Probabilities and Velocities

If two events, denoted by A and B, are mutually exclusive and have the individual probabilities P(A) and P(B), then the probability of "A or B" is just the sum of the individual probabilities, i.e.,

However, if the events A and B are not mutually exclusive, meaning that there is some non-zero probability P(A∩B) of both events occurring, then the above formula represents an over-estimate of their combined probability, because it counts the intersection P(A∩B) twice. To correct for this, the general expression for the probability of the union of two arbitrary events is

In particular, if A and B are independent, the probability of them both occurring equals the product of their individual probabilities, i.e., P(A∩B) = P(A) P(B), so the formula for the probability of independent events is

It's interesting to compare these formulas with the composition formulas for speeds with respect to different frames of reference. Of course, in ordinary units a speed can have a magnitude greater than 1, which makes it incommensurable with probabilities, which always have magnitude less than or equal to 1. However, if we agree to express all speeds as fractions of the speed of light, then the speed of every physical entity with respect to any inertial system of coordinates will be dimensionless and less than or equal to 1. Another potential difficulty is that speeds can be negative. In fact, we might even imagine complex speeds, similar to the complex values of the Schrödinger wave equation in quantum mechanics. Taking the same approach that Max Born proposed for the interpretation of the wave function, we could identify the squared norm of the speed v with the "probability". The squared norm of a complex number is simply the product of the number and its complex conjugate, i.e., . For ordinary real-valued speeds, this implies that we associate the "probability" with the value v².

Two speeds may be called independent if they are in orthogonal directions. For example, suppose a particle is moving with a speed v_y in the positive y direction of an inertial coordinate system S, and suppose the spatial origin of S is moving with a speed v_x in the positive x direction of another inertial coordinate system S′ whose axes are aligned with those of S. What is the combined speed V = v_x ∪ v_y of the particle with respect to the S′ system? According to Galilean kinematics, these orthogonal squared speeds are simply additive, so

which expresses the Pythagorean vector addition law

However, if the squared norm of a speed is to actually represent a probability, and if speeds in orthogonal directions are to be regarded as independent, then we can argue that the true law for the composition of orthogonal speeds should be

In terms of the actual speeds this represents the formula

Since these speeds are all expressed as fractions of the speed of light, it's clear that the term v_x²v_y² would be negligible unless at least one of the speeds involved was near the speed of light. Interestingly, this is precisely the correct formula for the composition of orthogonal speeds according to Einstein's theory of special relativity. To see this, let t,x,y,z denote the inertial coordinates of system S, and let T,X,Y,Z denote the (aligned) inertial coordinates of system S'. In S the particle is moving with speed v_y in the positive y direction so its coordinates are

The Lorentz transformation for a coordinate system S' whose spatial origin is moving with the speed v in the positive x (and X) direction with respect to system S is

so the coordinates of the particle with respect to the S′ system are

The first of these equations implies t = T(1 – v_x²)^1/2, so we can substitute for t in the expressions for X and Y to give

The total squared speed V² with respect to these coordinates is given by

Incidentally, subtracting 1 from both sides and factoring the right hand side, this relativistic composition rule for orthogonal speeds can be written in the form

This is essentially DeMorgan's Rule, which says that the negation of the logical OR of two events is the logical AND of the negations of the two events. Symbolically, this can be expressed as

The composition of three orthogonal speeds (such as in the x, y, and z directions) is likewise given by the probabilistic rule for independent events

Notice that the relativistic energy of a particle of rest mass m is m/(1–V²)^1/2, so this gives a decomposition of the kinetic components of the relativistic energy into orthogonal factors. Moreover, we can decompose the vibrational modes of a complex configuration into an infinite family of orthogonal components, and the relativistic energies combine in accordance with this rule.

Incidentally, the above formulas for the composition of orthogonal velocities can be generalized to cover any two velocities u and v. In terms of the dot product, the magnitude of the mutual velocity V between two particles that are moving with the velocities u and v relative to any given inertial coordinate system is

The denominator of the right hand side is unity if u and v are perpendicular, in which case this formula reduces to the previous expression.

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