# Relativistic Dynamics

*Michael Fowler*, *UVa
Physics, 3/1/2008*

### The Story So Far: A Brief Review

The first coherent statement of what physicists now call *relativity* was Galileo’s observation almost four hundred years ago that if you were
in a large closed room, you could not tell by observing how things move-living
things, thrown things, dripping liquids-whether the room was at rest in a
building, say, or below decks in a large ship moving with a steady velocity. More technically (but really saying the
same thing!) we would put it that *the laws of motion are the same in any
inertial frame.* That is, these
laws really only describe *relative* positions and velocities. In particular, they do not single out a
special inertial frame as the one that’s “really at rest”. This was later all written down more
formally, in terms of *Galilean transformations*. Using these simple linear equations,
motion analyzed in terms of positions and velocities in one inertial frame
could be translated into any other. When, after Galileo,

About two hundred years ago, it became clear that light was not just a
stream of particles (as
*wavelike* properties. This led naturally to the question of
what, exactly, was waving, and the consensus was that space was filled with an *aether*,
and light waves were ripples in this all-pervading aether analogous to sound
waves in air. Maxwell’s
discovery that the equations describing electromagnetic phenomena had wavelike
solutions, and predicted a speed which coincided with the measured speed of
light, suggested that electric and magnetic fields were stresses or strains in
the aether, and Maxwell’s equations were presumably only precisely
correct in the frame in which the aether was at rest. However, very precise experiments which
should have been able to detect this aether all failed.

About a hundred years ago, Einstein suggested that maybe *all* the laws
of physics were the same in *all *inertial frames, generalizing Galileo’s
pronouncements concerning motion to include the more recently discovered laws
of electricity and magnetism. This
would imply there could be no special “really at rest” frame, even
for light propagation, and hence no aether. This is a very appealing and very simple
concept: the same laws apply in all frames. What could be more reasonable? As we have seen, though, it turns out to
clash with some beliefs about space and time deeply held by everybody
encountering this for the first time. The central prediction is that since the
speed of light follows from the laws of physics (Maxwell’s equations) and
some simple electrostatic and magnetostatic experiments, which are clearly
frame-independent*, the speed of light is the same in all inertial frames*.
That is to say, the speed of a *particular
flash* of light will always be measured to be 3 10^{8} meters per
second even if measured by different observers moving rapidly relative to each
other, where each observer measures the speed of the flash relative to himself.
Nevertheless, experiments have show
again and again that Einstein’s elegant insight is right, and everybody’s
deeply held beliefs are wrong.

We have discussed in detail the *kinematical* consequences of Einstein’s
postulate: how measurements of
position, time and velocity in one frame relate to those in another, and how
apparent paradoxes can be resolved by careful analysis. So far, though, we have not thought much
about dynamics. We know that
*incorrect* except in the low speed
nonrelativistic limit. Therefore,
we had better look carefully at

###
Newton’s
Laws Revisited

*is* true, and
the content of special relativity is transformations between such frames.

*c/2* from rest. We
could fire them one after the other in a carefully timed way to generate a
continuous large force on the rocket, which would get it to *c/2* in the
first firing. If the acceleration
continued, the rocket would very soon be exceeding the speed of light. Yet we know from the addition of
velocities formula that in fact the rocket never reaches *c*. Evidently,

*A* on *B* is the opposite of the force of *B* on *A*,
at each instant of time, but that implies *simultaneous* measurements at
two bodies some distance from each other, and if it happens to be true in *A*’s
inertial frame, it won’t be in *B*’s.

### Conservation Laws

In nonrelativistic Newtonian physics, the Third Law tells us that two
interacting bodies feel equal but opposite forces from the interaction. Therefore from the Second Law, the rate
of change of momentum of one of the bodies is equal and opposite to that of the
other body, thus the *total* rate of change of momentum of the system
caused by the interaction is zero. Consequently,
for any *closed *dynamical system (no outside forces acting)* the* *total
momentum never changes*. This is
the *law of conservation of momentum*. It does *not* depend on the details
of the forces of interaction between the bodies, only that they be equal and
opposite.

The other major dynamical conservation law is the conservation of energy. This was not fully formulated until long after Newton, when it became clear that frictional heat generation, for example, could quantitatively account for the apparent loss of kinetic plus potential energy in actual dynamical systems.

Although these conservation laws were originally formulated within a
Newtonian worldview, their very general nature suggested to Einstein that they
might have a wider validity. Therefore, as a working hypothesis, he
assumed them to be satisfied *in all inertial frames*, and explored the
consequences. We follow that
approach.

### Momentum Conservation on the Pool Table

As a warm-up exercise, let us consider conservation of momentum for a
collision of two balls on a pool table. We draw a chalk line down the middle of
the pool table, and shoot the balls close to, but on opposite sides of, the
chalk line from either end, at the same speed, so they will hit in the middle
with a glancing blow, which will turn their velocities through a small angle. In other words, if initially we say their
(equal magnitude, opposite direction) velocities were parallel to the *x*-direction—the
chalk line—then after the collision they will also have equal and
opposite small velocities in the *y*-direction. (The *x*-direction velocities will
have decreased very slightly).

### A Symmetrical Spaceship Collision

Now let us repeat the exercise on a grand scale. Suppose somewhere in space, far from any
gravitational fields, we set out a string one million miles long. (It could be between our two clocks in
the time dilation experiment). This
string corresponds to the chalk line on the pool table. Suppose now we have two identical spaceships
approaching each other with equal and opposite velocities parallel to the
string from the two ends of the string, aimed so that they suffer a slight
glancing collision when they meet in the middle. It is evident from the symmetry of the
situation that momentum is conserved in both directions. In particular, the rate at which one
spaceship moves away from the string after the collision - its* y*-velocity
- is equal and opposite to the rate at which the other one moves away from the
string.

But now consider this collision as observed by someone in one of the
spaceships, call it *A*. (Remember,
momentum must be conserved in *all* inertial frames—they are all
equivalent—there is nothing special about the frame in which the string
is at rest.) Before the collision,
he sees the string moving very fast by the window, say a few meters away. After the collision, he sees the string
to be moving away, at, say, 15 meters per second. This is because spaceship *A* has
picked up a velocity perpendicular to the string of 15 meters per second. Meanwhile, since this is a completely
symmetrical situation, an observer on spaceship *B* would certainly deduce
that her spaceship was moving away from the string at 15 meters per second as
well.

### Just how symmetrical is it?

The crucial question is:* how fast does an observer in spaceship A see
spaceship B to be moving away from the string?* Let us suppose that relative to spaceship *A*, spaceship *B* is moving away (in the *x*-direction) at 0.6*c*.
First, recall that distances
perpendicular to the direction of motion are not Lorentz contracted. Therefore, when the observer in spaceship *B* says she has moved 15 meters further away from the string in a one
second interval, the observer watching this movement from spaceship *A* will agree on the 15 meters - but disagree on the one second! He will say her clocks run slow, so as
measured by his clocks 1.25 seconds will have elapsed as she moves 15 meters in
the *y*-direction.

It follows that, as a result of time dilation, this collision as viewed from
spaceship *A* does *not* cause equal and opposite velocities for the
two spaceships in the *y*-direction. Initially, both spaceships were moving
parallel to the *x*-axis - there was zero momentum in the *y*-direction.
Consider *y*-direction
momentum conservation in the inertial frame in which *A* was initially at
rest. An observer in that frame
measuring *y*-velocities after the collision will find *A* to be
moving at 15 meters per second, *B* to be moving at -0.8 x 15 meters per
second in the *y*-direction. So
how can we argue there is zero total momentum in the *y*-direction *after* the collision, when the identical spaceships do *not* have equal and
opposite velocities?

### Einstein rescues Momentum Conservation

Einstein was so sure that momentum conservation must always hold that he
rescued it with a bold hypothesis: the mass of an object must depend on its
speed! In fact, the mass must
increase with speed in just such a way as to cancel out the lower *y*-direction
velocity resulting from time dilation. That is to say, if an object at rest has
a mass *m*_{0},
moving at a speed *v* it must have mass

_{}

to conserve* y*-direction
momentum.

Note that this is an undetectably small effect at ordinary speeds, but as an object approaches the speed of light, the mass increases without limit!

Of course, we have taken a very special case here: a particular kind of collision. The reader might well wonder if the same mass correction would work in other types of collision, for example a straight line collision in which a heavy object rear-ends a lighter object. The algebra is straightforward, if tedious, and it is found that this mass correction factor does indeed ensure momentum conservation for any collision in all inertial frames.

### Mass Really* Does* Increase with Speed

Deciding that masses of objects must depend on speed like this seems a heavy price to pay to rescue conservation of momentum! However, it is a prediction that is not difficult to check by experiment. The first confirmation came in 1908, measuring the mass of fast electrons in a vacuum tube. In fact, the electrons in an old-fashioned color TV tube are about half a percent heavier than electrons at rest, and this must be allowed for in calculating the magnetic fields used to guide them to the screen.

Much more dramatically, in modern particle accelerators very powerful electric fields are used to accelerate electrons, protons and other particles. It is found in practice that these particles become heavier and heavier as the speed of light is approached, and hence need greater and greater forces for further acceleration. Consequently, the speed of light is a natural absolute speed limit. Particles are accelerated to speeds where their mass is thousands of times greater than their mass measured at rest, usually called the “rest mass”.

** Warning**:
It should be mentioned that some people don’t like the statement that
mass increases with speed, they feel that the word “mass” should be
restricted to the rest mass of an object, which we’ve called

*m*

_{0}. This difference of definition has no physical content, however—it’s just a matter of taste. We would write momentum as

*p*=

*mv*, they would write our

*m*

_{0}as

*m*, and say the formula for momentum in their notation is

_{}Either way, a fast electron is that much harder to deflect from a straight line.

### Mass and Energy Conservation: Kinetic Energy and Mass for Very Fast Particles

As everyone has heard, in special relativity mass and energy are not
separately conserved, in certain situations mass *m* can be converted to
energy *E *= *mc*^{2}. This equivalence is closely related to
the mass increase with speed, as we shall see. Suppose a constant force *F* accelerates a particle of rest mass *m*_{0 }in a straight line. The work done by the force in
accelerating the particle as it travels a distance *d* is *Fd*, and this work
has given the particle kinetic energy.

As a warm up, recall the elementary derivation of the kinetic energy
½*mv*² of an ordinary *non-relativistic* (i.e. slow
moving) object of mass *m*. Suppose it starts from rest. Then after
time *t*, it has traveled distance *d* = ½ *at*^{2}, and *v* = *at*. From
*F* = *ma*, the work done by the force *Fd* = *mad* = ½ *ma*^{2}*t*^{2} = ½ *mv*^{2}.

This won’t work if the mass is varying, because
*F* = *ma*,
for variable mass it’s

_{}

force = rate of change of momentum, and if the mass changes the momentum changes, even at constant velocity.

An instructive extreme case is the kinetic energy of a particle traveling close to the speed of light, as particles do in accelerators. In this regime, the change of speed with increasing momentum is negligible! Instead,

_{}

where as usual *c* is the speed of light. This is what happens in a particle
accelerator for a charged particle in a constant electric field, with *F* = *qE*.

Since the particle is moving at a speed very close to *c*, in time *dt* it will move *cdt* and
the force will do work *Fcdt*. The equation above can be rewritten

_{}

So the energy *dE* expended by the accelerating force in the time *dt* yields an increase in mass, and _{} Provided the
speed is close to *c*, this can of
course be integrated to an excellent approximation, to relate a finite particle
mass change to the energy expended in accelerating it.

### Kinetic Energy and Mass for Slow Particles

Recall that to get momentum to be conserved in all inertial frames, we had
to assume an increase of mass with speed by the factor _{} This necessarily implies that *even a
slow-moving object has a tiny mass increase if it is put in motion*.

How does this mass increase relate to the kinetic energy? Consider a mass *m*_{0}, moving at speed *v*,
much less than the speed of light. Its kinetic energy *E* =½*m*_{0}*v*², as discussed
above. Its mass is _{} which we can
write as *m*_{0 }+*dm, *so *dm* is the tiny mass increase we know must occur. It’s easy to calculate *dm*.

For _{} we can make the
approximations

_{}

and

_{}

So, for _{}

_{}

Again, the mass increase *dm* is related to the kinetic energy *KE* by *KE* = (*dm*)*c*^{2}*.* Having looked at two simple cases,
we’re ready to derive the general result, valid over the whole range of
possible speeds.

### Kinetic Energy and Mass for Particles of Arbitrary Speed

We have shown in the two sections above that (in the two limiting cases) when
a force does work to increase the kinetic energy of a particle it also causes
the mass of the particle to increase by an amount equal to the increase in
energy divided by *c*^{2}. In fact this result is exactly true
over the whole range of speed from zero to arbitrarily close to the speed of
light, as we shall now demonstrate.

For a particle of rest mass *m*_{0} accelerating along a straight line (from rest) under a
constant force *F*,

_{}

Therefore, the work done when the particle moves a distance *dx* is

_{}

using *v* = *dx*/*dt*.

Therefore the total work done from rest—the kinetic energy—is:

_{}

(The integral is easily done by making the substitution *y* = *v*^{2}/*c*^{2}.)

So we see that in the general case the work done on the body, by definition
its kinetic energy, is just equal to its mass increase multiplied by *c ^{2}*.

To understand why this isn’t noticed in everyday life, try an example,
such as a jet airplane weighing 100 tons moving at 2,000mph. 100 tons is 100,000 kilograms, 2,000mph
is about 1,000 meters per second. That’s a kinetic energy ½* mv*² of ½.10^{11}joules, but the corresponding mass
change of the airplane down by the factor *c*² = 9.10^{16},
giving an actual mass increase of about half a milligram, not too easy to
detect!

### Notation: *m* and *m*_{0}

As stated earlier, we use *m*_{0} to denote the “rest
mass” of an object, and *m* to denote its relativistic mass, _{}

In this notation, we follow French and Feynman*. Krane and Tipler, in contrast, use m for
the rest mass*. Using *m* as we do gives neater formulas for momentum and energy, but is not without its
dangers. One must remember that *m* is *not* a constant, but a function of speed. Also, *one must remember* that the
relativistic kinetic energy is (*m*-*m*_{0})*c*^{2},
and *not* equal to ½*mv*^{2}, even with the relativistic
mass!

*Example*: take *v*^{2}/*c*^{2} = 0.99, find
the kinetic energy, and compare it with ½*mv*^{2} (using
the relativistic mass).