8.2: Distortion of Space and Time
7.2.1 The Lorentz transformation
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 a .
A distortion of the Cartesian grid underlying the graph can arise for perfectly ordinary reasons that Isaac Newton would have readily accepted. For example, we can simply change the units used to measure time and position, as in figure b .
We're going to have quite a few examples of this type, so I'll adopt the convention shown in figure c for depicting them. Figure c summarizes the relationship between figures a and b in a more compact form. The gray rectangle represents the original coordinate grid of figure a , while the grid of black lines represents the new version from figure b . 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 c 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:
- No point in time or space has properties that make it different from any other point.
- Likewise, all directions in space have the same properties.
- Motion is relative, i.e., all inertial frames of reference are equally valid.
- Causality holds, in the sense described on page 381.
- Time depends on the state of motion of the observer.
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. 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.
If it were not for property 5, we could imagine that figure d 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.
Figure d 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 e 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 e 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.
The next type of transformation we want to kill off is shown in figure f , 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 rotating relative to the other, then there would be something special about the axis of rotation. But we're only talking about 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 f violates property 1.
The examples in figures e and f 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. Einstein wrote in his 1905 paper that “... on account of the property of homogeneity [property 1] which we ascribe to time and space, the [transformation] must be linear.” 1 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 g , that have this property. Case I is the Galilean transformation of figure d on page 386, which we've already ruled out.
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.
The only remaining possibility is case III, which I've redrawn in figure h with a couple of changes. This is the one that Einstein predicted in 1905. The transformation is known as the Lorentz transformation, after HendrikLorentz (1853-1928), 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 a - c on page 385 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 choosing different units of measurement for time and distance. In figure h 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.
As in our discussion of the Galilean transformation, slopes are interpreted as velocities, and the slope of the near-horizontal lines in figure i 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 figure i is extremely small, and the amount of distortion is tiny --- it would be much too small to see on this scale.
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 h 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 areas , as proved in the caption to figure j .
7.2.2 The \(\gamma\) factor
With a little algebra and geometry (homework problem 7 , page 439), one can use the equal-area property to show that the factor \(\gamma\) (Greek letter gamma) defined in figure k is given by the 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 special 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 \(c\), and the equation becomes
Here's why we care about \(\gamma\). Figure k defines it as the ratio of two times: the time between two events as expressed in one coordinate system, and the time between the same two events as measured in the other one. The interpretation is:
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 \(\gamma\).
As proved in figures l and m , lengths are also distorted:
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 \(\gamma\).
Exercise \(\PageIndex{1}\)
What is \(\gamma\) when \(v=0\)? What does this mean?
(answer in the back of the PDF version of the book)
Example \(\PageIndex{1}\): An 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.
Example \(\PageIndex{2}\): Large time dilation
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 \(\gamma\) 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 p shows a 1974 experiment 2 of a similar type which verified the time dilation predicted by relativity to a precision of about one part per thousand.
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 \(\mu\text{s}\) 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 p , circling around it until they underwent radioactive decay. At the speed at which these muons were traveling, they had \(\gamma=29.33\), 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 o 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.
Example \(\PageIndex{3}\): An example of length contraction
Figure q shows an artist's rendering of the length contraction for the collision of two gold nuclei at relativistic speeds in the 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.
Example \(\PageIndex{1}\): The garage paradox
One of the most famous of all the so-called relativity paradoxes has to do with our incorrect feeling that simultaneity is well defined. The idea is that one could 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, the bus would supposedly fit in the garage. The driver, however, will perceive the 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 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 r , 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.
7.2.3 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 me at 5 m/s relative to the sidewalk, and I run after it at 3 m/s, the dog's velocity in my 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 mine. 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 \(c\) (problem 1 , page 438). 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 s .
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.
Example \(\PageIndex{4}\): The ring laser gyroscope
If you've flown in a jet plane, you can thank relativity for helping you to avoid crashing into a mountain or an ocean.  A beam of light is split into two parts, sent around the perimeter of the device, and reunited. Since the speed of light is constant, we expect the two parts to come back together at the same time. If they don't, it's evidence that the device has been rotating. The plane's computer senses this and notes how much rotation has accumulated.
Example \(\PageIndex{6}\): No frequency-dependence
Relativity has only one universal speed, so it requires that all light waves travel at the same speed, regardless of their frequency and wavelength. Presently the best experimental tests of the invariance of the speed of light with respect to wavelength come from astronomical observations of gamma-ray bursts, which are sudden outpourings of high-frequency light, believed to originate from a supernova explosion in another galaxy. One such observation, in 2009, 3 found that the times of arrival of all the different frequencies in the burst differed by no more than 2 seconds out of a total time in flight on the order of ten billion years!
Discussion Questions
◊
Discussion question B.
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?
◊ On a spaceship moving at relativistic speeds, would a lecture seem even longer and more boring than normal?
◊ 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?
◊ Figure q from page 392, 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?
◊ The machine-gunner in the figure sends out a spray of bullets. Suppose that the bullets are being shot into outer space, and that the distances traveled are trillions of miles (so that the human figure in the diagram is not to scale). After a long time, the bullets reach the points shown with dots which are all equally far from the gun. Their arrivals at those points are events A through E, which happen at different times. The chain of impacts extends across space at a speed greater than \(c\). Does this violate special relativity?
7.2.4 No action at a distance
The Newtonian picture
The Newtonian picture of the universe has particles interacting with each other by exerting forces from a distance, and these forces are imagined to occur without any time delay. For example, suppose that super-powerful aliens, angered when they hear disco music in our AM radio transmissions, come to our solar system on a mission to cleanse the universe of our aesthetic contamination. They apply a force to our sun, causing it to go flying out of the solar system at a gazillion miles an hour. According to Newton's laws, the gravitational force of the sun on the earth will immediately start dropping off. This will be detectable on earth, and since sunlight takes eight minutes to get from the sun to the earth, the change in gravitational force will, according to Newton, be the first way in which earthlings learn the bad news --- the sun will not visibly start receding until a little later. Although this scenario is fanciful, it shows a real feature of Newton's laws: that information can be transmitted from one place in the universe to another with zero time delay, so that transmission and reception occur at exactly the same instant. Newton was sharp enough to realize that this required a nontrivial assumption, which was that there was some completely objective and well-defined way of saying whether two things happened at exactly the same instant. He stated this assumption explicitly: “Absolute, true, and mathematical time, of itself, and from its own nature flows at a constant rate without regard to anything external...”
Time delays in forces exerted at a distance
Relativity forbids Newton's instantaneous action at a distance. For suppose that instantaneous action at a distance existed. It would then be possible to send signals from one place in the universe to another without any time lag. This would allow perfect synchronization of all clocks. But the Hafele-Keating experiment demonstrates that clocks A and B that have been initially synchronized will drift out of sync if one is in motion relative to the other. With instantaneous transmission of signals, we could determine, without having to wait for A and B to be reunited, which was ahead and which was behind. Since they don't need to be reunited, neither one needs to undergo any acceleration; each clock can fix an inertial frame of reference, with a velocity vector that changes neither its direction nor its magnitude. But this violates the principle that constant-velocity motion is relative, because each clock can be considered to be at rest, in its own frame of reference. Since no experiment has ever detected any violation of the relativity of motion, we conclude that instantaneous action at a distance is impossible.
Since forces can't be transmitted instantaneously, it becomes natural to imagine force-effects spreading outward from their source like ripples on a pond, and we then have no choice but to impute some physical reality to these ripples. We call them fields, and they have their own independent existence. Gravity is transmitted through a field called the gravitational field. Besides gravity, there are other fundamental fields of force such as electricity and magnetism (ch. 10 - 11 ). Ripples of the electric and magnetic fields turn out to be light waves. This tells us that the speed at which electric and magnetic field ripples spread must be \(c\), and by an argument similar to the one in subsection 7.2.3 the same must hold for any other fundamental field, including the gravitational field.
Fields don't have to wiggle; they can hold still as well. The earth's magnetic field, for example, is nearly constant, which is why we can use it for direction-finding.
Even empty space, then, is not perfectly featureless. It has measurable properties. For example, we can drop a rock in order to measure the direction of the gravitational field, or use a magnetic compass to find the direction of the magnetic field. This concept made a deep impression on Einstein as a child. He recalled that as a five-year-old, the gift of a magnetic compass convinced him that there was “something behind things, something deeply hidden.”
More evidence that fields of force are real: they carry energy.
The smoking-gun argument for this strange notion of traveling force ripples comes from the fact that they carry energy. In figure x /1, Alice and Betty hold balls A and B at some distance from one another. These balls make a force on each other; it doesn't really matter for the sake of our argument whether this force is gravitational, electrical, or magnetic. Let's say it's electrical, i.e., that the balls have the kind of electrical charge that sometimes causes your socks to cling together when they come out of the clothes dryer. We'll say the force is repulsive, although again it doesn't really matter.
If Alice chooses to move her ball closer to Betty's, x /2, Alice will have to do some mechanical work against the electrical repulsion, burning off some of the calories from that chocolate cheesecake she had at lunch. This reduction in her body's chemical energy is offset by a corresponding increase in the electrical interaction energy.  She has to do a little extra work, but this is all properly accounted for in the interaction energy.
But now suppose, x /3, that Betty decides to play a trick on Alice by tossing B far away just as Alice is getting ready to move A. We have already established that Alice can't feel B's motion instantaneously, so the electric forces must actually be propagated by an electric field . Of course this experiment is utterly impractical, but suppose for the sake of argument that the time it takes the change in the electric field to propagate across the diagram is long enough so that Alice can complete her motion before she feels the effect of B's disappearance. She is still getting stale information about B's position. As she moves A to the right, she feels a repulsion, because the field in her region of space is still the field caused by B in its old position. She has burned some chocolate cheesecake calories, and it appears that conservation of energy has been violated, because these calories can't be properly accounted for by any interaction with B, which is long gone.
If we hope to preserve the law of conservation of energy, then the only possible conclusion is that the electric field itself carries away the cheesecake energy. In fact, this example represents an impractical method of transmitting radio waves. Alice does work on charge A, and that energy goes into the radio waves. Even if B had never existed, the radio waves would still have carried energy, and Alice would still have had to do work in order to create them.
Discussion Questions
◊ Use a graph to resolve the following relativity paradox. Relativity says that in one frame of reference, event A could happen before event B, but in someone else's frame B would come before A. How can this be? Obviously the two people could meet up at A and talk as they cruised past each other. Wouldn't they have to agree on whether B had already happened?
◊ 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.  In frame 2, B happens before A. Did the motion at the right end cause the person on the left to decide to pick up the hammer and use it?
7.2.5 The light cone
Given an event P, we can now classify all the causal relationships in which P can participate. In Newtonian physics, these relationships fell into two classes: P could potentially cause any event that lay in its future, and could have been caused by any event in its past. In relativity, we have a three-way distinction rather than a two-way one. There is a third class of events that are too far away from P in space, and too close in time, to allow any cause and effect relationship, since causality's maximum velocity is \(c\). Since we're working in units in which \(c=1\), the boundary of this set is formed by the lines with slope \(\pm1\) on a \((t,x)\) plot. This is referred to as the light cone, for reasons that become more visually obvious when we consider more than one spatial dimension, figure aa .
Events lying inside one another's light cones are said to have a timelike relationship. Events outside each other's light cones are spacelike in relation to one another, and in the case where they lie on the surfaces of each other's light cones the term is lightlike. \myoptionalsubsection[2]{The spacetime interval}
The light cone is an object of central importance in both special and general relativity. It relates the geometry of spacetime to possible cause-and-effect relationships between events. This is fundamentally how relativity works: it's a geometrical theory of causality.
These ideas naturally lead us to ask what fruitful analogies we can form between the bizarre geometry of spacetime and the more familiar geometry of the Euclidean plane. The light cone cuts spacetime into different regions according to certain measurements of relationships between points (events). Similarly, a circle in Euclidean geometry cuts the plane into two parts, an interior and an exterior, according to the measurement of the distance from the circle's center. A circle stays the same when we rotate the plane. A light cone stays the same when we change frames of reference. Let's build up the analogy more explicitly.
Measurement in Euclidean geometry
We say that two line segments are congruent, \(\text{AB}\cong \text{CD}\), if the distance between points A and B is the same as the distance between C and D, as measured by a rigid ruler.
Measurement in spacetime
We define \(\text{AB}\cong \text{CD}\) if:
- AB and CD are both spacelike, and the two distances are equal as measured by a rigid ruler, in a frame where the two events touch the ruler simultaneously.
- AB and CD are both timelike, and the two time intervals are equal as measured by clocks moving inertially.
- AB and CD are both lightlike.
The three parts of the relativistic version each require some justification.
Case 1 has to be the way it is because space is part of spacetime. In special relativity, this space is Euclidean, so the definition of congruence has to agree with the Euclidean definition, in the case where it is possible to apply the Euclidean definition. The spacelike relation between the points is both necessary and sufficient to make this possible. If points A and B are spacelike in relation to one another, then a frame of reference exists in which they are simultaneous, so we can use a ruler that is at rest in that frame to measure their distance. If they are lightlike or timelike, then no such frame of reference exists. For example, there is no frame of reference in which Charles VII's restoration to the throne is simultaneous with Joan of Arc's execution, so we can't arrange for both of these events to touch the same ruler at the same time.
The definition in case 2 is the only sensible way to proceed if we are to respect the symmetric treatment of time and space in relativity. The timelike relation between the events is necessary and sufficient to make it possible for a clock to move from one to the other. It makes a difference that the clocks move inertially, because the twins in example 1 on p. 391 disagree on the clock time between the traveling twin's departure and return.
Case 3 may seem strange, since it says that any two lightlike intervals are congruent. But this is the only possible definition, because this case can be obtained as a limit of the timelike one. Suppose that AB is a timelike interval, but in the planet earth's frame of reference it would be necessary to travel at almost the speed of light in order to reach B from A. The required speed is less than \(c\) (i.e., less than 1) by some tiny amount \(\epsilon\). In the earth's frame, the clock referred to in the definition suffers extreme time dilation. The time elapsed on the clock is very small. As \(\epsilon\) approaches zero, and the relationship between A and B approaches a lightlike one, this clock time approaches zero. In this sense, the relativistic notion of “distance” is very different from the Euclidean one. In Euclidean geometry, the distance between two points can only be zero if they are the same point.
The case splitting involved in the relativistic definition is a little ugly. Having worked out the physical interpretation, we can now consolidate the definition in a nicer way by appealing to Cartesian coordinates.
Cartesian definition of distance in Euclidean geometry
Given a vector \((\Delta x,\Delta y)\) from point A to point B, the square of the distance between them is defined as \(\overline{\text{AB}}^2=\Delta x^2+\Delta y^2\).
Definition of the interval in relativity
Given points separated by coordinate differences \(\Delta x\), \(\Delta y\), \(\Delta z\), and \(\Delta t\), the spacetime interval \(\mathcal I\) (cursive letter “I”) between them is defined as \(\mathcal I = \Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2\).
This is stated in natural units, so all four terms on the right-hand side have the same units; in metric units with \(c \ne 1\), appropriate factors of \(c\) should be inserted in order to make the units of the terms agree. The interval \(\mathcal I\) is positive if AB is timelike (regardless of which event comes first), zero if lightlike, and negative if spacelike. Since \(\mathcal I\) can be negative, we can't in general take its square root and define a real number \(\overline{\text{AB}}\) as in the Euclidean case. When the interval is timelike, we can interpret \(\sqrt{\mathcal I}\) as a time, and when it's spacelike we can take \(\sqrt{-\mathcal I}\) to be a distance.
The Euclidean definition of distance (i.e., the Pythagorean theorem) is useful because it gives the same answer regardless of how we rotate the plane. Although it is stated in terms of a certain coordinate system, its result is unambiguously defined because it is the same regardless of what coordinate system we arbitrarily pick. Similarly, \(\mathcal I\) is useful because, as proved in example 8 below, it is the same regardless of our frame of reference, i.e., regardless of our choice of coordinates.
Example \(\PageIndex{7}\): Pioneer 10
\(\triangleright\) The Pioneer 10 space probe was launched in 1972, and in 1973 was the first craft to fly by the planet Jupiter. It crossed the orbit of the planet Neptune in 1983, after which telemetry data were received until 2002. The following table gives the spacecraft's position relative to the sun at exactly midnight on January 1, 1983 and January 1, 1995. The 1983 date is taken to be \(t=0\).
| t (s) | x | y | z |
|---|---|---|---|
| 0 | 1.784 X 10 12 m | 3.951 X 10 12 m | 0.237 X 10 12 m |
| 3.7869120000 X 10 8 s | 2.420 X 10 12 m | 8.827 X 10 12 m | 0.488 X 10 12 m |
Compare the time elapsed on the spacecraft to the time in a frame of reference tied to the sun.
\(\triangleright\) We can convert these data into natural units, with the distance unit being the second (i.e., a light-second, the distance light travels in one second) and the time unit being seconds. Converting and carrying out this subtraction, we have:
| Δ t (s) | Δ x | Δ y | Δ z |
|---|---|---|---|
| 3.7869120000 X 10 8 s | 0.212 X 10 4 s | 1.626 X 10 4 s | 0.084 X 10 4 s |
Comparing the exponents of the temporal and spatial numbers, we can see that the spacecraft was moving at a velocity on the order of \(10^{-4}\) of the speed of light, so relativistic effects should be small but not completely negligible.
Since the interval is timelike, we can take its square root and interpret it as the time elapsed on the spacecraft. The result is \(\sqrt{\mathcal I}=3.786911996\times 10^8\ \text{s}\). This is 0.4 s less than the time elapsed in the sun's frame of reference.
Example \(\PageIndex{8}\): Invariance of the interval
In this example we prove that the interval is the same regardless of what frame of reference we compute it in. This is called “Lorentz invariance.” The proof is limited to the timelike case. Given events A and B, construct the light-rectangle as defined in figure ab /1. On p. 389 we proved that the Lorentz transformation doesn't change the area of a shape in the \(x\)-\(t\) plane. Therefore the area of this rectangle is unchanged if we switch to the frame of reference ab /2, in which A and B occurred at the same location and were separated by a time interval \(\tau\). This area equals half the interval \(\mathcal I\) between A and B. But a straightforward calculation shows that the rectangle in ab /1 also has an area equal to half the interval calculated in that frame. Since the area in any frame equals half the interval, and the area is the same in all frames, the interval is equal in all frames as well.
Example \(\PageIndex{9}\): A numerical example of invariance
Figure ac shows two frames of reference in motion relative to one another at \(v=3/5\). (For this velocity, the stretching and squishing of the main diagonals are both by a factor of 2.) Events are marked at coordinates that in the frame represented by the square are
The interval between these events is \(13^2-11^2=48\). In the frame represented by the parallelogram, the same two events lie at coordinates
Calculating the interval using these values, the result is
\(8^2-4^2=48\), which comes out the same as in the other frame.
Four-vectors and the inner product
Example 7 makes it natural that we define a type of vector with four components, the first one relating to time and the others being spatial. These are known as four-vectors. It's clear how we should define the equivalent of a dot product in relativity:
The term “dot product” has connotations of referring only to three-vectors, so the operation of taking the scalar product of two four-vectors is usually referred to instead as the “inner product.” The spacetime interval can then be thought of as the inner product of a four-vector with itself. We care about the relativistic inner product for exactly the same reason we care about its Euclidean version; both are scalars, so they have a fixed value regardless of what coordinate system we choose.
| Example 10: The twin paradox |
|---|
|
Alice and Betty are identical twins. Betty goes on a space voyage at relativistic speeds, traveling away from the earth and then turning around and coming back. Meanwhile, Alice stays on earth. When Betty returns, she is younger than Alice because of relativistic time dilation (example 1 , p. 391). But isn't it valid to say that Betty's spaceship is standing still and the earth moving? In that description, wouldn't Alice end up younger and Betty older? This is referred to as the “twin paradox.” It can't really be a paradox, since it's exactly what was observed in the Hafele-Keating experiment (p. 381). Betty's track in the \(x\)-\(t\) plane (her “world-line” in relativistic jargon) consists of vectors \(\mathbf{b}\) and \(\mathbf{c}\) strung end-to-end (figure ad ). We could adopt a frame of reference in which Betty was at rest during \(\mathbf{b}\) (i.e., \(b_x=0\)), but there is no frame in which \(\mathbf{b}\) and \(\mathbf{c}\) are parallel, so there is no frame in which Betty was at rest during both \(\mathbf{b}\) and \(\mathbf{c}\). This resolves the paradox. We have already established by other methods that Betty ages less that Alice, but let's see how this plays out in a simple numerical example. Omitting units and making up simple numbers, let's say that the vectors in figure ad are
\[\begin{align*} \mathbf{a} &= (6,1) \\ \mathbf{b} &= (3,2) \\ \mathbf{c} &= (3,-1) , \end{align*}\]
where the components are given in the order \((t,x)\). The time experienced by Alice is then
\[\begin{equation*} |\mathbf{a}| = \sqrt{6^2-1^2} =5.9 , \end{equation*}\]
which is greater than the Betty's elapsed time
\[\begin{equation*} |\mathbf{b}|+|\mathbf{c}| = \sqrt{3^2-2^2}+\sqrt{3^2-(-1)^2} = 5.1 . \end{equation*}\]
|
| Example 11: Simultaneity using inner products |
|---|
|
Suppose that an observer O moves inertially along a vector \(\mathbf{o}\), and let the vector separating two events P and Q be \(\mathbf{s}\). O judges these events to be simultaneous if \(\mathbf{o}\cdot\mathbf{s}=0\). To see why this is true, suppose we pick a coordinate system as defined by O. In this coordinate system, O considers herself to be at rest, so she says her vector has only a time component, \(\mathbf{o}=(\Delta t,0,0,0)\). If she considers P and Q to be simultaneous, then the vector from P to Q is of the form \((0,\Delta x,\Delta y,\Delta z)\). The inner product is then zero, since each of the four terms vanishes. Since the inner product is independent of the choice of coordinate system, it doesn't matter that we chose one tied to O herself. Any other observer \(\text{O}'\) can look at O's motion, note that \(\mathbf{o}\cdot\mathbf{s}=0\), and infer that O must consider P and Q to be simultaneous, even if \(\text{O}'\) says they weren't. |
Doppler shifts of light and addition of velocities
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 \(1/\gamma\) in frequency (or \(\gamma\) 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 af 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 by straightforward fiddling with the result of part c of problem 7 on p. 439 that the Doppler shift is
Here \(v>0\) is the case where the source and receiver are getting farther apart, \(v\lt0\) the case where they are approaching. (This is the opposite of the sign convention used in subsection 6.1.5 . It is convenient to change 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. When I first learned relativity, the thing that caused me the most pain was understanding how each observer could say that the other was the one whose time was slow. It seemed to me that if I could take a pill that would speed up my mind and my body, then naturally I 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 ah , 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.
self-check:
Turn your book upside-down and reinterpret figure ah .
(answer in the back of the PDF version of the book)
| Example 12: A symmetry property of the Doppler effect |
|---|
| Suppose that A and B are at rest relative to one another, but C is moving along the line between A and B. A transmits a signal to C, who then retransmits it to B. The signal accumulates two Doppler shifts, and the result is their product \(D(v)D(-v)\). But this product must equal 1, so we must have \(D(-v)D(v)=1\), which can be verified directly from the equation. |
| Example 13: The Ives-Stilwell experiment | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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The result of example 12 was the basis of one of the earliest laboratory tests of special relativity, by Ives and Stilwell in 1938. They observed the light emitted by excited by a beam of \(\text{H}_2^+\) and \(\text{H}_3^+\) ions with speeds of a few tenths of a percent of \(c\). Measuring the light from both ahead of and behind the beams, they found that the product of the Doppler shifts \(D(v)D(-v)\) was equal to 1, as predicted by relativity. If relativity had been false, then one would have expected the product to differ from 1 by an amount that would have been detectable in their experiment. In 2003, Saathoff et al. carried out an extremely precise version of the Ives-Stilwell technique with \(\text{Li}^+\) ions moving at 6.4% of \(c\). The frequencies observed, in units of MHz, were:
The results show incredibly precise agreement between \(f_\text{o}\) and \(\sqrt{f_\text{o}D(-v)\cdot f_\text{o} D(v)}\), as expected relativistically because \(D(v)D(-v)\) is supposed to equal 1. The agreement extends to 9 significant figures, whereas if relativity had been false there should have been a relative disagreement of about \(v^2=.004\), i.e., a discrepancy in the third significant figure. The spectacular agreement with theory has made this experiment a lightning rod for anti-relativity kooks. |
We saw on p. 394 that relativistic velocities should not be expected to be exactly additive, and problem 1 on p. 438 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; problem 17 on p. 442 guides you through the steps of this derivation, and the result is given on p. 936.
Contributors and Attributions
- Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


9.8: Physical Education and Physical Activity
According to evidence-based guidelines from the U.S. Department of Health and Human Services (HHS) and the CDC, children between the ages of 6-17 should be getting at least 60 minutes of moderate to vigorous physical activity each day. This can include age-appropriate activities such as sports, dance, exercise, and unstructured play. Any type of physical activity that is aerobic in nature, or promotes muscle or bone strengthening can be beneficial for both physical health and for cognitive skills like memory and focus (HHS, 2018). Childhood is also the best time to learn and develop motor skills such as running, skipping, jumping, throwing, catching, and balancing - foundational skills that increase the likelihood of participation in organized sports, dance, and exercise later in life. Although young children do tend to participate in physically active unstructured play “naturally”, the development of motor skills also requires instruction and practice, emphasizing the additional need for structured physical education during childhood. In fact, the combination of both structured physical education and opportunities for unstructured play allows for children to practice skills and improve their health and fitness (Moon et al., 2024). See Fig. \(\PageIndex{1}\) for more benefits of physical activity during childhood.
Physical activity also does not take away from academic time - in fact, it can play a supportive role. There is ample evidence at this point in time that physical activity can acutely improve the learning experience. Integrating physical activity into classroom lessons or creating activity breaks throughout the school day has been shown to improve students’ focus and educational outcomes (Lansky, 2022). Although there isn’t enough data to suggest that physical activity levels directly improve grades in high school, there is definitely a correlation between physical activity, less sedentary time, and higher grades earned ( Youth Risk Behavior Surveillance System (YRBSS) , 2023). The research on whether physical activity interventions improve academic performance is mixed, but mostly indicates a positive or neutral effect (Rasberry et al., 2011).
Current data suggest that less than a quarter (24%) of U.S. children and youth achieve the recommended minimum amount of daily physical activity. Just over 50% of U.S. high school students get daily physical activity and about the same percentage report attending physical education classes in an average week (CDC, 2022a). Total physical activity tends to decrease as children get older, with the sharpest declines happening in middle school. Girls tend to be less active than boys and experience a sharper decline as they enter higher grades (Pate et al., 2022). Perhaps this is due to increasing academic demands, whether or not students participate in organized sports, or even social and cultural pressures that cause a decrease in unstructured play. In the most recent Youth Risk Behavior Survey (YRBS) data from 2021, the percentage of high school students playing on at least one sports team dropped below 50% for the first time since 1999 - likely influenced by precautionary measures taken during the COVID-19 pandemic ( Youth Risk Behavior Surveillance System (YRBSS) , 2023).
Schools can provide opportunities for students to learn and practice physical skills, and engage in physical activity during the school day, and also before and after school hours. A comprehensive school physical activity program (CSAP) includes the community, staff, and families in promoting physical activity throughout the day. Installing safe playground equipment and maintaining school grounds and gymnasiums can encourage physical activity during recess. Physical education curriculum should teach students skills for lifelong physical activity, as well as supporting age-appropriate motor development, emotional intelligence, ethical behaviors in competition (aka “sportsmanship”), and self-efficacy. This requires a trained, certified physical education teacher. Partnerships with community resources (afterschool programs, sports clubs etc.) and keeping school yards and playgrounds open for the community can provide opportunities for physical activity outside of school hours. Safe sidewalks, paths, and street crossings can support active transport to and from school. Finally, emphasizing the importance of physical activity for children can also spill over into the rest of the community - whether via employee wellness programs or family activity events (CDC Healthy Schools, 2021, Seabert et al. 2021).


1.2: Demand States
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It is useful to think about different demand conditions or states. Specifically, there are three: effective demand, latent demand, and negative demand. Much of the class will emphasize effective demand because markets exist for products in a state of effective demand. For products in an effective demand state, you can see equilibrium prices and quantities. It is also possible estimate demand and understand how demand responds to changes in prices, consumer income, or other factors that affect the desirability of the product. In the other two demand states, markets do not yet exist, but there could or would be a market provided technological barriers can be addressed or consumer preferences can be changed. In fact, many product development efforts are aimed at overcoming some hurdle or limitation that would allow a product or service to move from a latent or negative demand state to an effective demand state.
Effective Demand
Normally, the word “demand” refers to a product in a state of effective demand. Effective demand refers to the demand for products that are commercially feasible. These products satisfy consumer wants and needs, and consumers (or some segments thereof) are willing to buy them at prices that justify their production. Automobiles, apples, vacation trips, restaurant meals, or any other good or service readily available in the economy is in a state of effective demand. Since market transactions are observed for products in a state of effective demand, it is possible to measure variables that influence demand. These measurements can be used to estimate and forecast demand to gain insight into business or policy decisions.
Latent Demand
A state of latent demand is said to exist in cases where a need or want exists but a product does not. For example, if there was a cure for cancer, there would be a demand for it. Because of this latent demand, biotechnology companies spend large amounts of money on research and development efforts to create new drugs to more effectively combat cancer. A latent demand state exists when the following conditions are present:
- the product in question is not technically feasible given the current state of science and engineering (as in the case of a complete cure to cancer) or has yet to be conceived by an inventor or entrepreneur, or
- the product is technically feasible but not commercially feasible given current technology, existing infrastructure, or the existing regulatory environment.
With latent demand, one does not actually see market transactions. For this reason, estimating the potential value of products in a latent demand state is often more difficult. Economists often use experimental methods to assess willingness to pay for products in a state of latent demand.
A change in technology, such as the introduction of the microwave oven, can unleash a number of new products that address latent demands and thereby bring them into a state of effective demand. The first link below points to an early television commercial for the microwave.  Today, much of what is in the frozen section of a supermarket is tailored to the microwave oven. Here is a link to an early microwave oven advertisement emphasizing how the microwave addresses latent demand ( visit youtube.com ).
Negative Demand
Negative demand refers to cases where there is a potential product that could meet a consumer want or need, but consumers are reluctant to purchase the product for one reason or another. For example, many insect species are edible and could be produced at a cost that compares favorably to other food sources. However, insects are not commonly being sold in attractive displays at the local supermarket. The reason for this is that cultural norms in the USA frown upon consumption of insects. This example is a bit “creepy,” and I hope it does not “bug” you too much, but the same general idea can apply to many actual or potential products that are viewed skeptically by consumers. When confronted with negative demand, it may be possible to change the image of the product so that consumers view it as desirable.
- National Geographic segment on edible insects ( visit youtube.com ).
- Fact sheet on entomophagy from the UN Food and Agriculture Organization (FAO) ( visit fao.org ).
Food Irradiation: A Case Example
Food irradiation is a process that many thought could potentially address latent demands for safer food. For example, irradiation could destroy Trichina parasites allowing pork to be cooked and served rare. In the late 1990’s, irradiation was approved by US regulatory agencies for use on raw meat and poultry. There were some predictions that this would become a standard process, akin to pasteurization. This turned out not to be the case. One reason is that irradiated foods also faced a negative demand problem. Consumers have been hesitant to embrace irradiated foods. Review the fact sheet and short video clip from the links below. Both are from reputable organizations: a land-grant university and the United Nations.
- Fact sheet from the University of Wisconsin on food irradiation (visit wisc.edu).
- Short video from International Atomic Energy Agency and FAO ( visit youtube.com ).
Now consider the following questions:
- What fundamental problem (or problems) can food irradiation address?
- Is food irradiation safe?
- Are there labeling requirements that would allow you tell if a food item in a retail store was irradiated?
- Why might consumers be reluctant to accept irradiated foods?
- Why is microwaveable food so widely accepted but irradiated food not?


9.2.1: Exposure to Radioactivity
- To know the differences between ionizing and nonionizing radiation and their effects on matter.
- To identify natural and artificial sources of radiation.
Because nuclear reactions do not typically affect the valence electrons of the atom (although electron capture draws an electron from an orbital of the lowest energy level), they do not directly cause chemical changes. Nonetheless, the particles and the photons emitted during nuclear decay are very energetic, and they can indirectly produce chemical changes in the matter surrounding the nucleus that has decayed. For instance, an α particle is an ionized helium nucleus (He 2 + ) that can act as a powerful oxidant. In this section, we describe how radiation interacts with matter and the some of the chemical and biological effects of radiation.
Ionizing versus Nonionizing Radiation
The effects of radiation on matter are determined primarily by the energy of the radiation, which depends on the nuclear decay reaction that produced it. Nonionizing radiation is relatively low in energy; when it collides with an atom in a molecule or an ion, most or all of its energy can be absorbed without causing a structural or a chemical change. Instead, the kinetic energy of the radiation is transferred to the atom or molecule with which it collides, causing it to rotate, vibrate, or move more rapidly. Because this energy can be transferred to adjacent molecules or ions in the form of heat, many radioactive substances are warm to the touch. Highly radioactive elements such as polonium, for example, have been used as heat sources in the US space program. As long as the intensity of the nonionizing radiation is not great enough to cause overheating, it is relatively harmless, and its effects can be neutralized by cooling.
In contrast, ionizing radiation is higher in energy, and some of its energy can be transferred to one or more atoms with which it collides as it passes through matter. If enough energy is transferred, electrons can be excited to very high energy levels, resulting in the formation of positively charged ions:
\[\mathrm{atom + ionizing\: radiation \rightarrow ion^+ + \, {e^-}\label{Eq1}} \]
Molecules that have been ionized in this way are often highly reactive, and they can decompose or undergo other chemical changes that create a cascade of reactive molecules that can damage biological tissues and other materials ( Figure \(\PageIndex{1}\) ). Because the energy of ionizing radiation is very high, we often report its energy in units such as megaelectronvolts (MeV) per particle:
\[\text{1 MeV/particle} = \text{96 billion J/mol}. \nonumber \]
The Effects of Ionizing Radiation on Matter
The effects of ionizing radiation depend on four factors:
- The type of radiation, which dictates how far it can penetrate into matter
- The energy of the individual particles or photons
- The number of particles or photons that strike a given area per unit time
- The chemical nature of the substance exposed to the radiation
The relative abilities of the various forms of ionizing radiation to penetrate biological tissues are illustrated in Figure \(\PageIndex{2}\) . Because of its high charge and mass, α radiation interacts strongly with matter. Consequently, it does not penetrate deeply into an object, and it can be stopped by a piece of paper, clothing, or skin. In contrast, γ rays, with no charge and essentially no mass, do not interact strongly with matter and penetrate deeply into most objects, including the human body. Several inches of lead or more than 12 inches of special concrete are needed to completely stop γ rays. Because β particles are intermediate in mass and charge between α particles and γ rays, their interaction with matter is also intermediate. Beta particles readily penetrate paper or skin, but they can be stopped by a piece of wood or a relatively thin sheet of metal.
Because of their great penetrating ability, γ rays are by far the most dangerous type of radiation when they come from a source outside the body. Alpha particles, however, are the most damaging if their source is inside the body because internal tissues absorb all of their energy. Thus danger from radiation depends strongly on the type of radiation emitted and the extent of exposure, which allows scientists to safely handle many radioactive materials if they take precautions to avoid, for example, inhaling fine particulate dust that contains alpha emitters. Some properties of ionizing radiation are summarized in Table \(\PageIndex{1}\) .
| Type | Energy Range (MeV) | Penetration Distance in Water* | Penetration Distance in Air* |
|---|---|---|---|
| *Distance at which half of the radiation has been absorbed. | |||
| α particles | 3–9 | < 0.05 mm | < 10 cm |
| β particles | ≤ 3 | < 4 mm | 1 m |
| x-rays | <10 −2 | < 1 cm | < 3 m |
| γ rays | 10 −2 –10 1 | < 20 cm | > 3 m |
There are many different ways to measure radiation exposure, or the dose. The roentgen (R) , which measures the amount of energy absorbed by dry air, can be used to describe quantitative exposure.Named after the German physicist Wilhelm Röntgen (1845–1923; Nobel Prize in Physics, 1901), who discovered x-rays. The roentgen is actually defined as the amount of radiation needed to produce an electrical charge of 2.58 × 10 −4 C in 1 kg of dry air. Damage to biological tissues, however, is proportional to the amount of energy absorbed by tissues, not air. The most common unit used to measure the effects of radiation on biological tissue is the rad (radiation absorbed dose) ; the SI equivalent is the gray (Gy). The rad is defined as the amount of radiation that causes 0.01 J of energy to be absorbed by 1 kg of matter, and the gray is defined as the amount of radiation that causes 1 J of energy to be absorbed per kilogram:
\[\mathrm{1\: rad = 0.010\: J/kg \hspace{25 pt} 1\: Gy = 1\: J/kg \label{Eq2}} \]
Thus a 70 kg human who receives a dose of 1.0 rad over his or her entire body absorbs 0.010 J/70 kg = 1.4 × 10 −4 J, or 0.14 mJ. To put this in perspective, 0.14 mJ is the amount of energy transferred to your skin by a 3.8 × 10 −5 g droplet of boiling water. Because the energy of the droplet of water is transferred to a relatively large area of tissue, it is harmless. A radioactive particle, however, transfers its energy to a single molecule, which makes it the atomic equivalent of a bullet fired from a high-powered rifle.
Because α particles have a much higher mass and charge than β particles or γ rays, the difference in mass between α and β particles is analogous to being hit by a bowling ball instead of a table tennis ball traveling at the same speed. Thus the amount of tissue damage caused by 1 rad of α particles is much greater than the damage caused by 1 rad of β particles or γ rays. Thus a unit called the rem (roentgen equivalent in man) was devised to describe the actual amount of tissue damage caused by a given amount of radiation. The number of rems of radiation is equal to the number of rads multiplied by the
RBE (relative biological effectiveness) factor, which is 1 for β particles, γ rays, and x-rays and about 20 for α particles. Because actual radiation doses tend to be very small, most measurements are reported in millirems (1 mrem = 10 −3 rem).Born in the Lower Rhine Province of Germany, Röntgen was the only child of a cloth manufacturer and merchant. His family moved to the Netherlands where he showed no particular aptitude in school, but where he was fond of roaming the countryside. Röntgen was expelled from technical school in Utrecht after being unjustly accused of drawing a caricature of one of the teachers. He began studying mechanical engineering in Zurich, which he could enter without having the credentials of a regular student, and received a PhD at the University of Zurich in 1869. In 1876 he became professor of physics.
Natural Sources of Radiation
We are continuously exposed to measurable background radiation from a variety of natural sources, which, on average, is equal to about 150–600 mrem/yr ( Figure \(\PageIndex{3}\) ). One component of background radiation is cosmic rays , high-energy particles and \(\gamma\) rays emitted by the sun and other stars, which bombard Earth continuously. Because cosmic rays are partially absorbed by the atmosphere before they reach Earth’s surface, the exposure of people living at sea level (about 30 mrem/yr) is significantly less than the exposure of people living at higher altitudes (about 50 mrem/yr in Denver, Colorado). Every 4 hours spent in an airplane at greater than 30,000 ft adds about 1 mrem to a person’s annual radiation exposure.
A second component of background radiation is cosmogenic radiation , produced by the interaction of cosmic rays with gases in the upper atmosphere. When high-energy cosmic rays collide with oxygen and nitrogen atoms, neutrons and protons are released. These, in turn, react with other atoms to produce radioactive isotopes, such as \(\ce{^{14}C}\):
\[\ce{^{14}_7 N + ^1_0 n \rightarrow ^{14}_6 C + ^1_1p }\label{Eq3} \]
The carbon atoms react with oxygen atoms to form CO 2 , which is eventually washed to Earth’s surface in rain and taken up by plants. About 1 atom in 1 × 10 12 of the carbon atoms in our bodies is radioactive 14 C, which decays by beta emission. About 5000 14 C nuclei disintegrate in your body during the 15 s or so that it takes you to read this paragraph. Tritium ( 3 H) is also produced in the upper atmosphere and falls to Earth in precipitation. The total radiation dose attributable to 14 C is estimated to be 1 mrem/yr, while that due to 3 H is about 1000 times less.
The third major component of background radiation is terrestrial radiation , which is due to the remnants of radioactive elements that were present on primordial Earth and their decay products. For example, many rocks and minerals in the soil contain small amounts of radioactive isotopes, such as \(\ce{^{232}Th}\) and \(\ce{^{238}U}\) as well as radioactive daughter isotopes, such as\(\ce{^{226}Ra}\). The amount of background radiation from these sources is about the same as that from cosmic rays (approximately 30 mrem/yr). These isotopes are also found in small amounts in building materials derived from rocks and minerals, which significantly increases the radiation exposure for people who live in brick or concrete-block houses (60–160 mrem/yr) instead of houses made of wood (10–20 mrem/yr). Our tissues also absorb radiation (about 40 mrem/yr) from naturally occurring radioactive elements that are present in our bodies. For example, the average adult contains about 140 g of potassium as the \(K^+\) ion. Naturally occurring potassium contains 0.0117% \(\ce{^{40}K}\), which decays by emitting both a β particle and a (\gamma\) ray. In the last 20 seconds, about the time it took you to read this paragraph, approximately 40,000 \(\ce{^{40}K}\) nuclei disintegrated in your body.
By far the most important source of background radiation is radon, the heaviest of the noble gases (group 18). Radon-222 is produced during the decay of 238 U, and other isotopes of radon are produced by the decay of other heavy elements. Even though radon is chemically inert, all its isotopes are radioactive. For example, 222 Rn undergoes two successive alpha-decay events to give 214 Pb:
\[\ce{^{222}_{86} Rn \rightarrow ^4_2\alpha + ^{218}_{84} Po + ^4_2\alpha + ^{214}_{82} Pb } \label{Eq4} \]
Because radon is a dense gas, it tends to accumulate in enclosed spaces such as basements, especially in locations where the soil contains greater-than-average amounts of naturally occurring uranium minerals. Under most conditions, radioactive decay of radon poses no problems because of the very short range of the emitted α particle. If an atom of radon happens to be in your lungs when it decays, however, the chemically reactive daughter isotope polonium-218 can become irreversibly bound to molecules in the lung tissue. Subsequent decay of \(\ce{^{218}Po}\) releases an α particle directly into one of the cells lining the lung, and the resulting damage can eventually cause lung cancer. The \(\ce{^{218}Po}\) isotope is also readily absorbed by particles in cigarette smoke, which adhere to the surface of the lungs and can hold the radioactive isotope in place. Recent estimates suggest that radon exposure is a contributing factor in about 15% of the deaths due to lung cancer. Because of the potential health problem radon poses, many states require houses to be tested for radon before they can be sold. By current estimates, radon accounts for more than half of the radiation exposure of a typical adult in the United States.
Artificial Sources of Radiation
In addition to naturally occurring background radiation, humans are exposed to small amounts of radiation from a variety of artificial sources. The most important of these are the x-rays used for diagnostic purposes in medicine and dentistry, which are photons with much lower energy than γ rays. A single chest x-ray provides a radiation dose of about 10 mrem, and a dental x-ray about 2–3 mrem. Other minor sources include television screens and computer monitors with cathode-ray tubes, which also produce x-rays. Luminescent paints for watch dials originally used radium, a highly toxic alpha emitter if ingested by those painting the dials. Radium was replaced by tritium ( 3 H) and promethium ( 147 Pr), which emit low-energy β particles that are absorbed by the watch crystal or the glass covering the instrument. Radiation exposure from television screens, monitors, and luminescent dials totals about 2 mrem/yr. Residual fallout from previous atmospheric nuclear-weapons testing is estimated to account for about twice this amount, and the nuclear power industry accounts for less than 1 mrem/yr (about the same as a single 4 h jet flight).
Calculate the annual radiation dose in rads a typical 70 kg chemistry student receives from the naturally occurring 40 K in his or her body, which contains about 140 g of potassium (as the K + ion). The natural abundance of 40 K is 0.0117%. Each 1.00 mol of 40 K undergoes 1.05 × 10 7 decays/s, and each decay event is accompanied by the emission of a 1.32 MeV β particle.
Given: mass of student, mass of isotope, natural abundance, rate of decay, and energy of particle
Asked for: annual radiation dose in rads
Strategy:
- Calculate the number of moles of 40 K present using its mass, molar mass, and natural abundance.
- Determine the number of decays per year for this amount of 40 K.
- Multiply the number of decays per year by the energy associated with each decay event. To obtain the annual radiation dose, use the mass of the student to convert this value to rads.
Solution
A The number of moles of 40 K present in the body is the total number of potassium atoms times the natural abundance of potassium atoms present as 40 K divided by the atomic mass of 40 K:
\[\textrm{moles }^{40}\textrm K= 140\textrm{ g K} \times \dfrac{0.0117\textrm{ mol }^{40}\textrm K}{100\textrm{ mol K}}\times\dfrac{1\textrm{ mol K}}{40.0\textrm{ g K}}=4.10\times10^{-4}\mathrm{\,mol\,^{40}K} \nonumber \]
B We are given the number of atoms of 40 K that decay per second in 1.00 mol of 40 K, so the number of decays per year is as follows:
C The total energy the body receives per year from the decay of 40 K is equal to the total number of decays per year multiplied by the energy associated with each decay event:
We use the definition of the rad (1 rad = 10 −2 J/kg of tissue) to convert this figure to a radiation dose in rads. If we assume the dose is equally distributed throughout the body, then the radiation dose per year is as follows:
This corresponds to almost half of the normal background radiation most people experience.
Because strontium is chemically similar to calcium, small amounts of the Sr 2 + ion are taken up by the body and deposited in calcium-rich tissues such as bone, using the same mechanism that is responsible for the absorption of Ca 2 + . Consequently, the radioactive strontium ( 90 Sr) found in fission waste and released by atmospheric nuclear-weapons testing is a major health concern. A normal 70 kg human body has about 280 mg of strontium, and each mole of 90 Sr undergoes 4.55 × 10 14 decays/s by the emission of a 0.546 MeV β particle. What would be the annual radiation dose in rads for a 70 kg person if 0.10% of the strontium ingested were 90 Sr?
- Answer
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5.7 × 10 3 rad/yr (which is 10 times the fatal dose)
One of the more controversial public policy issues debated today is whether the radiation exposure from artificial sources, when combined with exposure from natural sources, poses a significant risk to human health. The effects of single radiation doses of different magnitudes on humans are listed in Table \(\PageIndex{2}\) . Because of the many factors involved in radiation exposure (length of exposure, intensity of the source, and energy and type of particle), it is difficult to quantify the specific dangers of one radioisotope versus another. Nonetheless, some general conclusions regarding the effects of radiation exposure are generally accepted as valid.
| Dose (rem) | Symptoms/Effects |
|---|---|
| < 5 | no observable effect |
| 5–20 | possible chromosomal damage |
| 20–100 | temporary reduction in white blood cell count |
| 50–100 | temporary sterility in men (up to a year) |
| 100–200 | mild radiation sickness, vomiting, diarrhea, fatigue; immune system suppressed; bone growth in children retarded |
| > 300 | permanent sterility in women |
| > 500 | fatal to 50% within 30 days; destruction of bone marrow and intestine |
| > 3000 | fatal within hours |
Radiation doses of 600 rem and higher are invariably fatal, while a dose of 500 rem kills half the exposed subjects within 30 days. Smaller doses (≤ 50 rem) appear to cause only limited health effects, even though they correspond to tens of years of natural radiation. This does not, however, mean that such doses have no ill effects; they may cause long-term health problems, such as cancer or genetic changes that affect offspring. The possible detrimental effects of the much smaller doses attributable to artificial sources (< 100 mrem/yr) are more difficult to assess.
The tissues most affected by large, whole-body exposures are bone marrow, intestinal tissue, hair follicles, and reproductive organs, all of which contain rapidly dividing cells. The susceptibility of rapidly dividing cells to radiation exposure explains why cancers are often treated by radiation. Because cancer cells divide faster than normal cells, they are destroyed preferentially by radiation. Long-term radiation-exposure studies on fruit flies show a linear relationship between the number of genetic defects and both the magnitude of the dose and the exposure time. In contrast, similar studies on mice show a much lower number of defects when a given dose of radiation is spread out over a long period of time rather than received all at once. Both patterns are plotted in Figure \(\PageIndex{4}\) , but which of the two is applicable to humans? According to one hypothesis, mice have very low risk from low doses because their bodies have ways of dealing with the damage caused by natural radiation. At much higher doses, however, their natural repair mechanisms are overwhelmed, leading to irreversible damage. Because mice are biochemically much more similar to humans than are fruit flies, many scientists believe that this model also applies to humans. In contrast, the linear model assumes that all exposure to radiation is intrinsically damaging and suggests that stringent regulation of low-level radiation exposure is necessary. Which view is more accurate? The answer—while yet unknown—has extremely important consequences for regulating radiation exposure.
Summary
Nonionizing radiation is relatively low in energy and can be used as a heat source, whereas ionizing radiation, which is higher in energy, can penetrate biological tissues and is highly reactive. The effects of radiation on matter depend on the energy of the radiation. Nonionizing radiation is relatively low in energy, and the energy is transferred to matter in the form of heat. Ionizing radiation is relatively high in energy, and when it collides with an atom, it can completely remove an electron to form a positively charged ion that can damage biological tissues. Alpha particles do not penetrate very far into matter, whereas γ rays penetrate more deeply. Common units of radiation exposure, or dose, are the roentgen (R), the amount of energy absorbed by dry air, and the rad (radiation absorbed dose), the amount of radiation that produces 0.01 J of energy in 1 kg of matter. The rem (roentgen equivalent in man) measures the actual amount of tissue damage caused by a given amount of radiation. Natural sources of radiation include cosmic radiation, consisting of high-energy particles and γ rays emitted by the sun and other stars; cosmogenic radiation, which is produced by the interaction of cosmic rays with gases in the upper atmosphere; and terrestrial radiation, from radioactive elements present on primordial Earth and their decay products. The risks of ionizing radiation depend on the intensity of the radiation, the mode of exposure, and the duration of the exposure.


9.1: Contour Integrals
Recall that for a real function \(f(x)\) , the definite integral from \(x=a\) to \(x=b\) is the area under the curve between those two points. As discussed in Chapter 3, the integral can be expressed as a limit expression: we divide the interval into \(N\) segments of width \(\Delta x\) , take the sum of \(\Delta x\, f(x_n)\) , and go to the \(N \rightarrow \infty\) limit: \[\int_a^b dx\; f(x) \;=\; \lim_{N \rightarrow 0} \, \sum_{n=0}^{N} \Delta x\; f(x_n), \;\;\;\mathrm{where}\;\;\;x_n = a + n\Delta x, \;\;\Delta x = \frac{b-a}{N}.\] Now suppose \(f\) is a complex function of a complex variable. A straight-foward way to define the integral of \(f(z)\) is to adopt an analogous expression: \[\lim_{N \rightarrow 0} \, \sum_{n=0}^{N} \Delta z\; f(z_n)\] But there’s a conceptual snag: since \(f\) takes complex inputs, the values of \(z_n\) need not lie along the real line. In general, the complex numbers \(z_n\) form a set of points in the complex plane. To accommodate this, we can imagine chaining together a sequence of points \(z_1, z_2, \dots, z_N\) , separated by displacements \(\Delta z_1, \Delta z_2, \Delta z_3, \dots, \Delta z_{N-1}\) :
Then the sum we are interested in is \[\sum_{n=1}^{N-1} \Delta z_n\; f(z_n) \;=\; \Delta z_1\, f(z_1) + \Delta z_2\, f(z_2) + \cdots + \Delta z_{N-1}\, f(z_{N-1}).\] In the limit \(N \rightarrow \infty\) , each displacement \(\Delta z_{n}\) becomes infinitesimal, and the sequence of points \(\{z_1, z_2, \dots, z_N\}\) becomes a continuous trajectory in the complex plane (see Section 4.6). Such a trajectory is called a contour . Let us denote a given contour by an abstract symbol, such as \(\Gamma\) . Then the contour integral over \(\Gamma\) is defined as \[\int_\Gamma \, f(z)\, dz = \lim_{N \rightarrow \infty} \sum_{n=1}^{N-1} \Delta z_n\; f(z_n).\] The symbol \(\Gamma\) in the subscript of the integral sign indicates that the integral takes place over the contour \(\Gamma\) . When defining a contour integral, it is always necessary to specify which contour we are integrating over. This is analogous to specifying the end-points of the interval over which to perform a definite real integral. In the complex case, the integration variable \(z\) lies in a two-dimensional plane (the complex plane), not a line; therefore we cannot just specify two end-points, and must specify an entire contour.
Also, note that in defining a contour \(\Gamma\) we must specify not just a curve in the complex plane, but also the direction along which to traverse the curve. If we integrate along the same curve in the opposite direction, the value of the contour integral switches sign (this is similar to swapping the end-points of a definite real integral).
Note
A contour integral generally cannot be interpreted as the area under a curve, the way a definite real integral can. In particular, the contour should not be mistakenly interpreted the graph of the integrand! Always remember that in a contour integral, the integrand \(f(z)\) and the integration variable \(z\) are both complex numbers.
Moreover, the concept of an indefinite integral cannot be usefully generalized to the complex case.
Contour integral along a parametric curve
Simple contour integrals can be calculated by parameterizing the contour . Consider a contour integral \[\int_\Gamma \, dz \; f(z),\] where \(f\) is a complex function of a complex variable and \(\Gamma\) is a given contour. As discussed in Section 4.6, we can describe a trajectory in the complex plane by a complex function of a real variable, \(z(t)\) : \[\Gamma \equiv \Big\{z(t) \;\Big|\; t_1 < t < t_2\Big\}, \quad \mathrm{where}\;\; t \in \mathbb{R}, \,z(t) \in \mathbb{C}.\] The real numbers \(t_1\) and \(t_2\) specify two complex numbers, \(z(t_1)\) and \(z(t_2)\) , which are the end-points of the contour. The rest of the contour consists of the values of \(z(t)\) between those end-points. Provided we can parameterize \(\Gamma\) in such a manner, the complex displacement \(dz\) in the contour integral can be written as \[dz \rightarrow dt\, \frac{dz}{dt}.\] Then we can express the contour integral over \(\Gamma\) as a definite integral over \(t\) : \[\int_\Gamma dz\; f(z) = \int_{t_1}^{t_2} \; dt\; \frac{dz}{dt}\, f\big(z(t)\big).\] This can then be calculated using standard integration techniques. A simple example is given in the next section.
A contour integral over a circular arc
Let us use the method of parameterizing the contour to calculate the contour integral \[\int_{\Gamma[R, \theta_1,\theta_2]} dz\; z^n,\; n\in\mathbb{Z},\] where the trajectory \(\Gamma[R, \theta_1,\theta_2]\) consists of a counter-clockwise arc of radius \(R > 0\) , from the point \(z_1 = R e^{i\theta_1}\) to the point \(z_2 = R e^{i\theta_2}\) , as shown in the figure below:
We can parameterize the contour as follows: \[\Gamma[R, \theta_1,\theta_2] = \big\{z(\theta) \;\big|\; \theta_1 \le \theta \le \theta_2\big\}, \quad \mathrm{where}\;\, z(\theta) = R e^{i\theta}.\] Then the contour integral can be converted into an integral over the real parameter \(\theta\) : \[\begin{align} \int_{\Gamma[R, \theta_1,\theta_2]} dz\; z^n &= \int_{\theta_1}^{\theta_2} d\theta \; z^n \; \frac{dz}{d\theta} \\ &= \int_{\theta_1}^{\theta_2} d\theta \; \left( Re^{i\theta}\right)^n\, \left(iR\, e^{i\theta}\right) \\ &= i R^{n+1} \, \int_{\theta_1}^{\theta_2} d\theta \; e^{i(n+1)\theta}.\end{align}\] To proceed, there are two cases that we must treat separately. First, for \(n \ne -1\) , \[\int_{\theta_1}^{\theta_2} d\theta \; e^{i(n+1)\theta} = \left[\frac{e^{i(n+1)\theta}}{i(n+1)}\right]_{\theta_1}^{\theta_2} = \frac{e^{i(n+1)\theta_2} - e^{i(n+1)\theta_1}}{i(n+1)}.\] Second, we have the case \(n = -1\) . This cannot be handled by the above equations, since the factor of \(n + 1\) in the denominator would vanish. Instead, \[\int_{\theta_1}^{\theta_2} d\theta \; \left[e^{i(n+1)\theta} \right]_{n=-1} \; = \int_{\theta_1}^{\theta_2} d\theta \;=\; \theta_2 - \theta_1.\] Putting the two cases together, we arrive at the result \[\int_{\Gamma[\theta_1,\theta_2]} dz\; z^n = \left\{\begin{array}{ll}i(\theta_2 - \theta_1), & \mathrm{if}\;n = -1 \\ \displaystyle R^{n+1}\frac{e^{i(n+1)\theta_2} - e^{i(n+1)\theta_1}}{n+1},&\mathrm{if}\;n\ne -1.\end{array}\right. \label{arcresult}\] The case where \(\theta_2 = \theta_1 + 2\pi\) is of particular interest. Here, \(\Gamma\) forms a complete loop, and the result simplifies to \[\oint_{\Gamma} dz\; z^n = \left\{\begin{array}{ll}2\pi i, & \mathrm{if}\;n = -1 \\ \displaystyle 0,&\mathrm{if}\;n\ne -1,\end{array}\right. \label{simpleloop}\] which is independent of \(R\) as well as the choice of \(\theta_1\) and \(\theta_2\) . (Here, the special integration symbol \(\oint\) is used to indicate that the contour integral is taken over a loop.) Eq. \(\eqref{simpleloop}\) is a very important result that we will make ample use of later.
By the way, what if \(n\) is not an integer? In that case, the integrand \(z^n\) is a multi-valued operation (see Chapter 8), whereas the definition of a contour integral assumes the integrand is a well-defined function. To get around this problem, we can specify a branch cut and perform the contour integral with any of the branches of \(z^n\) (this is fine since the branches are well-defined functions). So long as the branch cut avoids intersecting with the contour \(\Gamma\) , the result \(\eqref{arcresult}\) remains valid. However, \(\Gamma\) cannot properly be taken along a complete loop, as that would entail crossing the branch cut.


1.14: Bravais flock
Space groups that are assigned to the same Bravais class belong to the same Bravais flock of space groups .
The introduction of the concept of Bravais flock is necessary in order to classify space groups on the basis of their Bravais type of lattice, independently from any accidental metric of the lattice.
An orthorhombic crystal may have accidentally a = b have a tetragonal lattice not because of symmetry restrictions but just by accident. If the Bravais types of lattices were used directly to classify space group, such a crystal would be go in another category with respect to an orthorhombic crystal without specialized metric. The concept of Bravais flock solves this ambiguity.
A space group G is assigned to a Bravais class on the basis of the corresponding point group P and the arithmetic crystal class associated with it:
- If the arithmetic crystal class of G is a Bravais class, then G is assigned to that Bravais class;
-
if the arithmetic crystal class of G is not a Bravais class, then the Bravais class to which G is assigned is obtained as follows:
- those Bravais classes are retained whose point group B is such that P is a subgroup of B;
- G is assigned to that Bravais class, among those selected above, for which the ratio of the order of the point groups of B and of P is minimal.
For example, a space group of type I 4 1 belongs to the arithmetic crystal class 4 I , to which two Bravais class can be associated, 4/ mmmI and \(m{\bar 3}mI\). The second condition uniquely assigns G to the Bravais flock of 4/ mmmI , despite the fact that the Bravais class of the lattice may be \(m{\bar 3}mI\) as a result of accidental symmetry.


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7.1
Physical Development and Health in Middle Childhood
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During middle childhood, growth rates slow down compared to earlier childhood, but children still increase in height and weight.
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Brain growth allows for improved sensory and motor skills, and faster reaction times and information processing abilities.
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Gross and fine motor skills continue to develop rapidly, allowing children to participate in organized sports activities, meet the demands of a school setting, and gain independence with improvements in self-care.
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Quality sleep should be a priority. A healthy diet and regular exercise also matter for general well-being as well as for cognitive and socioemotional development.
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Other health concerns during middle childhood include poor dental care, asthma, diabetes, accidents, and injuries.
7.2
Cognition in Middle Childhood
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Children reach a variety of cognitive milestones during middle childhood—such as problem solving, thinking logically and flexible, and language use—at different rates. Culture, family, and peers are important influences.
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Piaget believed that during middle childhood, children can use logic to solve concrete problems, such as conservation and seriation.
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The information processing approach to cognitive development focuses on the development of memory, attention, and executive function.
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Theory of mind becomes increasingly sophisticated as children gain more advanced awareness of how their thoughts and feelings differ from others.
7.3
Intelligence in Middle Childhood
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Intelligence is generally defined as the ability of an individual to adapt to the world around them.
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Intelligence is influenced by genetics as well as by environmental factors such as education, poverty, and nutrition.
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The general cognitive ability underlying specific cognitive skills is called general intelligence.
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Some psychologists have proposed that there are multiple types of intelligence.
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A measure of intelligence derived from intelligence tests is called the intelligence quotient (IQ). IQ scores are normally distributed, which means that most people have an IQ near the average (100), and many fewer have scores that are extremely high or extremely low.
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Variations in intelligence, such as intellectual disability and giftedness, can impact academic achievement.
7.4
Contexts: School and Learning Diversity in Middle Childhood
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Education takes place in both formal (schooling) and informal (home, extracurricular activities) settings. Formal education can include teacher-centered classrooms and student-centered classrooms.
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Teachers often use theories of education, including behaviorism and constructivism, to guide their students’ learning.
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Student academic achievement can be influenced by a variety of factors, including teacher expectancies and sociodemographic variables and resources.
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Learning disabilities are conditions that impact learning in specific academic areas and include dyslexia, dysgraphia and dyscalculia. One common neurodevelopmental disorder seen in children is attention-deficit/hyperactivity disorder (ADHD).
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Legislation in the United States provides children with disabilities with rights to a free and appropriate education.
7.5
Language in Middle Childhood
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During middle childhood, children have dramatic increases in vocabulary, knowledge of grammatical rules, and metalinguistic awareness.
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Children are learning how to change their speech in response to the needs of the listener. This is referred to as pragmatics.
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A major task during middle childhood is learning to read. There are two main approaches to teaching reading: the phonics-based approach and the whole language approach.
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Many children speak more than one language. Multilingualism has several positive cognitive and social benefits for development.
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English Language Learners (ELL) have special educational needs. Approaches to teaching ELL students include developmental bilingual programs, English as a Second Language programs, two-way immersion programs, and immersion programs.