page_title: Proper acceleration
section_title: Acceleration in (1+1)D
hierarchical_section_title: Proper acceleration / Viewed from a flat spacetime slice / Acceleration in (1+1)D
caption_attribution_description: This plot illustrates how a spaceship capable of 1 g acceleration for 100 years can power a round trip to most anywhere in the visible universe, and back in a lifetime or less. Additional time will have elapsed on earth by the time that you return. This advantage of constant proper acceleration in a relativistic world arises because proper velocity change is proportional to acceleration times the change in map rather than traveler time. As those two begin to differ, the advantage (over the dotted straight line above) emerges. Of course, building a spaceship that can do this is another story.
context_page_description: In relativity theory, proper acceleration is the physical acceleration experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration, since gravity acts upon the inertial observer that any proper acceleration must depart from. A corollary is that all inertial observers always have a proper acceleration of zero.
  Proper acceleration contrasts with coordinate acceleration, which is dependent on choice of coordinate systems and thus upon choice of observers.
  In the standard inertial coordinates of special relativity, for unidirectional motion, proper acceleration is the rate of change of proper velocity with respect to coordinate time.
  In an inertial frame in which the object is momentarily at rest, the proper acceleration 3-vector, combined with a zero time-component, yields the object's four-acceleration, which makes proper-acceleration's magnitude Lorentz-invariant. Thus the concept is useful: with accelerated coordinate systems, at relativistic speeds, and in curved spacetime.
context_section_description: In the unidirectional case i.e. when the object's acceleration is parallel or antiparallel to its velocity in the spacetime slice of the observer, proper acceleration α and coordinate acceleration a are related through the Lorentz factor γ by α=γ³a.  Hence the change in proper-velocity w=dx/dτ is the integral of proper acceleration over map-time t i.e. Δw=αΔt for constant α.  At low speeds this reduces to the well-known relation between coordinate velocity and coordinate acceleration times map-time, i.e. Δv=aΔt.
  For constant unidirectional proper-acceleration, similar relationships exist between rapidity η and elapsed proper time Δτ, as well as between Lorentz factor γ and distance traveled Δx.  To be specific:
  ,
  where the various velocity parameters are related by
  .
  These equations describe some consequences of accelerated travel at high speed.  For example, imagine a spaceship that can accelerate its passengers at "1 gee" (10 m/s² or about 1.0 light year per year squared) halfway to their destination, and then decelerate them at "1 gee" for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time.  For a map-distance of ΔxAB, the first equation above predicts a midpoint Lorentz factor (up from its unit rest value) of γmid=1+α(ΔxAB/2)/c².  Hence the round-trip time on traveler clocks will be Δτ = 4(c/α) cosh⁻¹(γmid), during which the time elapsed on map clocks will be Δt = 4(c/α) sinh[cosh⁻¹(γmid)].
  This imagined spaceship could offer round trips to Proxima Centauri lasting  about 7.1 traveler years (~12 years on Earth clocks), round trips to the Milky Way's central black hole of about 40 years (~54,000 years elapsed on earth clocks), and round trips to Andromeda Galaxy lasting around 57 years (over 5 million years on Earth clocks).  Unfortunately, sustaining 1-gee acceleration for years is easier said than done, as illustrated by the maximum payload to launch mass ratios shown in the figure at right.
caption: This plot shows how a spaceship capable of 1-gee (10 m/s2 or about 1.0 light year per year squared) acceleration for 100 years might power a trip to almost anywhere in the visible universe and back in a lifetime.
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