Special relativity introductions traditionally
begin with the Lorentz transform -- a relativistic analog to the
"Transforming Between Inertial Frames" or "Relative Velocity" section
of an introductory physics course. This particular part of
introductory courses is often described as the most difficult to
understand, as it involves description of the same set of events by
observers in two differently moving coordinate systems. It's not
surprising therefore that special relativity is considered abstract
and hard to understand as well. At the same time, few relativity
authors even mention constant acceleration problems. Some think
special relativity cannot cover accelerated motion, while others
think that relativistic acceleration is a messy and inelegant concept.
Hence students often are never taught how to work the kind of
acceleration problems in relativity which served as the mainstay of
their pre-relativity education in physics!
We are here to bring good news. A dynamical quantity called
proper acceleration has
wonderful and elegant properties, if we instead begin with
the "one-map two-clock" metric equation (Minkowski's
spacetime version of Pythagoras' theorem) that underlies
Einstein's extension of relativity to the treatment of
gravity with curved spacetime. In other words, as shown
below you can access spacetime either by the long trip over
"Multiple-Inertial-Frame (MIF) Mountain", or by the much shorter
and familiar path through "Minkowski Meadow". Moreover,
Minkowski's path leads to that side of "The Lake of SpaceTime"
from which general relativity and string
theory can be most easily accessed.
In relativistic research circles,
it is hence common
knowledge that constant acceleration problems can be neatly posed
and solved. We recently noticed that
even Galileo's familiar constant acceleration equations can be
put to work to describe proper acceleration at any speed,
at least for unidirectional motion, if one recognizes that inertial frame
and traveler clocks follow a cadence different from each
other (and from Galilean clocks) at high speed...
For conceptual clarity, we use T and V≡dx/dT to
denote Galilean "chase-plane" time and the non-coordinate
"Galilean-kinematic" velocity of our accelerated traveler
(map-distance traveled per unit chase-plane time),
t and v ≡ dx/dt for map-time and
coordinate-velocity (map distance traveled per unit map-time),
and τ (or tau) and w ≡ dx/dτ for the traveler's
"proper-time" and "proper-velocity" (map-distance traveled
per unit traveler time), respectively.
Notice that all of these velocities are defined in terms of
distances (dx) measured with respect to our map-frame.
Even though many of us seem to think routinely of
distances in this way, the simplification introduced here
has likely been missed in the teaching of special
relativity because of the laudable enthusiasm of its
inventors (Albert Einstein, in particular) for a
frame-independent perspective. To further maintain
clarity, we often quote map frame times in [years],
traveler times in [tyears], coordinate-velocities in
units of [c] (lightspeed in lightyears/year), and
in [ly/ty] or rodden-berries [rb].
every 1D constant acceleration problem,
these variables (11 in all) coexist and have values with
predictive power therein.
What to Start With?
Let's imagine that the proper acceleration felt by an accelerated
traveler, α, obeys the work-energy theorem
ΔE = W = F Δx = m α Δx,
where W is work, F is force, E is kinetic energy,
and Δ is "delta" in its usual meaning as "change in" or
"final minus initial". This in fact is a cool "time-independent"
way to define acceleration in special relativity! As we see below,
acceleration's relationship to time depends on whose time
we mean. Let's also define the dimensionless
"relativistic energy factor" gamma or
γ ≡ dt/dτ = E/mc2, and while we're
at it note (as Einstein suggested) that total energy
E = mc2 + K,
where K is the kinetic energy in Galilean terms exactly
equal to ½mV2. Finally, only two results from
special relativity as it is usually taught are needed. The
first is that, in terms of coordinate-velocity v,
γ = 1 / Sqrt[1-(v/c)2]. The second is that
the traveler's proper (or spatial 4-vector) velocity w = γ v.
Both of these equations follow from
metric equation and the definitions above,
which effectively define the relativistic behavior
of all clocks.
Using the given relationships and the definitions
of v and w
above, a bit of algebra will give you the "velocity interconversions":
This equation provides relationships between the various velocities
as well as to gamma. For example, some rearrangement of the terms
with w and V in them will yield the useful result that
w = V Sqrt[1+¼(V/c)2]. We can also infer
that v = w/γ = w / Sqrt[1+(w/c)2].
In this way, given a velocity in any of the three kinematics, it is a
simple matter (as shown in the figure) to convert it to the
corresponding velocity in another.
Velocity-Distance and Velocity-Time Relationships
A velocity-distance relationship follows easily from the work-energy
theorem mentioned above. Replacing E with γmc2
and solving for Δx, we get Δx =
(γf - γo) c2/α,
where the subscripts f and o denote final and
initial values, respectively. This equation can be used with
velocity variables from any of the three kinematics by substituting
the appropriate equation for gamma from equation 1 above.
Velocity-time relationships are only a bit harder to get. For
Galilean-kinematic variables T and V, note that
Note that proper acceleration is NOT a simple time derivative
of velocity in the inertial kinematic! If we put dt on the
left hand side of this equation, and write γ as a function of
v, we can integrate this equation to get
Once again, in the traveler kinematic acceleration is not a simple time
derivative of velocity. If we put dτ on the left side of this equation,
and write γ as a function of w, we can integrate this to
where η=ArcSinh(w/c) is called the "hyperbolic velocity angle"
or rapidity. Thus in terms of traveler time, acceleration IS a simple
time derivative of this velocity angle η (times the speed of light).
For more on this see SpaceTime Physics by Taylor and Wheeler
(W.H. Freeman, San Francisco, 1963 & 1992).
All of these velocity-distance/time relations can be summarized by one
big series of equations for the proper acceleration:
These expressions are effectively integrated, so that the lower case d's
have been replaced by delta's. A large number of differential
relations (i.e. not integrated) could also be added to this list.
If we note first that dw/dv = γ3 and
dt/dτ = γ, these include for example (these notation
conversions here still need to
From the velocity-time relationships summarized in equation 2, one can
integrate to determine distance traveled as a function of time in each
of the three kinematics. However, this is more simply accomplished
by calculating final velocity as a function of elapsed time, and then
putting the result into the velocity-distance relation for distance
traveled as a function of gamma. Thus we need to first calculate
gamma as a function of time-elapsed in each of the kinematics. From
the relationships given in equations 1 and 2, one can show that:
where ηo = ArcSinh[wo/c] and
γo = 1/Sqrt[1-(vo/c)2]
Distance traveled is then obtained in terms of time elapsed (and
initial velocity) in each of the kinematics using
Δx = (γf - γo)c2/α.
As a result, the distance is parabolic in chase-plane time
(as well as in Galilean-kinematic velocity),
hyperbolic in map-time (as well as in proper-velocity),
and exponential in proper-time, as shown in the graph
and in our equation
The graph also shows that inertial velocity is
asymptotic to lightspeed, and that gamma is
linear in distance traveled from rest.
Since distances are measured in units of
c2/α, times in units of c/α, and
speeds in units of lightspeed, such a graph can thus be
used to illustrate the relationships between all 11 variables
except acceleration, at all points in time, for any
1D constant acceleration problem which happens in flat
spacetime! It is convenient to note that when
acceleration is one earth gravity, to first order
time units are 1 year and distance units are 1 lightyear
on such a plot.
Note 01: Because of the utility of proper-velocity
and the absence for now of an official designation, we've been referring
to "one lightyear per traveler year" as a rodden-berry [rb] on
mnemonic grounds, since "hot rod" recalls high speed and "berry"
recalls a "self-contained unit". It's only ironic that for some
sci-fi fans the word "roddenberry" may also recall a TV series that,
like proper-velocity, is oblivious to the lightspeed limit to which
coordinate-velocity (v=dx/dt) is held.
This release dated 06 May 2005 (Copyright by
Phil Fraundorf 1988-2005)
There is also a (possibly less current) postscript version of this file.
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