Comment on higher derivative Lagrangians in relativistic theory

Abstract

We discuss the consequences of higher derivative Lagrangians of the form ${\alpha }_1{A}_{\mu }(x){\dot{x}}^{\mu }$, ${\alpha }_2{G}_{\mu }(x){\ddot{x}}^{\mu }$, ${\alpha }_3{B}_{\mu }(x){\stackrel{...}{x}}^{\mu }$, ${\alpha }_4{K}_{\mu }(x){\stackrel{....}{x}}^{\mu }$, $\cdots$, $U_{(n)\mu }(x)x^{(n)\mu }$ in relativistic theory. After establishing the equations of the motion of particles in these fields, we introduce the concept of the generalized induction principle assuming the coupling between the higher fields $U_{(n),\mu }(x),n\ge 1$ with the higher currents $j^{(n)\mu }=\rho (x)x^{(n)\mu }$, where $\rho (x)$ is the spatial density of mass or of electric charge. In addition, we discuss the analogy of the field $G_{\mu }(x)$ with the gravitational field and its inclusion in the general relativity framework in the last section. This letter is an invitation to reflect on a generalisation of the concept of inertia and we also discuss this problem in the last section.

References in corpus (1)

• Modifying gravity with the Aether: an alternative to Dark Matter