# Special Relativity

*Michael Fowler, UVa Physics 3/3/08*

## Galilean Relativity again

At this point in the course, we finally enter the twentieth century—Albert
Einstein wrote his first paper on relativity in 1905. To put his work in
context, let us first review just what is meant by “relativity” in
physics. The first example, mentioned in a previous lecture, is what is
called “Galilean relativity” and is nothing but Galileo’s
perception that by observing the motion of objects, alive or dead, in a closed
room there is no way to tell if the room is at rest or is in fact in a boat
moving at a steady speed in a fixed direction. (You *can* tell if
the room is accelerating or turning around.) Everything looks the same in
a room in steady motion as it does in a room at rest. After Newton formulated his Laws of Motion, describing how bodies move in response to forces and
so on, physicists reformulated Galileo’s observation in a slightly more
technical, but equivalent, way: they said* the laws of physics are the same
in a uniformly moving room as they are in a room at rest*. In other
words, the same force produces the same acceleration, and an object
experiencing no force moves at a steady speed in a straight line in either
case. Of course, talking in these terms implies that we have clocks and
rulers available so that we can actually time the motion of a body over a
measured distance, so the physicist envisions the room in question to have
calibrations along all the walls, so the position of anything can be measured,
and a good clock to time motion. Such a suitably equipped room is called
a “*frame of reference*”—the calibrations on the walls
are seen as a frame which you can use to specify the precise position of an
object at a given time. (This is the same as a set of
“coordinates”.) Anyway, the bottom line is that no amount of
measuring of motions of objects in the “frame of reference” will
tell you whether this is a frame at rest or one moving at a steady velocity.

What exactly do we mean by a frame “at rest” anyway? This
seems obvious from our perspective as creatures who live on the surface of the
earth—we mean, of course, at rest relative to fixed objects on the
earth’s surface. Actually, the earth’s rotation means this
isn’t quite a fixed frame, and also the earth is moving in orbit at 18
miles per second. From an astronaut’s point of view, then, a frame
fixed relative to the sun might seem more reasonable. But why stop there?
We believe the laws of physics are good throughout the universe. Let
us consider somewhere in space far from the sun, even far from our galaxy. We
would see galaxies in all directions, all moving in different ways. Suppose
we now set up a frame of reference and check that Newton’s laws still
work. In particular, we check that the First Law holds—that a body
experiencing no force moves at a steady speed in a straight line. *This
First law is often referred to as The Principle of Inertia, and a frame in
which it holds is called an Inertial Frame*. Then we set up another
frame of reference, moving at a steady velocity relative to the first one, and
find that Newton’s laws are o.k. in this frame too. The point to notice
here is that it is not at all obvious which—if either—of these
frames is “at rest”. We *can*, however, assert that they
are both *inertial* frames, after we’ve checked that in both of
them, a body with no forces acting on it moves at a steady speed in a straight
line (the speed could be zero). In this situation, Michelson would have
said that a frame “at rest” is one at rest relative to the aether. However,
his own experiment found motion through the aether to be undetectable, so how
would we ever know we were in the right frame?

As we mentioned in the last lecture, in the middle of the nineteenth century
there was a substantial advance in the understanding of electric and magnetic
fields. (In fact, this advance is in large part responsible for the improvement
in living standards since that time.) The new understanding was
summarized in a set of equations called Maxwell’s equations describing
how electric and magnetic fields interact and give rise to each other, just as,
two centuries earlier, the new understanding of dynamics was summarized in the
set of equations called Newton’s laws. The important thing about
Maxwell’s equations for our present purposes is that they predicted waves
made up of electric and magnetic fields that moved at 3×10^{8}
meters per second, and it was immediately realized that this was no
coincidence—light waves must be nothing but waving electric and magnetic
fields. (This is now fully established to be the case.)

It is worth emphasizing that Maxwell’s work predicted the speed of light from the results of experiments that were not thought at the time they were done to have anything to do with light—experiments on, for example, the strength of electric field produced by waving a magnet. Maxwell was able to deduce a speed for waves like this using methods analogous to those by which earlier scientists had figured out the speed of sound from a knowledge of the density and the springiness of air.

## Generalizing Galilean Relativity to Include Light: Special Relativity

We now come to Einstein’s major insight: the Theory of Special Relativity. It is deceptively simple. Einstein first dusted off Galileo’s discussion of experiments below decks on a uniformly moving ship, and restated it as :

*The Laws of
Physics are the same in all Inertial Frames.*

*Einstein then simply brought this up to date*, by pointing out that
the Laws of Physics must now include Maxwell’s equations describing
electric and magnetic fields as well as Newton’s laws describing motion
of masses under gravity and other forces. (*Note for experts and the
curious*: we shall find that Maxwell’s equations are completely
unaltered by special relativity, but, as will become clear later, Newton’s Laws do need a bit of readjustment to include special relativistic
phenomena. The First Law is still o.k., the Second Law in the form *F*
= *ma* is not, because we shall find mass varies; we need to equate force
to rate of change of momentum (Newton understood that, of course—that’s
the way he stated the law!). The Third Law, stated as action equals
reaction, no longer holds because if a body moves, its electric field, say,
does not readjust instantaneously—a ripple travels outwards at the speed
of light. Before the ripple reaches another charged body, the electric
forces between the two will be unbalanced. However, the crucial
consequence of the Third Law—the conservation of momentum when two bodies
interact, still holds. It turns out that the rippling field itself
carries momentum, and everything balances.)

Demanding that Maxwell’s equations be satisfied in all inertial frames
has one major consequence as far as we are concerned. As we stated above,
Maxwell’s equations give the speed of light to be 3×10^{8}
meters per second. Therefore, *demanding that the laws of physics are
the same in all inertial frames implies that the speed of any light wave,
measured in any inertial frame, must be *3×10^{8} *meters
per second*.

This then is the entire content of the Theory of Special Relativity: the
Laws of Physics are the same in any inertial frame, and, in particular, any
measurement of the speed of light in any inertial frame will always give 3×10^{8}
meters per second.

## You *Really* Can’t Tell You’re Moving!

Just as Galileo had asserted that observing gnats, fish and dripping
bottles, throwing things and generally jumping around would not help you to
find out if you were in a room at rest or moving at a steady velocity, Einstein
added that no kind of observation at all*, even measuring the speed of light
across your room* to any accuracy you like, would help find out if your room
was “really at rest”. This implies, of course, that the
concept of being “at rest” is meaningless. If Einstein is
right, there is* no* natural rest-frame in the universe. Naturally,
there can be no “aether”, no thin transparent jelly filling space
and vibrating with light waves, because if there were, *it* would provide
the natural rest frame, and affect the speed of light as measured in other
moving inertial frames as discussed above.

So we see the Michelson-Morley experiment was doomed from the start. There
never was an aether wind. The light was not slowed down by going
“upstream”—light *always* travels at the same speed,
which we shall now call *c*,

*c *= 3×10^{8}
meters per second

to save writing it out every time. *This now answers the question of
what the speed of light, c, is relative to*. We already found that it
is not like sound, relative to some underlying medium. It is also not
like bullets, relative to the source of the light (the discredited emitter
theory). *Light travels at c* *relative to the observer*, since
if the observer sets up an inertial frame (clocks, rulers, etc.) to measure the
speed of light he will find it to be *c*. (We always assume our
observers are very competent experimentalists!)

## Truth and Consequences

The Truth we are referring to here is the seemingly innocuous and plausible
sounding statement that all inertial frames are as good as each other—the
laws of physics are the same in all of them—and so the *speed of light*
is the same in all of them. As we shall soon see, this Special Theory of
Relativity has some surprising consequences, which reveal themselves most
dramatically when things are moving at relative speeds comparable to the speed
of light. Einstein liked to explain his theory using what he called
“thought experiments” involving trains and other kinds of
transportation moving at these speeds (technically unachievable so far!), and
we shall follow his general approach.

To begin with, let us consider a simple measurement of the speed of light carried out at the same time in two inertial frames moving at half the speed of light relative to each other. The setup is as follows: on a flat piece of ground, we have a flashlight which emits a blip of light, like a strobe. We have two photocells, devices which click and send a message down a wire when light falls on them. The photocells are placed 10 meters apart in the path of the blip of light, they are somehow wired into a clock so that the time taken by the blip of light to travel from the first photocell to the second, in other words, the time between clicks, can be measured. From this time and the known distance between them, we can easily find the speed of the blip of light.

Meanwhile, there is another observer, passing overhead in a spaceship
traveling at half the speed of light. She is also equipped with a couple
of photocells, placed 10 meters apart on the bottom of her spaceship as shown,
and she is able to measure the speed of the same blip of light, relative to her
frame of reference (the spaceship). *The observer on the spaceship will
measure the blip of light to be traveling at c relative to the spaceship, the
observer on the ground will measure the same blip to be traveling at c relative
to the ground*. That is the unavoidable consequence of the Theory of
Relativity.

(*Note*: actually the picture above is not quite the way it would
really look. As we shall find, objects moving at relativistic speeds are
contracted, and this combined with the different times light takes to reach the
eye from different parts of the ship would change the ship’s appearance.
But this does not affect the validity of the statements above.)