Rapidity is a physical quantity used in special relativity which represents the speed of a reference frame in a mathematically simplifying form.

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)(eθ - e)
cosh(θ) = (1/2)(eθ + e)

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 = cos(α) + isin(α),
where i = √-1,

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

eθ = cos(iθ) - isin(iθ),

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

and sinh(θ) = (1/2)(eθ - e) = -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, ut = c cosh(θ) and
the velocity through space, ux = c sinh(θ),

so that the velocity we would calculate for an object, v/c = ux/ut = 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.


*Sorry about the crappy gamma that looks like a v. I can't control everything.

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.

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