Deriving equations for anyspeed acceleration

Why am I here?

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...

~ ~ ~ Station Identification ~ ~ ~
  • Caution: This page is undergoing conversion to more standard notation, from...
  • Notation Key B: Galilean {time, velocity}={t, v}, coordinate {time, velocity}={b, w}, and proper {time, velocity}={T, u}.
  • Cite/Link:
    Thus anyspeed acceleration solver's may be built on the fact that constant acceleration problems involving motion in one spatial direction can be described exactly using either Galilean (low-velocity approximation), map (a local reference frame of yardsticks and synchronized clocks), or traveler-perspective (proper) times and speeds or "kinematics". These ideas are introduced in a paper accessible elsewhere (Gen. Rel. and Q. Cosm. eprint
    gr-qc/9512012), but the result is so simple that it can be understood without even using Lorentz transforms! Hence for prospective intro & modern physics textbook authors, and for students who want to understand accelerated twin-paradox problems without first learning to transform between inertial frames, this is for you now.

    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 proper-velocitiesNote01 in [ly/ty] or rodden-berries [rb]. For 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 the metric equation and the definitions above, which effectively define the relativistic behavior of all clocks.

    Velocity Conversions

    Using the given relationships and the definitions of v and w above, a bit of algebra will give you the "velocity interconversions":

    γ = 1 + ½(V/c)2 = 1/Sqrt[1-(v/c)2] = Sqrt[1+(w/c)2] = dt/dτ. (1)

    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

    α = c2dγ/dx = c2dγ/dV×dV/dx = c2(V/c2)dV/dx = VdV/dx = dx/dT×dV/dx = dV/dT.

    Hey -- this looks familiar! Hence ΔT = (Vf - Vo)/α, as we might have expected. This same trick works with the other kinematics.

    For the map-kinematic,

    α = c2dγ/dx = c2dγ/dv×dv/dx = c23w/c2)dv/dx = γ3v×dv/dx = γ3dx/dt×dv/dx = γ3dv/dt.

    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

    Δt = (γfvf - γovo)/α = (wf - wo)/α.

    For the traveler kinematic,

    α = c2dγ/dx = c2dγ/dw×dw/dx = c2(w/γc2)dw/dx = (1/γ)w dw/dx = (1/γ)dx/dτ×dw/dx = (1/γ)dw/dτ.

    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 get

    Δτ = (ArcSinh[wf/c] - ArcSinh[wo/c])c/&alpha = (ηf - ηo)c/α,

    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:

    α = F/m = (1/m)ΔE/Δx = c2Δγ/Δx = ΔV/ΔT = Δw/Δt = cΔη/Δτ. (2)

    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 be double-checked/pf...)

    α = d2x/dT2 = γ2d2x/dτdt = (1/γ)d2x/dτ2 = d2x/dtdτ = (c2/w)d2t/dτ2.


    Distance-Time Relationships

    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:

    γf = 1 + ½{(Vo + αdT)/c}2 = Cosh[ηo + αdτ/c] = Sqrt[1 + {(voγo + αdt)/c}2]. (3)

    where ηo = ArcSinh[wo/c] and γo = 1/Sqrt[1-(vo/c)2] from above. 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 group photo. 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 (w=dx/dτ) 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)
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