Proper acceleration

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.

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[7] 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:

${\displaystyle \alpha ={\frac {\Delta w}{\Delta t}}=c{\frac {\Delta \eta }{\Delta \tau }}=c^{2}{\frac {\Delta \gamma }{\Delta x}}}$,

where the various velocity parameters are related by

${\displaystyle \eta =\sinh ^{-1}\left({\frac {w}{c}}\right)=\tanh ^{-1}\left({\frac {v}{c}}\right)=\pm \cosh ^{-1}\left(\gamma \right)}$.

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.[8][9] For a map-distance of Δx_AB, the first equation above predicts a midpoint Lorentz factor (up from its unit rest value) of γ_mid=1+α(Δx_AB/2)/c². Hence the round-trip time on traveler clocks will be Δτ = 4(c/α) cosh⁻¹(γ_mid), during which the time elapsed on map clocks will be Δt = 4(c/α) sinh[cosh⁻¹(γ_mid)].

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.

Animation: roundtrip to a star 6.9 ly away

Map and traveler perspectives of a roundtrip at constant 1 gee proper-acceleration (red arrow in the traveler frame) between the sun (yellow) and a hypothetical star (cyan) 6.9 lightyears away. Proxima Centauri (orange) 4 lightyears from the sun is shown in orange toward the upper left. From each perspective a year should elapse about every two seconds or every 100/17.4 frames. After each round trip ship-pilots on this shuttle-run will have aged only half as much as colleagues stationed on earth. This is time dilation in action. Other differences include the distance changes between co-moving stars, seen in the traveler frame. This is length contraction in action. Coordinate acceleration (green) seen in the map frame is only significant in the year before and after each launch, while the proper-acceleration (red) felt by the traveler is significant throughout the voyage. Note also the trace of a light signal initiated from each launch point, but 0.886 map years after launch. This pulse reaches the traveler at the voyage midpoint to remind them to begin deceleration. In the map frame Proxima Centauri sees the turnaround pulse before the destination star does, but the converse is true in the traveler frame. This is relative simultaneity in action. Nonetheless both observers agree on the sequence of events along any time-like world line.

The above equation also offers some perspective on forces and the equivalence principle. Consider local book-keeper coordinates[4] for the metric (e.g. a local Lorentz tetrad[5] like that which global positioning systems provide information on) to describe time in seconds, and space in distance units along perpendicular axes. If we multiply the above equation by the traveling object's rest mass m, and divide by Lorentz factor γ = dt/dτ, the spacelike components express the rate of momentum change for that object from the perspective of the coordinates used to describe the metric.

This in turn can be broken down into parts due to proper and geometric components of acceleration and force. If we further multiply the time-like component by lightspeed c, and define coordinate velocity as v = dx/dt, we get an expression for rate of energy change as well:

${\displaystyle {\frac {dE}{dt}}={\vec {v}}\cdot {\frac {d{\vec {p}}}{dt}}}$ (timelike) and ${\displaystyle {\frac {d{\vec {p}}}{dt}}=\Sigma {\vec {f_{o}}}+\Sigma {\vec {f_{g}}}=m({\vec {a_{o}}}+{\vec {a_{g}}})}$ (spacelike).

Here a_o is an acceleration due to proper forces and a_g is, by default, a geometric acceleration that we see applied to the object because of our coordinate system choice. At low speeds these accelerations combine to generate a coordinate acceleration like a=d²x/dt², while for unidirectional motion at any speed a_o's magnitude is that of proper acceleration α as in the section above where α = γ³a when a_g is zero. In general expressing these accelerations and forces can be complicated.

Nonetheless if we use this breakdown to describe the connection coefficient (Γ) term above in terms of geometric forces, then the motion of objects from the point of view of any coordinate system (at least at low speeds) can be seen as locally Newtonian. This is already common practice e.g. with centrifugal force and gravity. Thus the equivalence principle extends the local usefulness of Newton's laws to accelerated coordinate systems and beyond.

For low speed observers being held at fixed radius from the center of a spherical planet or star, coordinate acceleration a_shell is approximately related to proper acceleration a_o by:

${\displaystyle {\vec {a}}_{shell}={\vec {a}}_{o}-{\sqrt {\frac {r}{r-r_{s}}}}{\frac {GM}{r^{2}}}{\hat {r}}}$

where the planet or star's Schwarzschild radius r_s=2GM/c². As our shell observer's radius approaches the Schwarzschild radius, the proper acceleration a_o needed to keep it from falling in becomes intolerable.

On the other hand, for r>>r_s, an upward proper force of only GMm/r² is needed to prevent one from accelerating downward. At the Earth's surface this becomes:

${\displaystyle {\vec {a}}_{shell}={\vec {a}}_{o}-g{\hat {r}}}$

where g is the downward 9.8 m/s² acceleration due to gravity, and ${\displaystyle {\hat {r}}}$ is a unit vector in the radially outward direction from the center of the gravitating body. Thus here an outward proper force of mg is needed to keep one from accelerating downward.

The spacetime equations of this section allow one to address all deviations between proper and coordinate acceleration in a single calculation. For example, let's calculate the Christoffel symbols:[11]

${\displaystyle \left({\begin{array}{llll}\left\{\Gamma _{tt}^{t},\Gamma _{tr}^{t},\Gamma _{t\theta }^{t},\Gamma _{t\phi }^{t}\right\}&\left\{\Gamma _{rt}^{t},\Gamma _{rr}^{t},\Gamma _{r\theta }^{t},\Gamma _{r\phi }^{t}\right\}&\left\{\Gamma _{\theta t}^{t},\Gamma _{\theta r}^{t},\Gamma _{\theta \theta }^{t},\Gamma _{\theta \phi }^{t}\right\}&\left\{\Gamma _{\phi t}^{t},\Gamma _{\phi r}^{t},\Gamma _{\phi \theta }^{t},\Gamma _{\phi \phi }^{t}\right\}\\\left\{\Gamma _{tt}^{r},\Gamma _{tr}^{r},\Gamma _{t\theta }^{r},\Gamma _{t\phi }^{r}\right\}&\left\{\Gamma _{rt}^{r},\Gamma _{rr}^{r},\Gamma _{r\theta }^{r},\Gamma _{r\phi }^{r}\right\}&\left\{\Gamma _{\theta t}^{r},\Gamma _{\theta r}^{r},\Gamma _{\theta \theta }^{r},\Gamma _{\theta \phi }^{r}\right\}&\left\{\Gamma _{\phi t}^{r},\Gamma _{\phi r}^{r},\Gamma _{\phi \theta }^{r},\Gamma _{\phi \phi }^{r}\right\}\\\left\{\Gamma _{tt}^{\theta },\Gamma _{tr}^{\theta },\Gamma _{t\theta }^{\theta },\Gamma _{t\phi }^{\theta }\right\}&\left\{\Gamma _{rt}^{\theta },\Gamma _{rr}^{\theta },\Gamma _{r\theta }^{\theta },\Gamma _{r\phi }^{\theta }\right\}&\left\{\Gamma _{\theta t}^{\theta },\Gamma _{\theta r}^{\theta },\Gamma _{\theta \theta }^{\theta },\Gamma _{\theta \phi }^{\theta }\right\}&\left\{\Gamma _{\phi t}^{\theta },\Gamma _{\phi r}^{\theta },\Gamma _{\phi \theta }^{\theta },\Gamma _{\phi \phi }^{\theta }\right\}\\\left\{\Gamma _{tt}^{\phi },\Gamma _{tr}^{\phi },\Gamma _{t\theta }^{\phi },\Gamma _{t\phi }^{\phi }\right\}&\left\{\Gamma _{rt}^{\phi },\Gamma _{rr}^{\phi },\Gamma _{r\theta }^{\phi },\Gamma _{r\phi }^{\phi }\right\}&\left\{\Gamma _{\theta t}^{\phi },\Gamma _{\theta r}^{\phi },\Gamma _{\theta \theta }^{\phi },\Gamma _{\theta \phi }^{\phi }\right\}&\left\{\Gamma _{\phi t}^{\phi },\Gamma _{\phi r}^{\phi },\Gamma _{\phi \theta }^{\phi },\Gamma _{\phi \phi }^{\phi }\right\}\end{array}}\right)}$

for the far-coordinate Schwarzschild metric (c dτ)² = (1−r_s/r)(c dt)² − (1/(1−r_s/r))dr² − r² dθ² − (r sinθ)² dφ², where r_s is the Schwarzschild radius 2GM/c². The resulting array of coefficients becomes:

${\displaystyle \left({\begin{array}{llll}\left\{0,{\frac {r_{s}}{2r(r-r_{s})}},0,0\right\}&\left\{{\frac {r_{s}}{2r(r-r_{s})}},0,0,0\right\}&\{0,0,0,0\}&\{0,0,0,0\}\\\left\{{\frac {r_{s}c^{2}(r-r_{s})}{2r^{3}}},0,0,0\right\}&\left\{0,{\frac {r_{s}}{2r(r_{s}-r)}},0,0\right\}&\{0,0,r_{s}-r,0\}&\left\{0,0,0,(r_{s}-r)\sin ^{2}\theta \right\}\\\{0,0,0,0\}&\left\{0,0,{\frac {1}{r}},0\right\}&\left\{0,{\frac {1}{r}},0,0\right\}&\{0,0,0,-\cos \theta \sin \theta \}\\\{0,0,0,0\}&\left\{0,0,0,{\frac {1}{r}}\right\}&\{0,0,0,\cot(\theta )\}&\left\{0,{\frac {1}{r}},\cot \theta ,0\right\}\end{array}}\right)}$.

From this you can obtain the shell-frame proper acceleration by setting coordinate acceleration to zero and thus requiring that proper acceleration cancel the geometric acceleration of a stationary object i.e. ${\displaystyle A^{\lambda }=\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }=\{0,GM/r^{2},0,0\}}$. This does not solve the problem yet, since Schwarzschild coordinates in curved spacetime are book-keeper coordinates[4] but not those of a local observer. The magnitude of the above proper acceleration 4-vector, namely ${\displaystyle \alpha ={\sqrt {1/(1-r_{s}/r)}}GM/r^{2}}$, is however precisely what we want i.e. the upward frame-invariant proper acceleration needed to counteract the downward geometric acceleration felt by dwellers on the surface of a planet.

A special case of the above Christoffel symbol set is the flat-space spherical coordinate set obtained by setting r_s or M above to zero:

${\displaystyle \left({\begin{array}{llll}\left\{0,0,0,0\right\}&\left\{0,0,0,0\right\}&\{0,0,0,0\}&\{0,0,0,0\}\\\left\{0,0,0,0\right\}&\left\{0,0,0,0\right\}&\{0,0,-r,0\}&\left\{0,0,0,-r\sin ^{2}\theta \right\}\\\{0,0,0,0\}&\left\{0,0,{\frac {1}{r}},0\right\}&\left\{0,{\frac {1}{r}},0,0\right\}&\{0,0,0,-\cos \theta \sin \theta \}\\\{0,0,0,0\}&\left\{0,0,0,{\frac {1}{r}}\right\}&\{0,0,0,\cot \theta \}&\left\{0,{\frac {1}{r}},\cot \theta ,0\right\}\end{array}}\right)}$.

From this we can obtain, for example, the centripetal proper acceleration needed to cancel the centrifugal geometric acceleration of an object moving at constant angular velocity ω=dφ/dτ at the equator where θ=π/2. Forming the same 4-vector sum as above for the case of dθ/dτ and dr/dτ zero yields nothing more than the classical acceleration for rotational motion given above, i.e. ${\displaystyle A^{\lambda }=\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }=\{0,-r(d\phi /d\tau )^{2},0,0\}}$ so that a_o=ω²r. Coriolis effects also reside in these connection coefficients, and similarly arise from coordinate-frame geometry alone.