General Relativity: An Introduction for Physicists · Problem 5.7

Special relativity revisited — Hobson, Efstathiou & Lasenby

Physics · constant-proper-acceleration-spaceship

Problem

A spaceship has constant acceleration

g

in the

x

-direction in its locally comoving frame, i.e. the IRF. Show that, in an inertial frame, the spaceship's 4-velocity

[u^\mu] = (u^0, u^1, 0, 0)

and 4-acceleration

[a^\mu] = (a^0, a^1, 0, 0)

satisfy

a^1 = gu^0/c

and

a^0 = gu^1/c

. Show also that

\frac{d^2 u^\mu}{d\tau^2} = \frac{g^2 u^\mu}{c^2},

where

\tau

is the proper time as measured by an occupant of the spaceship. A spaceship accelerates at a constant rate

g = 9.5\,\mathrm{m\,s^{-2}}

in its own locally comoving frame. It starts out towards the centre of the Galaxy

10\,\mathrm{kpc}

distant. After going

5\,\mathrm{kpc}

it decelerates at the same rate to come to rest again at the Galactic centre. The outward journey is then repeated in reverse to come back home. Show that, in the spaceship's frame, the elapsed travel time is

41.5

years. What is the elapsed time for the waiting observer (or descendants) on Earth?

Intuitive explanation

This problem combines the relativistic kinematics of constant proper acceleration with time dilation. The key insight is that "constant acceleration in the comoving frame" means the 4-acceleration has a special form—it's orthogonal to the 4-velocity and has constant magnitude $g$ —which leads to a differential equation whose solutions describe hyperbolic motion in spacetime. Once you solve that equation, you can integrate proper time along the worldline to find the spaceship's clock reading, then use time dilation to find what Earth observers measure.

Intuitive explanation

Related theory

Motion In A Plane With Constant Acceleration

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