The "four-vectors" for our "1D acceleration problems" of course have
zero *y* and *z* components. We will therefore only discuss
their coordinate time *t* and their position
*x* components. We then might write the four components of a
space-time displacement four-vector as **X**={*c*t,x,0,0}.
The *magnitude* or length of this 4-vector, which we might denote
as *c*τ where τ (tau) is a frame-invariant proper time,
refers to the square-root of the scalar product:
**X•X** = (*c*t)^{2} - x^{2} - y^{2} - z^{2} = (c τ)^{2}.
The negative signs in the scalar product sum extend *Pythagoras
theorem* to include time, with the result that proper time
intervals between events can be real (sum>0) or imaginary (sum<0).
In this latter case, some call the proper time interval "spacelike"
instead of "timelike".

The velocity four-vector associated with this displacement 4-vector is
**U** = d**X**/dτ = {*c*γ,w,0,0} =
{*c* cosh[η], *c* sinh[η], 0, 0}
where γ is energy factor gamma or *dt/dτ*, w is
proper velocity *dx/dτ*, and *η* is the
hyperbolic velocity angle eta or rapidity. The magnitude of
this four vector is lightspeed *c*, which if
multiplied by rest mass *m*, by *c*, and then squared,
verifies (or follows from) the relativistic mass-energy equation
(*mc*)^{2}**U•U** = (*mc*^{2})^{2} = (γ*mc*^{2})^{2} - (mwc)^{2} = E^{2} - (pc)^{2}
where p=*m*w is relativistic momentum and
E=γ*mc*^{2} is relativistic mass-energy.
If p=0, this is the familiar E=*mc*^{2}.

The acceleration 4-vector associated with this displacement 4-vector
is **A** = d**U**/dτ = {*c* dγ/dτ, dw/dτ, 0, 0} = {*c* sinh[η] dη/dτ, *c* cosh[η] dη/dτ, 0, 0}.
The magnitude-squared of this 4-vector is thus a negative number,
which we can define as -α^{2} where α
is the instantaneous proper acceleration. In other words,
**A•A** = -α^{2} = -(*c* dη/dτ)^{2},
so that dη/dτ = α/*c*.
The proper time derivative of eta, times *c*, is thus the
instantaneous proper acceleration.

If we multiply the acceleration 4-vector by rest mass, we get the
relativistic 4-vector version of Newton's law,
**F** = *m***A** = *m* d**U**/dτ =
{(1/*c*)dE/dτ, dp/dτ, 0, 0}. Note that if force is zero,
the spatial part of this 4-vector expresses conservation
of momentum, as is true for the 3-space version of
Newton's law. An added bonus with this 4-vector
form is that the time-component of the vector
(listed first) expresses conservation of energy.

None of the foregoing assumes that this proper
acceleration is constant. Even so, each of these 4-vectors
can be written in terms of the *velocities* associated
with any of the three kinematics: inertial, Galilean, or
traveler, using conversions derived
elsewhere.
For example, Galilean-kinematic velocity of our traveler is
defined as V ≡ dx/dT where T is time on the clocks of
chaseplane whose motion allows one to describe the traveler's
proper acceleration with Galileo's equations e.g. α = dV/dT.
In these terms the vector 4-velocity obeys
**U** = {*c* (1+½(V/*c*)^{2}), V Sqrt[1+¼(V/*c*)^{2}], 0, 0}.
This equation thus gives Galileo's acceleration equations predictive
power in the relativistic regime.

If proper acceleration α is constant, we can further integrate
these equations to predict **A**, **U**, and **X** as a
function of *time*. To make life simple, as in the
universal acceleration plots discussed elsewhere on these pages,
we write the result for the case when the zero of time corresponds
to the rest (zero-velocity) point along the constant
acceleration trajectory. We get η = ατ/*c*,
**A** = {α sinh[ατ/*c*], α cosh[ατ/*c*], 0, 0},
**U** = {*c* cosh[ατ/*c*], *c* sinh[ατ/*c*], 0, 0},
and
**X** = {sinh[ατ/*c*] *c*^{2}/α,
(cosh[ατ/*c*] - 1) *c*^{2}/α, 0, 0},
or...

Again, these can be re-expressed in terms of time in any of the three
kinematics. Note in particular that the components of **A** are not
constant, but are undergoing uniform "hyperbolic rotation" in the
velocity angle η, as the accelerated object speeds up.

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