THE LORENTZ TRANSFORMATION

In the preceding chapters I have shown you the kinematics of Special Relativity; that is, the part of Special Relativity that covers the relationships among space, time, and motion without reference to matter and its interactions (though some reference to dynamics did, in fact, creep in). Together we replaced Isaac Newton's static concept of space (an infinite array of points, all motionless relative to each other, through which bodies move) with Albert Einstein's dynamic concept (an infinite array of inertial frames, each comprising an infinite number of points that have no motion relative to each other, occupied and marked by bodies that are shifted from one frame to another by forces). It is upon this subtle change in worldview that Relativity is founded, the concept of the inertial frame of reference being as necessary to it and as fundamental to it as the concept of the straight line is to plane geometry.

Beginning with the proposition, derived from the observation that we exist, that the Universe cannot move, we deduced the conservation laws pertaining to linear and angular momenta. Those conservation laws, by way of the finite-value theorem, necessitate that space be finite and unbounded, which requires, in turn, that space be curved. The curvature of space raises the possibility of violating the conservation laws and thus necessitates that space and time be so shaped that every observer must conclude that they are at the center of an expanding space. That result led us to proclaim

COSMIC THEOREM 1: The results of any given experiment or observation and the laws of physics derived from them are the same for all experimenters or observers, regardless of how those experimenters or observers are positioned, oriented, or moving relative to each other.

COSMIC THEOREM 2: Any phenomenon that moves at the same speed at which the boundary of space moves passes all observers at that same speed, regardless of how those observers are positioned, oriented, or moving relative to each other.

These two theorems are the postulates of Relativity, although stated just a little more broadly than Einstein stated them. The speed at which the boundary of space moves is understood to be equal to the speed of light, which is 299,792.458 kilometers per second or 186,234.709 miles per second.

Using a series of imaginary experiments in a fantasy world, in which the speed of light is merely 100 miles per hour, we parlayed those theorems into five rules that govern the relationship between space and time in measurements made of any pair of events by observers who are moving relative to each other. To that set of rules we can add a "zeroth" rule for calculating the proportionality factor, the Lorentz factor, that appears in the rules. Thus, we have

LORENTZ RULE 0:(the Lorentz factor) If you are given two inertial frames of reference, take the velocity between the two inertial frames, square it, divide the square by the square of the speed of light, subtract the resulting fraction from the number one, extract the square root of the result, and divide that square root into the number one.

LORENTZ RULE 1:(invariance of lateral distances) In two inertial frames in relative motion, the distance between any two given points measured in a direction perpendicular to the relative motion will be the same for observers in both frames.

LORENTZ RULE 2:(time dilation) The time that an observer measures elapsed between two events that are touched by a moving clock is equal to the time interval measured by the moving clock multiplied by the Lorentz factor between the inertial frames occupied by the moving clock and the observer.

LORENTZ RULE 2B:(Lorentz-Fitzgerald contraction) An object moving relative to some observer appears to that observer to be shorter in the direction of relative motion than it does to an observer at rest in its inertial frame, the moving length being equal to the stationary length divided by the Lorentz factor between the two inertial frames.

LORENTZ RULE 3:(temporal offset) If two clocks are synchronized in the inertial frame in which they are at rest, then in any inertial frame in which they are moving the following clock will be found to be running "fast" relative to the leading clock by an interval equal to the product of the at-rest distance between the clocks (measured only in the direction of motion)and the speed at which the clocks are moving divided by the square of the speed of light.

LORENTZ RULE 4:(length dilation) If the distance between two events is measured in the frame in which they occur simultaneously and if that distance is measured parallel to the direction of motion of some other inertial frame, then the distance between the two events in the second frame will be equal to the distance measured in the first frame multiplied by the Lorentz factor between the two frames.

Using algebraic representations of those rules, physicists combine them in a different way and write them out as the four equations of the Lorentz Transformation. If you and I occupy two different inertial frames and measure the spatial and temporal intervals between the same two events, then those equations give us the recipes for combining the speed of light, the velocity between our inertial frames, and the measurements that you make and, by means of appropriate arithmetic, cooking up numbers equal to what I measure between those events. Because space has three dimensions, we superimpose upon it an artificial grid defined by three mutually perpendicular straight lines, which we label with the letters ex, wye, and zee. Conventionally, we so construct the grid that the velocity between the two inertial frames under consideration is oriented entirely in the x-direction. With that in mind and with the proviso that in my inertial frame you and your frame are moving in the direction in which the numbers on the mileposts set along the x-axis are increasing, we now have the four equations of the Lorentz Transformation:

LORENTZ TRANSFORMATION EQUATION 1: To calculate my measurement in the x-direction of the distance between two events take your measurement in the x-direction of the distance between the events, add the product of the velocity between our frames and the time that you measure between the events (by subtracting the time of the following event from the time of the leading event as measured on clocks synchronized in your frame), and multiply the sum by the Lorentz factor between our frames. This recipe combines Lorentz Rules 2 and 4 with Galilean Rule 1.

LORENTZ TRANSFORMATION EQUATION 2: My measurement in the y-direction of the distance between the two events is equal to your measurement in the y-direction of the distance between the two events. This is simply Lorentz Rule 1.

LORENTZ TRANSFORMATION EQUATION 3: My measurement in the z-direction of the distance between the two events is equal to your measurement in the z-direction of the distance between the two events. Again, this is simply Lorentz Rule 1.

LORENTZ TRANSFORMATION EQUATION 4: To calculate my measurement of the time interval between the events take your measurement in the x-direction of the distance between the events, multiply it by the speed between our frames, divide that product by the square of the speed of light, then add to that number the time that you measure between the events (again, by subtracting the time of the following event from the time of the leading event as measured on clocks synchronized in your frame), and multiply the result by the Lorentz factor. This recipe combines Lorentz Rules 2 and 3.

Lorentz Rule 2B does not appear in those equations directly because, although it is part of the deductive chain that leads to the Lorentz Transformation, it is not strictly a part of it. It is, rather, a consequence of applying the transformation equations to a particular situation. That situation is one in which I wish to measure the length of an object that's at rest in your frame by determining the distance between the events defined by the simultaneous emission of pulses of light from the ends of the object. The time that I measure, zero, is then put into the fourth equation and the equation solved for the time that you would measure, which turns out to be equal to the negative of the temporal offset between the clocks that you have placed at the ends of the object. When that result is incorporated into the first equation, the velocities are so combined that the multiplication of the length that you measure by the Lorentz factor between our frames is converted into a division of the length by that Lorentz factor, producing the description of a contraction of the object.

The first and fourth equations of the Lorentz Transformation can also be used to answer a question that often comes up in discussions of Relativity. In fact, I remember the exact content of that question as I put it to the school's physics teacher when I was in my junior year at Redwood High School: if a rocketship flies away from Earth at three-quarters of lightspeed and the crew of the rocketship then launches a small rocket at three-quarters of lightspeed, wouldn't that small rocket be moving away from Earth at one-and-a-half times the speed of light? I don't recall what Mr. Crane's response was, but I know now why the answer to the question must be no. In order to determine the velocity of the small rocket we must measure the distance and duration between two events at which the rocket is present and calculate their ratio. The events might, for example, be the rocket's passing Mars and the rocket's subsequently passing Jupiter.

If you are on the rocketship and I am on Earth, then the velocity that I calculate for the small rocket will be related to the velocity that you calculate for it by dividing the first equation of the Lorentz Transformation by the fourth equation. The result will resemble a fraction with the recipe of the first equation as its numerator and the recipe of the fourth equation as its denominator. Both of those recipes include the multiplication of everything else in the recipe by the Lorentz factor, so the Lorentz factor appears in our fraction as a common factor in the numerator and the denominator, which means, according to the arithmetic of fractions, that it can be canceled out of the fraction altogether. Further, we can represent the distance that you measure as the product of the velocity of the small rocket in your frame and the duration that you measure between its passing of the two planets. In making that representation we convert both of the recipes in our fraction into forms in which the duration that you measure is a common factor, so we can cancel that duration out of the fraction as well. What we have left is the following recipe for calculating the velocity of the small rocket in my frame:

1) Prepare a denominator by multiplying the velocity of the small rocket in your frame by the velocity between our frames, dividing the result by the square of the speed of light, and adding the number one to the result; and

2) Add together the velocity of the small rocket in your frame and the velocity between our frames and then divide the sum by the prepared denominator.

That formula is a description of how Reality stymies any effort to make a body move faster than light. When the summed velocities are small, the denominator is effectively indistinguishable from the number one and the compound velocity can be taken to be the simple sum: if I walk at three miles per hour on an airport conveyor that's moving at three miles per hour, we will all agree that I am progressing through the terminal at six miles per hour. But when the summed velocities are a substantial fraction of the speed of light, the denominator grows to keep their sum from exceeding that limit: thus, in my example above of a rocketship flying away from Earth at three-quarters of lightspeed and launching a small rocket at three-quarters of lightspeed the formula tells us that the small rocket will be flying away from Earth at ninety-six percent (twenty-four twenty-fifths) of lightspeed.

There's one more thing I want to do with the Lorentz Transformation before we get into relativistic dynamics. This is not, in my opinion, something that will significantly improve your understanding of Relativity. In fact, I believe that this feature of Relativity may have actually been more of a distraction than a help in the development of the complete theory of Relativity.

Multiply the fourth equation by the speed of light (which converts duration units into distance units), square it, then subtract the square from the sum of the squares of the other three equations. If the phrase "sum of the squares" seems familiar, it's because it's the key phrase in the conventional statement of the Pythagorean theorem. What we have just obtained from the procedure above is a version of what Pythagoras deduced over twenty-five centuries ago, a version known as Minkowski's theorem. The equation that corresponds to the theorem is called a metric equation because it reveals the "metric" or "measure" of the space in which measurements are made.

Consider the case of two right triangles that share a common hypotenuse. The sum of the squares of the sides of one triangle must equal the sum of the squares of the sides of the other triangle. As an example of the importance of that statement, consider how we might calculate the straight-line distance between the intersection of Sepulveda and National Boulevards and the intersection of Westwood and Pico Boulevards in West Los Angeles. You might measure along National Boulevard to Westwood Boulevard and thence along Westwood to Pico, square the measurements, add the squares, and extract the square root. I, on the other hand, might measure due east from Sepulveda and National to a point and thence due north to Westwood and Pico. Our measurements will differ from each other, but the distance that we calculate will be the same. We can devise transformation equations that will convert your measurements into mine (or vice versa) and we can test the validity of those equations by squaring them and adding them together: they are only valid transformations if that procedure yields the Pythagorean theorem as it was stated above. And that Pythagorean theorem, in the form of a metric equation, has the simple form of a sum of squares because our triangles are laid out on a surface that is more or less flat (albeit slightly tilted).

In a similar sense we can say that space and time as represented in Minkowski's theorem are also flat because the metric equation is also a simple sum and difference of squares. (The metric equations of General Relativity, in contrast, modify the squares with coefficients that represent a warping of space and time away from flatness.) The fact that his metric equation is analogous to the algebraic representation of the Pythagorean theorem led Hermann Minkowski to declare in 1908 that Special Relativity is analogous to Euclidean geometry. Just as Euclid's geometry, which includes Pythagoras' theorem, is worked out on flat planes, so Special Relativity must be worked out on a flat continuum analogous to a plane. Minkowski called that continuum "spacetime" and got so carried away in his enthusiasm that he predicted that "space and time by themselves are doomed to fade away...." That prediction has not come true to Reality, but that's not astonishing: there's nothing in Relativity that obliges us to give up our notion that space and time are separate things, so our experience of space and time as different things overrides any mystical impulse to see them unified.

There is another, more picturesque, if somewhat unenlightening, way in which we can represent a metric equation. Draw a straight line and then construct on that line all of the right triangles that have that line as their hypotenuse. The line that connects the free vertices of those triangles is a circle (Euclid proves a theorem to that effect). To make the equivalent construction of a diagram that represents the Minkowski metric you must select one direction on a piece of graph paper to be the x-direction and the other to be the time-direction (the y- and z-directions are usually suppressed in discussions of Relativity), pick a point to represent one event, and then plot all of the points that represent the distance and duration from that first point to a second event as they would be measured by observers in all the inertial frames moving in the x-direction. Those points will trace out an hyperbola, which can be seen as the line connecting the free vertices of a set of obtuse triangles whose common base represents the time interval between the two events in the inertial frame in which the events occur in the same place. It's true that the circle and the hyperbola, being conic sections, are related to each other, but they are also quite different from each other, one being closed on itself and the other being open to infinity. Likewise, Euclidean geometry and Special Relativity differ significantly from each other and an overemphasis on their similarities is not a good thing.

Why do I make such a claim? In 1907, the year before Minkowski presented his geometric interpretation of Special Relativity, Einstein conceived the basic idea behind his theory of General Relativity. He did so in response to reading about an incident that could have inspired a long list of imaginary experiments of the kind he had used to work out the features of Special Relativity. Those imaginary experiments would have led him to create a version of General Relativity that uses the same concepts and mathematics (essentially first-year calculus) that he used in Special Relativity, a version much like the one that I have been working out for the past eighteen years (as of May 2005). Instead, he was inspired by Minkowski's geometric vision and developed a version based on the use of non-Euclidean geometry and tensor calculus, thereby making it the most difficult theory in physics.

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