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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 In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
(a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has applications in areas such as the annihilation of
antiproton The antiproton, , (pronounced ''p-bar'') is the antiparticle of the proton. Antiprotons are stable, but they are typically short-lived, since any collision with a proton will cause both particles to be annihilated in a burst of energy. The exis ...
s, resonance of strange particles and radiation of an accelerated charge.


Four-acceleration in inertial coordinates

In inertial coordinates in special relativity, four-acceleration \mathbf is defined as the rate of change in four-velocity \mathbf with respect to the particle's
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
along its
worldline The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from con ...
. We can say: \begin \mathbf = \frac &= \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,\, \gamma_u^2\mathbf + \gamma_u^4\frac\mathbf \right) \\ &= \left( \gamma_u^4\frac,\, \gamma_u^4\left(\mathbf + \frac\right) \right), \end where * \mathbf a = \frac , with \mathbf a the three-acceleration and \mathbf u the three-velocity, and * \dot\gamma_u = \frac \gamma_u^3 = \frac \frac, and * \gamma_u is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
for the speed u (with , \mathbf, = u). A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time \tau (in other terms, \dot\gamma_u = \frac). In an instantaneously co-moving inertial reference frame \mathbf u = 0, \gamma_u = 1 and \dot\gamma_u = 0, i.e. in such a reference frame \mathbf = \left(0, \mathbf a\right) . Geometrically, four-acceleration is a curvature vector of a worldline. 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'') 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: F^\mu = m A^\mu , where 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
. 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 In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
with respect to proper time. A^\lambda := \frac = \frac + \Gamma^\lambda _U^\mu U^\nu In inertial coordinates the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
\Gamma^\lambda _ 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 In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
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 In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
are no longer all zero.


See also

* Four-vector * Four-velocity * Four-momentum * Four-force *
Four-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties ...
* Proper acceleration


References

* * *


External links


Curvature vector
on Britannica {{DEFAULTSORT:Four-Acceleration Four-vectors Acceleration