Proper acceleration

Proper acceleration

Proper acceleration [Edwin F. Taylor & John Archibald Wheeler (1966 1st ed. only) "Spacetime Physics" (W.H. Freeman, San Francisco) ISBN 0-7167-0336-X] is the physical acceleration experienced by an object. It equals the rate of change of proper velocity with respect to coordinate time. It equals the coordinate acceleration if you're using an inertial coordinate system in flat spacetime, provided the object's proper-velocity [Francis W. Sears & Robert W. Brehme (1968) "Introduction to the theory of relativity" (Addison-Wesley, NY) [ LCCN 680019344] , section 7-3] (momentum per unit mass) is much less than lightspeed "c". The proper acceleration 3-vector, combined with a null time-component, yields the object's four-acceleration (as measured by the object itself) which makes proper-acceleration's magnitude Lorentz-invariant. Thus it comes in handy: (i) with accelerated coordinate systems, (ii) at relativistic speeds, and (iii) in curved spacetime.

For instance, when holding onto a carousel that turns at constant angular velocity you experience a radially-inward (centripetal) proper-acceleration due to the interaction between the hand-hold and your hand. This cancels the radially outward "geometric acceleration" associated with your spinning coordinate frame. This outward acceleration (from the spinning frame's perspective) will become the coordinate acceleration when you let go, causing you to fly off along a zero proper-acceleration (geodesic) path. Unaccelerated observers, of course, in their frame simply see your equal proper and coordinate accelerations vanish when you let go.


Thus the distinction between proper-acceleration and coordinate accelerationcf. C. W. Misner, K. S. Thorne and J. A. Wheeler (1973) "Gravitation" (W. H. Freeman, NY) ISBN 0-7167-0334-0, section 1.6] allows one to track the experience of accelerated travelers from various non-Newtonian perspectives. These perspectives include those of accelerated coordinate systems (like a carousel), of high speeds (where proper time differs from coordinate time), and of curved spacetime (like that associated with gravity on earth).

Classical applications

At low speeds in the inertial coordinate systems of Newtonian physics, proper acceleration simply equals the coordinate acceleration a=d2x/dt2. As reviewed above, however, it differs from coordinate acceleration if one chooses (against Newton's advice) to describe the world from the perspective of an accelerated coordinate system like a motor vehicle accelerating from rest, or a stone being spun around in a slingshot. If one chooses to recognize that gravity is caused by the curvature of spacetime (see below), proper acceleration also differs from coordinate acceleration in a gravitational field.

For example, an object subjected to physical or proper acceleration ao will be seen by observers in a coordinate system undergoing constant acceleration aframe to have coordinate acceleration::vec{a}_{acc} = vec{a}_{o} - vec{a}_{frame}.Thus if the object is accelerating with the frame, observers fixed to the frame will see no acceleration at all.


In curved spacetime

In the language of general relativity, the components of an object's acceleration four-vector A (whose magnitude is proper acceleration) are related to elements of the four-velocity via a covariant derivative D with respect to proper time τ:

:A^lambda := frac{DU^lambda }{d au} = frac{dU^lambda }{d au } + Gamma^lambda {}_{mu u}U^mu U^ u

Here U is the object's four-velocity, and Γ represents the coordinate system's 64 connection coefficients or Christoffel symbols. Note that the Greek subscripts take on four possible values, namely 0 for the time-axis and 1-3 for spatial coordinate axes, and that repeated indices are used to indicate summation over all values of that index. Trajectories with zero proper acceleration are referred to as geodesics.

The left hand side of this set of four equations (one each for the time-like and three spacelike values of index λ) is the object's proper-acceleration 3-vector combined with a null time component as seen from the vantage point of a reference or book-keeper coordinate system. The first term on the right hand side lists the rate at which the time-like (energy/mc) and space-like (momentum/m) components of the object's four-velocity U change, per unit time τ on traveler clocks.

Let's solve for that first term on the right since at low speeds its spacelike components represent the coordinate acceleration. More generally, when that first term goes to zero the object's coordinate acceleration goes to zero. This yields...

:frac{dU^lambda }{d au } =A^lambda - Gamma^lambda {}_{mu u}U^mu U^ u.

Thus, as exemplified with the first two animations above, coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by the connection (or "geometric acceleration") term on the far right [cf. R. J. Cook (2004) Physical time and physical space in general relativity, "Am. J. Phys." 72:214-219] . "Caution:" This term may be a sum of as many as sixteen separate velocity and position dependent terms, since the repeated indices μ and ν are by convention summed over all pairs of their four allowed values.

Force and equivalence

The above equation also offers some perspective on forces and the equivalence principle. Consider "local" book-keeper coordinates ] for the metric (e.g. a local Lorentz tetrad ] 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:

:frac{dE}{dt}=vec{v}cdotfrac{dvec{p{dt} (timelike) and frac{dvec{p{dt}=Sigmavec{f_o}+Sigmavec{f_g}=m(vec{a_o}+vec{a_g}) (spacelike).

Here ao is an acceleration due to proper forces and ag 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" ao's magnitude is that of proper acceleration α as in the section above where α=γ3a when ag 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.

urface dwellers on a planet

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

:vec{a}_{shell} = vec{a}_o - sqrt{frac{r}{r-r_s frac{G M}{r^2} hat{r}

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

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

:vec{a}_{shell} = vec{a}_o - g hat{r}

where g is the downward 9.8 m/s2 acceleration due to gravity, and 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.

Four-vector derivations

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 [Hartle, James B. (2003). Gravity: an Introduction to Einstein's General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8662-9.] :

:left(egin{array}{llll} left{Gamma _{t,t}^t,Gamma _{t,r}^t,Gamma _{t, heta }^t,Gamma _{t,phi }^t ight} & left{Gamma _{r,t}^t,Gamma _{r,r}^t,Gamma _{r, heta }^t,Gamma _{r,phi }^t ight} & left{Gamma _{ heta ,t}^t,Gamma _{ heta ,r}^t,Gamma _{ heta , heta }^t,Gamma _{ heta ,phi }^t ight} & left{Gamma _{phi ,t}^t,Gamma _{phi ,r}^t,Gamma _{phi , heta }^t,Gamma _{phi ,phi }^t ight} \ left{Gamma _{t,t}^r,Gamma _{t,r}^r,Gamma _{t, heta }^r,Gamma _{t,phi }^r ight} & left{Gamma _{r,t}^r,Gamma _{r,r}^r,Gamma _{r, heta }^r,Gamma _{r,phi }^r ight} & left{Gamma _{ heta ,t}^r,Gamma _{ heta ,r}^r,Gamma _{ heta , heta }^r,Gamma _{ heta ,phi }^r ight} & left{Gamma _{phi ,t}^r,Gamma _{phi ,r}^r,Gamma _{phi , heta }^r,Gamma _{phi ,phi }^r ight} \ left{Gamma _{t,t}^{ heta },Gamma _{t,r}^{ heta },Gamma _{t, heta }^{ heta },Gamma _{t,phi }^{ heta } ight} & left{Gamma _{r,t}^{ heta },Gamma _{r,r}^{ heta },Gamma _{r, heta }^{ heta },Gamma _{r,phi }^{ heta } ight} & left{Gamma _{ heta ,t}^{ heta },Gamma _{ heta ,r}^{ heta },Gamma _{ heta , heta }^{ heta },Gamma _{ heta ,phi }^{ heta } ight} & left{Gamma _{phi ,t}^{ heta },Gamma _{phi ,r}^{ heta },Gamma _{phi , heta }^{ heta },Gamma _{phi ,phi }^{ heta } ight} \ left{Gamma _{t,t}^{phi },Gamma _{t,r}^{phi },Gamma _{t, heta }^{phi },Gamma _{t,phi }^{phi } ight} & left{Gamma _{r,t}^{phi },Gamma _{r,r}^{phi },Gamma _{r, heta }^{phi },Gamma _{r,phi }^{phi } ight} & left{Gamma _{ heta ,t}^{phi },Gamma _{ heta ,r}^{phi },Gamma _{ heta , heta }^{phi },Gamma _{ heta ,phi }^{phi } ight} & left{Gamma _{phi ,t}^{phi },Gamma _{phi ,r}^{phi },Gamma _{phi , heta }^{phi },Gamma _{phi ,phi }^{phi } ight}end{array} ight)

for the far-coordinate Schwarzschild metric (c dτ)2 = (1-rs/r)(c dt)2 - (1/(1-rs/r))dr2 - r22 - (r sin [θ] )22, where rs is the Schwarzschild radius 2GM/c2. The resulting array of coefficients becomes::left(egin{array}{llll} left{0,frac{r_s}{2 r (r - r_s)},0,0 ight} & left{frac{r_s}{2 r (r - r_s)},0,0,0 ight} & {0,0,0,0} & {0,0,0,0} \ left{frac{r_s c^2 (r-r_s)}{2 r^3},0,0,0 ight} & left{0,frac{r_s}{2 r (r_s-r)},0,0 ight} & {0,0,r_s-r,0} & left{0,0,0,(r_s-r) sin ^2( heta ) ight} \ {0,0,0,0} & left{0,0,frac{1}{r},0 ight} & left{0,frac{1}{r},0,0 ight} & {0,0,0,-cos ( heta ) sin ( heta )} \ {0,0,0,0} & left{0,0,0,frac{1}{r} ight} & {0,0,0,cot ( heta )} & left{0,frac{1}{r},cot ( heta ),0 ight}end{array} ight).

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. A^lambda = Gamma^lambda {}_{mu u}U^mu U^ u = {0,GM/r2,0,0}. This does not solve the problem yet, since Schwarzschild coordinates in curved spacetime are book-keeper coordinates ] but not those of a local observer. The magnitude of the above proper acceleration 4-vector, namely α=Sqrt [1/(1-rs/r)] GM/r2, 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 rs or M above to zero:

:left(egin{array}{llll} left{0,0,0,0 ight} & left{0,0,0,0 ight} & {0,0,0,0} & {0,0,0,0} \ left{0,0,0,0 ight} & left{0,0,0,0 ight} & {0,0,-r,0} & left{0,0,0,-r sin ^2( heta ) ight} \ {0,0,0,0} & left{0,0,frac{1}{r},0 ight} & left{0,frac{1}{r},0,0 ight} & {0,0,0,-cos ( heta ) sin ( heta )} \ {0,0,0,0} & left{0,0,0,frac{1}{r} ight} & {0,0,0,cot ( heta )} & left{0,frac{1}{r},cot ( heta ),0 ight}end{array} ight).

From this we can obtain, for example, the centri"petal" proper acceleration needed to cancel the centri"fugal" 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. A^lambda = Gamma^lambda {}_{mu u}U^mu U^ u = {0,-r(dφ/dτ)2,0,0} so that ao2r. Coriolis effects also reside in these connection coefficients, and similarly arise from coordinate-frame geometry alone.

ee also

*Kinematics: for studying ways that position changes with time
*Uniform acceleration: holding coordinate acceleration fixed
*Four-vector: making the connection between space and time explicit
*Fictitious force: one name for mass times "geometric acceleration"


External links

* Excerpts from the first edition of "Spacetime Physics", and other [ resources posted by Edwin F. Taylor]
* [ James Hartle's gravity book page] including Mathematica programs to calculate Christoffel symbols.
* Andrew Hamilton's [ notes and programs] for working with local tetrads at U. Colorado, Boulder.

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