories in my third year of high school - "One, Two, Three,...,Infinity," "Einstein and the Universe," and other books. All I needed was algebra, and the Lorenz Transformations yielded to my study. I loved it.

A number of years ago I started on Quantum Mechanics, as I did for Relativity so long ago. In the same way, little by little, some small understanding begins to emerge. Each tiny step of a consciousness that I have learned something new and important is exciting and humbling at the same time. I'd say that it was better than sex, but then I'd be getting carried away with myself.

Posted by: Norman Costa | Friday, March 15, 2013 at 03:57 PM

Two other things I find fascinating with Einstein's Special Relativity.

1. The title of his 1905 paper on SR gives no clue, to the naive reader, as to the momentous nature of his work - "On the electrodynamics of moving bodies." Also, for such a 'game-changing' publication in the field of physics, there is not a single reference in this paper.

2. Friedrich Hasenöhrl published the formula for mass-energy equivalence - a very slight variation of it - nine months before Einstein. However, Hasenöhrl did not appreciate the full implications of the formula E = mc^2.

Posted by: Norman Costa | Friday, March 15, 2013 at 04:16 PM