L’ horizon


Imagine that it’s possible to build a spaceship that constantly accelerates. We don’t really care how it does this; this is a thought experiment in physics, not an exercise in interstellar spacecraft engineering. Because we’re dealing with relativistic velocities, we need to be precise and say that the spaceship undergoes constant proper acceleration: passengers will measure a constant inertial force pushing them towards the back of the ship, whereas observers in motion relative to the ship will necessarily describe its acceleration differently.

If we start out by treating the spaceship as a single point, we can describe its world line in an inertial reference frame with coordinates (t,x) as:

x(τ) = (t(τ), x(τ)) = (sinh(a τ)/a, cosh(a τ)/a)

where τ is proper time measured by a clock in the spaceship (set to zero when t=0), a is the acceleration the passengers feel, and we have chosen coordinates so that x=1/a when t=τ=0. (We are using units where the speed of light is set equal to 1, and this means that whenever we talk about an inverse acceleration being equal to a distance, or vice versa, their product will equal c2 in conventional units. For example 1/g, where g is the gravitional acceleration at the Earth’s surface, corresponds to a distance of 9.18 x 1015m, or about a light-year, while 1/x, where x is one metre, corresponds to an acceleration of 9.18 x 1015g.)

The world line of the spaceship is a hyperbola:

Constant aceleration world line

This hyperbola can also be described by the equation:

a2x2 – a2t2 – 1 = 0, or
x.x = 1/a2

The best style is the style you don’t notice. Somerset Maugham


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