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Here we discuss integrating the differential relationships between proper-acceleration and proper-velocity to come up with some constant-acceleration equations which work, at least to good approximation under some well-defined special conditions.
The primary precise-solution focus will be on the integrals of constant proper-acceleration in (1+1)D and (3+1)D flat spacetime. As discussed in the main paper, these also underpin local approximations (e.g. use of the Galilean kinematic with help from geometric ``forces") in selected curved-space and accelerated frame settings.
Minkowski's version of the flat-space metric equation in (3+1)D might be written:
.
Here c is the spacetime-constant relating time and distance units (often called "lightspeed"), r ≡ Sqrt[x2+y2+z2], δτ is the proper-time elapsed on the clocks of a traveler when |dr/dt| is less than c, and δσ is the proper-distance between two spatially separated events when |dr| is greater than |cdt|.
Proper-acceleration α is the acce
leration felt by accelerometers (broadly defined) that are moving with a traveling object. Its frame-invariant magnitude is that of the object's acceleration 4-vector.
This 4-vector becomes a 3-vector in the tangent free-float-frame of the traveling object itself. The 3-vector direction of the proper-acceleration α is therefore defined only locally in the frame of the traveling object, much as is the magnitude of the proper-time elapsed on a traveling clock.
In flat spacetime, coordinate-acceleration a ≡ d2x/dt2 = α||w/γ3+α⊥w/γ2 scales differently with proper-acceleration components parallel and perpendicular to the direction of our traveling object's proper-velocity w≡ dx/dt. If we stick to (1+1)D i.e. unidirectional motion, this yields some wonderfully simple integrals for constant proper-acceleration, namely α = Δw/Δt = cΔη/Δτ = c2Δγ/Δx, where rapidity η = asinh[w/c] = atanh[v/c].
These reduce to the familiar conceptual-physics relationships a = Δv/Δt = ½Δ(v2)/Δx at low speed. However they allow beginning students to easily explore interstellar constant proper-acceleration round-trip problems. In (3+1)D, the use of minimally frame-variant quantities means that one can also predict trajectories from the map point of view using acceleration directions referenced to the traveler, with no need to consider explicitly the "Wigner rotation" of velocity directions between frames!
Question: Although it's easy to derive relationships between velocity-measures w, v, γ and η from the flat-space metric equation (cδτ)2 = (cδt)2-(δx)2, is there also a simple way to derive the (1+1)D constant acceleration integrals?
One might start by solving the flat-spacetime metric's differential relation which defines proper-acceleration's frame-invariant magnitude, namely -α2 ≡ (cδγ/δτ)2 - (δw/δτ)2 for simplicity using the initial condition w[0]=0, where γ = Sqrt[1+(w/c)2] from the flat-space metric equation, to get w ≡ δx/δτ = c Sinh[ατ/c]. This is much like solving the low-speed relation a ≡ δv/δt with initial condition v[0]=0 to get v = at. It is also the method we used elsewhere for the more complicated (3+1)D case. Does it work here?
Yes, because then the metric equation (along with w ≡ δx/δτ = c Sinh[ατ/c] from above) says that γ ≡ δt/δτ = Sqrt[1+Sinh[ατ/c]2] = Cosh[ατ/c]. Much as when integrating v ≡ δx/δt = at with x[0]=0 to get x = ½at2, in (1+1)D the map-position x can be obtained by integrating w ≡ δx/δτ with x[0]=0 to get x = (c2/α)(Cosh[ατ/c]-1), and map-time t can be obtained by integrating γ ≡ δt/δτ with t[0]=0 to get t = (c/α)Sinh[ατ/c].
In a world where locations and simultaneity are defined by map or bookkeeper coordinates x and t from the metric, frame-invariant proper-time τ, synchrony-free proper-velocity w, and frame-invariant proper-acceleration α's magnitude thus provide us with robust tools for describing motion at any speed from the vantage point of an accelerated traveler[1]. The same tools[2] also help us, via the distinction between proper and geometric forces, to extend the utility of Newton's laws (at least locally) into our everyday world of accelerated frames and curved spacetime. This metric-first[3] or one-map two-clock[4] approach is thus of interest to engineers[5] working with accelerated frames and high speed particles for a variety of reasons.
The foregoing also illustrates how hyperbolic velocity angle (or rapidity) η ≡ asinh[w/c] assumes a natural role in (1+1)D constant proper-acceleration as something which obeys Δη = αΔτ/c. In (3+1)D, we've shown that η' ≡ asinh[w||/(γ+c)] plays a similar role by obeying Δη' = αΔτ/(γ+c), where γ+ ≡ Sqrt[(γo+1)/2] with γo the "minimum Lorentz factor which persists" when the constant proper acceleration is extrapolated to "turnaround" i.e. the Lorentz factor or "speed of map time" δt/δτ ≥ 1 for a fly ball when it reaches the peak of its arc.
A blackboard summary of the (1+1)D integrals might look like this:
In (3+1)D the sequence of derivatives, cast in terms of proper velocity and minimum (starting) Lorentz factor, look like:
A blackboard summary of the corresponding (3+1)D integrals might instead look like this:
A sample problem with solution using these equations may be found on this page about accelerated roundtrips.
References:
P. Fraundorf (2016/2017) "Traveler-point dynamics", hal-01503971 working draft.
P. Fraundorf (2016) "The proper-force 3-vector", hal-01344268 current pdf.
Edwin F. Taylor and John Archibald Wheeler (2000) Exploring Black Holes (Addison-Wesley-Longman) 2nd edition link.
P. Fraundorf (1996) "A one-map two-clock approach to teaching relativity in introductory physics", arXiv:physics/9611011.
David G. Messerschmitt (2017) “Relativistic timekeeping, motion, and gravity in distributed systems,” Proceedings of the IEEE, 105, 1511–1573, link.
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