Georgi, Howard. Physics 16 - Mechanics and Special Relativity (англ.)

component but no space component — the frame in which the clock is at rest. In fact, any space

and time interval with components

(∆t, ∆~r ) (28)

that satisﬁes

∆t >

¯

¯∆~r

¯

¯ (29)

has the property that there is a frame in which it looks like (∆τ, 0) with ∆τ > 0. Why? One way

to see this is to note that any interval that satisﬁes (29) could be the interval associated with a clock

moving with velocity

~v =

∆~r

∆t

(30)

simply because if the clock moves with ~v for a time ∆t, it ends up translated by the vector ∆~r,

reproducing (28). Thus if we go to a frame moving with velocity ~v, we will be moving along with

the clock and in this new frame the intervals will have the form

(∆τ, 0) where ∆τ =

r

³

∆t

´

2

−

¯

¯∆~r

¯

2

(31)

This relation incorporates the statement of time dilation. ∆τ is always less than or equal to ∆t.

Two events that are separated by a time-like interval are closest together

in time in the frame in which they occur at the same point in space.

(32)

The frame in which the two events occur at the same point in space is the frame in which the two

events could be two ticks of the same clock. Thus (32) is equivalent to the statement that a moving

clock ticks slowly. Two events that are separated by a time-like interval are sometimes referred to

as “time-like separated.”

On the other hand, if

∆t <

¯

¯∆~r

¯

¯ (33)

then it cannot represent the interval between two points on the path of a clock. As we will see in

more detail later, we cannot go to a frame moving with velocity ~v = ∆~r/∆t, because |~v | > 1, so

the frame would have to moving at a speed greater than the speed of light, which is impossible for

clocks and meter sticks and all the other things we need to have a frame of reference. Then what

is this interval? Suppose that we look in a frame of reference moving with velocity

~u = ∆~r

∆t

¯

¯∆~r

¯

2

(34)

To see what happens, let us rotate our coordinate system until ∆~r is in the x direction

∆~r =

¯

¯∆~r

¯

¯ˆx (35)

so that

~u = u ˆx where u =

∆t

¯

¯∆~r

¯

(36)

10

Now in a frame moving with velocity ~u, the y and z components of the interval remain zero, and

we can compute what the time and x components become using the Lorentz transformation,

∆t

0

=

1

√

1 − u

2

³

∆t − u

¯

¯∆~r

¯

´

= 0

∆~r

0

=

1

√

1 − u

2

³

¯

¯∆~r

¯

¯− u ∆t

´

=

1

√

1 − u

2

¯

¯∆~r

¯





1 − u

∆t

¯

¯∆~r

¯





=

1

√

1 − u

2

¯

¯∆~r

¯

³

1 − u

2

´

=

√

1 − u

2

¯

¯∆~r

¯

¯ =

r

¯

¯∆~r

¯

2

−

³

∆t

´

2

(37)

This is called a “space-like” interval, because there is a frame of reference in which it has a

space component but no time component. A very important example of a space-like interval is

the interval between two ends of a measuring stick at ﬁxed time. This is what we deﬁne to be a

measurement of distance. Two objects are a distance one meter apart in a given reference frame at

time t if we can line up a meter stick so that the objects are at opposite ends of the stick at time t.

This distance is associated with a space-time interval with is the difference between two “events”

which are the space-time coordinates of the two ends of the meter stick at time t. This is obviously

a space-like interval, because the time component is zero. This corresponds to the primed frame in

(37). The properties of space-like intervals are then responsible for the phenomenon of “Lorentz

contraction”. In any other frame, the two events occur at different times. Therefore, (37) implies

that

¯

¯∆~r

¯

¯ >

¯

¯∆~r

0

¯

¯. The two events are closest together in the frame in which they occur at the

same time. This is generally true for any space-like interval.

Two events that are separated by a space-like interval are closest together

in space in the frame in which they occur at the same time.

(38)

Thus for example, if we measure the length of a moving train, the length we measure is the distance

between two events describing the positions of the front and back of the train at the same time in

our frame. But in the train frame, these two events do not occur at the same time, and thus the

distance between them is greater than what we measure (by a factor of 1/

√

1 − v

2

). This is Lorentz

contraction.

Two events that are separated by a space-like interval are sometimes referred to as “space-like

separated.”

The tip of tomorrow

Why is it that you are not used to thinking about space-time as a 4-dimensional object. I think that

once again it is because you are slow. You are used to thinking of time and space very differently.

Space is spread out around you to inﬁnity in all directions. Time is not spread out at all. Time

11

passes, and you live in the present on your way from the past to the future. Here is a cartoon of our

subjective picture of space and time, plotted just in the x − t plane.

.

..

.

...

.

..........................................................................................................................................................................................................................................................................................................................

.

..........................................................................................................................................................................................................................................................................................................................

.

..

.

spacespace

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

future future future future future future future

past past past past past past past

But much of the difference between space and time is an illusion. Time and space are different, to

be sure, because of that cricial minus sign in the invariant interval. The passing of time is real. But

the spreading out of space is a ﬁction. It is a drastic oversimpliﬁcation that has been built into our

brains and our sensory apparatus over millions of generations because we are slow. What we don’t

think about or notice is that when we look out at the space spread out around us, we are actually

looking back into the past. And we cannot access the space around us instantaneously. The real

story looks more like this.

.

..

.

...

.

..

.

..

.

..

.

..

.

..

.

space?space?

↑

future

↑

past

The slight slope of the arrows is the non-zero value of 1/c - the effect of the ﬁnite speed of light.

If we were faster, we would recognize intuitively that the “space” that is spread out on all sides of

us at this moment in time is just as inaccesible to us as the future. This is the region of space-time

events from which we are separated by space-like intervals. We can’t get to them, nor they to us.

12

Then the picture would look more like this.

.

..

.

...

.

..

.

..

.

..

.

..

.

l

i

g

h

t

c

o

n

e

l

i

g

h

t

c

o

n

e

n

o

c

t

h

g

i

l

e

n

o

c

t

h

g

i

l

.

..

.

spacelikespacelike

timelike

↑

future

↑

past

The boundary between the space-like separate region and the time-like separated regions of our

past and future are separated by a cone called our light cone. Our light cone consists of all the

events in space-time that we could communicate with or receive communication from by means of

a single light signal, moving at the speed of light. In the ﬁgure, it looks like two crossed lines, but

we can rotate in space. If we rotate it just in the x − y plane and keep z (for example) ﬁxed, we

get an ordinary three dimensional cone in the three dimensions, x, y and t. The light cone is like

funnel from out past to our future. Of course, if we make an arbitrary space rotation, we get a four

dimensional cone, which is harder to visualize, so I recommend just ignoring

z

for now.

The light-cone separates space-time into three regions, the time-like past which has affected us,

the time-like future that we can affect, and space-like unknown on all sides that can neither affect

us nor be affect by us. This picture doesn’t matter much to us in our everyday life, because we are

so slow. But it will be crucial when we try to understand the large-scale structure of the universe

later in the course.

For now, I hope that it help you to understand why relativity seem so strange. Seen properly,

through the lens of relativity, the “the present” is not in any real sense an inﬁnite three dimensional

13

space. The present is both NOW and HERE!

.

..

.

..

.

..

.

..

.

l

i

g

h

t

c

o

n

e

l

i

g

h

t

c

o

n

e

n

o

c

t

h

g

i

l

e

n

o

c

t

h

g

i

l

here

and now

the tip of

tomorrow

.................................................................................................................................................................................

.

.................................................................................................................................................................................

.

↑

future

↑

past

If you were fast, you would recognize that the here and now is just another space-time event, at the

singular tip of our light cone, poised on the tip of tomorrow.

14

lecture 11

Topics:

World lines and proper time

Massive particles

Energy and momentum

Lorentz transformation of energy-momentum

Energy, momentum, velocity and mass

Massless particles

4-vectors and the invariant product

World lines and proper time

There is a nice relativistic analog of the concept of a trajectory in Newtonian physics - based on

the concept of a world line. The idea is that as any massive particle evolves with time, whether

it is sitting still or moving, free or accelerating, can always be described by some curve in 4-

dimensional space-time that is just the collection of the space-time events

µ

t, ~r(t)

¶

that describe

where the particle is at every time. The reason that it is useful to think about this in this funny

4-dimensional space (even though it is kind of hard to visuallize) is that the world line itself has

an invariant meaning because it is a collection of space time events that have invariant meaning

even though their coordinates will change depending on the inertial frame. The situation is similar

to that of a curve in three dimensional space. The curve consists of points that have an invariant

meaning, but their coordinates change depending on the coordinate system. Also, just as we can

label the points on a curve in three dimensional space by the distance along the curve from some

arbitrary point, so we can measure the distance along world line of a particle by measuring the

time ticked on the particles internal clock. This is called the proper time, τ . The change in proper

time dτ along any short segment of the world line is just

dτ = dt/γ = dt

√

1 − v

2

=

√

dt

2

− d~r

2

(1)

where t and v are measured in whatever coordinate system and inertial frame you like. You can see

that this is independent of the inertial frame and the coordinate system by looking at the last term,

which is simply the invariant interval for the short line segment. It is crucial here that every short

line segment along the world line of a massive particle is time-like. This is true because massive

particles can never travel as fast as light. We will return to this again and again and understand it

various ways.

A good way of describing the world line is to give both t and ~r as functions of τ:

t(τ) ~r(τ) (2)

For example, for a particle at rest

t(τ) = τ ~r(τ) = 0 (3)

1

and for a particle with velocity ~v

t(τ) = γτ ~r(τ) = γ~vτ (4)

For travel in one dimension, we can plot world lines on a plane. For example, consider the twin

paradox. In their own frame of reference, the world lines of earth (and twin 2 on earth) and planet

X are vertical lines. The world line of twin 1 in the rocket has a kink where the rocket turns around

- in the curious geometry of spacetime - the kinked line takes the shorter time.

↑

t

x →

.

..

.

planet X

earth

worldline

of twin 2

twin

1

Massive particles

I now want to change gears from talking about space and time to talking about energy and momen-

tum. I am going to write down the Lagrangian for a free relativistic massive particle. Because this

is a free particle, the Euler-Lagrange equation is not all that interesting. We know without thinking

about it that it is going to tell us that the particle moves at constant velocity. Nevertheless, ﬁnding

the right form for the Lagrangian can be instructive. It will allow to ask questions about what

would happen if we put in forces, for example. Also, it allows us to identify the quantities that we

expect to be conserved because of Noether’s theorem and translation invariance in time and space.

In fact, once we write down these quantities — better known as the energy and the momentum —

we will largely forget about the Lagrangian formulation. So don’t panic if this Lagrangian looks

so strange that you can’t deal with it. What will really matter is that you learn to deal with the

relativistic energy and momentum that come out of it from Noether’s theorem.

What principles should we use to write down the Lagrangian for the free massive particle

moving at relativistic speeds? One is that the Action should look the same in all reference frames.

This is necessary, because if it were not true, Hamilton’s principle would not give us equations of

motion that give the same trajectories in all reference frames. In addition, if we call the position of

the particle ~r, we expect that the Lagrangian is independent of ~r and of t, and depends only on

˙

~r,

because it should be invariant under translations in space and time. In a moment, I will show that

2

the following Lagrangian gives rise to an action that is independent of the reference frame:

L(r, ˙r) = −m

q

1 −

˙

~r

2

. (5)

We are using units with c = 1 as usual. Here are a few things to note. This looks a bit funny

because of the square-root. But as we will see, that is what relativity requires us to write down,

so we will just have to live with it. The constant m, which we will see in a moment is the mass

of the particle, must be there so that the Lagrangian has units of energy. The same thing happens

in the Lagrangian for the free non-relativistic particle, which is just

1

2

m

˙

~r

2

. We can see that m

corresponds to the mass of the particle by looking at the Lagrangian for small velocities — which

we can do by Taylor expanding (5):

−m

q

1 −

˙

~r

2

= −m +

1

2

m

˙

~r

2

+ ··· (6)

The constant −m in (6) is irrelevant, so at small velocities, this reduces to the non-relativistic

Lagrangian, as we expect.

Now let us show what happens to the action

S = −m

Z

dt

q

1 −

˙

~r

2

(7)

under a Lorentz transformation. This is a little complicated because both the integrand and the

dt change under a Lorentz transformation. But we can make what is going on more obvious by

writing (7) as follows:

S = −m

Z

dt

q

1 − (d~r/dt)

2

= −m

Z

q

dt

2

− (d~r )

2

(8)

The second form is a very funny looking integral, but it makes sense because it is equivalent to

the previous form. The important point is that the last form is evidently unchanged by Lorentz

transformations. The inﬁnitesimal interval

(dt,

~

dr) (9)

is just a space-time interval. It transforms under Lorentz transformations just like (∆t, ∆~r). And

therefore, the combination

dt

2

− (d~r )

2

(10)

is just an inﬁnitesimal version of the invariant interval, and it has the same value in all inertial

frames. To get something proportional to dt (so that we can put it under an integral sign and get a

ﬁnite result), we must take the square-root of (10). That is why (7) looks the way it does.

Energy and momentum

Now that we have a Lagrangian, we can construct the conserved energy and momentum that we

expect for a relativistic particle. The energy is

E =

˙

~r ·

∂L

∂

˙

~r

− L = m

˙

~r

2

q

1 −

˙

~r

2

+ m

q

1 −

˙

~r

2

=

m

q

1 −

˙

~r

2

=

m

√

1 − v

2

= mγ . (11)

3

The momentum is

~p =

∂L

∂

˙

~r

= m

˙

~r

q

1 −

˙

~r

2

=

m ~v

√

1 − v

2

= m~vγ . (12)

Here is an example. Suppose we have a particle with mass m traveling at v =

4

5

. Then

γ =

1

q

1 − (4/5)

2

=

5

√

5

2

− 4

2

=

5

√

25 − 16

=

5

√

9

=

5

3

(13)

so

E = γm =

5

3

m p = vγm =

4

3

m (14)

You will observe that these do not look like the expressions for energy and momentum of a

Newtonian particle. Nevertheless, these are the objects that are really conserved. It is instructive

to look at them with the factors of c put back in:

E =

m c

2

q

1 − v

2

/c

2

(15)

~p =

m ~v

q

1 − v

2

/c

2

(16)

In this form, it is also useful to Taylor expand in powers of v/c:

E =

m c

2

q

1 − v

2

/c

2

= m c

2

+

1

2

m v

2

+ ··· (17)

~p =

m ~v

q

1 − v

2

/c

2

= m ~v + ··· (18)

As we expected, the Newtonian expressions for the kinetic energy and the momentum appear.

In the expression for energy, (17), there is also a constant term, the famous m c

2

. What is

actually important about this is not so much the constant m c

2

, but the fact that mass plays a very

different role in relativistic processes than it does at lower speeds. In Newtonian mechanics, mass,

kinetic energy, and momentum are separately conserved. But in relativistic physics, it is only the

momentum ~p and the total energy E that are actually conserved in arbitrary collisions of elementary

particles at any speed — even in processes in which particles are created or annihilated. Mass is

not conserved. The mass of any one kind of particle is always the same, so mass is conserved in

collisions that don’t change the type of particle involved. For example if you accelerate an electron

to speed v and it collides with an electron at rest, some of the time you will get a collision in which

you end up with two electrons going off in different directions and no other particles. We might

represent this process by the schematic “equation”

e

−

+ e

−

→ e

−

+ e

−

(19)

4

In this case the sum of the masses of the particles before the collision is the same as the sum of

the masses of the particles after the collision. But other things can happen that don’t conserve

mass. For example, in the process of the collision, you may produce an extra electron (so there are

three electrons in the ﬁnal state) and a positron (an anti-electron - with the same mass but opposite

charge).

e

−

+ e

−

→ e

−

+ e

−

+ e

−

+ e

+

(20)

Here the sum of the masses before the collision is 2m

e

, and the sum after the collision is 4m

e

.

Mass is not conserved. In fact, there is no reason to compute the sum of the masses at all. It is just

not an interesting quantity.

There is a related issue that sometimes causes confusion. Some of you have probably seen

relativity before, and you may have been exposed to the rather idiotic notion of a “rest mass” that

is the actual mass of the particle and a “relativistic mass” that depends on velocity. This is not

useful! If you ask a physicist what the mass of the electron is, the response will be

m

e

≈ 9.11 × 10

−28

g (21)

or

m

e

≈ 0.511 MeV (22)

The response will certainly not be “Do you mean the rest mass of the electron?” or “How fast is

your electron moving?”

Not only do these silly notions of “rest mass” and “relativistic mass” not correspond to the way

physicists actually talk about these things, but the motivation for them (such as it is) is philosophi-

cally ﬂawed. I think that the idea was to preserve the form of the equation

~

F = m~a (23)

for large velocities. There are two problems with this. One is that it doesn’t work, even if you allow

m to depend on ~v. You can still only preserve this form in certain special cases. But more impor-

tantly, you shouldn’t want to preserve this form anyway. We have seen that the more fundamental

form that arises naturally in a Lagrangian description of mechanics is

~

F =

d

dt

~p (24)

We will see next week that this relation survives intact in relativistic physics. No silly deﬁnitions

are required.

So if you are used to using the term “rest mass” and “relativistic mass,” you should try to get

over this as soon as possible. They will only cause you grief and confusion in this course and

beyond. If all else fails, perhaps hypnosis might help.

Lorentz transformation of energy-momentum

One of the important things about energy and momentum is that they behave under Lorentz trans-

formations very much like a space and time interval. Remember that this has to do with what

5

Georgi, Howard. Physics 16 - Mechanics and Special Relativity (англ.)

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