Four-acceleration

In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particle's proper time:

$\mathbf{A} =\frac{d\mathbf{U}}{d\tau}=\left(\gamma_u\dot\gamma_u c,\gamma_u^2\mathbf a+\gamma_u\dot\gamma_u\mathbf u\right) =\left(\gamma_u^4\frac{\mathbf{a}\cdot\mathbf{u}}{c},\gamma_u^2\mathbf{a}+\gamma_u^4\frac{\left(\mathbf{a}\cdot\mathbf{u}\right)}{c^2}\mathbf{u}\right)$,

where

$\mathbf a = {d\mathbf u \over dt}$

and

$\dot\gamma_u = \frac{\mathbf{a \cdot u}}{c^2} \gamma_u^3 = \frac{\mathbf{a \cdot u}}{c^2} \frac{1}{\left(1-\frac{u^2}{c^2}\right)^{3/2}}$

and γ_u is the Lorentz factor for the speed u. A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time τ.

In an instantaneously co-moving inertial reference frame $\mathbf u = 0$, γ_u = 1 and $\dot\gamma_u = 0$, i.e. in such a reference frame

$\mathbf{A} =\left(0, \mathbf a\right)$

Geometrically, four-acceleration is a curvature vector of world line.

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)

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

F^μ = mA^μ

where m is the invariant mass of a particle.

In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant derivative with respect to proper time.

$A^\lambda := \frac{DU^\lambda }{d\tau} = \frac{dU^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu$

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.

See also

• four-vector
• four-velocity
• four-momentum
• four-force

Further reading

• Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5.