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Covariant Four-Dimensional Relativistic Dynamics Equation
The standard 3D relativistic dynamics equation
 * $$ \mathbf{F} = \frac{d\mathbf{p}}{dt} $$

is an adaptation of Newton's Second Law $$ \mathbf{F}=m\mathbf{a} $$ to special relativity. As above,$$ \mathbf{F}$$  is the 3D force, and  $$\mathbf{p}=m\gamma\mathbf{v}$$ is the 3D relativistic momentum. When the 3D force $$\mathbf{F}$$ is constant, the solutions to the above equation are traditionally called uniformly accelerated motion. This equation, however, is covariant only with respect to the little Lorentz group andnot covariant with respect to the full Lorentz group.

As a 4D covariant extension, we have
 * $$ F_\nu=\frac{dp_\nu}{d\tau},$$

where $$F$$ is the four-force, $$p$$ is the four-momentum, and $$\tau$$ is proper time. Unfortunately, this equation is also covariant only with respect to the little Lorentz group. Moreover, when $$F$$ is a constant, as in a homogeneous gravitational field, this equation has no solution! This follows from the fact that the four-velocity and the four-acceleration are perpendicular. This was noticed by Planck, who wrote to Einstein about it. This, in turn, prompted Einstein to submit a ``correction" to . In the correction, he states that the ``concept `uniformly accelerated' needs further clarification." This was a call for a fully Lorentz covariant relativistic dynamics equation and for a better definition of ``uniform acceleration."

It was clear, even in 1908, that the physical definition of uniformly accelerated motion is motion whose acceleration is constant in the comoving frame. This definition is found widely in the literature, as early as and, again in , and as recently as and. This definition is natural, since the acceleration in the comoving frame is "precisely the push we feel when sitting in an accelerating rocket" or automobile. Similarly, "by the equivalence principle, the gravitational field in our terrestrial lab is the negative of our proper acceleration, our instantaneous rest frame being an imagined Einstein cabin falling with acceleration g." (\cite{Rindler}, page 71).

If the acceleration is constant in the comoving frame, then the length of the four-acceleration a is constant:

a^\mu a_\mu= \hbox{constant}.$$

This equation is a good candidate to replace $$ F_\nu=\frac{dp_\nu}{d\tau}$$. It's even fully Lorentz covariant. However, existing techniques have produced only one-dimensional hyperbolic motion as solutions. There are clearly some missing solutions, since this equation is covariant, while the class of 1D hyperbolic motions is not.

In Friedman and Scarr introduce a new covariant four-dimensional Relativistic Dynamics Equation:

c\frac{du^{\mu}}{d\tau}=A^{\mu}_{\nu}u^{\nu}, $$ where $$u$$ is the four-velocity, $$\tau$$ is proper time, and $$A_{\mu\nu}$$ is a rank 2 antisymmetric tensor, or, equivalently, $$A^\mu_\nu$$ is skew adjoint with respect to the Minkowski inner product $$\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$$. The Friedman-Scarr equation has the following advantages: c\frac{du^{\mu}}{d\tau}=A^{\mu}_{\nu}u^{\nu}, $$ have constant acceleration in the comoving frame. Y. Friedman, Yu. Gofman, Physica Scripta, 82 (2010) 015004 <\ref> Y. Friedman, Ann. Phys. (Berlin) 523 (2011) 408 <\ref> Y. Friedman and E. Resin Dynamics of hydrogen-like atom bounded by maximal acceleration, Physica Scripta,  86 (2012) 015002 <\ref>
 * It canonically extends $$ \mathbf{F} = \frac{d\mathbf{p}}{dt} $$ and is covariant with respect to the full Lorentz group
 * By redefining uniformly accelerated motion as the solutions to the Friedman-Scarr equation, we obtain the clarification that Einstein was looking for
 * It admits four Lorentz-invariant classes of solutions: null acceleration, linear acceleration, rotational acceleration, and general acceleration. The null, rotational, and general classes were previously unknown. The linear class is a covariant extension of 1D hyperbolic motion and contains the motion of an object in a homogeneous gravitational field
 * It can be extended in a straightforward manner to obtain a covariant definition of the "comoving frame" of a uniformly accelerated observer. All of the solutions of $$
 * It can be modified to accommodate a universal maximal acceleration. Thus, this paper is an important step in the study of evidences for and implications of the existence of a universal maximal acceleration