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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). 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>
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== Four-acceleration in inertial coordinates ==
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In inertial coordinates in [[special relativity]], four-acceleration is defined as the change in [[four-velocity]] over the particle's [[proper time]]:
 
: <math>\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{\left(\mathbf{a}\cdot\mathbf{u}\right)}{c^2}\mathbf{u}\right)</math>,
 
where
 
: <math>\mathbf a = {d\mathbf u \over dt}</math>
and
: <math>\dot\gamma_u = \frac{\mathbf{a \cdot u}}{c^2} \gamma_u^3 = \frac{\mathbf{a \cdot 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>.  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 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>\mathbf{A} =\left(0, \mathbf a\right)</math>
 
Geometrically, four-acceleration is a [[curvature vector]] of a [[world line]].<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=149, 153 and 170}}</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 [[world line]].
The world lines having constant magnitude of four-acceleration are Minkowski-circles i.e. hyperbolas (see [[hyperbolic motion (relativity)|''hyperbolic motion'']])
 
The [[scalar product]] of a [[four-velocity]] and the corresponding four-acceleration is always 0.
 
Even at relativistic speeds four-acceleration is related to the [[four-force]] such that
 
: <math> F^\mu = mA^\mu</math>
 
where ''m'' is the [[invariant mass]] of a particle.
 
In special relativity the coordinates are those of a rectilinear inertial frame, so the [[Christoffel symbols]] term vanishes, but sometimes when authors uses 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.
 
When the [[four-force]] is zero one has gravitation acting alone, and the four-vector version of Newton's second law above reduces to the [[geodesic equation]]. A particle executing geodesic motion has  zero value for each component of the acceleration four vector.This conforms to the fact that Gravity is not a force. The radial component four-acceleration of a body under free-fall,incidentally, is zero.This identifies the fact that the spatial part of four-acceleration is different from what we understand by acceleration in common day to day experience for example in the case of an apple falling from a tree.
 
== Four-acceleration in non-inertial coordinates ==
 
In non-inertial coordinates, which include accelerated coordinates in special relatrivity 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>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.
 
==See also==
* [[four-vector]]
* [[four-velocity]]
* [[four-momentum]]
* [[four-force]]
 
== 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}}
*
 
== External links ==
*[http://www.britannica.com/EBchecked/topic/147246/curvature-vector Curvature vector] on [[Britannica]]
 
{{DEFAULTSORT:Four-Acceleration}}
[[Category:Minkowski spacetime]]
[[Category:Theory of relativity]]
[[Category:Acceleration]]

Latest revision as of 22:16, 12 January 2015

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