A Crash Course in Relativity

[Simon Bridge]

Several threads have shown a need for something like this.

Introduction:
The purpose here is to produce a simple introduction to special relativity in a way that can be understood, with some work, by folk who have only high-school maths and basic Newtonian physics.

Given this, there will be things left out or perhaps not quite as well described as they may be. Since this is an interactive forum, I am sure that others will be quick to point these things out.

Mostly, though, I'd like to hear from those who normally struggle with this and how they are coping with the methods shown here.

Caveat: there ain't no such thing as a free lunch - be prepared to work at it.

.....

1. Wierdness occurs without invoking Einstein...

Newtonian Relativity

two observers pass each other (Anne and Bo) and a stationary object (a power pole).

According to the power pole;
Anne passes at speed u
Bo passes at speed v

According to Anne
Bo passes at speed u+v
The pole passes at speed u

According to Bo
Anne passes at speed u+v
The pole passes at speed v

Note that Anne and Bo will disagree on the speed of the pole.

One consequence of Newtons Laws is that you cannot tell what speed your doing (though you will notice accelerations).

However, we are used to being able to tell who is doing the moving. In this case, common sense tells Anne and Bo that the pole is stationary and they use this to determine their "true" speed (with respect to the pole). This will give them a picture of the world which both of them will agree with.

This agreement is reached because of two observations.
1. The pole is stationary with respect to the ground - which common experience says is also stationary.
2. Both Anne and Bo are exerting themselves. If they were "really" stationary, they argue, they wouldn't be getting so tired.

Removing these references: put all three on identical platforms which are in motion (so Anne and Bo do not need to exert themselves and cannot tell if the platform is working), and remove line of sight to the ground/background references.

In this case, there is no objective physical experiment which can be performed during the motion which will inform Anne, Bo, or the pole, which is "really" moving. The situation is identical to the case where (with respect to the ground) Anne has speed 0, the pole has speed u towards Anne, and Bo is chasing the pole at speed w=u+v. There is nothing Anne or Bo or the pole can do to determine that this is not the case.

To be continued:

 

[Simon Bridge]

... still talking classically:
Light
I'll be talking a lot about light beams, here's light in the Newtonian context:
A light beam (this need not be a laser; a bright lamp at the focal point of a converging lense will suffice) shone from behind Anne and in the direction of her motion can have it's speed measured by each observer. From Newtonian relativity: if the pole measures a speed of c, then Anne sees c-u, and Bo sees c+v ... however, there is a problem with this.

In the special case that u=c (Anne moves at the speed of light) then Anne could observe that the light is stationary. In this event, she would observe the beam as a standing electromagnetic feild whose amplitude varies, sinusoidally, in space. Upon observing such a weird feild, she would deduce that she is travelling at the speed of light[1]. However, knowledge of ones speed is forbidden in Newtonian relativity...

There are philosophical fudges around this. But more importantly, when the experiment is actually carried out, each observer actually measures the same speed for light. All observers will agree about the speed of light, even though they disagree about the speed of everything else.

The historical experiment most quoted is known as the "Michealson-Morely" experiment. Here, interference methods are used to measure the speed of the Earth wrt the Universe. The text-book result was that the Earth is stationary wrt the Universe.

Perhaps the Earth is the center of everything?
Perhaps the Aether changes it's refractive index close to moving objects so that the Michealson-Morely Apparatus regesters no difference? (i.e. lightspeed is not a constant wrt the Universe, and the expermenters had discovered a way of detecting Aether!)
Perhaps lightspeed is constant for all observers?

The third accumulated evidence from other experiments, and was the least unweildy mathematically. Further, Aetheric description increasingly lead to questions about how the Aether could have two different lightspeeds in the same region of space. As the required, yet undetectable, properties of Aether multiplied, Occams razor starts it's ruthless cull.

The "all observers" explaination is currently the accepted one. Aetheric theories still abound, usually among theists and proponents of perpetual motion.[2]

---------------
[1] The speed of light with respect to space/time or the Aether. The constancy of the speed of light has been well known. But constant with respect to what? Postulating an absolute reference (the very fabric of the universe itself, against which all things move) this question is answered... sort of. By observing this singular EM feild, presumably with instruments, Anne knows her speed with respect to the Universe. By knowing everyone elses reports of the speed of light, she can deduce everyone elses speed wrt the Universe. And it is this knowledge which is forbidden under Newton. So, Newton must be wrong somehow. The actual observation that light-speed is the same for everyone, regardless of their motion, suggests quite a lot of work is needed. Or, perhaps, everyone is actually stationary and all motion is an illusion?

[2] If you think relativity is weird, just take a gander at the consequences of various Aetheric theories. After that, relativity comes as a relief similar to the adoption of decimal currency.

 

[Simon Bridge]

... introduced the constancy of the speed of light, but still in the classical realm. (Classical mechanics is used to measure the speed of light - this is just a startling result.) Staying with classical realms, but feeling on shaky ground:

Simultaniety:
If two events are simultainious, then they occur at the same time. Easy.
However, I may observe distant thunder to come some seconds after the lightning flash. Or I see smoke from a starter's pistol before I hear the "bang". In these cases, and many more, two events which are simultanious to some other observer are not simultanious to me.

However, I would like to assert that the flash and the thunder are "really" simultanious.

The situation becomes very clear when considering observations of cause and effect. It is possible for me to be wounded by a high-speed bullet and then hear the shot. From this observation, and it is repeatable, the bullet caused the shot. However, I would tend to think of this the other way around regardless (and despite the pain). Anyway, I'd like to blame the gunman - if me getting hurt caused the shot then he isn't to blame.

So, we can modify the definition to read that events are simultanious if an observer on the spot thinks they are.

Yes but...

It is possible to arrainge for two events in different places... who can be said to be "on the spot" in this case? Perhaps the observer needs to be exactly half way between the events? But if one event is a sound and the other a flash of light...

(I had a recent experience of observing a thunderstorm some 4 sound-seconds away. Sometimes two lightning flashes would occur 4 seconds apart... as a result, I would sometimes hear thunder at the same time as I saw a lightning flash! A creepy feeling.)

Hopefully you are getting the idea that we have to very careful about how we determine that two events take place at the same time, especially if they occur in different places.

Perhaps we can exploit the fact that all observers agree about the speed of light?

A careful definition of "simultanious" would go like this...
For any set of events, we need to know the time that the light from each event got to the observer, and the distance to the position of each event. From this, the observer can deduce the time of each event. Those events with the same time (after the flight-time of the light is accounted for) are simultanious.

Aside: it has been argued, most recently by Charles Fort[3], that light does not travel. You switch on a lamp and the room is illuminated. This would mean that the event of a flash of light would be simultanious to all observers and nobody would be able to go faster than the speed of light. This would also eliminate the objections to Newtonian relativity given above.

Notice: so far I havn't introduced Einstein or anything that is outide of normal classical mechanics. Even the speed of light can be verified without resorting to relativity or quantum mechanics.

---------------
[3] New Lands - though Charlies aim was more to expose wooly thinking and complacency in the scientific community of his day rather than to expound a serious theory. The challenge is to overturn the idea (speed of light = infinity) in a non-esoteric way that a layman can understand. (The observed behaviour of the Galilean moons, or the use of rapidly spinning wheels, is deemed too complicated.)

 

[Simon Bridge]

So now we've seen one way of using the constancy of light to clear up ideas about timing. That's probably the hardest bit to think about. We are now motivated to see if there are other consequences... go through this slowly, you'll probably want to write out the equations longhand. (If any are unclear I can LaTeX them for you... just say the word.)

Enter Einstein.
If the speed of light is the same for all observers, then observers moving at different speeds will disagree about the time that things happen. The argument proceeds from classical mechanics as follows:

Set up the observers carefully as before. This time Anne is in a box of height y moving at speed v with respect to Bo, who is outside. In the box is the light-beam projector from the last example, and a mirror is mounted on the ceiling. Both Bo and Anne can see the light apparatus.

Their watches are synchonised so that at t=0 for both of them, the light beam is switched on. Now we get to the fun bit.

The apparatus is set up so that, according to Anne, the beam fires virtically, hits the mirror, and reflects down, then strikes the floor after time a. The light, as it reflects off dust particles in the air, looks like a single beam. So:

a = 2y/c (from speed=distance over time)

Bo is also watching. According to him, the box has moved between the initial firing and the reflection and hitting the floor at time b. The beam looks like a triangle[4].

The box moves a total distance of x=bv in this time. This is the base of the triangle. The light traverses the upper slopes; a total distance of d=cb. Notice that b > a because Bo sees the light traverse a longer distance than Anne. (Remember, both agree about the speed of this light. No relativity has been invoked at all. This is a direct consequence of the classical mechanics and the observed constancy of the speed of light.)

Having observed this, it would be nice to work out the relationship between a and b.

Treat the triangle observed by Bo as two right-angled triangles back-to-back.

For each, the base length is bv/2, the height is y and the hypotenuse is cb/2. By pythagoras:

(cb/2)^2 = y^2 + (bv/2)^2 (messy, so I'll tidy it up: multiply through by 4)

=> (cb)^2 = (2y)^2 + (vb)^2 (subtract (vb)^2 from both sides, leaves the y term by itself for later)

=> (2y)^2 = (cb)^2 - (vb)^2 = (c-v)^2b^2

(the point of the last step was to separate out the time term (b) so it can be used for comparison.)

from Anne, 2y = ac so that:

(ac)^2 = (c-v)^2b^2

This can be written in the form a = kb if k^2 = (c-v)^2/c^2

So if Anne's watch ticks off 1 second, Bo's watch will tick off 1/k seconds. This effect is called "time dilation".

The proportionality, k, is usually represented by it's inverse and assigned the greek letter gamma: gamma = 1/k

It is also useful to represent speeds in terms of the speed of light (0.998c for atmospheric muons). So defining w=v/c, gamma^2 = 1/(1-w^2).

So text book tend to write the time dilation relation as follows:

b = gamma.a

All the other relativity transformations follow from this one.

eg. Length Contraction:

According to Bo, the box has moved a distance x = vb, but according to Anne, the ground (and by common sence, the box) has moved a distance z = va ... these lengths clearly disagree. Since we know the relationship between b and a we can find the relationship between the lengths.

b=ga and b/x = v = a/z (where g = gamma)

so - ga/x = a/z => x = gz ... so when Anne measures 1 meter along the ground, Bo measures (between the same two point's on the same ground!) gamma meters ... that is, moving observers see a shorter length. If Anne measures out 1m on the floor of her box, Bo will see that as 1/g meters.

Note: this is not an optical illusion. Anne's 1m length isn't any more "real" than Bo's 1/g meter length here. i.e. Anne could be holding paddles 1m apart so they strike Bo in the head, one after the other. The time between head-thwacks will be consistent with a seperation of 1/g for Bo and a seperation of 1m for Anne. As you can imagine, Bo is utterly convinced.

(Bo could reason thus: "I see 1/g meters but I deduce from relativity that it is really 1m, so I can afford to wait a bit longer before I duck." The subsequent whack on the head will correct any such notions.)

Anne is similarily convinced from the experience of malicious glee. That's little sisters for you.

The relativity introduced here is Special Relativity. From this we get the total energy of a particle to be E = gamma.mc^2, which shows that even with no motion and at reference potential the particle has energy E=mc^2 (the mass-energy relation).

And that is really all there is to special relativity.

---------------
[4] This is quite in keeping with Newton. The box gives the light some horizontal component to it's velocity when viewed from Bo's POV. If Anne had tossed a ball, what appears a straight up-and-down motion to her would look like a parabola to Bo. Classically, light moves in lines, so Bo sees a triangle.

 

[Simon Bridge]

So now I have, hopefully, shown special relativity as a consequence of Newtonian realtivity and the observation that the speed of light is constant.

General relativity is harder - however, some appreciation is possible if you consider that no observer can tell the difference between an acceleration and gravity.

Anything else?

 

[.13.]

riangle observed by Bo as two right-angled triangles back-to-back.

For each, the base length is bv/2, the height is y and the hypotenuse is cb/2. By pythagoras:

(cb/2)^2 = y^2 + (bv/2)^2 (messy, so I'll tidy it up: multiply through by 4)

=> (cb)^2 = (2y)^2 + (vb)^2 (subtract (vb)^2 from both sides, leaves the y term by itself for later)

=> (2y)^2 = (cb)^2 - (vb)^2 = (c-v)^2b^2

(the point of the last step was to separate out the time term (b) so it can be used for comparison.)

from Anne, 2y = ac so that:

I like your presentation and by first glance didn't notice anything wrong with it. My knowledge of these things is limited though.

But I noticed a typo. That emphasised bit should be (c^2-v^2)*2b^2. Right?

Edited to add:

from Anne, 2y = ac so that:

(ac)^2 = (c-v)^2b^2

This can be written in the form a = kb if k^2 = (c-v)^2/c^2

So if Anne's watch ticks off 1 second, Bo's watch will tick off 1/k seconds. This effect is called "time dilation".

The first emphasis is the previous typo haunting. And the second emphasis should be (c^2-v^2). Right?

 

[epepke]

So now I have, hopefully, shown special relativity as a consequence of Newtonian realtivity and the observation that the speed of light is constant.

General relativity is harder - however, some appreciation is possible if you consider that no observer can tell the difference between an acceleration and gravity.

Anything else?

Nice one. You've done the equation route; I prefer geometrical arguments for tyros.

But I am going to steal from you the idea that "weird things happen without Einstein." I came up with another example.

Love and marriage may not go together like a horse and carriage, but juggling and unicycles do. Practically everyone who learns to juggle, at first, throws the balls forward, because their brains are wired to think that they'll catch up. Of course, they have to throw them up.

People's brains just aren't wired for relativity, even the Galilean kind. They're wired for walking and running. Too bad it was so long ago that we swung through the trees; we might remember something about inertia.

 

[Jimbo07]

I like your presentation and by first glance didn't notice anything wrong with it. My knowledge of these things is limited though.

But I noticed a typo. That emphasised bit should be (c^2-v^2)*2b^2. Right?

The first emphasis is the previous typo haunting. And the second emphasis should be (c^2-v^2). Right?

You're right.

The argument (removing the k step) should go:

[latex]$(2y)^2=(cb)^2-(vb)^2\\
(2y)^2=b^2(c^2-v^2)\\
setting $2y=ac$ gives\\
$(ac)^2=b^2(c^2-v^2)\\
divide through\\
$a^2c^2/(c^2-v^2)=b^2\\
therefore\\
$b=a/sqrt(1-v^2/c^2) or\\
$t=t'/sqrt(1-v^2/c^2)$[/latex]

or, and here simon is correct, b=gamma*a. Gamma is (by definition) > 0, so for any delta t' measured by Anne, Bo's will be a longer delta t. BTW, Anne's is called the proper time for the light beam fired off in Anne's reference frame (that is, separate in time, but in the same place).

 

[Soapy Sam]

I often think most of high school dynamics could have been better taught by piling the class into a van and driving around the playground. You quickly acquire an intuitive grasp of concepts like acceleration.

Just a pity we don't have an FTL starship to hand.

 

[Simon Bridge]

.13. and Jimbo07: you are correct

Jimbo07: a wee LaTeX lesson ... if you write text in math mode, spaces won't be rendered. If you want to include text and math in the same line you go like this:

setting $2y=ac$ gives:

... when you stick this inside the LaTeX bv boxes, it comes out as:
[latex]setting $2y=ac$ gives:[/latex]

 

[Simon Bridge]

The equation route is nice and descriptive. Geometry is best with pictures. However, I'd be interested in seeing a geometrical route for this which is suitable for beginners.

After all the above, the next step is to introduce space-time graphs (assuming d-t and v-t graphs have already been covered). They are constructed exploiting the discussion on simultaniety.

This can be quite useful in illustrating the difference between a Lorentz transformation and a rotation.

General Relativity would normally start, after some exercizes and a breither, with the introduction of the metric tensor. Quite a jump but there is a way...