In our previous episodes we have discussed the notion of length and time. Now it’s time to start writing some equations. You might have noticed that the title of the post has the letter c prominently displayed. In physics letters usually stand for variables or constants in a given situation. The letter ‘a’ usually stands for acceleration, ‘F’ for force, ‘E’ for energy or electric field, ‘P’ for pressure, ‘V’ for volume, and you might have noticed that there is a pattern of naming variables in a mnemonic way after the initial of the word you are describing.
If you were doing a physics alphabet, you would start with ‘a’ is for acceleration, ‘b’ is for belocity (the number of b’s you can type on your keyboard per second, not really a physics term), and ‘c’ is for cookie (you know the song). Incidentally, even though physics does not happen without coffee, cookies are also an important part of the activities that take place in a physics department. Cookie time is a time to get together and catch up on what’s going on and of course, free cookies are a must. Now, back to the letter ‘c’. It stands for the speed of light, so this post will be about the special theory of relativity. This is one of the cornerstones of modern physics. WARNING: LONG POST WITH EQUATIONS, jump to the next red piece of text for some conclusions if you want to skip the argument.
So let us start with the notion of speed. You probably have noticed that the cars you drive have speedometers and that if you are traveling too fast a policeman might stop you. I’ll leave the description of how a radar gun works for some other day. The notion of speed is how much distance can you travel in a given alloted time. In the international units system, it is usually written in meters per second. Of course, in cars, it is given in kilometers per hour or miles per hour depending on where you live. The letter we use to describe speed is ‘v’. This one stands short for velocity.
The typical equation describing a problem involving velocity is
Which tells you that if someone gives you a car at velocity and an amount of time equal to , then you can cover a distance as described by the equation. Many freshman problems in physics involve manipulating the symbols of that equation to compute the velocity, the distance, or the time given some other information.
Now, a little bit more work shows that if you have two cars moving at each other at speeds and , then the speed at which the distance between them shrinks is . If one is chasing the other one, the speed at which the distance between them shrinks is . This is true for an observer that measures both and . So far so good. This is not mysterious at all. And even though sometime physics problems seem to be designed to confuse the student, it pretty much boils down to those equations.
What would someone driving the cars see? Would they agree with you that the relative velocity between them is either of these numbers? The answer is actually no. Although if the cars are moving slow enough, they would need really good speedometers to tell the difference.
To understand why this somewhat bizarre conclusion follows, we need to ask a different set of questions. The question we will ask is does light travel at some given speed, and is the speed the same in all directions? This question was answered experimentally in the affirmative, in the experiment of Michelson and Morley. This is not what people were expecting. People thought that light was pretty much like waves in water, and that if the earth is a boat moving through the water, we would notice that the speed of the waves was different in different directions. Here is a nice link I found with some simulation of how the MichelsonMorley experiment was supposed to measure the speed of earth relative to this ‘water’. The official name for it was the aether, a substance filling all of space whose ripples were what we call light.
It was found instead that no matter in which direction you point the light, it always, invariably, travels at the same speed. It seems that Einstein arrived at this conclusion independently just based on understanding the theory of electromagnetism better. It has been speculated often on whether he knew or not about the experimental result of Michelson and Morley before he publishes his papers in 1905. He postulated that ‘c’, the speed of light is universal, no matter what state of motion you are in. One can ask, what are the consequences of this?
Well, let us build a clock that uses light. We just put a pair of mirrors at some fixed distance from each other, and we bounce light back and forth between them. We will record times at both places. If the distance between them is , then the time elapsed for one full bounce is
One of the assumptions of physics is that if you can do something once, then you can do it many times. So let us make a factory of these clocks and let us have as many as we need. These clocks being ACME brand (the one of the WB cartoons), will perform as specified with arbitrary precision and they will be perfect. They will work just as well when you make them move with respect to one another and if you rotate them. Seeing as the speed of light is the same for everyone, everyone will agree that these clocks are doing the same.
The time each clock measure in each cycle is the same, always, and that can be broken arbitrarily precisely into smaller fractions of the given by means of placing more mirrors in the middle as often as we need them. At least in principle, we can measure time arbitrarily well with these devices.
So let us consider two of these devices (labelled by A and B. If you prefer to name them after people, call them Alice and Bob, I will do so), with perpendicular arms and in motion relative to each other as shown in the figure. The relative velocity of these Alice and Bob will be .
What does Alice see, when she looks at Bob’s clock? Since the clock is moving, she will see that as the light goes up the vertical arm, it also goes some horizontal distance. The up and down motion being symmetrical, she would conclude that it takes half the time to go up, and half the time to go down.
However, for the horizontal line, she would notice that the light is chasing the mirror, so the time to get to the right should be the apparent length divided by , while the time to move from right to left should be the apparent length divided by . These two are clearly different. Notice that I am using the fact that light travels at speed c. It’s just the chasing and running away from that I am taking into account inthese equations.

Thus, she must conclude that the time of arrival on the right is different than the time of arrival on the upper mirror
. But wait a minute! For Bob, these two events happen at the same time, seeing as he is carrying both clocks and they have the same arm length!
The conclusion is that the notion of simultaneity is nonuniversal. It depends on your state of motion.
But if Bob finds that the light should return to the initial point at the same time from the vertical as from the horizontal clock, then we must conclude that Alice saw the same thing as well. The reason is that Bob can have a set of fireworks fired if these two events coincide, and both Alice and Bob would agree on whether the fireworks were fired or not. Being at the same place at the same time is universal. Otherwise it would make no sense: the history of the world would not be consistent with your neighbor if you move a little bit. That has never happened that we know of.
Something that happens someplace at some time is called an event. All observers must agree that events happen. They might not agree on the time that their clocks measure, but they will agree on the fact that the events happened and that a consistent history can be put together.
Alice would also find that the upward arm of Bob’s clock matches her own. This is easy to establish.
She and Bob both carry a marker at the top end of their clocks. If the bottom of the clocks coincides at some time, then at that same time Alice would mark a dot on Bob’s upper arm and Bob would do the same.
If Alice finds that she marked Bob’s arm higher than his, then by symmetry Bob would figure out the same with respect to Alice, and again, the consistent history would not allow them to disagree. We find that having either mark the other at a different length is an untenable position.
Alice would find this way that the time that it takes for Bob’s light to go up and down is longer than the time it takes on her clock, because the light is also moving horizontally. Thus, two events that happen in the same place in Bob’s look of the world (the base of his clock), take longer in Alice’s point of view. This phenomenon is called time dilation. After a bit of algebra and trigonometry, one finds that
where is the time that Alice observes, when Bob sees . More algebra shows that the horizontal arm of Bob’s clock has apparent length
This phenomenon is called length contraction.
What can we conclude? Well, for one, simultaneity depends on your state of motion. One says this by saying ‘it is relative’. Also, the length of a rod and the time something takes is relative. Unfortunately, taken out of context, this is taken as `So everything is relative and maybe we should not even bother’, or it is used to make statements stating ‘relativity tells you that there is no morality because it all depends on your point of view’ and a whole bunch of other nonsense, which is fine if one is doing sociology, but don’t blame space and time fer cryin out loud!
As a matter of fact, Einstein’s theory of relativity is the opposite: it is about how to reconcile measurements between two different observers in motion so that all of them agree that a particular history happened. They all agree on events. What this synchronization of data between observers entitles goes under the rubric of Lorentz transformations. In this sense, the name ‘theory of relativity’ is not that well chosen. It should be the theory of absolutism and the tyranny of history. Also notice that the equations make no sense if the observers are moving with respect to each other faster than the speed of light (one would find square roots of negative numbers for measuring something quite real: a time in a clock). This can really be understood by realizing that one can not go faster than the speed of light. No one. No matter what anyone tells you, in the theory of relativity this can not happen. This is a bummer for Science fiction becoming a reality, as if you have an intergalactic empire and you get attacked at the other side of the Galaxy, it will take you thousands of years to get the message and send reinforcements, and by that time it’s too late.
Now, I mentioned some cars above and their relative speed. You realize that their relative velocity can not be just . All you have to do is assume that their apparent velocity from where you are standing is larger than the speed of light. But light emitted from car one will make it to car two even when they are moving apart. Thus, the relative velocity that they themselves see is smaller than the speed of light.
The modern viewpoint is that space and time are part of one bigger structure called spacetime. Time and space mix when you move through them. They are unified mathematically into one bigger entity. Thus, we can say that they are made from the same thing, yet they are very different: it is obvious from our experience. Also, the universality of the speed of light can serve as a universal conversion factor between time and length. Thus you can use seconds to measure distance. If one is pedantic, one would call them lightseconds: the amount of distance traveled by light in one second. One hears units like this a lot more often in Astronomy, where measuring distance in lightyears is quite the norm. Phew. I’m exhausted. I need some cookies.
I’ve nothing of relevance to add to this post (other than… hurray, accessible physics! I will be passing your blog address on to all of my nonscienceeducatedbutinterested friends). Just wanted to let you know I (and friends) are very glad you’ve decided to join the blogosphere.
Don’t forget the naughty bits! For velocities at an arbitrary angle theta,
u_parallel = (u’_parallel + v)/(1+(v dot u’)/c^2)
u_perp = u’_perp/(gamma_v(1+(v dot u’)/c^2))
http://arXiv.org/abs/0706.2031
MichelsonMorley on steroids
Rules arise from symmetries, observations from symmetry breakings. All the real fun is in the footnotes.
Hi,
I wonder if you could comment on the following which has bugged me for some time:
It is stated that the speed of light in a vacuum is a constant.
It is also known that the speed of light in a vacuum is directly related to the permittivity and permeability of free space – which I think of in my mind as the ability of free space to “hold” an electric and magnetic field.
From this, I get the following:
1. It takes an electromagnetic field a finite time to dissipate in free space – which sort of makes intuitive sense as long as you don’t go down the “what’s keeping it there?” road.
2. You would expect this rate of dissipation to be universal and not to rely on the observer’s speed.
Therefore, would it not be more correct to talk of the universality of the permittivity and permeability of free space and the consequence of this upon the speed of light?
Hi Bill:
That is a good question. I would like to say that the notion of permitivity and permeability of space is in a certain sense a legacy concept in physics (much like backwards compatibility in software). Before electromagnetism was unified and before relativity was formulated, these were useful concepts. In the end, they just describe our normalization of units for what we call charge and magnetic field and electric field.
The modern viewpoint is that the speed of light is not just about light, but it affects everything: all matter is subject to it. In that sense it is more universal than just light. Then the notion of constant permitivity and permeability of space are a corollary of that fact in that the electromagnetic field should be intrinsically compatible with this (symmetry) structure of spacetime.
This reminds me a little bit of the age old question: what came first, the chicken or the egg? Evolution tells us that it is the egg, unless you define egg to be an egg laid by a chicken, in which case it is the chicken.
I hope this helps.
Thanks, that clears thing up for me.
I should explain that I’m an electrical engineer, so we started with Maxwell’s Equations and moved on from there which has obviously moulded my thinking since – I’ll now look at it from the other side.
On a related note, I think that I understand that the magnetic field is merely a manifestation of relativistic effects upon the electric field. That is, a magnetic field manifests itself due to the apparent – or is it real? – increase in electric charge density that is present due to relative motion past it – if you know what I mean.
Is my understanding correct?
Bill:
I don’t know. I don’t think of it in those terms because one can have a magnetic field without an electric field. I’m sure that there are situations where that is a convenient point of view, but I wouldn’t qualify it as an ‘intrinsically correct’ statement.
I prefer to think of the unified electromagnetic field as a whole entity, where electric and magnetic fields are convenient labels for some components in a given frame of reference.
Gee, frustratingly I follow your explanation up until here:
“Alice would also find that the upward arm of Bob’s clock matches her own. This is easy to establish.
She and Bob both carry a marker at the top end of their clocks. If the bottom of the clocks coincides at some time, then at that same time Alice would mark a dot on Bob’s upper arm and Bob would do the same.
If Alice finds that she marked Bob’s arm higher than his, then by symmetry Bob would figure out the same with respect to Alice, and again, the consistent history would not allow them to disagree. We find that having either mark the other at a different length is an untenable position.”
I know nothing is more confusing to a smart person than someone who doesn’t understand something, but if anyone would care to reword the above / explain it differently I’d be grateful — I just can’t really follow what’s being described, but it’s the crucial, final logical bridge to the whole point of the post, time dilation and length contraction.
Cheer,
Oss
Hi Oss:
I agree that it was not the best explanation (some important details are skipped sometimes because we think everyone already gets them).
So let us imagine a situation that would make it better. Assume that Alice and Bob are cycling on a tightrope, each of them carrying a balancing beam and that each of them is at the exact center of their balancing beam. Let us also assume that Alice and Bob are moving towards each other, each with the same speed and that you are overseeing the circus.
The idea of this situation is that clearly Alice and Bob appear symmetrically. When they cross each other, let us assume that they are carrying objects that need to be transfered between the ends of the beams and that one of them has some hooks (let us say Alice), while the other has some rings that need to be put in the hooks.
From your point of view, when they cross, Bob can clearly put the rings in the hooks and he does it for both at the same time so he does it and clearly everyone in the Circus applauds.
Now, look at it from Alice’s point of view. If the horizontal bar that Bob is carrying appears shorter than hers, he will miss and Alice’s history would be inconsistent with yours. Also if it is larger. But if you already agree that the trick was a success, it must be the case that Bob’s balance beam is the exact same length as Alice’s as far as Alice is concerned.
Now, do some hocuspokus and imagine that the beams are magically converted into the mirror clocks I described. I know in the picture it does not seem that the vertical clock are symmetrical, so that might be confusing you. But then move them so that they are centered and maybe the rest of the argument will make more sense.
Put in other words: there is no Lorentz contraction in directions orthogonal to motion.
Let me know if this helps.
[...] I discussed previously, once you find that the speed of light is the same for everyone, your units of time and length are [...]