Introduction to Classical Mechanics · Problem 13.4

General Relativity — Morin

Physics — Mechanics · acceleration-linear-motion-4-vectors

Problem

Acceleration for linear motion * A spaceship starts at rest with respect to frame

S

and accelerates with constant proper acceleration

a

. In Section 11.9, we showed that the speed of the spaceship with respect to

S

is given by

v(\tau) = \tanh(a\tau)

, where

\tau

is the spaceship's proper time (we have dropped the

c

's). Let

V

be the spaceship's 4-velocity, and let

A

be its 4-acceleration. In terms of the proper time

\tau

: (a) Find

V

and

A

in frame

S

, by explicitly using

v(\tau) = \tanh(a\tau)

. (b) Write down

V'

and

A'

in the spaceship's frame,

S'

. (c) Verify that

V

and

V'

transform like 4-vectors between the two frames. Likewise for

A

and

A'

Intuitive explanation

In special relativity, 4-vectors encode spacetime information in a frame-independent way, but their components change predictably under Lorentz transformations. Here you'll compute the 4-velocity and 4-acceleration explicitly using the given relativistic velocity formula, then verify they transform correctly—the key insight is that the Lorentz transformation rules for 4-vectors are the same as for position and time, so checking this consistency confirms your calculations and deepens your understanding of relativistic kinematics.

Intuitive explanation

Related theory

Motion In A Plane With Constant Acceleration

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