Editing Four-acceleration

In the [[theory of relativity]], '''four-acceleration''' is a [[four-vector]] (vector in four-dimensional [[spacetime]]) that is analogous to classical [[acceleration]] (a three-dimensional vector, see [[acceleration (special relativity)|three-acceleration in special relativity]]).  Four-acceleration has applications in areas such as the annihilation of [[antiproton]]s, resonance of [[strangeness|strange particles]] and radiation of an accelerated charge.<ref>{{cite book|title=Special Relativity|author=Tsamparlis M.|year=2010|page=185|edition=Online|publisher=Springer Berlin Heidelberg|isbn=978-3-642-03837-2}}</ref>

== Four-acceleration in inertial coordinates ==

In inertial coordinates in [[special relativity]], four-acceleration <math>\mathbf{A}</math> is defined as the rate of change in [[four-velocity]] <math>\mathbf{U}</math> with respect to the particle's [[proper time]] along its [[worldline]].  We can say:
<math display="block">\begin{align}
  \mathbf{A} = \frac{d\mathbf{U}}{d\tau}
    &= \left(\gamma_u\dot\gamma_u c,\, \gamma_u^2\mathbf a + \gamma_u\dot\gamma_u\mathbf u\right) \\
    &= \left(
         \gamma_u^4\frac{\mathbf{a}\cdot\mathbf{u}}{c},\,
         \gamma_u^2\mathbf{a} + \gamma_u^4\frac{\mathbf{a}\cdot\mathbf{u}}{c^2}\mathbf{u}
       \right) \\
    &= \left(
         \gamma_u^4\frac{\mathbf{a}\cdot\mathbf{u}}{c},\,
         \gamma_u^4\left(\mathbf{a} + \frac{\mathbf{u}\times \left(\mathbf{u}\times\mathbf{a}\right)}{c^2}\right)
       \right),
\end{align}</math>
where
* <math>\mathbf a = \frac{d\mathbf u}{dt}</math> , with <math>\mathbf a </math> the three-acceleration and <math>\mathbf u </math> the three-velocity, and 
* <math>\dot\gamma_u
  = \frac{\mathbf a \cdot \mathbf u}{c^2} \gamma_u^3
  = \frac{\mathbf a \cdot \mathbf u}{c^2} \frac{1}{\left(1 - \frac{u^2}{c^2}\right)^{3/2}},
</math> and
* <math>\gamma_u</math> is the [[Lorentz factor]] for the speed <math>u</math> (with <math>|\mathbf{u}| = u</math>).  A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time <math>\tau</math> (in other terms, <math display="inline">\dot\gamma_u = \frac{d\gamma_u}{dt}</math>).

In an instantaneously co-moving inertial reference frame <math>\mathbf u = 0</math>, <math>\gamma_u = 1 </math> and <math>\dot\gamma_u = 0</math>, i.e. in such a reference frame 
<math display="block">\mathbf{A} = \left(0, \mathbf a\right) .</math>

Geometrically, four-acceleration is a [[curvature vector]] of a worldline.<ref>{{cite book| author=Pauli W.|title=Theory of Relativity |edition=1981 Dover|publisher=B.G. Teubner, Leipzig|year=1921|pages=74|isbn=978-0-486-64152-2}}</ref><ref>{{cite book| author1=Synge J.L.|author2=Schild A.|title=Tensor Calculus|edition=1978 Dover|publisher=University of Toronto Press| year=1949| isbn=0-486-63612-7|pages=[https://archive.org/details/tensorcalculus00syng/page/149 149, 153 and 170]|url-access=registration| url=https://archive.org/details/tensorcalculus00syng/page/149}}</ref>

Therefore, the magnitude of the four-acceleration (which is an invariant scalar) is equal to the [[proper acceleration]] that a moving particle "feels" moving along a worldline. A worldline having constant four-acceleration is a Minkowski-circle i.e. hyperbola (see [[hyperbolic motion (relativity)|''hyperbolic motion'']])

The [[scalar product]] of a particle's [[four-velocity]] and its four-acceleration is always 0.

Even at relativistic speeds four-acceleration is related to the [[four-force]]: <math display="block"> F^\mu = m A^\mu ,</math> where {{mvar|m}} is the [[invariant mass]] of a particle.

When the [[four-force]] is zero, only gravitation affects the trajectory of a particle, and the four-vector equivalent of Newton's second law above reduces to the [[geodesic equation]]. The four-acceleration of a particle executing geodesic motion is zero.  This corresponds to gravity not being a force.  Four-acceleration is different from what we understand by acceleration as defined in Newtonian physics, where gravity is treated as a force.

== Four-acceleration in non-inertial coordinates ==

In non-inertial coordinates, which include accelerated coordinates in special relativity and all coordinates in [[general relativity]], the acceleration four-vector is related to the [[four-velocity]] through an [[absolute derivative]] with respect to proper time.

<math display="block">A^\lambda := \frac{DU^\lambda }{d\tau} = \frac{dU^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu </math>

In inertial coordinates the [[Christoffel symbols]] <math>\Gamma^\lambda {}_{\mu \nu}</math> are all zero, so this formula is compatible with the formula given earlier.

In special relativity the coordinates are those of a rectilinear inertial frame, so the [[Christoffel symbols]] term vanishes, but sometimes when authors use curved coordinates in order to describe an accelerated frame, the frame of reference isn't inertial, they will still describe the physics as special relativistic because the metric is just a frame transformation of the [[Minkowski space]] metric. In that case this is the expression that must be used because the [[Christoffel symbols]] are no longer all zero.

==See also==
* [[Four-vector]]
* [[Four-velocity]]
* [[Four-momentum]]
* [[Four-force]]
* [[Four-gradient]]
* [[Proper acceleration]]

== References ==
{{reflist}}

*{{cite book|author=Papapetrou A.|title=Lectures on General Relativity|publisher=D. Reidel Publishing Company|year=1974|isbn=90-277-0514-3}}
* {{cite book| author=Rindler, Wolfgang| title=Introduction to Special Relativity (2nd)| publisher=Oxford: Oxford University Press| year=1991| isbn=0-19-853952-5| url-access=registration| url=https://archive.org/details/introductiontosp0000rind}}
*

== External links ==
*[http://www.britannica.com/EBchecked/topic/147246/curvature-vector Curvature vector] on [[Britannica]]

{{DEFAULTSORT:Four-Acceleration}}
[[Category:Four-vectors]]
[[Category:Acceleration]]