The Lorentz Transformation
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In the preceding essays I have shown you the kinematics of Special Relativity; that is, the part of Special Relativity that pertains purely to 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 are motionless relative to each other, occupied and marked by bodies that are shifted from one frame to another by forces). Upon this subtle change in worldview Albert Einstein founded Relativity, 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 comprise the postulates of Relativity, although stated more broadly than Einstein stated them. The speed at which the boundary of space moves, in our current understanding, corresponds to the speed of light, which flies 299,792.458 kilometers per second or 186,234.709 miles per second.
Using a series of imaginary experiments in a fantasy world, one in which light flies merely 100 miles per hour, we parlayed those theorems into five rules. Each of those rules reveals one aspect of the relationship between space and time that observers who are moving relative to each other find in measurements that they make of any pair of events. 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 consider 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. If we represent the Lorentz factor with the letter L and the relative velocity between the frames with the letter V, then we have
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. We orient the relative motion in the x-direction, so we also orient both the y-direction and the z-direction perpendicular to the relative motion. If an observer who is moving relative to us measures distances y and z, then we will measure distances Y and Z in accordance with
LORENTZ RULE 2:(time dilation) If a moving clock touches two events, then the time that any observer measures elapsed between those two events equals 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. If the moving clock measures the interval represented by t, then the observer will measure an interval represented by T in accordance with
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. If the object has a length of x as measured by observers moving with it, then we will observe a length X in accordance with
LORENTZ RULE 3:(temporal offset) If someone synchronizes two clocks in the inertial frame in which they lie at rest, then in any inertial frame in which they move the following clock will run "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 move divided by the square of the speed of light. Representing that offset by )t, we have
LORENTZ RULE 4:(length dilation) If someone measures the distance between two events in the frame in which those events occur simultaneously and if they measure that distance parallel to the direction of motion of some other inertial frame, then the distance between the two events in the second frame (which I take as our frame in this case) will equal the distance measured in the first frame multiplied by the Lorentz factor between the two frames. To distinguish this rule from Lorentz Rule 2B we add apostrophes to X and x to give us X' and x' (read "eks prime") that are related through Lorentz Rule 4 in accordance with
Using the 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 have superimposed upon it an artificial grid defined by three mutually perpendicular straight lines, which we label with the letters ex, wye, and zee. Following convention, we so construct the grid that the velocity between the two inertial frames under consideration points entirely in the x-direction. In these equations I will use upper-case letters to represent the measurements of distance and duration made in my frame and lower-case letters to represent the measurements of distance and duration made in your frame. With that in mind and with the proviso that in my inertial frame you and your frame move in the direction in which the numbers on the mileposts set along the x-axis increase, 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, subtract 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 from the essay on inertial frames, which rule is expressed algebraically in Equation 1 of that essay. We thus have
LORENTZ TRANSFORMATION EQUATION 2: My measurement in the y-direction of the distance between the two events equals your measurement in the y-direction of the distance between the two events. This is simply Lorentz Rule 1 and the equation describing it is simply Equation 2.
LORENTZ TRANSFORMATION EQUATION 3: My measurement in the z-direction of the distance between the two events equals your measurement in the z-direction of the distance between the two events. Again, this is simply Lorentz Rule 1 and the equation describing it is simply Equation 3.
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 subtract that number from 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. The resulting equation is
Lorentz Rule 2B does not appear in those equations directly because, although we made it part of the deductive chain that leads to the Lorentz Transformation, it is not strictly a part of it. It comes to us, rather, as a consequence of applying the transformation equations to a particular situation. In that particular situation I wish to measure the length of an object that lies at rest in your frame by determining the distance between two events defined by the simultaneous emission of pulses of light from the ends of the object. We put the time that I measure, T = 0, into the fourth equation and solve the equation for the time that you would measure, which turns out to equal the positive of the temporal offset between the clocks that you have placed at the ends of the object. When we incorporate that result into the first equation, the velocities so combine that the multiplication of the length that you measure by the Lorentz factor between our frames becomes a division of the length by that Lorentz factor, producing the description of a contraction of the object.
Now that we have done it the hard way, I'll show you a simpler and easier way to deduce the whole Lorentz Transformation directly from the perpendicular distances theorem and the postulates by means of a modified Feynman clock. In the conventional deduction we used the Feynman clock only in the deduction of the time dilation theorem, but if we turn the clock ninety degrees, we can do much more with it.
The modified Feynman clock consists of a laser mounted in the center of a long, straight tube that has mirrors and photocells mounted side by side at both of its ends. This clock differs from the ones that we have used so far in that it has time counters at both its ends. In this clock we must so position the laser that the two pulses that it emits reach the photocells simultaneously in any inertial frame in which the clock does not move. Thus the time counters at both ends of the clock will always display the same numbers: that, at least, is the intent. We will then discern the structure of space and time in the way in which that intent fails.
Let's imagine that we are sitting on a bench by the little whistle-stop station of Caballero Blanco on the X-Axis Line of the Huffenpuff Railroad and that, after setting our new Feynman clock, we sit and admire its operation. Soon a track speeder whirs by at some speed v and we notice that it carryies two of the new Feynman clocks, one oriented vertically, parallel to our Y-axis, and the other lying down, parallel to our X-axis. Both clocks are running, so we take the appropriate series of photographs and then wave gaily to the little blond girl in the blue dress and white apron who is driving the speeder like the proverbial bat out of The Hot Place. When we develop the photographs, we analyze them as we did before.
Because we hold our clock upright, we represent the length of its tube, from photocell to photocell, by the letter Y. We thus represent the length of the tube of the vertical clock on the speeder by the letter y and we know already that
We also know that the time required for a pulse to go from one laser to the opposite photocell equals the round-trip distance in the tube divided by the speed of light; that is,
Of course we have a similar equation in the track speeder's frame for the motion of the pulses in its vertical clock. Equivalently, we also have an equation describing the distance that a pulse travels in the clock in the time it takes to go from one laser to the opposite photocell; that equation gives the distance as the product of the elapsed time and the speed of light,
using quantities y and t measured in the speeder's frame.
As before, we notice that the pulses
traveling in the vertical clock on the speeder trace out a sawtooth pattern
(actually a double sawtooth pattern because we have two pulses traveling in
opposite directions at any one time). And again we can describe the distance,
measured in the X-direction, between the emission of a pulse and its arrival at
the opposite mirror with the equation X = vT. Again we take a portion of the
sawtooth for convenience, draw the appropriate right triangle, and measure the
Y-ward distance between the emission and absorption of the pulse as Y, so the
distance that the pulse travels comes from the Pythagorean theorem as
But the second postulate of Relativity tells us that the pulse travels the distance L at the same speed of light at which it crosses all distances, so we have L = cT to substitute into the equation for L. We can also replace Y by y and then replace y by its equivalent from Equation 14.11 and thereby transform Equation 12 into
We subtract v2T2 from both sides of that equation, divide the whole equation by cee-squared, and extract the square root of the equation to obtain the time dilation theorem,
The clock on the speeder shows time elapsing at a rate slower than the rate at which time elapses on our clock; that is, it shows fewer seconds elapsing between any two events than our clock shows.
Now we examine our photographs of the clock that lies horizontally on the speeder with its tube oriented in the X-direction. For convenience we have labeled the forward photocell as P-1 and the aftward photocell as P-2. In the speeder's frame of reference we describe the time a pulse takes to go from the laser at P-2 to P-1 as
in which we have replaced the letter eks, representing the distance between the laser and the photocell, by wye, which represents the measured distance between the laser and the photocell in the speeder's frame (which is the same for both clocks). The corresponding time for a pulse to go from the laser at P-1 to P-2 is
In our frame the distance that a pulse must travel from the laser at P-2 to P-1 equals X+vT1 (the length of the optical path in the clock plus the distance that the clock moves while the pulse is in flight), so we have the time that the pulse takes to cross that distance (at the speed of light, of course) as
In similar manner we can determine the time T2 that a pulse requires to go from the laser at P-1 to P-2 as
We can solve both of those equations for their respective times and then give the results a common denominator;
A little thought should be sufficient to convince us that the sum T3 = T1+T2 simply gives us the time that a pulse needs to go from one laser to the opposite laser and then back to the first laser. But in the speeder's frame the time required by the pulse to make that traverse is simply t = 2x/c = 2y/c, which we then convert by time dilation to
Adding Equations 19 and 20 gives us
Logic makes those two equations mutually true to Reality if and only if it also makes
true to Reality. This describes, of course, the Lorentz-Fitzgerald contraction.
Alternatively, if we subtract Equation 20 from Equation 19, we discover that in our frame the time counter associated with P-2 differs from the time counter associated with P-1 by
To see what that means suppose that two events occur in such a way that the first one occurs at a point whose X-component matches that of P-1 when the event occurs and that the second event occurs at a point whose X-component matches that of P-2 when the event occurs. In our frame the counter associated with P-2 runs "fast" relative to the counter associated with P-1; that is, it displays a higher number, representing a later time. In essence P-2 lies a certain interval in the future relative to P-1, so as we see it an observer in the speeder's frame measures the time of the second event with a clock that runs fast relative to the clock that they have used to measure the time of the first event. In order to calculate the time difference between the events in our frame from the measured time difference t and the distance x' = 2x measured in the speeder's frame, we must subtract the temporal offset of Equation 24 from the dilated time to obtain
That coincides perfectly with the time
equation of the Lorentz Transformation, Equation 9.
Finally we want to calculate the distance between the two events as we might measure it in our frame from the equivalent measurements made in the speeder's frame. Because we have assumed that the event occurring adjacent to P-1 occurs before the event occurring adjacent to P-2, the distance between them must equal the distance between P-1 and P-2 minus the distance that the clock moves in the time interval between the events. We have, then,
If we give all the terms on the right side of the equality sign a common denominator and carry out the subtraction of the terms involving x'v2/c2, we obtain
With that equation, which coincides with
Equation 8, we complete this derivation of the Lorentz Transformation.
This little derivation doesn't add
anything new to Relativity, but it does have the advantage of making the process
of deriving the Lorentz Transformation more transparent. Where the vertically
oriented clock shows us time dilation, the horizontally oriented clock
complements it by giving us the temporal offset, the Lorentz-Fitzgerald
contraction, and the dilation of space. It does not really differ all that much
from the conventional derivation and like the conventional derivation it
emanates directly from Einstein's postulates.
We can also use the first and fourth
equations of the Lorentz Transformation 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 how Mr. Crane responded, but I know now why we must answer no to the
question. In order to determine the velocity of the small rocket we must measure
the distance and duration between two events that occur arbitrarily close to the
rocket and calculate their ratio. Such events might, for example, comprise the
rocket's crossing the orbit of Mars and the rocket's subsequently crossing the
orbit of Jupiter.
If you make your measurements on board the rocketship (moving away from Earth with speed V in the positive x-direction, the velocity between our respective inertial frames) and I make the corresponding measurements from Earth, then we will relate the velocity that I calculate for the small rocket (represented by lower-case double yu) to the velocity that you calculate for it (represented by lower-case vee) 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 we can cancel it out of the fraction altogether. We then obtain
But x/t = v, so we have, after
substitution and then cancellation of the tees,
Can we find anything in that equation to
assure us that w can never exceed the speed of light? To answer that question
I'll start by multiplying the numerator and the denominator of the formula by
the square of lightspeed, in essence multiplying and dividing both sides of the
equation by the same number, which leaves it unchanged. We have then
That equation tells us that w will not
exceed c if the inequality
holds true for all allowed values of v and V.
We can manipulate an inequality just as we
manipulate an equation, so I start by multiplying both sides by the denominator
in Inequality 1:
I then subtract Vc and vV from both sides
of the inequality sign and obtain
I then factor both sides of the inequality
and divide both sides by c-V to obtain
Thus we prove and verify the proposition
that w never exceeds c so long as v never exceeds c, which we assumed as a
Equation 29 thus describes how Reality
stymies any effort to make a body move faster than light. When the summed
velocities yield a number much smaller than the speed of light, then the
denominator comes close to the number one and we can take the compound velocity
to equal the simple sum: if I walk at three miles per hour on an airport
conveyor that moves at three miles per hour, we will all agree that I progress
through the terminal at six miles per hour. But when summing the velocities
yields a substantial fraction of the speed of light, then 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 fly away from Earth at ninety-six percent (twenty-four
twenty-fifths) of lightspeed.
The final speed of the second stage is still less than the speed of light. It doesn't matter what velocities we use in Equation 29 (as long as they are all less than the speed of light, which is what we would have to begin with in any case) or how many times we repeat the procedure that led us to Equation 28 (that is, no matter how many stages we give our rocket), the result will always come out less than the speed of light. Equation 29, then, gives us one mathematical statement telling us that nothing can fly faster than light, at least by direct addition of velocities (which is what straightforward acceleration is). If we want to fly faster than light, as the characters in "Star Trek" do, we will have to find a way to fly outside our space and time, but that's a subject for a different book.
But there's one other possibility that may come to mind, a way in which we might fool Nature into letting us go faster than light. Perhaps all we need to do is get a spaceship up to a speed close to the speed of light along the x-axis and then accelerate in the y-direction. That possibility, too, will go away and it will do so for us when we consider the Rindler-Shaw paradox.
Is Mathematics True To
In our analysis of Equation 29 above we
used a series of inequalities to show that no combination of subluminal
velocities can ever add up to a superluminal velocity. In carrying out that
analysis I made the assumption that we can treat inequalities as we treat
equations, applying Euclid's second and third common notions and their
extensions to create a solution, which isolates one variable on one side of the
equality/inequality sign and all of the other variables and constants in the
original formulae on the other side. Our progression from Inequality 1 to
Inequality 4 provides a simple example of that process.
But I could have worked that progression
another way, one that calls my assumption into serious question. Let's start
again with Inequality 1:
Again multiply both sides by the
denominator to get
Now subtract vc and c2 from both sides to
Factor that inequality and divide both
sides by their common factor (V-c) and we end up with
Using perfectly legitimate algebraic
operations, I have thus obtained from Inequality 1 two mutually exclusive
results. Can I call "reductio ad absurdum" or have I missed something? Have I
found an incompatibility between logic and mathematics (highly unlikely) or have
I made a error somewhere in my manipulation (almost certainly)?
I assumed that I could treat an inequality just like an equation. In so doing I missed a subtle, but important proviso and had to spend a full day in figuring out what I had done wrong. We can, indeed, apply the usual techniques of algebra to "solve" an inequality, but we must acknowledge a basic difference between an equality and an inequality when we do so. In particular, in this case, we must note that changing the algebraic sign of our formulae has no effect upon an equality, but it reverses an inequality (for example, we know that 3<5, but that -3>-5). When we divided Inequality 5 by V-c we divided it by a negative number and thus changed the algebraic sign. So I should have made the inequality sign in Inequality 6 point from left to right, rather than right to left (as it had done in the preceding inequalities).
So, to answer the question with which I titled this section, mathematics does indeed conform truly to logic, at least as far as I have used it in these derivations.
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