Electromagnetism

The magnetic field

No magnetic monopoles

If you could play with a handful of electric dipoles and a handful of bar magnets, they would appear very similar. For instance, a pair of bar magnets wants to align themselves head-to-tail, and a pair of electric dipoles does the same thing. (It is unfortunately not that easy to make a permanent electric dipole that can be handled like this, since the charge tends to leak.)

You would eventually notice an important difference between the two types of objects, however. The electric dipoles can be broken apart to form isolated positive charges and negative charges. The two-ended device can be broken into parts that are not two-ended. But if you break a bar magnet in half, a, you will find that you have simply made two smaller two-ended objects.

The reason for this behavior is not hard to divine from our microscopic picture of permanent iron magnets. An electric dipole has extra positive “stuff” concentrated in one end and extra negative in the other. The bar magnet, on the other hand, gets its magnetic properties not from an imbalance of magnetic “stuff” at the two ends but from the orientation of the rotation of its electrons. One end is the one from which we could look down the axis and see the electrons rotating clockwise, and the other is the one from which they would appear to go counterclockwise. There is no difference between the “stuff” in one end of the magnet and the other, b.

Nobody has ever succeeded in isolating a single magnetic pole. In technical language, we say that magnetic monopoles do not seem to exist. Electric monopoles do exist --- that's what charges are.

Electric and magnetic forces seem similar in many ways. Both act at a distance, both can be either attractive or repulsive, and both are intimately related to the property of matter called charge. (Recall that magnetism is an interaction between moving charges.) Physicists's aesthetic senses have been offended for a long time because this seeming symmetry is broken by the existence of electric monopoles and the absence of magnetic ones. Perhaps some exotic form of matter exists, composed of particles that are magnetic monopoles. If such particles could be found in cosmic rays or moon rocks, it would be evidence that the apparent asymmetry was only an asymmetry in the composition of the universe, not in the laws of physics. For these admittedly subjective reasons, there have been several searches for magnetic monopoles. Experiments have been performed, with negative results, to look for magnetic monopoles embedded in ordinary matter. Soviet physicists in the 1960s made exciting claims that they had created and detected magnetic monopoles in particle accelerators, but there was no success in attempts to reproduce the results there or at other accelerators. The most recent search for magnetic monopoles, done by reanalyzing data from the search for the top quark at Fermilab, turned up no candidates, which shows that either monopoles don't exist in nature or they are extremely massive and thus hard to create in accelerators.

Definition of the magnetic field

Since magnetic monopoles don't seem to exist, it would not make much sense to define a magnetic field in terms of the force on a test monopole. Instead, we follow the philosophy of the alternative definition of the electric field, and define the field in terms of the torque on a magnetic test dipole. This is exactly what a magnetic compass does: the needle is a little iron magnet which acts like a magnetic dipole and shows us the direction of the earth's magnetic field.

To define the strength of a magnetic field, however, we need some way of defining the strength of a test dipole, i.e., we need a definition of the magnetic dipole moment. We could use an iron permanent magnet constructed according to certain specifications, but such an object is really an extremely complex system consisting of many iron atoms, only some of which are aligned. A more fundamental standard dipole is a square current loop. This could be little resistive circuit consisting of a square of wire shorting across a battery.

We will find that such a loop, when placed in a magnetic field, experiences a torque that tends to align plane so that its face points in a certain direction. (Since the loop is symmetric, it doesn't care if we rotate it like a wheel without changing the plane in which it lies.) It is this preferred facing direction that we will end up defining as the direction of the magnetic field.

Experiments show if the loop is out of alignment with the field, the torque on it is proportional to the amount of current, and also to the interior area of the loop. The proportionality to current makes sense, since magnetic forces are interactions between moving charges, and current is a measure of the motion of charge. The proportionality to the loop's area is also not hard to understand, because increasing the length of the sides of the square increases both the amount of charge contained in this circular “river” and the amount of leverage supplied for making torque. Two separate physical reasons for a proportionality to length result in an overall proportionality to length squared, which is the same as the area of the loop. For these reasons, we define the magnetic dipole moment of a square current loop as $$D_m = IA \;\;\;\;\;\; {\text{[definition of the magnetic}}$$ $$\text{ dipole moment of a square current loop]}$$

We now define the magnetic field in a manner entirely analogous to the second definition of the electric field:

We find from this definition that the magnetic field has units of \(N \cdot m/A \cdot m^2 = N/A \cdot m\). This unwieldy combination of units is abbreviated as the tesla, \( T=1 1 N/A \cdot m\). Refrain from memorizing the part about the counterclockwise direction at the end.

The nonexistence of magnetic monopoles means that unlike an electric field, d/1, a magnetic one, d/2, can never have sources or sinks. The magnetic field vectors lead in paths that loop back on themselves, without ever converging or diverging at a point.

Calculating magnetic fields and forces

Magnetostatics

Our study of the electric field built on our previous understanding of electric forces, which was ultimately based on Coulomb's law for the electric force between two point charges. Since magnetism is ultimately an interaction between currents, i.e., between moving charges, it is reasonable to wish for a magnetic analog of Coulomb's law, an equation that would tell us the magnetic force between any two moving point charges.

Such a law, unfortunately, does not exist. Coulomb's law describes the special case of electrostatics: if a set of charges is sitting around and not moving, it tells us the interactions among them. Coulomb's law fails if the charges are in motion, since it does not incorporate any allowance for the time delay in the outward propagation of a change in the locations of the charges.

A pair of moving point charges will certainly exert magnetic forces on one another, but their magnetic fields are like the v-shaped bow waves left by boats. Each point charge experiences a magnetic field that originated from the other charge when it was at some previous position. There is no way to construct a force law that tells us the force between them based only on their current positions in space.

There is, however, a science of magnetostatics that covers a great many important cases. Magnetostatics describes magnetic forces among currents in the special case where the currents are steady and continuous, leading to magnetic fields throughout space that do not change over time.

The magnetic field of a long, straight wire is one example that we can say something about without resorting to fancy mathematics. The electric field of a uniform line of charge is E=2kq/Lr, where r is the distance from the line and q/L is the charge per unit length. In a frame of reference moving at velocity v parallel to the line, this electric field will be observed as a combination of electric and magnetic fields. It therefore follows that the magnetic field of a long, straight, current-carrying wire must be proportional to 1/r. We also expect that it will be proportional to the Coulomb constant, which sets the strength of electric and magnetic interactions, and to the current I in the wire. The complete expression turns out to be \(B=(k/c^2 )(2I/r) \). This is identical to the expression for E except for replacement of q/L with I and an additional factor of \(1/c^2\). The latter occurs because magnetism is a purely relativistic effect, and the relativistic length contraction depends on \(v^2 /c^2\).

Figure e shows the equations for some of the more commonly encountered configurations, with illustrations of their field patterns. They all have a factor of \(k /c^2\) in front, which shows that magnetism is just electricity (k) seen through the lens of relativity (\(1 /c^2\)). A convenient feature of SI units is that \(k /c^2\) has a numerical value of exactly \(10^{-7}\), with units of \(N/m^2\)

Field created by a long, straight wire carrying current I: $$B = \frac{k}{c^2}\cdot\frac{2I}{r}$$

Here r is the distance from the center of the wire. The field vectors trace circles in planes perpendicular to the wire, going clockwise when viewed from along the direction of the current.

Field created by a single circular loop of current: The field vectors form a dipole-like pattern, coming through the loop and back around on the outside. Each oval path traced out by the field vectors appears clockwise if viewed from along the direction the current is going when it punches through it. There is no simple equation for a field at an arbitrary point in space, but for a point lying along the central axis perpendicular to the loop, the field is $$B = \frac{k}{c^2}\cdot 2\pi Ib^2\left(b^2+z^2\right)^{-3/2} $$

where b is the radius of the loop and z is the distance of the point from the plane of the loop.

Field created by a solenoid (cylindrical coil): The field pattern is similar to that of a single loop, but for a long solenoid the paths of the field vectors become very straight on the inside of the coil and on the outside immediately next to the coil. For a sufficiently long solenoid, the interior field also becomes very nearly uniform, with a magnitude of

$$B = \frac{k}{c^2}\cdot 4\pi I N/\ell $$

where N is the number of turns of wire and ℓ is the length of the solenoid. The field near the mouths or outside the coil is not constant, and is more difficult to calculate. For a long solenoid, the exterior field is much smaller than the interior field.

Force on a charge moving through a magnetic field

We now know how to calculate magnetic fields in some typical situations, but one might also like to be able to calculate magnetic forces, such as the force of a solenoid on a moving charged particle, or the force between two parallel current-carrying wires.

We will restrict ourselves to the case of the force on a charged particle moving through a magnetic field, which allows us to calculate the force between two objects when one is a moving charged particle and the other is one whose magnetic field we know how to find. An example is the use of solenoids inside a TV tube to guide the electron beam as it paints a picture.

Experiments show that the magnetic force on a moving charged particle has a magnitude given by $$|\boldsymbol{F}| = q|\boldsymbol{v}|sin\theta $$

where \(v\) is the velocity vector of the particle, and θ is the angle between the \(v\) and \(B\) vectors. Unlike electric and gravitational forces, magnetic forces do not lie along the same line as the field vector. The force is always perpendicular to both \(v\) and \(B\). Given two vectors, there is only one line perpendicular to both of them, so the force vector points in one of the two possible directions along this line. For a positively charged particle, the direction of the force vector can be found as follows. First, position the \(v\) and \(B\) vectors with their tails together. The direction of \(F\) is such that if you sight along it, the \(B\) vector is clockwise from the \(v\) vector; for a negatively charged particle the direction of the force is reversed. Note that since the force is perpendicular to the particle's motion, the magnetic field never does work on it.

Energy in the magnetic field

Provided previously are equations for the energy stored in the gravitational and electric fields. Since a magnetic field is essentially an electric field seen in a different frame of reference, we expect the magnetic-field equation to be closely analogous to the electric version, and it is:

$$\text{(energy stored in the gravitational field per} \; m^3) = - \frac{1}{8 \pi G} |\boldsymbol{g}|^2$$ $$\text{(energy stored in the electric field per} \; m^3) = \frac{1}{8 \pi k} |\boldsymbol{E}|^2$$ $$\text{(energy stored in the magnetic field per} \; m^3) = \frac{c^2}{8 \pi k} |\boldsymbol{B}|^2$$

The idea here is that \(k/c^2\) is the magnetic version of the electric quantity k, the \(1/c^2\) representing the fact that magnetism is a relativistic effect.

The universal speed c

Let's think a little more about the role of the 45-degree diagonal in the Lorentz transformation. Slopes on these graphs are interpreted as velocities. This line has a slope of 1 in relativistic units, but that slope corresponds to c in ordinary metric units. We already know that the relativistic distance unit must be extremely large compared to the relativistic time unit, so c must be extremely large. Now note what happens when we perform a Lorentz transformation: this particular line gets stretched, but the new version of the line lies right on top of the old one, and its slope stays the same. In other words, if one observer says that something has a velocity equal to c, every other observer will agree on that velocity as well. (The same thing happens with -c.)

Velocities don't simply add and subtract.

This is counterintuitive, since we expect velocities to add and subtract in relative motion. If a dog is running away from you at 5 m/s relative to the sidewalk, and you run after it at 3 m/s, the dog's velocity in your frame of reference is 2 m/s. According to everything we have learned about motion, the dog must have different speeds in the two frames: 5 m/s in the sidewalk's frame and 2 m/s in yours. But velocities are measured by dividing a distance by a time, and both distance and time are distorted by relativistic effects, so we actually shouldn't expect the ordinary arithmetic addition of velocities to hold in relativity; it's an approximation that's valid at velocities that are small compared to c.

A universal speed limit

For example, suppose Janet takes a trip in a spaceship, and accelerates until she is moving at 0.6c relative to the earth. She then launches a space probe in the forward direction at a speed relative to her ship of 0.6c. We might think that the probe was then moving at a velocity of 1.2c, but in fact the answer is still less than c. This is an example of a more general fact about relativity, which is that c represents a universal speed limit. This is required by causality, as shown in figure i.

Light travels at c.

Now consider a beam of light. We're used to talking casually about the “speed of light,” but what does that really mean? Motion is relative, so normally if we want to talk about a velocity, we have to specify what it's measured relative to. A sound wave has a certain speed relative to the air, and a water wave has its own speed relative to the water. If we want to measure the speed of an ocean wave, for example, we should make sure to measure it in a frame of reference at rest relative to the water. But light isn't a vibration of a physical medium; it can propagate through the near-perfect vacuum of outer space, as when rays of sunlight travel to earth. This seems like a paradox: light is supposed to have a specific speed, but there is no way to decide what frame of reference to measure it in. The way out of the paradox is that light must travel at a velocity equal to c. Since all observers agree on a velocity of c, regardless of their frame of reference, everything is consistent.

The Michelson-Morley experiment

The constancy of the speed of light had in fact already been observed when Einstein was an 8-year-old boy, but because nobody could figure out how to interpret it, the result was largely ignored. In 1887 Michelson and Morley set up a clever apparatus to measure any difference in the speed of light beams traveling east-west and north-south. The motion of the earth around the sun at 110,000 km/hour (about 0.01% of the speed of light) is to our west during the day. Michelson and Morley believed that light was a vibration of a mysterious medium called the ether, so they expected that the speed of light would be a fixed value relative to the ether. As the earth moved through the ether, they thought they would observe an effect on the velocity of light along an east-west line. For instance, if they released a beam of light in a westward direction during the day, they expected that it would move away from them at less than the normal speed because the earth was chasing it through the ether. They were surprised when they found that the expected 0.01% change in the speed of light did not occur.

Induction

The principle of induction

Physicists of Michelson and Morley's generation thought that light was a mechanical vibration of the ether, but we now know that it is a ripple in the electric and magnetic fields. With hindsight, relativity essentially requires this:

1. Relativity requires that changes in any field propagate as waves at a finite speed.

2. Relativity says that if a wave has a fixed speed but is not a mechanical disturbance in a physical medium, then it must travel at the universal velocity c.

What is less obvious is that there are not two separate kinds of waves, electric and magnetic. In fact an electric wave can't exist without a magnetic one, or a magnetic one without an electric one. This new fact follows from the principle of induction, which was discovered experimentally by Faraday in 1831, seventy-five years before Einstein. Let's state Faraday's idea first, and then see how something like it must follow inevitably from relativity:

The induced field tends to have a whirlpool pattern, as shown in figure k, but the whirlpool image is not to be taken too literally; the principle of induction really just requires a field pattern such that, if one inserted a paddlewheel in it, the paddlewheel would spin. All of the field patterns shown in figure l are ones that could be created by induction; all have a counterclockwise “curl” to them.

Figure m shows an example of the fundamental reason why a changing B field must create an E field. We established that according to relativity, what one observer describes as a purely magnetic field, an observer in a different state of motion describes as a mixture of magnetic and electric fields. This is why there must be both an E and a B in observer 2's frame. Observer 2 cannot explain the electric field as coming from any charges. In frame 2, the E can only be explained as an effect caused by the changing B.

Observer 1 says, “2 feels a changing B field because he's moving through a static field.” Observer 2 says, “I feel a changing B because the magnet is getting closer.”

Although this argument doesn't prove the “whirlpool” geometry, we can verify that the fields drawn in figure m are consistent with it. The \(\Delta \boldsymbol{B}\) vector is upward, and the electric field has a curliness to it: a paddlewheel inserted in the electric field would spin clockwise as seen from above, since the clockwise torque made by the strong electric field on the right is greater than the counterclockwise torque made by the weaker electric field on the left.

Electromagnetic waves

The most important consequence of induction is the existence of electromagnetic waves. Whereas a gravitational wave would consist of nothing more than a rippling of gravitational fields, the principle of induction tells us that there can be no purely electrical or purely magnetic waves. Instead, we have waves in which there are both electric and magnetic fields, such as the sinusoidal one shown in the figure. Maxwell proved that such waves were a direct consequence of his equations, and derived their properties mathematically.

A sinusoidal electromagnetic wave has the geometry shown above in figure p. The E and B fields are perpendicular to the direction of motion, and are also perpendicular to each other. If you look along the direction of motion of the wave, the B vector is always 90 degrees clockwise from the E vector. The magnitudes of the two fields are related by the equation \(|\boldsymbol{E}| = c|\boldsymbol{B}|\).

How is an electromagnetic wave created? It could be emitted, for example, by an electron orbiting an atom or currents going back and forth in a transmitting antenna. In general any accelerating charge will create an electromagnetic wave, although only a current that varies sinusoidally with time will create a sinusoidal wave. Once created, the wave spreads out through space without any need for charges or currents along the way to keep it going. As the electric field oscillates back and forth, it induces the magnetic field, and the oscillating magnetic field in turn creates the electric field. The whole wave pattern propagates through empty space at the velocity c.

Polarization

Two electromagnetic waves traveling in the same direction through space can differ by having their electric and magnetic fields in different directions, a property of the wave called its polarization.

Light is an electromagnetic wave

Once Maxwell had derived the existence of electromagnetic waves, he became certain that they were the same phenomenon as light. Both are transverse waves (i.e., the vibration is perpendicular to the direction the wave is moving), and the velocity is the same.

Heinrich Hertz (for whom the unit of frequency is named) verified Maxwell's ideas experimentally. Hertz was the first to succeed in producing, detecting, and studying electromagnetic waves in detail using antennas and electric circuits. To produce the waves, he had to make electric currents oscillate very rapidly in a circuit. In fact, there was really no hope of making the current reverse directions at the frequencies of \(10^{15}\) Hz possessed by visible light. The fastest electrical oscillations he could produce were \(10^{9}\) Hz, which would give a wavelength of about 30 cm. He succeeded in showing that, just like light, the waves he produced were polarizable, and could be reflected and refracted (i.e., bent, as by a lens), and he built devices such as parabolic mirrors that worked according to the same optical principles as those employing light. Hertz's results were convincing evidence that light and electromagnetic waves were one and the same.

The electromagnetic spectrum

Today, electromagnetic waves with frequencies in the range employed by Hertz are known as radio waves. Any remaining doubts that the “Hertzian waves,” as they were then called, were the same type of wave as light waves were soon dispelled by experiments in the whole range of frequencies in between, as well as the frequencies outside that range. In analogy to the spectrum of visible light, we speak of the entire electromagnetic spectrum, of which the visible spectrum is one segment.

The terminology for the various parts of the spectrum is worth memorizing, and is most easily learned by recognizing the logical relationships between the wavelengths and the properties of the waves with which you are already familiar. Radio waves have wavelengths that are comparable to the size of a radio antenna, i.e., meters to tens of meters. Microwaves were named that because they have much shorter wavelengths than radio waves; when food heats unevenly in a microwave oven, the small distances between neighboring hot and cold spots is half of one wavelength of the standing wave the oven creates. The infrared, visible, and ultraviolet obviously have much shorter wavelengths, because otherwise the wave nature of light would have been as obvious to humans as the wave nature of ocean waves. To remember that ultraviolet, x-rays, and gamma rays all lie on the short-wavelength side of visible, recall that all three of these can cause cancer. (There is a basic physical reason why the cancer-causing disruption of DNA can only be caused by very short-wavelength electromagnetic waves. Contrary to popular belief, microwaves cannot cause cancer, which is why we have microwave ovens and not x-ray ovens!)

Momentum of light

Newton defined momentum as mv, and that would lead us to believe that light, which has no mass, should have no momentum. However, Newton's laws only work at velocities that are small compared to the speed of light, and light travels at the speed of light, so there is no reason to trust Newton here. In fact, it's straightforward to show that electromagnetic waves have momentum. If a light wave strikes an ohmic surface, as in figure s, the wave's electric field causes charges to vibrate back and forth in the surface. These currents then experience a magnetic force from the wave's magnetic field, and application of the geometrical rule shows that the resulting force is in the direction of propagation of the wave. Thus the light wave acts as if it has momentum and inertia.

Symmetry and handedness

Imagine that you establish radio contact with an alien on another planet. Neither of you even knows where the other one's planet is, and you aren't able to establish any landmarks that you both recognize. You manage to learn quite a bit of each other's languages, but you're stumped when you try to establish the definitions of left and right (or, equivalently, clockwise and counterclockwise). Is there any way to do it?

If there was any way to do it without reference to external landmarks, then it would imply that the laws of physics themselves were asymmetric, which would be strange. Why should they distinguish left from right? The gravitational field pattern surrounding a star or planet looks the same in a mirror, and the same goes for electric fields. However, the field patterns shown previously seem to violate this principle, but do they really? Could you use these patterns to explain left and right to the alien? In fact, the answer is no. If you look back at the definition of the magnetic field, it also contains a reference to handedness: the counterclockwise direction of the loop's current as viewed along the magnetic field. The aliens might have reversed their definition of the magnetic field, in which case their drawings of field patterns would look like mirror images of ours.

Until the middle of the twentieth century, physicists assumed that any reasonable set of physical laws would have to have this kind of symmetry between left and right. An asymmetry would be grotesque. Whatever their aesthetic feelings, they had to change their opinions about reality when experiments showed that the weak nuclear force violates right-left symmetry! It is still a mystery why right-left symmetry is observed so scrupulously in general, but is violated by one particular type of physical process.

Doppler shifts and clock time

Figure p shows our now-familiar method of visualizing a Lorentz transformation, in a case where the numbers come out to be particularly simple. This diagram has two geometrical features that we have referred to before without digging into their physical significance: the stretch factor of the diagonals, and the area.

Doppler shifts of light

When Doppler shifts happen to ripples on a pond or the sound waves from an airplane, they can depend on the relative motion of three different objects: the source, the receiver, and the medium. But light waves don't have a medium. Therefore Doppler shifts of light can only depend on the relative motion of the source and observer.

One simple case is the one in which the relative motion of the source and the receiver is perpendicular to the line connecting them. That is, the motion is transverse. Nonrelativistic Doppler shifts happen because the distance between the source and receiver is changing, so in nonrelativistic physics we don't expect any Doppler shift at all when the motion is transverse, and this is what is in fact observed to high precision. For example, the photo shows shortened and lengthened wavelengths to the right and left, along the source's line of motion, but an observer above or below the source measures just the normal, unshifted wavelength and frequency. But relativistically, we have a time dilation effect, so for light waves emitted transversely, there is a Doppler shift of 1/γ in frequency (or γ in wavelength).

The other simple case is the one in which the relative motion of the source and receiver is longitudinal, i.e., they are either approaching or receding from one another. For example, distant galaxies are receding from our galaxy due to the expansion of the universe, and this expansion was originally detected because Doppler shifts toward the red (low-frequency) end of the spectrum were observed.

Nonrelativistically, we would expect the light from such a galaxy to be Doppler shifted down in frequency by some factor, which would depend on the relative velocities of three different objects: the source, the wave's medium, and the receiver. Relativistically, things get simpler, because light isn't a vibration of a physical medium, so the Doppler shift can only depend on a single velocity v, which is the rate at which the separation between the source and the receiver is increasing.

The square in figure u is the “graph paper” used by someone who considers the source to be at rest, while the parallelogram plays a similar role for the receiver. The figure is drawn for the case where v=3/5 (in units where c=1), and in this case the stretch factor of the long diagonal is 2. To keep the area the same, the short diagonal has to be squished to half its original size. But now it's a matter of simple geometry to show that OP equals half the width of the square, and this tells us that the Doppler shift is a factor of 1/2 in frequency. That is, the squish factor of the short diagonal is interpreted as the Doppler shift. To get this as a general equation for velocities other than 3/5, one can show that the Doppler shift is

$$D(v) = \sqrt{\frac{1-v}{1+v}}$$

Here v>0 is the case where the source and receiver are getting farther apart, v<0 the case where they are approaching. (It is convenient to change sign conventions here so that we can use positive values of v in the case of cosmological red-shifts, which are the most important application.)

Suppose that Alice stays at home on earth while her twin Betty takes off in her rocket ship at 3/5 of the speed of light. The thing that caused the most pain is understanding how each observer could say that the other was the one whose time was slow. It seemed that if you could take a pill that would speed up mind and body, then naturally you would see everybody else as being slow. Shouldn't the same apply to relativity? But suppose Alice and Betty get on the radio and try to settle who is the fast one and who is the slow one. Each twin's voice sounds slooooowed doooowwwwn to the other. If Alice claps her hands twice, at a time interval of one second by her clock, Betty hears the hand-claps coming over the radio two seconds apart, but the situation is exactly symmetric, and Alice hears the same thing if Betty claps. Each twin analyzes the situation using a diagram identical to r, and attributes her sister's observations to a complicated combination of time distortion, the time taken by the radio signals to propagate, and the motion of her twin relative to her.

We saw that relativistic velocities should not be expected to be exactly additive, and example 1 verifies this in the special case where A moves relative to B at 0.6c and B relative to C at 0.6c --- the result not being 1.2c. The relativistic Doppler shift provides a simple way of deriving a general equation for the relativistic combination of velocities.

Clock time

We proved that the Lorentz transformation doesn't change the area of a shape in the x-t plane. We used this only as a stepping stone toward the Lorentz transformation, but it is natural to wonder whether this kind of area has any physical interest of its own.

The equal-area result is not relativistic. But the area does have a nice interpretation in the relativistic case. Suppose that we have events A (Charles VII is restored to the throne) and B (Joan of Arc is executed). Now imagine that technologically advanced aliens want to be present at both A and B, but in the interim they wish to fly away in their spaceship, be present at some other event P (perhaps a news conference at which they give an update on the events taking place on earth), but get back in time for B. Since nothing can go faster than c (which we take to equal 1 in appropriate units), P cannot be too far away. The set of all possible events P forms a rectangle, figure v/1, in the x-t plane that has A and B at opposite corners and whose edges have slopes equal to ± 1. We call this type of rectangle a light-rectangle, because its sides could represent the motion of rays of light.

The area of this rectangle will be the same regardless of one's frame of reference. In particular, we could choose a special frame of reference, panel 2 of the figure, such that A and B occur in the same place. (They do not occur at the same place, for example, in the sun's frame, because the earth is spinning and going around the sun.) Since the speed c, which equals 1 in our units, is the same in all frames of reference, and the sides of the rectangle had slopes ± 1 in frame 1, they must still have slopes ± 1 in frame 2. The rectangle becomes a square with its diagonals parallel to the x and t axes, and the length of these diagonals equals the time τ elapsed on a clock that is at rest in frame 2, i.e., a clock that glides through space at constant velocity from A to B, meeting up with the planet earth at the appointed time. As shown in panel 3 of the figure, the area of the gray regions can be interpreted as half the square of this gliding-clock time. If events A and B are separated by a distance x and a time t, then in general \(t^2-x^2\) gives the square of the gliding-clock time.

When |x| is greater than |t|, events A and are so far apart in space and so close together in time that it would be impossible to have a cause and effect relationship between them, since c=1 is the maximum speed of cause and effect. In this situation \(t^2-x^2\) is negative and cannot be interpreted as a clock time, but it can be interpreted as minus the square of the distance between A and B as measured by rulers at rest in a frame in which A and B are simultaneous.

No matter what, \(t^2-x^2\) is the same as measured in all frames of reference. Geometrically, it plays the same role in the x-t plane that ruler measurements play in the Euclidean plane. In Euclidean geometry, the ruler-distance between any two points stays the same regardless of rotation, i.e., regardless of the angle from which we view the scene; according to the Pythagorean theorem, the square of this distance is \(x^2+y^2\). In the x-t plane, \(t^2-x^2\) stays the same regardless of the frame of reference.