- #1

- 560

- 2

The covariant form of the Lorentz force can be written as

[tex]m \ddot x^\mu =q F^{\mu \nu} \dot x_\nu [/tex]

and such a relation should prove by the quotient rule that F is indeed a tensor.

But what kind of tensor is it? One can show that it transforms from an unprimed

to a primed system like

[tex] F'^{\mu \nu} = \Lambda^\mu_{\ \alpha} \Lambda^{\ \nu}_{\beta} F^{\alpha \beta} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta}[/tex],

where [tex]\Lambda^{\ \nu}_{\beta}[/tex] is the inverse Lorentz transformation matrix. But what kind of tensors transforms like this? Does it have a name? I know about covariant, contravariant and mixed tensors. The closest I get is a mixed tensor but it would transform like

[tex] T^\mu_\nu = \frac{\partial x'^\mu}{\partial x^\rho }\frac{\partial x^\sigma}{\partial x'^\nu} T^\rho_\sigma[/tex].

[tex]m \ddot x^\mu =q F^{\mu \nu} \dot x_\nu [/tex]

and such a relation should prove by the quotient rule that F is indeed a tensor.

But what kind of tensor is it? One can show that it transforms from an unprimed

to a primed system like

[tex] F'^{\mu \nu} = \Lambda^\mu_{\ \alpha} \Lambda^{\ \nu}_{\beta} F^{\alpha \beta} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta}[/tex],

where [tex]\Lambda^{\ \nu}_{\beta}[/tex] is the inverse Lorentz transformation matrix. But what kind of tensors transforms like this? Does it have a name? I know about covariant, contravariant and mixed tensors. The closest I get is a mixed tensor but it would transform like

[tex] T^\mu_\nu = \frac{\partial x'^\mu}{\partial x^\rho }\frac{\partial x^\sigma}{\partial x'^\nu} T^\rho_\sigma[/tex].

Last edited: