Rapidity

Here is a basic Lorentz transformation of space and time, to a reference frame moving in the x-direction at velocity v:

t' = γ(t - vx/c²)
x' = γ(x - vt)

Where γ (the greek letter gamma*) = 1/√(1-v²/c²) is the constant of proportionality representing time dilation and length contraction, the vx/c² term represents the relativity of simultaneity, and the vt term simply gives us our classical galilean transformation, x' = x - vt, the transformation one would expect in absence of relativistic effects.

These equations look a little bit nicer if, instead of working directly with the velocity v, we work with β = v/c:

ct' = γ(ct - βx)
x' = γ(x - βct)

This is still a rather awkward way to represent the transformation, so let's add some simplifying mathematical notation. Define the rapidity, θ, to be the hyperbolic angle which satisfies:

tanh(θ) = β

where tanh(θ) = sinh(θ)/cosh(θ)
sinh(θ) = (1/2)(e^θ - e^-θ)
cosh(θ) = (1/2)(e^θ + e^-θ)

Then with some basic algebra, using cosh²(θ) - sinh²(θ) = 1, we can find a simpler formula for γ:

γ = 1/√(1-tanh²(θ)) = 1/√((cosh²(θ) - sinh²(θ))/cosh²(θ)) = 1/√(1/cosh²(θ)) = cosh(θ)

γ = cosh(θ)

and therefore

γβ = sinh(θ)

Going back to our transformation equations above,

ct' = ct cosh(θ) - x sinh(θ)
x' = x cosh(θ) - ct sinh(θ)

This almost looks exactly like a rotation in spacetime, except that the trigonometric functions are replaced with hyperbolic trig functions. A normal rotation in the x-y plane looks like:

y' = y cos(α) + x sin(α)
x' = x cos(α) - y sin(α)

This is very close to the transformation above, and we can get even closer by looking at the relationship between exponentials and trig functions in a complex number space. Using Euler's formula,

e^iα = cos(α) + isin(α),
where i = √-1,

and making the substitution iθ = α, or θ = -iα,

e^θ = cos(iθ) - isin(iθ),

Then cosh(θ) = (1/2)(e^θ + e^-θ) = cos(iθ)

and sinh(θ) = (1/2)(e^θ - e^-θ) = -isin(iθ)

Now we make this substitution in our x-y rotation formula above, so that we're rotating by an imaginary angle:

y' = y cosh(θ) + ix sinh(θ)
x' = x cosh(θ) - iy sinh(θ)

Now set y = -ict, so that t = iy/c is like an "imaginary" coordinate. Now our rotation looks like:

-ict' = -ict cosh(θ) + ix sinh(θ)
x' = x cosh(θ) - ct sinh(θ)

Multiplying by i in the first equation, we recover the complete lorentz transformation:

ct' = ct cosh(θ) - x sinh(θ)
x' = x cosh(θ) - ct sinh(θ)

Thus, a transformation from one moving reference frame to another can be thought of as a rotation by an imaginary angle into imaginary time. The angle is what we have been calling "rapidity".

Unlike a normal trigonometric angle, where the sine, cosine and tangent are periodic, repeating themselves after going from zero to 2π (or zero to 360⁰), The sinh, cosh, and tanh functions are not periodic. They always increase, but

as θ → +∞, tanh(θ) → +1, and
as θ → -∞, tanh(θ) → -1.

In other words, the velocity of the reference frame can only take on values between -c and +c, which makes sense.

All of this discussion provides one way of interpreting the motion of objects: that "stationary" objects are actually moving through time at the speed of light, and that moving objects are also moving at the speed of light, but through a "rotated" combination of space and time, at an "angle" θ with respect to the time axis. This is almost correct. Theta is really a hyperbolic angle. In a regular rotation, one would expect motion through time to reduce in lieu of motion through space increasing. When rotating by a hyperbolic angle, however, motion through time actually increases as motion through space increases, in such a way that the velocity, Δx/Δt is kept smaller than the speed of light. Specifically,

the velocity through time, u_t = c cosh(θ) and
the velocity through space, u_x = c sinh(θ),

so that the velocity we would calculate for an object, v/c = u_x/u_t = tanh(θ), recovering our definition of rapidity.

Really, this is all just a repackaging of the theory, but it helps to provide insight into different (equivalent) ways of interpreting things.

Ra*pid"i*ty (?), n. [L. rapiditas: cf. F. rapidit'e.]

The quality or state of being rapid; swiftness; celerity; velocity; as, the rapidity of growth or improvement.

Syn. -- -- Rapidness; haste; speed; celerity; velocity; swiftness; fleetness; quickness; agility.

© Webster 1913.