fix problems with html output

[light-and-matter.git] / lm / em / c.rbtex

<%
  require "../eruby_util.rb"
%>

<% if false then figure_in_toc("einstein") end %>

<%
  chapter(
    '23',
    %q{Relativity and magnetism},
    'ch:relativity',
    '',
    {'opener'=>''}
  )
%>

Many people imagine Einstein's theory of relativity as something exotic and speculative. It's certainly
not speculative at this point in history, since you use it every time you use a GPS receiver. But it's
even less exotic than that. Every time you stick a magnet to your refrigerator, you're making use of
relativity. The title of this chapter is ``Electromagnetism,'' but we'll start out by digging a little deeper
into relativity as preparation for understanding what magnetism is and where it comes from.

<% begin_sec("Relativistic distortion of Space and Time") %>\label{sec:x-t-distortion}
<% begin_sec("Time distortion arising from motion and gravity") %>
Let's refer back to the results of the Hafele-Keating experiment described on
p.~\pageref{hafele-keating}. Hafele and Keating were testing specific quantitative predictions of relativity, and they verified them to within
their experiment's error bars. Let's work backward instead, and
inspect the empirical results for clues as to how time works.

The east-going clock lost time, ending up off by $-59\pm10$ nanoseconds, while the west-going one gained $273\pm7$ ns.
Since two traveling clocks experienced effects in opposite directions,
we can tell that the rate at which time flows depends on the motion
of the observer. The east-going clock was moving in the same direction as the earth's rotation, so its velocity
relative to the earth's center was greater than that of the clock that remained in Washington, while the west-going clock's velocity was
correspondingly reduced. The fact that the east-going clock fell behind, and the west-going one got ahead,
shows that the effect of motion is to make time go more slowly. This effect of motion on time was predicted by
Einstein in his original 1905 paper on relativity, written when he was 26.
<% marg(70) %>
<%
  fig(
    'hafele-keating-directions',
    %q{All three clocks are moving to the east. Even though the west-going plane is moving to the west relative to the air, the air is moving
           to the east due to the earth's rotation.}
  )
%>
<% end_marg %>

If this had been the only effect in the Hafele-Keating experiment, then we would have expected to see effects on the
two flying clocks that were equal in size. Making up some simple numbers to keep the arithmetic transparent, suppose that the earth rotates
from west to east at 1000 km/hr, and that the planes fly at 300 km/hr. Then the speed of the clock on the ground
is 1000 km/hr, the speed of the clock on the east-going plane is 1300 km/hr, and that of the west-going clock 700 km/hr.
Since the speeds of 700, 1000, and 1300 km/hr have equal spacing on either side of 1000, we would expect the discrepancies
of the moving clocks relative to the one in the lab to be equal in size but opposite in sign.

In fact, the two effects are unequal in size: $-59$ ns and 273 ns. This
implies that there is a second effect involved, a speeding up of time
simply due to the planes' being up in the air. 
This was verified more directly in a 1978 experiment by Iijima and Fujiwara, figure \figref{iijima}, in which identical atomic
clocks were kept at rest at the top and bottom of a mountain near Tokyo.
This experiment, unlike the Hafele-Keating one, isolates one effect on time, the gravitational one: time's
rate of flow increases with height in a gravitational field. Einstein didn't figure out
how to incorporate gravity into relativity until 1915, after much frustration and many false starts. The
simpler version of the theory without gravity is known as special relativity, the full version as general
relativity. We'll restrict ourselves to special relativity until chapter \ref{ch:genrel}, and that means that what we want to
focus on right now is the distortion of time due to motion, not gravity.\label{iijima}

<%
  fig(   
    'iijima',
    %q{%
       A graph 
       showing the time difference between two atomic clocks. One clock was kept at Mitaka Observatory, at 58 m above sea level.
       The other was moved back and forth to a second observatory, Norikura Corona Station, at the peak of the Norikura volcano, 2876 m above sea level.
       The plateaus on the graph are data from the periods when the clocks were compared side by side at Mitaka. The difference between one plateau and the next
       shows a gravitational effect on the rate of flow of time, accumulated during the period when the mobile clock was at the top of Norikura.
       Cf.~problem \ref{hw:pound-rebka}, p.~\pageref{hw:pound-rebka}.
    },
    {
      'width'=>'wide','sidecaption'=>false
    }
  )
%>

We can now see in more detail how to apply the correspondence principle. The behavior of the three clocks in the
Hafele-Keating experiment shows that the amount of time distortion increases as the speed of the clock's motion
increases. Newton lived in an era when the fastest
mode of transportation was a galloping horse, and the best
pendulum clocks would accumulate errors of perhaps a minute over the course of several days.
A horse is much slower than a jet plane, so the
distortion of time would have had a relative size of only $\sim10^{-15}$ --- much smaller than the clocks were capable of detecting.
At the speed of a passenger jet, the effect is about $10^{-12}$,
and state-of-the-art atomic clocks in 1971 were capable of measuring that.
A GPS satellite travels much faster than a jet airplane, and the effect on the satellite
turns out to be $\sim10^{-10}$. The general idea here is that all physical laws are approximations, and
approximations aren't simply right or wrong in different situations. Approximations are better or worse
in different situations, and the question is whether a particular approximation is good enough in a given
situation to serve a particular purpose. The faster the motion, the worse the Newtonian approximation of
absolute time. Whether the approximation is good enough depends on what you're trying to accomplish.
The correspondence principle says that the approximation must have been good enough to explain
all the experiments done in the centuries before Einstein came up with relativity.
<% marg(80) %>
<%
  fig(
    'correspondence-dramatized',
    %q{The correspondence principle requires that the relativistic distortion of time become small for small velocities.}
  )
%>
<% end_marg %>

By the way, don't get an inflated idea of the importance of these atomic clock
experiments. Special relativity had already been confirmed by a vast and varied body of experiments decades
before the 1970's. The only reason I'm giving such a prominent role to these experiments, which were actually more important as tests of
general relativity, is that they were conceptually very direct.
It would be nice to have an equally simple and transparent atomic clock experiment 
in which only the effect of motion was singled out, with no gravitational effect.
Example \ref{eg:chou} on page \pageref{eg:chou} describes how something along these lines was eventually
carried out, forty years after the Hafele-Keating experiment.
<% end_sec %> % Time distortion arising from motion and gravity

<% begin_sec("The Lorentz transformation") %>\label{sec:lorentz}
Relativity says that when two observers are in different frames of reference, each observer considers
the other one's perception of time to be distorted. We'll also
see that something similar happens to their observations of distances, so both space and
time are distorted.
What exactly is this distortion? How do we even conceptualize it?

The idea isn't really as radical as it might seem at first. We can visualize the structure of space
and time using a graph with position and time on its axes. These graphs are familiar by now, but
we're going to look at them in a slightly different way. Before, we used them to describe the motion
of objects. The grid underlying the graph was merely the stage on which the actors played their parts.
Now the background comes to the foreground: it's time and space themselves that we're studying.
We don't necessarily need to have a line or a curve drawn on top of the grid to represent a particular
object. We may, for example, just want to talk about events, depicted as points on the graph as in
figure \figref{joan-of-arc}. A distortion of the Cartesian grid underlying the graph can arise for
perfectly ordinary reasons that Newton would have readily accepted. For example, we can simply
change the units used to measure time and position, as in figure \figref{change-of-units}. 
<% marg(50) %>
<%
  fig(
    'joan-of-arc',
    %q{Two events are given as points on a graph of position versus time. Joan of Arc helps to restore Charles VII to the throne.
         At a later time and a different position, Joan of Arc is sentenced to death.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'change-of-units',
    %q{A change of units distorts an $x$-$t$ graph. This graph depicts exactly the same events as figure \figref{joan-of-arc}.
          The only change is that the $x$ and $t$ coordinates are measured using different units, so the grid is compressed
          in $t$ and expanded in $x$.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'change-of-units-convention',
    %q{A convention we'll use to represent a distortion of time and space.}
  )
%>
<% end_marg %>

\enlargethispage{-3\baselineskip}

We're going to
have quite a few examples of this type, so I'll adopt the convention shown in figure \figref{change-of-units-convention}
for depicting them. Figure \figref{change-of-units-convention} summarizes the relationship between figures
\figref{joan-of-arc} and \figref{change-of-units} in a more compact form. The gray rectangle represents the
original coordinate grid of figure \figref{joan-of-arc}, while the grid of black lines represents the new version
from figure \figref{change-of-units}. Omitting the grid from the gray rectangle makes the diagram easier
to decode visually.

Our goal of unraveling the mysteries of special relativity amounts to nothing more than finding out how to
draw a diagram like \figref{change-of-units-convention} in the case where the two different sets of coordinates represent
measurements of time and space made by two different observers, each in motion relative to the other.
Galileo and Newton thought they knew the answer to this question, but their answer turned out to be
only approximately right. To avoid repeating the same mistakes, we need to clearly spell out what we think are
the basic properties of time and space that will be a reliable foundation for our reasoning. I want to emphasize
that there is no purely logical way of deciding on this list of properties. The ones I'll list are simply a summary of the
patterns observed in the results
from a large body of experiments. Furthermore, some of them are only approximate. For example, property 1 below
is only a good approximation when the gravitational field is weak, so it is a property that applies to
special relativity, not to general relativity.

Experiments show that:\label{spacetime-properties}
\begin{enumerate}
\item No point in time or space has properties that make it different from any other point.
\item Likewise, all directions in space have the same properties.
\item Motion is relative, i.e., all inertial frames of reference are equally valid.
\item Causality holds, in the sense described on page \pageref{causality-defined}.
\item Time depends on the state of motion of the observer.
\end{enumerate}

Most of these are not very subversive. Properties 1 and 2 date back to the time when Galileo and Newton started
applying the same universal laws of motion to the solar system and to the earth; this contradicted
Aristotle, who believed that, for example, a rock would naturally want to move in a certain special
direction (down) in order to reach a certain special location (the earth's surface).
Property 3 is the reason that Einstein called his theory ``relativity,'' but Galileo and Newton
believed exactly the same thing to be true, as dramatized by Galileo's run-in with the Church over
the question of whether the earth could really be in motion around the sun. Example
\ref{eg:clock-comparison-inertia} on p.~\pageref{eg:clock-comparison-inertia} describes a modern, high-precision
experiment that can be interpreted as a test of this principle.
Property 4 would probably surprise most people only because it asserts in such a weak and specialized
way something that they feel deeply must be true. The only really strange item on the list is 5,
but the Hafele-Keating experiment forces it upon us.

\enlargethispage{-2\baselineskip}

If it were not for property 5, we could imagine that figure \figref{galilean-boost} would
give the correct transformation between frames of reference in motion relative to one another.
Let's say that observer 1, whose grid coincides with the gray rectangle, is a hitch-hiker standing
by the side of a road. Event A is a raindrop hitting his head, and event B is another raindrop hitting
his head. He says that A and B occur at the same location in space. Observer 2 is a motorist who
drives by without stopping; to him, the passenger compartment of his car is at rest, while the
asphalt slides by underneath. He says that A and B occur at different points in space, because
during the time between the first raindrop and the second, the hitch-hiker has moved backward.
On the other hand, observer 2 says that events A and C occur in the same place, while the hitch-hiker
disagrees. The slope of the grid-lines is simply the velocity of the relative motion of each observer
relative to the other.
<% marg(70) %>
<%
  fig(
    'galilean-boost',
    %q{A Galilean version of the relationship between two frames of reference. As in all such graphs in
      this chapter, the original coordinates, represented by the gray rectangle, have a time axis that
      goes to the right, and a position axis that goes straight up.}
  )
%>
<% end_marg %>

Figure \figref{galilean-boost} has familiar, comforting, and eminently sensible
behavior, but it also happens to be wrong, because it violates property 5. The distortion of
the coordinate grid has only moved the vertical lines up and down, so both observers agree
that events like B and C are simultaneous. If this was really the way things worked, then
all observers could synchronize all their clocks with one another for once and for all, and
the clocks would never get out of sync. This contradicts the results of the Hafele-Keating
experiment, in which all three clocks were initially synchronized in Washington, but later
went out of sync because of their different states of motion.

It might seem as though we still had a huge amount of wiggle room available for the correct
form of the distortion. It turns out, however, that properties 1-5 are sufficient to prove that there
is only one answer, which is the one found by Einstein in 1905. To see why this is, let's work by
a process of elimination.

Figure \figref{bowtie} shows a transformation that might
seem at first glance to be as good a candidate as any other,
but it violates property 3, that motion is relative, for the following
reason. In observer 2's frame of reference, some of the grid lines cross one another.
This means that observers 1 and 2 disagree on whether or not certain events are the same.
For instance, suppose that event A marks the arrival of an arrow at the bull's-eye of a
target, and event B is the location and time when the bull's-eye is punctured.
Events A and B occur
at the same location and at the same time. If one observer says that A and B coincide, but another
says that they don't, we have a direct contradiction. Since the two frames of reference in figure
\figref{bowtie} give contradictory results, one of them is right and one is wrong. This violates
property 3, because all inertial frames of reference are supposed to be equally valid.
To avoid problems like this, we clearly need to make sure that none of the grid lines ever cross
one another.

<% marg(70) %>
<%
  fig(
    'bowtie',
    %q{A transformation that leads to disagreements about whether two events occur at the same time and place.
       This is not just a matter of opinion. Either the arrow hit the bull's-eye or it didn't.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'nonlinear-transformation',
    %q{A nonlinear transformation.}
  )
%>
<% end_marg %>

The next type of transformation we want to kill off is shown in figure \figref{nonlinear-transformation},
in which the grid lines curve, but never cross one another.
The trouble with this one is that
it violates property 1, the uniformity of time and space. The transformation is unusually
``twisty'' at A, whereas at B it's much more smooth. This can't be correct, because the transformation
is only supposed to depend on the relative state of motion of the two frames of reference, and
that given information doesn't single out a special role for any particular point in spacetime.
If, for example, we had one frame of reference \emph{rotating} relative to the other, then there
would be something special about the axis of rotation. But we're only talking about \emph{inertial}
frames of reference here, as specified in property 3, so we can't have rotation; each frame of reference
has to be moving in a straight line at constant speed.
For frames related in this way, there is nothing that could single out an event like A for special
treatment compared to B, so transformation \figref{nonlinear-transformation} violates property 1.

\enlargethispage{-3\baselineskip}

The examples in figures \figref{bowtie} and \figref{nonlinear-transformation} show that the transformation
we're looking for must be linear, meaning that it must transform lines into lines, and furthermore that
it has to take parallel lines to parallel lines.\index{homogeneity of spacetime}
Einstein wrote in his 1905 paper that ``\ldots on account of the property of homogeneity [property 1] which we ascribe to time and space,
the [transformation] must be linear.''\footnote{A. Einstein, ``On the Electrodynamics of Moving Bodies,''
\emph{Annalen der Physik 17} (1905), p. 891, tr. Saha and Bose.}
% Shamos, p. 323
Applying this to our diagrams,
the original gray rectangle, which is a special type of parallelogram containing right angles,
must be transformed into another parallelogram.
There are three types of transformations, figure \figref{three-cases}, that have this property.
Case I is the Galilean transformation of figure \figref{galilean-boost} on page \pageref{fig:galilean-boost},
which we've already ruled out.
<%
  fig(   
    'three-cases',
    %q{%
      Three types of transformations that preserve parallelism. Their distinguishing feature is what they do
      to simultaneity, as shown by what happens to the left edge of the original rectangle. In I, the left edge remains
      vertical, so simultaneous events remain simultaneous. In II, the left edge turns counterclockwise. In III, it turns clockwise.
    },
    {
      'width'=>'wide','sidecaption'=>false
    }
  )
%>

Case II can also be discarded. Here every point on the grid rotates counterclockwise. What physical parameter would
determine the amount of rotation? The only thing that could be relevant would be
$v$, the relative velocity of the motion of the two frames of reference with respect to one
another. But if the angle of rotation was proportional to $v$, then for large enough velocities
the grid would have left and right reversed, and this would violate property 4, causality: one observer
would say that event A caused a later event B, but another observer would say that B came first
and caused A.
<% marg(-21) %>
<%
  fig(
    'smooshing',
    %q{In the units that are most convenient for relativity, the transformation has symmetry about a 45-degree diagonal line.}
  )
%>
<% end_marg %>

\enlargethispage{-3\baselineskip}

\vspace{15mm}

The only remaining possibility is case III, which I've redrawn in figure \figref{smooshing} with a couple
of changes. This is the one that Einstein predicted in 1905. The transformation is known as the Lorentz transformation,
after Hendrik Lorentz (1853-1928),\index{Lorentz, Hendrik}\index{Lorentz transformation}
who partially anticipated Einstein's work, without arriving at the correct interpretation.
The distortion is a kind of smooshing and stretching, as suggested by the hands. Also, we've already seen in figures
\figref{joan-of-arc}-\figref{change-of-units-convention} on page \pageref{fig:joan-of-arc} that we're free to stretch
or compress everything as much as we like in the horizontal and vertical directions, because this simply corresponds
to changing the units of measurement for time and distance. In figure \figref{smooshing} I've chosen units that
give the whole drawing a convenient symmetry about a 45-degree diagonal line. Ordinarily it wouldn't make sense to
talk about a 45-degree angle on a graph whose axes had different units. But in relativity, the symmetric appearance of
the transformation tells us that space and time ought to be treated on the same footing, and measured in the same units.

\pagebreak

As in our discussion of the Galilean transformation, slopes are interpreted as velocities, and
the slope of the near-horizontal lines in figure \figref{lorentz-slope} is interpreted as the relative velocity of the two observers.
The difference between the Galilean version and the relativistic one is that now there is smooshing happening from the
other side as well. Lines that were vertical in the original grid, representing simultaneous events, now slant over to
the right. This tells us that, as required by property 5, different observers do not agree on whether events that occur in different places are
simultaneous. The Hafele-Keating experiment tells us that this non-simultaneity effect is fairly small, even when the velocity is as
big as that of a passenger jet, and this is what we would have anticipated by the correspondence principle. The way that
this is expressed in the graph is that if we pick the time unit to be the second, then the distance unit turns out to be hundreds of thousands of miles.
In these units, the velocity of a passenger jet is an extremely small number, so the slope $v$ in a figure like \figref{lorentz-slope}
is extremely small, and the amount of distortion is tiny --- it would be much too small to see on this scale.
<% marg(100) %>
<%
  fig(
    'lorentz-slope',
    %q{Interpretation of the Lorentz transformation. The slope indicated in the figure gives the relative velocity of the two frames of reference.
         Events A and B that were simultaneous in frame 1 are not simultaneous in frame 2, where event A occurs to the right of the $t=0$ line represented
         by the left edge of the grid, but event B occurs to its left.}
  )
%>
<% end_marg %>

The only thing left to determine about the Lorentz transformation is the size of the transformed parallelogram relative to the
size of the original one. Although the drawing of the hands in figure \figref{smooshing} may suggest that the grid deforms like
a framework made of rigid coat-hanger wire, that is not the case. If you look carefully at the figure, you'll see that the edges
of the smooshed parallelogram are actually a little longer than the edges of the original rectangle. In fact what stays the same
is not lengths but \emph{areas}, as proved in the caption to figure \figref{area-proof}.

<% end_sec %> % The Lorentz transformation

<%
  fig('area-proof',
    %q{Proof that Lorentz transformations don't change area: We first subject a square to a transformation with velocity $v$, and this increases its area by a factor $R(v)$, which
       we want to prove equals 1. We chop the resulting parallelogram up into little squares and finally apply a $-v$ transformation;
       this changes each little square's area by a factor $R(-v)$, so the whole figure's area is also scaled by $R(-v)$.
       The final result is to restore the square to its original shape and area, so $R(v)R(-v)=1$. But $R(v)=R(-v)$ by property 2 of spacetime
       on page \pageref{spacetime-properties}, which states that all directions in space have the same properties, so $R(v)=1$.
     },
    {
      'width'=>'fullpage'
    }
  )
%>

<% begin_sec("The $\\mygamma$ factor") %>\label{sec:gamma}

Figure \figref{lorentz-slope} showed us that observers in different frames disagree on whether different events are simultaneous.
This is an indication that time is not absolute, so we shouldn't be surprised that time's rate of flow is also different for different observers.
We use the symbol $\mygamma$ (Greek letter gamma)
defined in the figure \figref{gamma-as-projection} to measure this unequal rate of flow.
With a little algebra and geometry (homework problem \ref{hw:gamma-derivation}, page \pageref{hw:gamma-derivation}),
one can use the equal-area property to show that this ratio is given by
\begin{equation*}
  \mygamma = \frac{1}{\sqrt{1-v^2}} \qquad .
\end{equation*}
If you've had good training in physics, the first thing you probably think when you look at this equation is that it must be
nonsense, because its units don't make sense. How can we take something with units of velocity squared, and subtract it from
a unitless 1? But remember that this is expressed in our new relativistic units, in which the same units are used for
distance and time. In this system, velocities are always unitless. This sort of thing happens frequently in physics. For
instance, before James Joule discovered conservation of energy, nobody knew that heat and mechanical energy were different
forms of the same thing, so instead of measuring them both in units of joules as we would do now, they measured heat in
one unit (such as calories) and mechanical energy in another (such as foot-pounds). In ordinary metric units, we just need
an extra conversion factor, called $c$, and the equation becomes
\begin{equation*}
  \mygamma = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} \qquad .
\end{equation*}
<% marg(200) %>
<%
  fig(
    'gamma-as-projection',
    %q{The clock is at rest in the original frame of reference, and it measures a time interval $t$.
       In the new frame of reference, the time interval is greater by a factor that we notate as $\mygamma$.}
  )
%>
<% end_marg %>

When we say, ``It's five hours from LA to Vegas,'' we're using a unit of time as a unit of distance. This works because
there is a standard speed implied: the speed of a car on the freeway. Similarly, the conversion factor $c$ can be interpreted
as a speed, so that $v/c$ is the unitless ratio of two speeds. As argued on p.~\pageref{subsec:time-delays},
cause and effect can never be propagated instantaneously; $c$ turns out to be the specific numerical speed limit on
cause and effect. In particular, we'll see in section \ref{sec:universal-speed-c} that light travels at $c$, which has a numerical
value of $3.0\times10^8\ \munit/\sunit$.\index{speed of light}
<% marg(20) %>
<%
  fig(
    'gamma-graph-small',
    %q{A graph of $\gamma$ as a function of $v$.}
  )
%>
<% end_marg %>


\pagebreak

Because $\mygamma$ is always greater than 1, we have the following interpretation:

\begin{lessimportant}[Time dilation]
A clock runs fastest in the frame of reference of an observer who is at rest relative to the clock. An observer
in motion relative to the clock at speed $v$ perceives the clock as running more slowly by a factor of $\mygamma$.
\end{lessimportant}

<% marg(100) %>
<%
  fig(
    'length-contraction',
    %q{The ruler is moving in frame 1, represented by a square, but at rest in frame 2, shown as a parallelogram.
       Each picture of the ruler is a snapshot taken at a certain moment as judged according to frame 2's
       notion of simultaneity. An observer in frame 1 judges the ruler's length instead according to
       frame 1's definition of simultaneity, i.e., using points that are lined up vertically on the graph.
       The ruler appears shorter in the frame in which it is moving.
       As proved in figure \figref{length-contraction-proof}, the length contracts from $L$ to $L/\gamma$.}
  )
%>
<% end_marg %>

<%
  fig('length-contraction-proof',
    %q{This figure proves, as claimed in figure \figref{length-contraction}, that the length contraction is $x=1/\gamma$.
       First we slice the parallelogram vertically like a salami and slide the slices down, making the
       top and bottom edges horizontal. Then we do the same in the horizontal direction, forming a rectangle
       with sides $\gamma$ and $x$. Since both the Lorentz transformation and the slicing processes leave
       areas unchanged, the area $\gamma x$  of the rectangle must equal the area of the original square, which is 1.
     },
    {
      'width'=>'wide'
      #'sidecaption'=>true
    }
  )
%>

\noindent As proved in figures \figref{length-contraction} and \figref{length-contraction-proof}, lengths are also distorted:

\begin{lessimportant}[Length contraction]
A meter-stick appears longest to an observer who is at rest relative to it. An observer moving relative to the
meter-stick at $v$ observes the stick to be shortened by a factor of $\mygamma$.
\end{lessimportant}

\vfill

<% self_check('gammaatvzero',<<-'SELF_CHECK'
What is $\mygamma$ when $v=0$? What does this mean?
  SELF_CHECK
  ) %>

\vfill

\pagebreak

<%
  fig('interstellar-road-trip',
    'Example \ref{eg:interstellar-road-trip}.',
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

\begin{eg}{An interstellar road trip}\label{eg:interstellar-road-trip}
Alice stays on earth while her twin Betty
heads off in a spaceship for Tau Ceti, a nearby star. Tau Ceti is 12 light-years
away, so even though Betty travels at 87\% of the speed of light, it will take
her a long time to get there: 14 years, according to Alice.

Betty
experiences time dilation. At this speed, her $\gamma$ is 2.0, so that the voyage will
only seem to her to last 7 years. But there is perfect symmetry between Alice's
and Betty's frames of reference, so Betty agrees with Alice on their relative speed;
Betty sees herself as being at rest, while the sun and Tau Ceti both move backward
at 87\% of the speed of light. How, then, can she observe Tau Ceti to get to her
in only 7 years, when it should take 14 years to travel 12 light-years at this speed?

We need to take into account length contraction.
Betty sees the distance between the sun and Tau Ceti
to be shrunk by a factor of 2. The same thing occurs for Alice, who observes
Betty and her spaceship to be foreshortened.
\end{eg}


\pagebreak

\begin{eg}{The correspondence principle}\label{eg:gamma-for-low-v}
The correspondence principle requires that $\mygamma$ be close to 1 for the velocities much less than $c$ encountered
in everyday life. Let's explicitly find the amount $\epsilon$ by which $\mygamma$ differs from 1, when $v$ is small. Let $\mygamma=1+\epsilon$.
The definition of $\mygamma$ gives $1=\mygamma^2(1-v^2/c^2)$, so $1=(1+2\epsilon+\epsilon^2)(1-v^2/c^2)\approx 1+2\epsilon-v^2/c^2$, where
the approximation comes from discarding very small terms such as $\epsilon^2$ and $\epsilon v^2/c^2$. We find $\epsilon=v^2/2c^2$.
As expected, this will be small when $v$ is small compared to $c$.
\end{eg}

\vfill

Figure \figref{gamma-graph-small} shows that the approximation found
in example \ref{eg:gamma-for-low-v} is \emph{not} valid for large values of $v/c$. In fact, $\gamma$
blows up to infinity as $v$ gets closer and closer to $c$.

\vfill

<% marg(130) %>
<%
  fig(
    'atomic-clock-gamma',
    %q{Time dilation measured with an atomic clock at low speeds. The theoretical curve, shown with a dashed line,
       is calculated from $\mygamma=1/\sqrt{1-(v/c)^2}$; at these small velocities, the approximation of example
       \ref{eg:gamma-for-low-v} is an excellent one, so $\mygamma\approx 1+v^2/2c^2$, and the graph is indistinguishable
       from a parabola. This graph corresponds to an extreme close-up view of the lower left corner of 
       figure \figref{gamma-graph-small}.  The error bars on the experimental points are about the same size as the dots.}
  )
%>
<% end_marg %>

\begin{eg}{A moving atomic clock}\label{eg:chou}
Example \ref{eg:gamma-for-low-v} shows that when
$v$ is small, relativistic effects are approximately proportional to $v^2$, so it is very
difficult to observe them at low speeds. For example, a car on
the freeway travels at about 1/10 the speed of a passenger jet, so the resulting
time dilation is only 1/100 as much. For this reason, it was not until four decades after Hafele and Keating
that anyone did a conceptually simple atomic clock experiment
in which the only effect was motion, not gravity; it is difficult to
move a clock at a high enough velocity without putting it in some kind of aircraft, which then has to fly at some
altitude. In 2010, however, Chou \emph{et al.}\footnote{Science 329 (2010) 1630} succeeded in building an atomic
clock accurate enough to detect time dilation at speeds as low
as 10 m/s. Figure \figref{atomic-clock-gamma} shows their results.
Since it was not practical to move the entire clock, the experimenters only moved the aluminum atoms inside the
clock that actually made it ``tick.''
\end{eg}


<%
  fig(
    'cern-muon-storage-ring',
    %q{Apparatus used for the test of relativistic time dilation described in example \ref{eg:cern-muons}. 
       The prominent black and white blocks are large magnets surrounding a circular pipe
       with a vacuum inside. \linebreak (c) 1974 by CERN.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true,
      'sidepos'=>'t'
    }
  )
%>

\begin{eg}{Large time dilation}\label{eg:cern-muons}
The time dilation effect in the Hafele-Keating experiment was very small. If we want to see a large time dilation
effect, we can't do it with something the size of the atomic clocks they used; the kinetic energy would be
greater than the total megatonnage of all the world's nuclear arsenals. We can, however, accelerate subatomic particles
to speeds at which $\mygamma$ is large. For experimental particle physicists, relativity is something you do all day
before heading home and stopping off at the store for milk. An early, low-precision experiment of this kind was
performed by Rossi and Hall in 1941, using naturally occurring cosmic rays. Figure \figref{cern-muon-storage-ring} shows
a 1974 experiment\footnote{Bailey at al., Nucl. Phys. B150(1979) 1}
of a similar type which verified the time dilation predicted by relativity to a precision of about
one part per thousand.
<% marg(-4) %>
<%
  fig(
    'cern-muon-graph',
    %q{Example \ref{eg:cern-muons}: Muons accelerated to nearly $c$ undergo radioactive decay much more slowly than they would according to
       an observer at rest with respect to the muons. The first two data-points (unfilled circles) were subject
       to large systematic errors.},
    {
      'anaonymous'=>true
    }
  )
%>
<% end_marg %>

\vspace{30mm}

Particles called muons (named after the Greek letter $\mu$, ``myoo'') were produced by an
accelerator at CERN, near Geneva. A muon is essentially a heavier version
of the electron. Muons undergo radioactive decay,
lasting an average of only 2.197 $\mu\sunit$ before they evaporate into an electron and two neutrinos.
The 1974 experiment was actually built in order to measure the magnetic properties of muons, but it produced a high-precision
test of time dilation as a byproduct. Because muons have the same electric charge as electrons, they can be trapped using
magnetic fields. Muons were injected into the ring shown in figure \figref{cern-muon-storage-ring}, circling around it until they underwent radioactive decay.
At the speed at which these muons were traveling, they had $\mygamma=29.33$, so on the average they lasted 29.33 times
longer than the normal lifetime. In other words, they were like tiny alarm clocks that self-destructed at a randomly
selected time. Figure \figref{cern-muon-graph} shows the number of radioactive decays counted, as a function of the
time elapsed after a given stream of muons was injected into the storage ring. The two dashed lines show the rates
of decay predicted with and without relativity. The relativistic line is the one that agrees with experiment.
\end{eg}

\pagebreak

<%
  fig(
    'schoolbus-with-x-t',
    %q{%
      Example }+ref_workaround('eg:garage-paradox')+%q{: In the garage's frame of reference, the bus
      is moving, and fits in the garage due to length contraction. In the bus's frame,
      the garage is moving, and can't hold the bus due to \emph{its} length contraction.
    },
    {
      # 'anonymous'=>true,
      'width'=>'fullpage'
    }
  )
%>

\begin{eg}{The garage paradox}\label{eg:garage-paradox}\index{garage paradox}
Suppose we
take a schoolbus and drive it at relativistic
speeds into a garage of ordinary size, in which it normally
would not fit. Because of the length contraction, it fits.
But the driver will perceive the
\emph{garage} as being contracted and thus even less able to
contain the bus. 

The paradox is
resolved when we recognize that the concept of fitting the
bus in the garage ``all at once'' contains a hidden
assumption, the assumption that it makes sense to ask
whether the front and back of the bus can \emph{simultaneously} be
in the garage. Observers in different frames of reference
moving at high relative speeds do not necessarily agree on
whether things happen simultaneously. As shown in figure \figref{schoolbus-with-x-t}, the person in the
garage's frame can shut the door at an instant B he perceives
to be simultaneous with the front bumper's arrival A at the
back wall of the garage, but the driver would not agree
about the simultaneity of these two events, and would
perceive the door as having shut long after she plowed
through the back wall.
\end{eg}

\pagebreak

\begin{eg}{An example of length contraction}\label{eg:rhic}
Figure \figref{rhic} shows an
artist's rendering of the length contraction for the collision of two
gold nuclei at relativistic speeds in the RHIC accelerator\index{RHIC accelerator}
in Long Island, New York, which went on line in 2000.
The gold nuclei would appear nearly spherical (or just
slightly lengthened like an American football) in frames
moving along with them, but in the laboratory's frame, they
both appear drastically foreshortened as they approach the
point of collision. The later pictures show the nuclei
merging to form a hot soup, in which experimenters hope to
observe a new form of matter.

<%
  fig(
    'rhic',
    %q{Example \ref{eg:rhic}: Colliding nuclei show relativistic length contraction.},
    {
      'width'=>'wide',
      'sidecaption'=>true,
      'anonymous'=>true
    }
  )
%>
\end{eg}

<% end_sec %> % The $\mygamma$ factor

<%
def relativity_dq_space
  print "\\vspace{7mm}\n"
end
%>

\vfill

\startdqs
\begin{dq}
A person in a spaceship moving at 99.99999999\% of the
speed of light relative to Earth shines a flashlight forward
through dusty air, so the beam is visible. What does she
see? What would it look like to an observer on Earth?
\end{dq}
<% marg(-30) %>
<%
  fig(
    'dqillusion',
    %q{Discussion question \ref{dq:illusion}},
    {
      'anonymous'=>true
    }
  )
%>
<% end_marg %>

<% relativity_dq_space %>

\begin{dq}\label{dq:illusion}
A question that students often struggle with is whether
time and space can really be distorted, or whether it just
seems that way. Compare with optical illusions or magic
tricks. How could you verify, for instance, that the lines
in the figure are actually parallel? Are relativistic
effects the same, or not?
\end{dq}

<% relativity_dq_space %>

\begin{dq}
On a spaceship moving at relativistic speeds, would a
lecture seem even longer and more boring than normal?
\end{dq}

<% relativity_dq_space %>

\begin{dq}
Mechanical clocks can be affected by motion. For example,
it was a significant technological achievement to build a
clock that could sail aboard a ship and still keep accurate
time, allowing longitude to be determined. How is this
similar to or different from relativistic time dilation?
\end{dq}

\pagebreak

\begin{dq}\label{dq:rhic}
Figure \figref{rhic} from page \pageref{fig:rhic}, depicting the collision of
two nuclei at the RHIC accelerator, is reproduced below.
What would the shapes of the two nuclei
look like to a microscopic observer riding on the
left-hand nucleus? To an observer riding on the right-hand
one? Can they agree on what is happening? If not, why not
--- after all, shouldn't they see the same thing if they
both compare the two nuclei side-by-side at the same instant in time?
\end{dq}

<%
  fig(
    'rhic',
    %q{\hspace{-2mm}Discussion question \ref{dq:rhic}: colliding nuclei show relativistic length contraction.},
    {
      'width'=>'wide',
      'sidecaption'=>false,
      'suffix'=>'2',
      'anonymous'=>true
    }
  )
%>

<% relativity_dq_space %>

\begin{dq}\label{dq:foam-rubber}
If you stick a piece of foam rubber out the window of
your car while driving down the freeway, the wind may
compress it a little. Does it make sense to interpret the
relativistic length contraction as a type of strain that
pushes an object's atoms together like this? How does this
relate to discussion question \ref{dq:rhic}?
\end{dq}

<% relativity_dq_space %>

<% marg(-50) %>
<%
  fig(
    'dq-pole-paradox',
    %q{Discussion question \ref{dq:pole-paradox}.}
  )   
%>
<% end_marg %>

\begin{dq}\label{dq:pole-paradox}
The rod in the figure is perfectly rigid. At event A, the hammer strikes one end of the rod.
At event B, the other end moves. Since the rod is perfectly rigid, it can't compress, so A
and B are simultaneous. In frame 2, B happens before A. Did the motion at the right end \emph{cause}
the person on the left to decide to pick up the hammer and use it?
\end{dq}

<% end_sec %> % Distortion of Space and Time

\vfill

<% begin_sec("Magnetic Interactions",4,'magnetic-interactions') %>
        \epigraph{Think not that I am come to destroy the law, or the prophets:
        I am not come to destroy, but to fulfill.}{Matthew 5:17}

At this stage, you understand roughly as much about the classification of interactions as physicists
understood around the year 1800. There appear to be three fundamentally different types
of interactions: gravitational, electrical, and magnetic. Many types of interactions that appear superficially to be
distinct --- stickiness, chemical interactions, the energy an archer stores in a bow --- are
really the same: they're manifestations of electrical interactions between atoms.
Is there any way to shorten the list any further? The prospects seem dim at first. For instance,
we find that if we rub a piece of fur on a rubber rod, the fur does not attract or repel a magnet.
The fur has an electric field, and the magnet has a magnetic field. The two are completely separate,
and don't seem to affect one another. Likewise we can test whether magnetizing a piece of iron
changes its weight. The weight doesn't seem to change by any measurable amount, so magnetism and
gravity seem to be unrelated.

That was where things stood until 1820, when the Danish physicist Hans Christian\index{Oersted, Hans Christian}
Oersted was delivering a lecture at the University of Copenhagen, and he wanted to give his
students a demonstration that would illustrate the cutting edge of research. He generated
a current in a wire by making a short circuit across a battery, and held the wire near a
magnetic compass. The ideas was to give an example of how one could search for a previously undiscovered
link between electricity (the electric current in the wire) and magnetism. One never knows how much
to believe from these dramatic legends, but the story is\footnote{Oersted's paper
describing the phenomenon says that ``The first experiments on the subject \ldots
were set on foot in the classes for electricity, galvanism, and magnetism, which were
held by me in the winter just past,'' but that doesn't tell us whether the result was
really a surprise that occurred in front of his students.} that the experiment he'd expected to turn out
negative instead turned out positive: when he held the wire near the
compass, the current in the wire caused the compass to twist!
<% marg(60) %>
<%
  fig(
    'oersted',
    %q{1. When the circuit is incomplete, no current flows through the wire, and the magnet is
unaffected. It points in the direction of the Earth's magnetic field. 2. The circuit is completed, and
current flows through the wire. The wire has a strong effect on the magnet, which turns almost perpendicular
to it. If the earth's field could be removed entirely, the compass would point exactly perpendicular to the
wire; this is the direction of the wire's field.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'magnetized',
    %q{A schematic representation of an unmagnetized material, 1, and a magnetized one, 2.}
  )
%>

<% end_marg %>

People had tried similar experiments before, but only with static electricity, not with
a moving electric current. For instance, they had hung batteries so that they were free to
rotate in the earth's magnetic field, and found no effect; since the battery was not connected
to a complete circuit, there was no current flowing. With Oersted's own setup, \figref{oersted},
the effect was only produced when the ``circuit was closed, but not
when open, as certain very celebrated physicists in vain attempted several years ago.''\footnote{All
quotes are from the 1876 translation by J.E. Kempe.}

Oersted was eventually
led to the conclusion that magnetism was an interaction between moving charges and
other moving charges, i.e., between one current and another.  \index{magnetism!caused by moving charges}
A permanent magnet, he inferred, contained currents on a microscopic
scale that simply weren't practical to measure with an ammeter. Today this seems natural
to us, since we're accustomed to picturing an atom as a tiny solar system, with the electrons
whizzing around the nucleus in circles. As shown in figure \figref{magnetized},
a magnetized piece of iron is different from an
unmagnetized piece because the atoms in the unmagnetized piece are jumbled in random
orientations, whereas the atoms in the magnetized piece are at least partially organized
to face in a certain direction.

<% marg(0) %>
<%
  fig(
    'magdeflects',
    %q{Magnetism is an interaction between moving charges and moving charges. The moving
charges in the wire attract the moving charges in the beam of charged particles in the vacuum tube.}
  )
%>
<% end_marg %>


Figure \figref{magdeflects} shows an example that is conceptually
simple, but not very practical. If you try this with a typical vacuum tube, like a TV
or computer monitor, the current in the wire probably won't be enough to produce a visible
effect. A more practical method is to hold a magnet near the screen. We still have
an interaction between moving charges and moving charges, but the swirling electrons
in the atoms in the magnet are now playing the role played by the moving charges in the wire
in figure \figref{magdeflects}. Warning: if you do this, make sure your monitor has a
demagnetizing button! If not, then your monitor may be permanently ruined.

<% begin_sec("Relativity requires magnetism") %>\index{magnetism!and relativity}\index{relativity!and magnetism}\label{relativity-requires-magnetism}
So magnetism is an interaction between moving charges and moving charges. But how
can that be?
Relativity tells us that
motion is a matter of opinion. Consider figure \figref{fulfill}. In this figure and in figure
\figref{magrelativity}, the dark and light coloring of the particles represents the fact that
one particle has positive charge and the other negative.
Observer \figref{fulfill}/2 sees the two particles as flying through space side by side, so they
would interact both electrically (simply because they're charged) and magnetically
(because they're charges in motion). But an observer moving along with them,  \figref{fulfill}/1, would
say they were both at rest, and would expect only an electrical interaction. This seems
like a paradox.
Magnetism, however, comes not to destroy relativity but to fulfill it. Magnetic interactions
        \emph{must} exist according to the theory of relativity. To understand how this can be,
        consider how time and space behave in relativity. Observers in different frames of reference
        disagree about the lengths of measuring sticks and the speeds of clocks, but the laws
        of physics are valid and self-consistent in either frame of reference.
        Similarly, observers in different frames of reference disagree about what electric and magnetic
        fields there are, but they agree about concrete physical events.
        An observer in frame of reference \figref{fulfill}/1
        says there are electric fields around the particles, and predicts that as time goes on, the
        particles will begin to accelerate towards one another, eventually colliding. She explains the
        collision as being due to the electrical attraction between the particles.
        A different observer, \figref{fulfill}/2, says the particles are moving. This observer
        also predicts that the particles will collide, but explains their motion in terms of both
        an electric field and a magnetic field. As we'll see shortly, the
        magnetic field is \emph{required} in order to maintain consistency between the predictions made
        in the two frames of reference.
        
<% marg(40) %>
<%
  fig(
    'fulfill',
    %q{One observer sees an electric field, while the other sees both an electric field and a magnetic one.}
  )
%>
\spacebetweenfigs
<% fig(
    'magrelativity',
     %q{A model of a charged particle and a current-carrying wire, seen in
        two different frames of reference. The relativistic length contraction is highly
        exaggerated. The force on the lone particle is purely
        magnetic in 1, and purely electric in 2.}
   )
%>
<% end_marg %>
        To see how this really works out, we need to find a nice simple example.
        An example like figure \figref{fulfill} is \emph{not} easy
        to handle, because in the second frame of reference, the moving charges
        create fields that change over time at any given location, like when the V-shaped wake of a speedboat
        washes over a buoy. Examples like
        figure \figref{magdeflects} are easier, because there is a steady flow of charges, and
        all the fields stay the same over time.
        Figure \figref{magrelativity}/1 shows a simplified and idealized model of figure \figref{magdeflects}.
        The charge by itself is like one of the charged particles
        in the vacuum tube beam of figure \figref{magdeflects}, and instead of the wire, we have
        two long lines of charges moving in opposite directions. Note that,
        as discussed in discussion question \ref{dq:signsofcurrent} on page \pageref{dq:signsofcurrent},
        the currents of the two lines of charges do not cancel out. The dark and light balls represent particles with
        opposite charges. Because of this, the total current in
        the ``wire'' is double what it would be if we took away one line.

        As a model of figure \figref{magdeflects}, figure \figref{magrelativity}/1 is partly realistic and
        partly unrealistic. In a real piece of copper wire, there are indeed charged particles of both types,
        but it turns out that the particles of one type (the protons) are locked in place, while only some 
        of the other type (the electrons) are free to move. The model also shows the particles moving in
        a simple and orderly way, like cars on a two-lane road, whereas in reality most of the particles are
        organized into copper atoms, and there is also a great deal of random thermal motion.
        The model's unrealistic features aren't a
        problem, because the point of this exercise is only to find one particular situation that shows
        magnetic effects must exist based on relativity.

        What electrical force does the lone particle in figure \figref{magrelativity}/1 feel? Since the
        density of ``traffic'' on the two sides of the ``road'' is equal, there is zero overall
        electrical force on the lone particle. Each ``car'' that attracts the lone particle is paired with a partner on the other
        side of the road that repels it. If we didn't know about magnetism, we'd think this
        was the whole story: the lone particle feels no force at all from the wire.

        Figure \figref{magrelativity}/2
        shows what we'd see if we were observing all this from a frame of reference moving
        along with the lone charge.
        Here's where the relativity comes in. Relativity tells us that moving objects
        appear contracted to an observer who is not moving along with them.
        Both lines of charge are in motion in both frames of reference, but in frame 1
        they were moving at equal speeds, so their contractions were equal.
        In frame 2, however, their speeds are unequal. The dark
        charges are moving more slowly than in frame 1, so in frame 2 they are less contracted.
        The light-colored charges are moving more quickly, so their contraction is greater now.
        The ``cars'' on the two sides of the ``road'' are no longer paired off, so the electrical
        forces on the lone particle no longer cancel out as they did in \figref{magrelativity}/1.
        The lone particle is attracted to the wire, because the particles attracting it are more
        dense than the ones repelling it. Furthermore, the attraction felt
        by the lone charge must be purely electrical, since the lone charge is at rest in this
        frame of reference, and magnetic effects occur only between moving charges and other
        moving charges.

        Now observers in frames 1 and 2 disagree about many things, but they do agree on
        concrete events. Observer 2 is going to see the lone particle drift toward the wire
        due to the wire's electrical attraction, gradually speeding up, and eventually hit
        the wire. If 2 sees this collision, then 1 must as well. But 1 knows that the total
        electrical force on the lone particle is exactly zero. There must be some new type
        of force. She invents a name for this new type of force: magnetism. This was a particularly
        simple example, because the force was purely magnetic in one frame of reference, and
        purely electrical in another. In general, an observer in a certain frame of reference
        will measure a mixture of electric and magnetic fields, while an observer in another
        frame, in motion with respect to the first, says that the same volume of space contains a different mixture.

We therefore arrive at the conclusion that electric and magnetic phenomena aren't
separate. They're different sides of the same coin. We refer to electric and magnetic interactions
collectively as electromagnetic interactions. Our list of the fundamental interactions
of nature now has two items on it instead of three: gravity and electromagnetism.\index{magnetism!related to electricity}\index{electromagnetism}

Oersted found that magnetism was an interaction between moving charges and other moving charges.
We can see this in the situation described in figure \subfigref{magrelativity}{1}, in which the result of the argument depended 
on the fact that both the lone charge and the charges in the wire were moving. To see this in a different way, we can
apply the result of example \ref{eg:gamma-for-low-v} on p.~\pageref{eg:gamma-for-low-v}, that for small velocities
the $\mygamma$ factor differs from 1 by about $v^2/2c^2$. Let the lone charge in figure \subfigref{magrelativity}{1} have
velocity $u$, the ones in the wire $\pm v$. As we'll see on p.~\pageref{relativistic-combination-of-vel}, velocities in
relative motion don't exactly add and subtract relativistically, but as long as we assume that $u$ and $v$ are small, the correspondence
principle guarantees that they will approximately add and subtract. Then the velocities in the lone charge's rest frame,
\subfigref{magrelativity}{2}, are approximately 0, $v-u$, and $-v-u$. The nonzero charge density of the wire in frame
\subfigref{magrelativity}{2} is then proportional to the difference in the length contractions $\mygamma_{-v-u}-\mygamma_{v-u}\approx 2uv/c^2$.
This depends on the product of the velocities $u$ and $v$, which is as expected if magnetism is an interaction of moving charges
with moving charges.
<% marg(30) %>
<% fig(
   'magtwobody',
   %q{Magnetic interactions involving only two particles at a time. In these figures, unlike figure
    \figref{magrelativity}/1, there are electrical forces as well as magnetic ones. The electrical forces are
    not shown here. Don't memorize these rules!}
   )
%>
<% end_marg %>

The basic rules for magnetic attractions and repulsions, shown in figure \figref{magtwobody}, aren't
quite as simple as the ones for gravity and electricity. Rules \figref{magtwobody}/1 and
\figref{magtwobody}/2 follow directly from our previous analysis of figure \figref{magrelativity}.
Rules 3 and 4 are obtained by flipping the type of charge
that the bottom particle has. For instance, rule 3 is like rule 1, except that the bottom charge
is now the opposite type. This turns the attraction into a repulsion. (We know that flipping the charge
reverses the interaction, because that's the way it works for electric forces, and magnetic forces
are just electric forces viewed in a different frame of reference.)

\begin{eg}{A magnetic weathervane placed near a current.}\label{eg:weathervane}
Figure \figref{weathervane} shows a magnetic weathervane, consisting of two charges that spin
in circles around the axis of the arrow. (The magnetic field doesn't cause them to spin; a motor
is needed to get them to spin in the first place.) Just like the magnetic compass in figure \figref{oersted},
the weathervane's arrow tends to align itself in the direction perpendicular to the wire. This
is its preferred orientation because the charge close to the wire is attracted to the
wire, while the charge far from the wire is repelled by it.
\end{eg}

\startdq
\begin{dq}
Resolve the following paradox concerning the argument given in this section.
We would expect that at any given time,
electrons in a solid would be associated with protons in a definite way.
For simplicity, let's imagine that the solid is made out of hydrogen (which
actually does become a metal under conditions of very high pressure).
A hydrogen atom consists of a single proton and a single electron.
Even if the electrons are moving and forming an electric current, we would
imagine that this would be like a game of musical chairs, with the protons
as chairs and the electrons as people. Each electron has a proton that is its
``friend,'' at least for the moment. This is the situation shown in figure
\subfigref{magrelativity}{1}. How, then, can an observer in a different frame
see the electrons and protons as not being paired up, as in \subfigref{magrelativity}{2}?
\end{dq}

<% marg(80) %>
<% fig(
     'weathervane',
     %q{Example \ref{eg:weathervane}},
     {'anonymous'=>true}
   )
%>
<% end_marg %>


<% end_sec() %> % Relativity requires magnetism
<% end_sec() %> % Magnetic Interactions



 %%===============================================================================
\begin{summary}

\begin{notation}
\notationitem{$\mygamma$}{an abbreviation for $1/\sqrt{1-v^2/c^2}$}
\end{notation}

\begin{summarytext}

Experiments show that space and time do not have the properties claimed by Galileo and Newton.
Time and space as seen by one
observer are distorted compared to another observer's perceptions if they are moving
relative to each other. This distortion is quantified by the factor
\begin{equation*}
  \mygamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \qquad ,
\end{equation*}
where $v$ is the relative velocity of the two observers, and $c$ is a universal velocity
that is the same in all frames of reference. Light travels at $c$. A clock appears to run fastest to an
observer who is not in motion relative to it, and appears to run too slowly by a factor of
$\mygamma$ to an observer who has a velocity $v$ relative to the clock. Similarly, a meter-stick
appears longest to an observer who sees it at rest, and appears shorter to other observers.
Time and space are relative, not absolute.

As a consequence of relativity, we must have not just electrical interactions
of charges with charges, but also an additional magnetic interaction of
moving charges with other moving charges.
\end{summarytext}

\end{summary}

 %%===============================================================================

<% begin_hw_sec %>
<% begin_hw('gammafornegativev') %>__incl(../../share/relativity/hw/gammafornegativev)<% end_hw() %>

\vfill

<% begin_hw('gamma-derivation') %>__incl(../../share/relativity/hw/gamma-derivation)<% end_hw() %>
<% marg(50) %>
<%
  fig(
    'hw-gamma-derivation',
    %q{Problem \ref{hw:gamma-derivation}.}
  )
%>
<% end_marg %>

\vfill

<% begin_hw('agreeontime') %>__incl(../../share/relativity/hw/agreeontime)<% end_hw() %>

\vfill

<% begin_hw('voyagergamma') %>__incl(../../share/relativity/hw/voyagergamma)<% end_hw() %>

\vfill

<% begin_hw('earth-lorentz-contraction') %>__incl(../../share/relativity/hw/earth-lorentz-contraction)<% end_hw() %>

\pagebreak

<% begin_hw('sr-from-length-contraction') %>__incl(../../share/relativity/hw/sr-from-length-contraction)<% end_hw() %>



<% end_hw_sec %>



<% end_chapter() %>

%<% figure_in_toc("rhic") %>