Special Relativity

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7/21/2019 Special Relativity

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Chapter 37 … sort of

Special Relativity

1900: Lord Kelvin Gives a Talk …

"Nineteenth-Century Clouds overthe Dynamical Theory of Heat andLight "

Problems with … – The Michelson-Morley experiment – Black body radiation

"Galilean" Relativity

Suppose you're playing pool on a train. – The train is traveling with speed v relative to the ground. – The ball is traveling with speed u’ relative to the train.

Obviously, the speed of the ball relative to the ground mustbe

Now, question: do the laws of mechanics work in the movingtrain? – Let's test three big ones: conservation of mass, momentum, and

energy.

Suppose one billiard ball collides with another. – Assume elastic, but don't assume the balls stay intact.

– Remember, u is the speed relative to the ground.

Do these same laws hold for an observer on the train?

Use u = u' + v to write conservation of momentum as

– Conservation of mass gets rid of the terms with v in them:

– This is conservation of momentum relative to the train.So, yes – the laws of mechanics work just as well on themoving train.

This is, in fact, what we now call Galileo's principle ofrelativity:

What about electrodynamics?

The laws of classical mechanics apply in all inertial

reference systems.

Recall that an inertial reference frameis one in which Newton’s laws hold – that is, it isn’t an acceleratingreference frame.

Consider again an experiment with a train.

Put some charge in the train … – Then, to an observer on the ground, there would be a magnetic field

generated. – But to someone on the train, there wouldn't be -- just an electric

field.

So it looks like electrodynamics doesn’t obey Galileo'sprinciple.

But wait! – What reference frame do we use to measure the velocities of

charges? The ground? – But the ground is moving, too …

Before 1905, physicists thought electromagnetic fieldsmoved through a medium called the ether . – So we must measure the velocities with respect to the ether.

Question: is the Earth moving with respect to the ether? – If so, how fast is it moving through? – We better find out!

How do we measure the "etherwind" speed?Michelson and Morley did anexperiment that measured thedifference in the speed of light indifferent directions.

The Michelson-Morley experiment:Suppose the apparatus is moving throughthe ether. – Then the travel times of light going to each

mirror and back will differ. – And we should get an interference pattern

produced.

Michelson and Morley found that thespeed of light was exactly the same in all

directions. – It's hard to understand how perplexing this

was to physicists back then; all other wavestravel with respect to a medium.

Can we explain this lack of speed difference of light? – Sure -- various explanations were proposed. – But they resulted in new predictions of electrodynamics that were

never found.

Here's a possibility: – Lorentz suggested that the ether made objects shrink a little bit. – If the effect was just enough, and in just the right direction, it would

compensate for the ether wind.

A final example: motional emf versus Faraday's law.

Consider our electrodynamic train experiment, except nowwe have a loop of wire.

First, a person on the ground watches the loop travel downthe track. – As it passes a magnet, the magnetic field exerts a force on the

charges in the wire. – We get a motional emf:

Now consider the point of view of someone on the train. – The loop is at rest, so no motional emf. – Instead, the changing magnetic field induces an electric field in the

loop, producing an emf

The important thing to realize is that, before 1905, physiciststhought that it was a coincidence that the two emfs were thesame.

Einstein's Postulates

Enter Einstein in 1905.

In relative isolation, he producedfour ground breaking papers inone year: – On the photoelectric effect (particle

theory of light!) – On Brownian motion – On the electrodynamics of moving

bodies (special relativity) – And the one on E = mc 2

What did Einstein do differently than everybody else? – The battle here was between Newtonian mechanics and

electrodynamics; he chose electrodynamics. – He took the Michelson-Morley experiment at face value -- the speed

of light was the same in all directions. – From two postulates (only!), he derived the special theory of

relativity: this described, essentially, the arena in which all physicalphenomena take place.

The Principle of Relativity: the laws of physics applyin all inertial reference frames.

The universal speed of light: the speed of light invacuum is the same for all inertial observers,regardless of the motion of the source, the

observer, or any assumed medium of propagation.

Actually, this follows from applying the firstprinciple to Maxwell’s laws of electrodynamics.

Consequences of the Two Postulates

The Relativity of SimultaneityConsider our train once again … – At the centre is a light bulb. Let's switch it on.

– An observer on the train measures two events:(a) Light reaches the front end of the train car(b) Light reaches the back end of the train car

– And finds they occur simultaneously .

The Relativity of SimultaneityWhat about an observer on the ground? – They'll find that event (b) -- light hitting the back end -- occurs first!

Two events which are simultaneous in oneinertial system are not, in general,

simultaneous in another.

This assumes thespeed of light is thesame for bothobservers.

Be careful! – We're not talking about the same thing as, say, thunder and

lightning. – We're assuming the observers are smart enough to take the time for

the signal to reach you into account. – Sometimes we'll need multiple assistants, each with synchronized

clocks, making measurements in various places.

The relativity of simultaneity is a genuine discrepancy

between measurements made by competent observers inrelative motion.

Time DilationNow let's consider a light ray that leaves the bulb and strikesthe floor of the car directly below.

– How long does it take light to make this trip?For an observer on the train, it's easy:

Time DilationFor an observer on the ground, the same ray must travelfurther because the train itself is moving.

– This distance is

– So the travel time of the light ray is

Time DilationThe two travel times are related by

– Thus the time elapsed by the same two events -- (a) light leaves bulb(b) light hits floor -- is different for the two observers.

– The interval recorded on the train clock, Δt', is shorter:

Moving clocks run slow.

Time DilationThe square root term is called the Lorentz factor :

– It'll show up again.

Time DilationThe effect of time dilation has been well tested. – Consider a muon.

– It decays, on average, after 2 × 10-6

seconds have passed. – But -- that's for a muon at rest. – One that is moving close to the speed of light will actually last much

longer -- because its moving clock is running slower.

Time DilationConsider a muon moving through the lab at 3/5 the speed oflight. How long does it last?

– Well, the Lorentz factor of the muon is

– So it'll live 5/4 times longer than it would at rest:

The Twin ParadoxIt goes like this … – On her 21 st birthday, an astronaut takes off in a rocket ship at a

speed of 12/13 c. After 5 years have elapsed on her watch, she turnsaround and heads back at the same speed to rejoin her twin brother,who stayed at home.

– How old is each twin at their reunion?

The Twin ParadoxThe traveling twin, of course, has aged 10 years; she is 31when she arrives home.

However, as viewed from earth, her moving clock has beenrunning slow by a factor

So the time elapsed on earth-bound clocks is

The Twin Paradox So her twin brother is 47 when she arrives home! – It's not just her watch that runs slow -- it's time itself. – So her biological processes are slower, too.

Okay, where's the paradox? – Well, let's look at it from her point of view, rather than her brother's

on earth. – She sees the earth fly off at 12/13 c, turn around after 5 years, and

return. – So the earth's clocks must be moving slower, not hers -- and so she

should be 16 years older, not younger.

The Twin Paradox What's the resolution of the paradox? – I won't tell you.

– Think about Einstein's two postulates, and see if you can come upwith the answer.

Lorentz ContractionBack to the train! Now we've set up a lamp at one end ofthe car, and a mirror at the other.

How long does the signal take to complete the round trip? – To an observer on the train, it's easy:

Lorentz ContractionFor the observer on the ground, it's a bit trickier. – The round-trip time will be

Remember, the same two intervals are related by the timedilation formula:

Lorentz ContractionInserting our expressions for Δt' and Δt into the time dilationformula gives us

Moving objects are shortened.

The Barn and Ladder ParadoxThere once was a farmer who had a ladder too long to storein his barn.

– Luckily, he knew some relativity. – So he had his son run with the ladder really fast , so that the ladder

Lorentz contracts to a size small enough to fit into the barn.

However, his lazy son argued that, from his point of view,the barn will be moving and so will become smaller. – Then there's no way the ladder will fit.

The Barn and Ladder ParadoxSo, who right? Will the ladder fit or not?

They're both right! There are really two events we're talking

about here:(a) Back end of ladder makes it in the door(b) Front end of ladder hits the far wall.

The farmer says (a) occurs before (b); the son says (b) occurs

before (a) -- and they're both right. – Remember, the order of events depends on the observer.

The Lorentz Transformations

To be careful about measuring things that move relative toeach other, we'll define an event as something that takesplace at a particular place ( x , y , z) and time t .

Now suppose we describe an event in one inertial system(say, the ground) as ( x , y , z, t ). – What are the coordinates of the event in another coordinate system

that is moving relative to the ground?

The Lorentz Transformations

Before Einstein, any physicist would have said that thecoordinates are related by

Einstein instead said that we must use the Lorentztransformations :

A taste of more advanced relativity …

Four-VectorsEinstein's theory puts time on equal footing with space; infact, things look simpler when expressed in terms of four-

vectors :

Then the Lorentz transformations read ( β = v / c)

MomentumMomentum is no longer simply mu ; the relativisticallycorrect equation is

Energy The rest energy of an object is given by Einstein's famousequation,

– So the total energy will be the rest mass plus the kinetic energy:

It turns out you can write this simply as

Energy Einstein’s equation implies an equivalence between massand energy.

Consider the following inelastic collision:

Before:

After:

Looks like energy isn’t conserved!

Energy In fact, energy is conserved! The final product doesn’t havea mass of 2 m – equating energies gives

Before:

After:

Mass isn’t conserved!

Energy Really?

Or, fission and fusion.

A Final Note

Careful! There is a lot of subtlety to special relativity, and alot of stuff I glossed over.

Also, special relativity is special because is only deals with aspecial case : – Inertial reference frames.

It took Einstein ten years to publish the general theory,which holds for noninertial systems as well. – It ends up relating the matter of the system to its curvature. – And so is a theory of gravity.

But the mathematical language is formidable.

Special Relativity

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