For more on the variables here, see this slide,
Anyspeed kinematics: times and velocities
Times don't behave the way Galileo expected at high speeds, i.e at speeds
speed of light c. His acceleration equations, which are simpler
than any at low speeds, continue to work for unidirectional motion at
any speed for clocks monitoring the acceleration from a special
Galilean-kinematic (GK) "chase plane". At high speed, added equations
are nonetheless needed to predict motion as a function of
time-elapsed on other physical clocks. We are particularly interested
in the yardsticks and clocks of a reference "map-frame" (which the
equivalence principle incidentally tells us can provide a "locally
Newtonian" base of operations even in accelerated frames with high
spacetime curvature). Of course, we're also interested in the behavior of
clocks on board the traveling object itself. In context of simultaneity
defined by our reference map frame, traveler "proper-time" is conveniently
given by Minkowski's space-time version of Pythagoras' theorem.
This metric equation (which contains all of special relativity and
can be used to describe the structure even of curved spacetime) can be
where c (lightspeed) is the number of meters (∼3×108) in
If we measure all distances x from the vantage point of a
mapframe with interlinked yardsticks and
synchronized clocks, along with chase-plane time T
and GK-velocity V≡dx/dT
it is traditional to define map-time t and coordinate-velocity
as well as traveler (proper) time τ and proper-velocity
w≡dx/dτ. The Galilean
kinematic provides the simplest expressions at low speed,
but the traveler-kinematic (proper-time/velocity/acceleration) is
easier to calculate other stuff with at high speeds.
Proper-velocity w is proportional to the momentum of a moving object,
and can be determined from GK-velocity V using the relation
w = V Sqrt[1+¼(V/c)2].
Coordinate-velocity v in turn relates to proper-velocity via
v = w/Sqrt[1+(w/c)2].
Can you show from this last expression (which follows directly
from the metric equation) that finite proper-velocities imply
coordinate-velocities which are always less than lightspeed
We thus have three familiar ways to
describe a traveler's velocity at any speed, with respect
to a chosen reference or map frame. To minimize
confusion when talking about inter-convertable
velocities, defined with
reference to distances measured in a single inertial frame
but differing based on whose time
is being considered, we find it convenient to report
coordinate-velocity v (distance
traveled per unit map time) in units of [lightyears per coordinate year]
or [c], GK-velocities V are reported simply
in [lightyears per chase-plane year] or [ly/gy], and
proper-velocity w (distance traveled per unit
proper time) in [lightyears per traveler year] or [rb*].
Units of years and lightyears are used here since
a typical acceleration involving
people, namely the acceleration due to gravity on earth,
is 1.03 [ly/yr^2]. For example, a proper-velocity of
1 [lightyear per traveler year] marks the transition between
relativistic and non-relativistic behaviors. (Q2)
With a bit of manipulation of the equations above, can you show that
1[rb] = 0.7071[c] = 0.9102[ly/gy]?
During unidirectional motion of constant acceleration
α, defined below as change in
energy per unit rest mass per unit of distance traveled so as to
make the definition
independent of whose time we are using, the map
(stationary clock) time-elapsed
can be figured from initial and final proper velocities
(wf and wo) using
Δt = (wf-wo)/α.
This is reminescent of the familiar GK relation which also works at
ΔT = (Vf-Vo)/α.
The traveler (e.g. rocket-ship) or proper-time elapsed, by
Δ τ = (ArcSinh[wf/c] - ArcSinh[wo/c]) c/α.
We use a symbol α to represent "proper acceleration" experienced
by a traveler because at high speeds it is not equal to the
coordinate-acceleration a, the second maptime derivative of
map position. In fact, for unidirectional motion
α = γ3a.
After all, the first coordinate-time
derivative of x (i.e. coordinate-velocity v) has little room
for increase when it gets close to
c, even though one's energy and momentum can continue
upward without bound. (Q3)
Given this, how much traveler time Δτ elapses
during constant 1-gee
acceleration for a million lightyears distance?
Also, what's the final proper
velocity wf, and coordinate-velocity vf,
which results therefrom? (Hint: You can get final
Galilean velocity Vf from the familiar classical
equations for constant acceleration.)
Another feature of special relativity
is the "rest energy" mc2
associated with the rest mass
m of a moving object. The total energy of a moving object is
therefore E = mc2 + K where
K is an object's kinetic energy. Energy is connected to the
various relativistic velocity
types mentioned above through the dimensionless energy factor
γ ≡ dt/dτ =
Since we defined proper acceleration above in terms of the
energy increase per unit distance, the relativistic equation
for forced motion in one
direction (or Newton's second law) becomes
F = dE/dx = mα =
mc2 Δ γ/ Δ x =
Similarly, momentum in collisions is conserved in the absence of
external forces, provided
that momentum at high speeds for this calculation is written as
p = m w
(thanks to the traveler-independent nature of t).
is the proper-velocity "land-speed record" for objects accelerated by
man? This was probably attained in November 1995 by electrons
in the LEP2 accelerator
at CERN in Geneva, which were accelerated through an electric potential
of 70 billion
volts and hence had a kinetic energy of K = 70 GeV?
Note: The rest energy of an
electron is mc^2 = 511,000 electron volts.
When it comes to comparing distances, as well as times, measured in
frames moving at
high speeds with respect to one another, things get more complicated.
and a new form of Pythagoras theorem involving time needs to be developed, and
phenomena like length contraction and frame-dependent simultaneity need to be
understood. Although we don't have time to treat these here, one of the
that relative velocities can no longer be calculated by simple addition.
In fact, only in this
way is it possible for light in a vacuum to travel at the speed of light
relative to all
observers, even if the observers are traveling at high speeds with respect
to each other.
There is a simple way to keep track of these effects. Note from above
that proper-velocity w can be written in terms of coordinate
w = γv =
object is moving rightward with coordinate speed
v1 in the frame in which you measure
distances, and a second object is moving leftward toward the first
with speed v2 in that
same frame, then the proper-velocity of the first object
in the frame of the second is w12 =
In other words, when calculating relative proper velocities, the coordinate
velocities add while the gamma values have to be multiplied.
This expression then allows
one to calculate the relative speeds and energies attainable when
throwing objects (like
elementary particles) at each other at relativistic speeds from
opposite directions. (Q5)
What is the relative proper-velocity, in lightyears per traveler year,
of two 70 GeV electrons in head-on collision trajectories?
This may be
the "relative-speed record" for objects accelerated by man.
* Because of the utility of proper-velocity w and the
absence for now of an official
designation, we refer to "one lightyear per traveler year" as
a "rodden-berry" [rb] on
mnemonic grounds, since "hot rod" recalls high speed,
"berry" recalls a "self-contained unit". It's ironic
that for some "roddenberry" may also recall a TV series that,
like proper-velocity, is oblivious to the lightspeed limit to
which coordinate-velocity is held.
From notes on "Introducing Newcomers to the 21st Century"
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