Transformation of Lateral Velocities

In the essay on The Lorentz Transformation I showed you how two observers would determine the velocity of a body moving in a direction parallel to the relative motion between them. Now I want to consider the velocity of a body that moves in a direction perpendicular to the direction of relative motion between two observers.

Imagine that Dave sits on a bench beside a straight stretch of railroad track that runs due east-west and that Dora rides a track speeder eastward at speed U. As before I assert that the speed of light in this imaginary world is a mere 100 miles per hour. Imagine further that just before Dora passes his location Dave throws a baseball due north with a speed V. How fast does that baseball move in Dora's frame of reference?

Let's start to answer that question by asking how Dave knows that the baseball moves at speed V. How does he know what number to put in place of V? In accordance with physicists' convention we say that the track lies parallel to the X-axis of a coordinate frame and that the baseball moves parallel to the Y-axis of that frame. We can now readily mathematize what Dave does. Now Dave marks off a distance in the Y-direction, say from the north rail of the track to a conveniently located backstop, and calls that distance Y. Then he puts a clock at each end of that distance, making certain to synchronize them, and notes the time that each clock displays when the baseball passes it. Subtracting the earlier time from the later time gives Dave the time required for the baseball to cross the distance Y, which time interval Dave calls T. Dave determines the speed V as the ratio of distance crossed to time elapsed; that is,

(Eq'n 1)

What speed would Dora measure for the baseball? In the X-direction it has the speed -U that Dave has relative to Dora's inertial frame. In the Y-direction it has a speed that Dora will have to determine in the same way that Dave did, she will determine the ratio between the distance y and the time interval t that she measures between the baseball's crossing the north rail and hitting the backstop. Or she could simply calculate the speed from Dave's data after transforming it appropriately.

She doesn't need to transform the distance, since it lies oriented perpendicular to the direction of relative motion between Dave and her. She knows that y = Y.

She needs to think about the transformation of time measured on Dave's clocks. She knows that she has no temporal offset to account between the two clocks because the clocks have no component of separation from each other in the x-direction. She has only to take into account the fact that Dave's clocks tick off dilated time in her frame, that for any interval T that Dave's clocks count off her clocks will count off t = LT, in which the Lorentz factor comes from

(Eq'n 2)

In the y-direction, then, Dora infers the baseball's speed as

(Eq'n 3)

In Dora's frame the baseball appears to fly more slowly than it does in Dave's frame, at least in the y-direction.

Dora can now use the Pythagorean Theorem to calculate the baseball's full speed in her frame, the speed that comes from its motion in both the x- and y-directions. If Dora represents that speed with w, then she calculates

(Eq'n 4)

We know that neither V nor U can exceed the speed of light, so now we want to ask whether their combination given in Equation 4 could ever lead to w exceeding the speed of light.

If we square Equation 4, then we have the inequality

(In'q 1)

which expresses our proposition that w never
exceed c. To prove and verify that proposition we must solve that inequality as
if it were an equation and obtain a statement that has the self-evident truth of
an axiom. I can achieve that proof by subtracting U^{2} from both sides
of the inequality sign, giving both remaining terms on the left side a common
denominator (c^{2} in this case), and then dividing both sides by c^{2}-U^{2}.
I then obtain

(In'q 2)

That statement has the axiomatic property that I want and completes my proof. We now know that we cannot make any object move faster than the speed of light by accelerating it in one direction and then accelerating it in a direction perpendicular to the first direction.

Now I want to consider one final check on the derivation of the lateral velocity above. Imagine that Dave's clocks don't work. In order to calculate the speed of the baseball in his frame, Dave will have to use the times shown on Dora's clocks.

But we won't accept the readings on Dora's clocks unless we see them right next to the baseball at the appropriate instants. That means that Dora must mount her southern clock a distance x = Ut east of her northern clock in the x-direction so that her eastward motion relative to Dave will bring her clocks, one after the other, into near coincidence with the baseball. That x-ward separation obliges Dave to compensate a temporal offset, even though Dora has carefully synchronized her clocks with each other.

If he takes a snapshot of Dora going by his
location, Dave will measure the x-ward distance between Dora's clocks as X =
x/L. He knows that relative motion has shoved Dora's northern clock into the
future relative to her southern clock by the offset xU/c^{2}. To obtain
the correct time of flight for the baseball he must subtract that number from
the difference between the readings on Dora's clocks and then, because Dora's
clocks tick off dilated time in his frame, multiply the result by the Lorentz
factor between his and Dora's inertial frames. He thus obtains

(Eq'n 5)

He then calculates the speed of the baseball as

(Eq'n 6)

That equation agrees with Equation 3, as
it should.

So now Dave and Dora have solved a problem in relativistic kinematics that involves bodies moving in two different dimensions of space. Surely this means that they can readily extend their derivation to a fully general two-dimensional version of the Lorentz Transformation. How hard can it be? (Hint: what's the difference between smashing a tyrannosaur's egg and taking on the fully grown Tyrannosaurus rex Herself?) I will answer that question in the essay on the Rindler-Shaw paradox.

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