lecture 9 theoretical mechanics

8/13/2019 lecture 9 theoretical mechanics

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lecture 9

Topics:

The structure of science and common sense

The speed of light

Time dilation

The twin paradox

The Doppler effect

The twin paradox and inertial frames

The structure of science and common sense

from www.raremaps.com

The beautiful edifice of Newtonian mechanics, which we have seen a bit of in the last few

weeks, provides a wonderful precise mathematical description of most of the things we see in our

everyday world. It is obviously right. But it is also wrong. We have discussed qualitatively the

underlying quantum mechanical reality from which Newtons mechanics emerges as an approx-

imation. In the next few weeks, we will discuss in quantitative detail the bizarre things that go

wrong with Newtons picture at large velocities. What is going on here? Newtonian mechanics

beautifully captures the mathematical essence of what we know about the world we have grown

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up it. Once we get used to the mathematical language, it is perfectly in accord with our common

sense understanding of the world. We feel in our bones that it is right. How can it be wrong?

But it is wrong. I am going to tell you today that it is wrong and what is right and you will

not understand me or believe. Even if you have heard this before, and you think that you have

internalized it, you are still not going to really understand it or believe it. In fact, you will not evenhave any sense of what it would mean to understand it or believe it. It is that strange. It doesnt

make any sense.

The first thing to say is that there is no reason why our sense should have anything at all to do

with what happens at extreme conditions, far from what we are used to in everyday life. We have

some direct intuition about things that are about our size and maybe a few powers of ten bigger

and smaller. We have feeling in our bones what happens for accelerations not much different

thang, and velocities like those we are used to. But if we go far outside this familiar range of

parameters, it would be rather surprising if our common sense worked very well. We should be

prepared for surprises. If anything, what should surprise us is that our common sense works as far

as it does. We have to go to really enormous velocities, on our everyday scale, before Newtonian

kinematics starts to break down. And atoms, which exhibit quantum behavior in all its glory, are

very small. This is a theme that we will return to several times.

The wonderful thing about the discipline of modern science is that we can say sensible things

about phenomena even when our sense doesnt work. We do this by keeping ourselves firmly

grounded in what we understand, but at the same time recognizing the limitations of our knowl-

edge. It is useful to think of science as a map of a peculiar space - the space of parameters that

describe physical phenomena. We expand our knowledge in much the same way that ancient ex-

plorers improved their maps of the known world. We work our way out from what we know into

the unknown, pushing farther and farther in different directions away from the range of phenomenathat we see in the everyday world. In the next few weeks, we are going to discuss one of these

directions the realm of the very fast. When it is strange, dont be surprised. The reason that you

dont understand is very simple. It is that you are slow! Not mentally slow, but physically slow.

You have spent all your life moving at speeds very very tiny compared to the speed of light, so

nothing in your experience has prepared you for the phenomena that happen all the time at large

speeds (which we call relativistic speeds a rather bizarre grammatical construction if you

think about it, but standard). Try to bear that in mind when it seems that what we are doing doesnt

make any sense.

The speed of light

The speed of light is exactly 299,792,458 m/s. What exactly means in this case is just what

it says. Because the speed of light, as we will see, is built into the structure of space and time,

it makes sense to use it to define our unit of length (the meter) in terms of our unit of time (the

second). This is what is done in SI, the International System of Units. It is no longer necessary

to keep a standard 1 meter bar in a vault someplace. The second is now defined in terms of

a particular oscillation of an atom in an atomic clock. The meter is then defined as the distance

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that light travels in 1/299,792,458th of a second. I should say that when I talk about the speed of

light, I always mean the speed of light in vacuum that is in empty space. 1 Things get more

complicated in material like glass because the interactions of the light with the material can slow

the light down.

Now 299,792,458 m/s is fast. It is a heck of a lot faster than we can actually move ourselves.But it is certainly not infinitely fast. With modern electronics, we can measure very short times, so

it is not impossible to see the effect of the finite speed of light even over fairly short distances. The

point I am trying to make here is that while motion at close to the speed of light is far beyond our

everyday experience, it is not science fiction. In fact, we routinely measure the speed of light, and

routinely see things (small things like electrons, but things nevertheless) moving at speeds very

close to the speed of light.

But the surprising thing about light in a vacuum is that the speed of light that we measure

doesnt depend on the velocity of the object that produced the light, and it doesnt depend on the

velocity of the measuring apparatus. Now if you think that you understand this, you obviously

have not been listening carefully enough, because this doesnt make any sense at all. Nevertheless,

it is true. If, for example, I am running towards a light-bulb at speedv carrying a light-speed

meter, a device to measure the speed of light, all of you sitting at rest see the light from the bulb

approaching me at a speedv+ c. But when I do the measurement, I get the same value for the

speed of light that I would get if I were standing still. In fact, I get the same value that you would

get measuring the same light beam in about the same place at about the same time, but standing

still. The same thing happens if I am running away from the light source.

c=299792458 v

c=299792458 vc=299792458

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(1)

This is absolutely crazy. Surely if I am moving towards the light beam, I should register a larger

speed on my light-speed meter. That is what common sense would say. However, that is not the

way the world works. The way the world works is that the speed of light in vacuum is constant,

period! It is not that something goes wrong with my light-speed meter. This bizarre fact is built

into the way the world works.

The full power of this remarkable fact, the constancy of the speed of light, is unleashed when

we combine it with another, much more reasonable fact about the way the world works the

principle of relativity. The principle of relativity says simply that all uniform motion is relative.

There is no absolute sense in which I can say I am moving. There is no preferred notion of standing

still. In a moment, we will formalize this idea with the notion of an inertial frame of reference.

Note that we can tell if our motion is not uniform. Acceleration is accompanied by forces that we

1The notion of empty space is itself rather problematic. Even classically, space is only completely empty at

absolute zero. And when we include the effects of quantum mechanics, as we will see much later, empty space begins

to look anything but empty. Nevertheless, there is a well-defined meaning to the notion of the speed in light in vacuum.

Its role as a cosmic speed limit survives all this extra complication.

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can feel in our bones. But uniform motion is not detectable, so long as everything else we need

is moving along with us. This, of course, is something that feel in our bones for the slow motions

that we are used to. We all know this very well from travel in vehicles, cars, trains, planes, and

whatnot. We are going to assume, with Einstein, that it remains true at relativistic speeds.

Inertial frames

The idea of an inertial frame of reference or just inertial frame for short, is one that already

plays an important role in non-relativistic mechanics. It is an attempt to formalize the notion that

motion is relative in an operational way. To do this, we must carefully describe what velocity

means by describing precisely what we need to measure it.

On the surface, the speed of light does not seem to be a complicated concept. You measure

it in the obvious way with clocks and meter sticks, by dividing the distance traveled by the time

taken. But first, you have to synchronize your clocks! This is where the idea of an inertial frame

comes in. Aninertial frameis a real or imaginary collection of clocks that are fixed with respectto one another and synchronized, for example by requiring that some signal that originates midway

between each pair of clocks arrives at the two clocks at the same time. 2 In addition, an inertial

framemust not be accelerating, which is easy to check because you can just demand that Newtons

laws hold for small velocities free particles travel in straight lines, that sort of thing.

So we have two fundamental principles.

A. That the laws of physics are the same in all inertial frames, and

B. That one of the laws of physics is that the velocity of light is a constant with the same value

in all inertial frames,

As you will see in more detail in the notes, these two principles are amazingly powerful. They will

revolutionize our picture of space and time. Now lets see some of the consequences of putting

these two ideas about the world together.

Time dilation

Lets start with one of the strangest and most trivial of the consequences of relativity time

dilation. The phenomena of time dilation can be stated precisely as follows. Observations done on

a single clock moving with speedv with respect to a number of clocks fixed in an inertial frameshow the ticking of the moving clock slowed down by a factor of

1 v2/c2. The standard way

of deriving this result is to consider an idealized clock made out of two parallel mirrors and a pulse

2It doesnt matter for this purpose whether we are using a light signal or the Pony Express, as long as the signal

travels at the same speed in both directions!

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able to distinguish between the moving frame and the frame in which the light clock is fixed. But

this violates the principle that all frames are equivalent. Every kind of clock must tick out seconds

at the same rate in all inertial frames.

Incidentally, this factoris going to reappear all the time, so it pays to actually either memorize

it, or to be able to reproduce the light-clock argument in real time so you can get it whenever youneed it.

It is quite easy with modern electronics and atomic clocks to see relativistic effects like time

dilation. In fact, both special and general relativistic effects are very important in one very practical

application the Global Positioning System which is based on a system of atomic clocks aboard

satelites. The relativistic corrections are small, because the satellites are traveling at only about

4000 m/s, but enormous accuracy is required to make GPS work and the relativistic effects must

be properly included. See for example http://www.physicscentral.com/writers/writers-00-2.html.

Even more dramatic examples of time dilation occur all the time with elementary particles.

That seems like a lot to ask of tiny particles that are supposed to be elementary and have no

internal structure. But the fact is that quantum mechanics provides us with internal clocks for

many elementary particles because they are unstable, and when they are sitting still and evolving

in time, they have a constant probability per unit time of decaying into other lighter particles. We

can actually see these internal clocks ticking (at least in an average sense) by watching the particles

decay. The observed lifetime of unstable particles is a tangible measure of how fast these internal

clocks are ticking. We see this all the time in particle experiments. But we are relying on another

fact all particles of a particular kind are exactly the same. We never actually measure the decay

rate of the same decaying particle in two different frames. But we can quite easily measure the

lifetime of a particle at rest, and then measure the lifetime of the same TYPE of particle in a

moving frame. We find that the ratio of the lifetimes is. Since all particles of a particular typeare identical, this is just as good.

Another important thing about time dilation is that although it is strange, it is probably the

easiest of the relativity principles to remember and use. The thing to remember is

The single clock measures the shorter time. (6)

If you keep this in mind, and just remember that >1, you will always be able to reconstruct the

right formula. Use time dilation whenever you can to solve problems!

The twin paradox

Now you may very well be thinking, at this point, that once you define what you are talking about

carefully, with inertial frames, that there isnt anything particularly strange about motion at rela-

tivistic speeds, but that we have just confused the issue with a bizarre definition of measurement.

Even our experiment on decays of elementary particles might be just a matter of a bad definition

of what we mean by the ticking of their internal clocks. Perhaps, you think, that there is some

other way of constructing our light-speed meters so that the speed of light is not constant and the

bizarre features of relativity go away. Think again! Perhaps the simplest way of making clear that

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http://www.physicscentral.com/writers/writers-00-2.html


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The distance between crests for the moving train is

v =Vs v

0(10)

Thus the wavelength of the sound as recorded at the sound meter is reduced by the nonrelativisticDoppler factor

Vs v

Vs(11)

Because (7) must be satisfied, the frequency is increased by the inverse of (11), and the train whistle

has a higher pitch when it is moving towards us.

If the train is moving away, the argument is exactly the same we just have to replacev v

in (11).

Now suppose we do a similar thing, but replace the train with a rocket moving at relativistic

speed, and replace sound with light. The speed of light is the product of the frequency and the

wavelength:

c (m/s) =0(m/cycle) 0(cycles/s) (12)

The inverse of the frequency is the period of the light wave, which is the time between successive

crests of the wave. In the diagram, almost everything is the same, except that because of time

dilation, the time between the emission of successive crests of the wave is longer than1/0by the

ubiquitous factor of, because the moving clock ticks more slowly. Thus the picture looks like

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c

0

v

0

(cv)0

light meter ..............................................................................

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............................................. | t= 0rocket

light meter .......................................................................................................................................................................... | .......................................................................................................................................................................... | t= /0rocket

(13)

Because the rocket is moving forward as it emits the wave, the crests are closer together than they

would be if the rocket were standing still. The distance between crests for the rocket at rest is just

the wavelength,

0 = c

0(14)

The distance between crests for the moving rocket is

v =(c v)

0(15)

Thus the wavelength of the light as recorded at the light meter is reduced by the relativistic Doppler

factor

(c v)

c =

11 v2/c2

(1 v/c) =

1 v/c1 +v/c

(16)

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Because (12) must be satisfied, the frequency is increased by the inverse of (16),1 +v/c1 v/c

(17)

and the light has higher frequency when the rocket is moving towards us. This is called blue-shiftbecause raising frequency in the optical spectrum is a shift towards the blue.

Again, if the rocket is moving away, the argument is exactly the same we just have to replace

v v everywhere. This is called red-shift because lowering frequency in the optical spectrum

is a shift towards the red.

There is one very important distinction to note about the relativistic Doppler effect versus the

nonrelativistic version. In the relativistic version, it doesnt matter whether the rocket is approach-

ing the observer at speedv or the observer is approaching the rocket at speed v . It cant, because

of the principle of relativity. This is not true for the nonrelativistic Doppler effect because the air

in which sound moves defines a special frame.

The twin paradox and inertial frames

We can now use the Doppler effect to understand time dilation and the twin paradox, by keeping

track of every tick of the moving clock. Imagine that twin 1 takes a trip to planet X, at distance L

from earth, but stays in constant communication with twin 2 behind on earth using radio waves, or

some other electromagnetic waves that have fixed frequency in twin 1s frame (the rocket frame).

We will assume the typical form of the twin paradox that we discussed above, where twin 1 goes

out to planet X at speedv, quickly turns around, and returns at the same speed.

Here are a series of snapshots showing the important times during the trip. Twin 1 leaves twin2 at t = 0traveling at constantv staying in constant radio communication

t= 02

1v

X

Twin 1 arrives at planet X at t = L/v, turns around (quickly) and sends a turn-around signal to

twin 2, indicating that he has reached the planet.

t= L/v 2

v 1X

The turn-around signal reaches twin 2 att = tX=L/v+L/c

t= tX2

v1

X

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Twin 1 and twin 2 are reunited att = 2L/v

t= 2L/v

2

1

X

The radio transmitter is a clock number of ticks (cycles) ist whereis frequency of trans-

mitter. Thus the number of ticks sent by twin 1 isT , whereT is the total time he aged on the

trip.

But twin 2 receives red shifted photons for a time tX = L/v + L/c and blue shifted for

2L/v tX=L/v L/c. Therefore, the number of cycles received by the twin 2 is

1 v/c

1 +v/c

L

v +

L

c

+

1 +v/c

1 v/c

L

v

L

c

(18)

=

1 v/c1 +v/c

L

1 +v/c

v

+

1 +v/c1 v/c

L

1 v/c

v

=2L

v

1 v2/c2 = T (19)

Thus

T =2L

v

1 v2/c2 (20)

Nonrelativistically, we would have expected

T =2L

v (21)

Thus twin 2 sees rocket clocks ticking more slowly by a factor of

1 v2/c2 = 1/ (22)

This is time dilation. The moving clock ticks more slowly as seen by the clock at rest. This is also

the twin paradox. Because twin 1 has sent out fewer ticks, he has also aged less. He is younger

than twin 2 when he returns.

There are a couple of other things to notice about (22). First note that the two terms in (22) areequal. This had to be the case, because twin 1 sent the same number of ticks on the way to planet

X as on the way back, so twin 2 received the same number of ticks in the red-shifted signal from

the trip out as in the blue-shifted signal from the trip back.

But now, you say, why isnt the situation symmetrical? After all, from twin 1s point of view,

twin 2 (along with the rest of the earth) has moved away at speed v and then come back at the

same speed. Why is it that twin 1 is younger at the end, rather than twin 2? Lets look at the trip

from twin 1s point of view, assuming that it is twin 2 who is sending out radio signals the whole

time. Now things look a bit different. Here is the chronology. Att = 0, twin 1 watches twin 2 and

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the earth recede at constantv . Twin 1 receives radio signals from twin 2 that are red-shifted until

planet X appears. Planet X is shown as dashed in the figure because it is not in the same inertial

frame as twin 1s ship, so twin 1 has to be a whiz at relativity to calculate its position.

t= 0 v 2

1X

Planet X reaches twin 1 at t = T /2, turns around (quickly) and starts to recede again. From this

point on, twin 1 receives blue-shifted signals until earth reappears and he is reunited with twin 2.

t= T /22

1

Xv

Twin 1 and twin 2 are reunited att = T

t= T2

1

X

Now we can check that the two pictures are consistent. The number of cycles that twin 1

receives is

T2

1 v/c

1 +v/c+1 +v/c

1 v/c

= T2

1 v/c

1 +v/c

1 v/c1 v/c

+1 +v/c

1 v/c

1 +v/c1 +v/c

(23)

T

2

1 v/c

1 v2/c2+

1 +v/c1 v2/c2

= T 1

1 v2/c2(24)

which using (20) is2L

v (25)

which is exactly what we expected. I hope it is clear from this discussion what the asymmetry is.

It should be clear that I have not quite described this process the way twin 1 experiences it. Hedoesnt just watch earth receding at t = 0. He blasts off and accelerates. Similarly, he doesnt

just watch planet X turn around. He decelerates and accelerates again in the opposite direction.

He feels these accelerations in his bones! From twin 1s point of view, the switch from red-shift

to blue-shift occurs the moment he turns around. This makes sense. He knows that he has turned

around. But from twin 2s point of view, nothing special happens when twin 1 reaches the planet.

He has to wait until the turn-around signal arrives to see the shift from red-shift to blue-shift.

Here is another way of saying what the difference is. Twin 2, remaining on earth, is at rest in

a single inertial frame the whole time. Twin 1 is not. He is in one frame on the way out, and in a

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different frame on the way back. It is the fact that twin 1 must switch from one inertial frame to

the other that makes his experience different.

Relativity may be strange, but it is consistent.

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