Lorentz Transformation from Light Alone
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We know that space and time consist of an infinite ordered set of points and instants. In order to describe that set mathematically we assign quartets of real numbers to the points and instants in a way that matches the order of those points and instants. The set of all points that remain motionless relative to each other constitutes a unique inertial frame of reference if the points do not suffer any acceleration. We use three coordinates (three of the numbers assigned to the point) to locate that point relative to the others. The incrementing of the fourth coordinate yields the elapse of time in that frame.
Imagine two inertial frames of reference. Establish in each frame a coordinate grid in four dimensions by laying out a Cartesian spatial grid marked by an array of rulers and marking the temporal dimension by an array of synchronized clocks. Specify that the axes of one frame’s grid lie parallel to the corresponding axes of the other frame’s grid and that the frames move, one relative to the other, in the x-direction only. Observers in one frame (the U-frame) use upper-case letters to represent the results of measurements that they make and observers in the other frame (the L-frame) use lower-case letters to represent the results of measurements that they make.
In accordance with theorems already proven and verified, we assert that light travels at the same speed (c=299,792.458 kilometers per second) in both frames. That fact means that an instantaneous flash of light emanating from a point source forms a spherical shell that expands away from a point that remains stationary for whomever observes the light, regardless of how that observer moves relative to other observers or to the source of the light. Algebraically we express that fact through the statement that
for all observers. We then assert the proposition that this fact alone determines the form of the geometry of space and time as expressed through transformation equations that relate measurements made in one inertial frame to the corresponding measurements made in another frame. Thus from the fundamental properties of light alone we seek to deduce those transformation equations, the four equations of the Lorentz Transformation. To that end we need to deduce several phenomena that the observers in the two frames will infer from their communications.
Imagine that a team of observers occupy the U-frame at or near the point where the Y-axis crosses the X-axis and the Z-axis (the point that we call the origin of the coordinate grid). A powerful flashlamp lies on the positive Y-axis a very large distance from the grid’s origin and at a certain instant emits an instantaneous pulse of light in all directions as a sphere of light. Part of that light moves down the Y-axis at the speed c=dY/dT. By the time the light reaches the U-frame observers at the origin the light sphere’s radius of curvature has become so large that, over the distances that our observers typically measure, it has become effectively indistinguishable from a perfectly flat plane of light. That sheet of light illuminates the part of the X-Z plane on which the U-frame observers conduct their measurements over a brief interval essentially indistinguishable from a single instant.
If the L-frame moves through the U-frame in the negative X-direction at the speed v (as measured from the L-frame), a team of observers occupying that frame will see the light emanating from a stationary point that lies some distance from them in both the x-direction and the y-direction. That fact necessitates that the light received by the L-frame observers move in a direction that makes a nonzero angle with the y-axis. Because the light moves in a direction perpendicular to the plane tangent to the light sphere, the almost-plane sheet of light comes to the L-frame observers tilted out of the x-z plane, the parts of the sheet ahead of the observers (assuming that they face in the direction in which the L-frame moves through the U-frame) striking the x-axis before the parts behind the observers.
To calculate the angle at which the sheet of light crosses their x-axis the L-frame observers exploit the Pythagorean Theorem. At the instant that the flashlamp emits its pulse the distance between it and the observers has components in the x-direction and in the y-direction, components that form the sides of a right triangle. The hypotenuse of that triangle lies in the same orientation as does the velocity vector representing the motion of the sheet of light that the observers detect, so we have a similar right triangle whose sides represent the components of the light’s velocity. The angleθ that the motion of the light makes with the y-axis, which equals the angle at which the sheet of light crosses the x-axis, conforms to the relation
Suppose that the U-frame observers have set up a series of mutually synchronized clocks on the X-axis. A photograph taken by those observers when the sheet of light strikes the clocks will show all of the clocks displaying exactly the same time. On the other hand, the L-frame observers won’t make a single photograph, but will make a series of them, making a short little movie, much as Eadweard Muybridge did in 1878 of a galloping horse. When the observers project that short film, we will see the clocks illuminated one after another, but all nonetheless displaying the same time. We infer that the interval between the illumination of two successive clocks indicates the existence of a temporal offset that the observers would measure between clocks synchronized in the U-frame and clocks synchronized in the L-frame.
The L-frame observers pick two of the U-frame clocks and measure the time interval between the sheet of light illuminating one and then the other asΔt. In that elapse of time the light moves a distance cΔt. The x-ward distance between the locations that the clocks occupy at the instants when the light illuminates them consists of the instantaneous distance x between the clocks plus the distance vΔt that the second clock moves between the two illumination events. That distance equals the length of the hypotenuse of a right triangle whose short side has a length equal to cΔt; thus, the L-frame observer can write
Equating the right side of that equation to the right side of Equation 2 yields
which we can solve forΔt, getting
in which we have the square of the Lorentz factor between the L-frame and the U-frame as
Thus we obtain a description of the temporal offset between the clocks.
Imagine an emitter sending out a pulse of light that expands as a spherical shell of illumination. At a time T after the emission of the light in the U-frame the surface area A of the expanding light balloon conforms to
in which r represents the radius of the spherical shell occupied by the light. At that same time T the light illuminates a stationary clock separated from the emitter only in the Y-direction and lying on our two inertial frames’ common x-axis.
In the moving frame, the L-frame, the light illuminates the same clock at a time t. At that instant the light balloon has a surface area A’ such that
We know that equation stands true to Reality because when the tilted light plane illuminates the moving frame’s clock part of the light balloon has already crossed the x-axis and propagated further. Simple geometry then lets us calculate the radius r’=ct in terms of the radius r in Equation 7 and the relative velocity between the frames.
In Equation 8 the only variables consist of the velocity of the stationary frame through the moving frame and the times measured in both frames between the emission of the pulse and the illumination of a clock at rest in the stationary frame. That fact enables us to solve the equation with variables independent of the other variables involved in our description of the light pulse and its circumstances. We thus get
which encodes the phenomenon of time dilation. That fact means that for any observer a moving clock ticks off time more slowly than does a clock at rest relative to the observer.
Invariant Lateral Distance
We want to prove and verify that dY=dy (and that dZ=dz) and to do so without using the usual imaginary mechanical experiment and the Principle of Relativity. In the L-frame we have the y-ward component of the speed of light as
We also have c=dY/dT in the U-frame, so we can rewrite that equation as
Making the substitution dt=γdT and multiplying both sides by dt transforms that equation into
which represents the invariance of distances measured perpendicular to the direction of relative motion.
If we have two events occurring at the same point in the U-frame and separated from each other by the interval dT, then in the L-frame the events occur at different points separated from each other by a distance dx=vdt=vγdT, the distance the U-frame moves through the L-frame in the time interval dt. Thus, between any two events separated from each other in time we expect to observe a spatial offset, analogous to the temporal offset deduced above.
Imagine that on the surface of a light balloon there exist two bright spots a minuscule distance dX apart. Those spots strike the X-axis simultaneously and thereby mark off a line segment dX long. We also specify that dX is extremely small compared to the distance the light has traveled since it was emitted; thus, over relatively long distances the bright spots follow paths that differ negligibly from parallel lines.
In the L-frame both of those paths tilt to the same degree, so when the wave front between the bright spots crosses the x-axis the distance between the spots differs negligibly from dX. If we posit the line segment running across the wavefront between the spots as one side of a right triangle, then we have the segment of the x-axis between the points where the spots strike that axis as the hypotenuse of that triangle. We calculate the length of that hypotenuse as
That equation tells us that an object that has a certain length in the X-direction in the U-frame will extend over a greater x-ward length in the L-frame. But one of the other phenomena that we have deduced will alter that fact and change what our observers see.
If we want to measure the length of an object, especially an object moving in the x-direction, we must measure the distance between two events that occur simultaneously at the ends of the object. The placement of the points of a calipers against the ends of the object would work nicely for that purpose. If the object remains stationary in our frame, then the clocks on the ends of the object and on the points of the calipers all show the same time, so we can put the calipers against the object and then use a ruler to measure the distance dX between the points. If we occupy a frame in which the object moves in the x-direction at some velocity v, then we must us a slightly different procedure.
In that case we must place the calipers’ points as the object moves past us. Assuming that we can achieve that placement, we measure the length of the object as dx. Again the clocks on the calipers’ points display the same time, so we have measured the relative positions of the object’s endpoints simultaneously, as required.
But on the object itself the clocks on the endpoints, synchronized in the frame in which the object lies at rest, do not appear synchronized (in accordance with Equation 5). That temporal offset combined with the object’s motion affects the distance between the events that we use to determine the object’s length. In fact, the distance that we measure conforms to the statement that
in which dX represents the length of the object measured at rest and dT represents the interval that elapses between the two events as the times would be measured on the object’s clocks. The second term on the right side of that equation represents a spatial offset between the events, analogous to the temporal offset in the time equation,
In the above case we have the interval between the clocks on the calipers’ points as dt=0, so we must thus have dT=vdX/c2. Substituting that result into Equation 14 gives us then
That equation tells us that the temporal offset combined with the object’s motion makes the object’s rear clock partly overtake the object’s front clock. Pro rated over the length of the object that effect causes the object to shrink in the direction of relative motion, an effect that we call the Lorentz-Fitzgerald contraction.
And thus we get dX=γdx, which we substitute into Equation 5 in order to obtain the proper algebraic form of the temporal offset.
So far we have used the velocity v at which the U-frame moves through the L-frame, noting that v, the relative velocity between the observers, lies entirely in the x-direction. But we know that observers in the U-frame see objects in the L-frame moving past them at a velocity V, also oriented in the X-direction. Now we want to work out the relationship between those velocities.
We measure the velocity of an object, and thus of the frame it occupies, by laying out a straight course between two points X0 and X1, and measure how long it takes the object to traverse that course. If the course has length dX and the object takes dT to traverse it, then we calculate the object’s velocity as the ratio V=dX/dT. Thus, observers in the U-frame can determine how fast the L-frame moves through their frame.
Observers in the L-frame use a slightly different method to measure how fast the U-frame moves through their frame: they measure how long it takes an object of a given length occupying the U-frame to pass a single fixed point in their frame. In this case the object consists of the course defined above. As seen from the L-frame, it has length
in accordance with the Lorentz-Fitzgerald contraction.
We can determine the proper time dT that elapses on the object’s clocks as the object passes the fixed point in the L-frame by using the relation dX=vdT. The validity of that relation comes from the fact that the observers in the U-frame measure dT as the fixed point in the L-frame traverses the length of the object, so observers in the L-frame must see the same readings on the U-frame’s clocks and rulers. The relation gives us directly v=dX/dT=V. If that seems too easy, we can use the relation to check our result by substituting it into the temporal offset term in the temporal equation to obtain
Taking the ratio of that equation with Equation 17 gives us
That fact, combined with the facts that dy=dY and dz=dZ, gives us the axiom-like foundation on which we can deduce Einstein’s first postulate, the principle of Relativity. That deduction goes beyond the scope of this essay, so I won’t produce it here.
The Lorentz Transformation
Finally we put it all together and get four simple equations:
Thus we deduce the Lorentz Transformation from only Einstein’s second postulate and basic logic.
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