Covariant Uniform Acceleration

8th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields

Friedman, Yaakov^1 ; Scarr, Tzvi^1

Jerusalem College of Technology, Departments of Mathematics and Physics, P.O.B. 16031, Jerusalem 91160, Israel^1

关键词: Accelerated observers;    Acceleration transformations;    Constant acceleration;    Linear accelerations;    Relativistic dynamics;    Relativistic momentum;    Rotational acceleration;    Space-time transformations;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/437/1/012009/pdf
DOI  :  10.1088/1742-6596/437/1/012009

来源: IOP

【 摘 要 】

We derive a 4D covariant Relativistic Dynamics Equation. This equation canonically extends the 3D relativistic dynamics equation F = dp/dt, where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation F = dp/dτ is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck. We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational, and general. For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion. We use Generalized Fermi-Walker transport to construct a uniformly accelerated family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K′ to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame. We obtain velocity and acceleration transformations from a uniformly accelerated system K′ to an inertial frame K. We introduce the 4D velocity, an adaptation of Horwitz and Piron s notion of "off-shell." We derive the general formula for the time dilation between accelerated clocks. We obtain a formula for the angular velocity of a uniformly accelerated object. Every rest point of K′ is uniformly accelerated, and its acceleration is a function of the observer's acceleration and its position. We obtain an interpretation of the Lorentz-Abraham-Dirac equation as an acceleration transformation from K′ to K.