In this paper, we give the covariant formulation of second gradient electrodynamics, which is a generalized electrodynamics of second order including derivatives of higher order. The relativistic form of the field equations, the energy-momentum tensor and the Lorentz force density are presented. For an electric point charge, the generalized Lienard-Wiechert potentials and the corresponding electromagnetic field strength tensor are given as retarded integral expressions. Explicit formulas for the electromagnetic potential vector and electromagnetic field strength tensor of a uniformly moving point charge are found without any singularity and discontinuity. In addition, a world-line integral expression for the self-force of a charged point particle is given. The relativistic equation of motion of a charged particle coupled with electromagnetic fields in second gradient electrodynamics is derived, which is an integro-differential equation with nonlocality in time. For a uniformly accelerated charge, explicit formulas of the self-force and the electromagnetic mass, being non-singular, are given. Moreover, the wave propagation and the dispersion relations in the vacuum of second gradient electrodynamics are analyzed. Three modes of waves were found: one non-dispersive wave as in Maxwell electrodynamics, and two dispersive waves similar to the wave propagation in a collisionless plasma.

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第二梯度电动力学：非奇异相对论场论

在本文中，我们给出了二阶梯度电动力学的协变公式，它是包括高阶导数在内的二阶广义电动力学。给出了场方程的相对论形式、能量-动量张量和洛伦兹力密度。对于点电荷，广义的 Lienard-Wiechert 势和相应的电磁场强度张量作为延迟积分表达式给出。找到了均匀运动点电荷的电磁势矢量和电磁场强度张量的明确公式，没有任何奇异性和不连续性。此外，还给出了带电点粒子自力的世界线积分表达式。推导出第二梯度电动力学中带电粒子与电磁场耦合运动的相对论方程，该方程是时间非定域性的积分微分方程。对于均匀加速的电荷，给出了非奇异的自力和电磁质量的明确公式。此外，分析了二次梯度电动力学在真空中的波传播和色散关系。发现了三种波模式：一种是麦克斯韦电动力学中的非色散波，另一种类似于无碰撞等离子体中的波传播的色散波。是非奇异的，是给定的。此外，分析了二次梯度电动力学在真空中的波传播和色散关系。发现了三种波模式：一种是麦克斯韦电动力学中的非色散波，另一种类似于无碰撞等离子体中的波传播的色散波。是非奇异的，是给定的。此外，分析了二次梯度电动力学在真空中的波传播和色散关系。发现了三种波模式：一种是麦克斯韦电动力学中的非色散波，另一种类似于无碰撞等离子体中的波传播的色散波。