arXiv:physics/9611011 (xxx.lanl.gov archive, Los Alamos, NM, 1996).

Because of this, relativistic acceleration is seldom discussed in
introductory courses. Can relativistic equations for constant acceleration,
instead, be cast in familiar form? The answer is yes: a frame-invariant
3-vector acceleration, with simple integrals, arises naturally in one-map
two-clock relativity. Although the development of the (3+1)D equations is
tedious, we show that this acceleration bears a familiar relationship to the
frame-independent rate of momentum change (i.e. the force) *felt* by an
accelerated traveler.

Before considering integrals of the motion for constant proper acceleration
**alpha**, let's review the classical integrals of motion
for constant acceleration **a**. These can be written as
*a* ~ Delta[*v*_{||}]/Delta[*t*] ~
(1/2) Delta[*v*^2]/Delta[*x*_{||}].
The first of these is associated with conservation of
momentum in the absence of acceleration, and the second with the work-energy
theorem. These may look more familiar in the form *v*_{||f} ~
*v*_{||i}+*a* Delta[*t*], and
*v*_{f}^2 ~ *v*_{i}^2+2*a*
Delta[*x*_{||}]. Given that coordinate velocity has an upper
limit at the speed of light, it is easy to imagine why holding coordinate
acceleration constant in relativistic situations requires forces which
change even from the traveler's point of view, and is not possible at all
for Delta[*t*] > (*c*-*v*_{||i})/*a*.

Provided that proper time *tau*, proper velocity *w*, and time-speed
*gamma* can be used as variables, three simple integrals of the proper
acceleration can be obtained by a procedure which works for integrating
other non-coordinate velocity/time expressions as well {Noncoord}. The
resulting integrals are summarized in compact form, like those above, as

In classical kinematics, the rate at which traveler energy *E* increases
with time is frame-dependent, but the rate at which momentum *p* increases
is invariant. In special relativity, these rates (when figured with respect
to proper time) relate to each other as time and space components,
respectively, of the acceleration 4-vector. Both are frame-*dependent*
at high speed. However, we can define proper force separately as the force
*felt* by an accelerated object. We show in the
Appendix that this is
simply **F** = *m* **alpha**. That is, all
accelerated objects *feel* a frame-invariant 3-vector force
**F** in the direction of their acceleration. The magnitude of
this force can be calculated from any inertial frame, by multiplying the
rate of momentum change *in the acceleration direction* times
*gamma*_{perp}, or by multiplying mass times
the proper acceleration *alpha*.
The classical relation *F* ~ d*p*/d*t* ~
*m* d*v*/d*t* = *m*d^2*x*/d*t*^2
= *ma* then
becomes:

Although they depend on the observer's inertial frame, it is instructive to
write out the components of momentum and energy rate-of-change in terms of
proper force magnitude *F*. The classical equation relating rates of
momentum change to force is d**p**/d*t* ~ **F** ~
*ma* **i**_{||}, where **i**_{||}
is the unit vector in the direction of acceleration. This
becomes

As mentioned above, the rate at which traveler energy increases with time
classically depends on traveler velocity through the relation
d*E*/d*t* ~ *Fv*_{||} ~
*m* (**a * v**).
Relativistically, this becomes

Similarly, the classical relationship between work, force, and impulse can
be summarized with the relation d*E*/d*x*_{||} ~ *F*
~ d*p*_{||}/d*t*. Relativistically, this becomes

The development above is of course too complicated for an introductory class. However, for the case of unidirectional motion, and constant acceleration from rest, the Newtonian equations have exact relativistic analogs except for the changed functional dependence of kinetic energy on velocity. These equations are summarized in Table II.

For a numerical example, imagine trying to predict how far one might travel
by accelerating at one earth gravity for a fixed traveler-time, and then
turning your thrusters around and decelerating for the same traveler-time
until you are once more at rest in your starting or ``map'' frame. To be
specific, consider the 14.2 proper-year first half of such a trip all the
way to the Andromeda galaxy {
LagouteDavoust}, one of the most distant
(and largest) objects visible to the naked eye. From Eqn. 6,
the maximum (final) rapidity is *eta*_{||} = *alpha tau*/*c*
=14.7. Hence the final proper velocity is *w* = Sinh[*alpha tau*/*c*] =
1.2 10^6[ly/tyr]. From Eqn. 2
this means that *gamma*
= Sqrt[1+(*w*/*c*)^2] = 1.2 10^6, and the coordinate velocity
*v* = *w*/Sqrt[1+(*w*/*c*)^2] = 0.99999999999963[ly/yr].
Going back to Eqn. 6, this means that coordinate time elapsed is
*t* = *w*/*alpha c* = 1.1 10^6[years], and distance
traveled *x* = (*gamma* -1)*c*^2/*alpha* = 1.1 10^6[ly]. Few
might imagine, from typical intro-physics treatments of relativity, that one
could travel over a million lightyears in less than 15 years on the
traveler's clock!

From Eqn. 5, the coordinate acceleration falls
from 1[gee] at the start of the leg to *a* = *alpha*/*gamma* ^3 = 6
10^{-19}[gee] at maximum speed. The forces, energies, and momenta of course
depend on the spacecraft's mass. At any given point along the trajectory
from the equations above, *F* is of course just *m alpha*, d*E*/d*x* is
*gamma*_{perp} *F* = *F*, d*p*/d*t* is
*F*/*gamma*_{perp} = *F*, and d*E*/d*t* is
*gamma*_{perp} *Fv*_{||} = *Fv*.
Note that all except the last of these
are constant if mass is constant, albeit dependent on the reference frame
chosen. However, the 4-vector components d*p*/d*tau* and
d*E*/d*tau* are not
constant at all, showing in another way the pervasive frame-dependences
mentioned above.

The foregoing solution may seem routine, as well it should be. It is not. Note that it was implemented using distances measured (and concepts defined) in context of a single map frame. Moreover, the 3-vector forces and accelerations used and calculated have frame-invariant components, i.e. those particular parameters are correct in context of all inertial frames.

The mass of the ship in the problem above may vary with time. For example,
if the spacecraft is propelled by ejecting particles at velocity *u*
opposite to the acceleration direction, the force felt in the frame of the
traveler will be simply *m alpha* = -*u*d*m*/d*tau*.
Hence in terms of traveler
time the mass obeys *m* = *m*_{o}
Exp [-*alpha tau* /*u*]. In terms of coordinate
time, the differential equation becomes *m alpha* =
-*u gamma* d*m*/d*t*. This can
be solved to get the solution derived with significantly more trouble in the
reference above {LagouteDavoust}.

For more, see our web construction table of contents on this subject. Please share your thoughts using our review template, or send comments, answers to problems posed, and/or complaints, to philf@newton.umsl.edu. This page contains original material, so if you choose to echo in your work, in print, or on the web, a citation would be cool.

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