A uni-directional anyspeed-motion primer

For more on the variables here, see this slide, this PDF, or

Anyspeed kinematics: times and velocities

Times don't behave the way Galileo expected at high speeds, i.e at speeds approaching the 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 written as properTime²=mapTime²-(mapDistance/c)² where c (lightspeed) is the number of meters (∼3×10⁸) in one second.

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 v≡dx/dt, 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]. (Q1) 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 c?

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]?

Stop by the spacetime explorer to let nature speak for herself.

Anyspeed kinematics: acceleration

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 (w_f and w_o) using Δt = (w_f-w_o)/α. This is reminescent of the familiar GK relation which also works at any speed: ΔT = (V_f-V_o)/α. The traveler (e.g. rocket-ship) or proper-time elapsed, by comparison, is Δ τ = (ArcSinh[w_f/c] - ArcSinh[w_o/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 w_f, and coordinate-velocity v_f, which results therefrom? (Hint: You can get final Galilean velocity V_f from the familiar classical equations for constant acceleration.)

For more check this anyspeed weblist, and proper acceleration: the calculator/the movie.

Anyspeed dynamics

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 = E/mc² = 1+½(V/c)² = Sqrt[1+(w/c)²] = 1/Sqrt[1-(v/c)²]. 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 = m Δw/Δt. 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).

(Q4) What 
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.

For a closer look at electrons, check here.

Relative speeds

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. Lorentz transforms 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 complications is 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 velocity as w = γv = v/Sqrt[1-(v/c)²]. If one 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 = γ₁γ₂(v₁+v₂). 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.

Related papers: anyspeed modeling, modernizing Newton, and one-map two-clocks

Footnotes * 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"

At UM-StLouis see also: accel1, cme, programs, stei-lab, & wuzzlers.

A table of contents for these "frame-dependent relativity" pages.

For source, cite URL at http://www.umsl.edu/~fraundor/primer.html

Version release date: 1 May 2005.

Mindquilts site page requests ~2000/day approaching a million per year.

Requests for a "stat-counter linked subset of pages" since 4/7/2005: .

Send comments, your 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. (Thanks. /philf :)