Uniformly Accelerated Motion

Determine the relativistic uniformly accelerated motion (i.e., the rectilinear motion) for which the acceleration ω₀ in the proper reference frame (at each instant of time) remains constant.

a) Show that the 4-velocity

b) Show that the condition for such a motion is

where ω₀ is the usual three dimensional acceleration .

c) Show that in a fixed frame (b) reduces to

d) Show that

Do these expressions have the correct classical behavior as c → ∞.

SOLUTION

a) The 4-velocity, by definition

where  Therefore,

or

b) For arbitrary velocity v, the 4-acceleration

For μ = 0

For μ = 1, 2, 3

Therefore, we find

(1)

In the proper frame of reference, where the velocity of the particle v = 0 at any given moment, and assuming

Using ω² = ω^μω_μ, we have

c) ω^μ ω_μ from (1) may be written in the form

(2)

Using the identity (A × B)² = A² B² – (A.B)², we may rewrite (2)

(3)

In the fixed frame, since the acceleration is parallel to the velocity, (3) reduces to

So, given that ω² is a relativistic invariant,  Now, differentiating

(4)

d) Integrating (4), we have

(5)

Taking v = 0 at t = 0, we obtain

and so

(6)

As ω₀t → ∞, v → c. Integrating (6) with yields x(0) = 0

(7)

As c → ∞ (classical limit), (6) and (7) become

appropriate behavior for a uniformly accelerated classical particle.