The electromagentism can be formulated in terms of differential forms, defining de electromagnetic four-potential $A$ as a 1-form, the electromagnetic 2-form (to give it a name) $F=dA$, etc. And the Lorentz force is defined as the inner product of the electromagentic tensor, which is the 2-form as a 2-rank tensor (I think you understand what I'm trying to say with that), with the four-velocity(and the charge), but it can be defined directly like some operation using the 2-form $F$ and the four-velocity?


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A $p$-form is, in tensor terms, a tensor whose components have exactly $p$ lower indices that are antisymmetric. For example, the electromagnetic 2-form $F=dA$ can be written in terms of its components as follows:

$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$

Meanwhile, a vector is, in tensor terms, a tensor whose components have exactly one upper index. For example, the relativistic 4-velocity has components $u^\mu$.

There is a way to take the product of a vector and a $p$-form and get a $p-1$-form from it; it's called the interior product. The equations on the Wikipedia page may look formidable, but in terms of the components, you simply contract the (upper) vector index with the first (lower) index of the $p$-form:

$$(\iota_u F)_\nu = u^\mu F_{\mu\nu},$$

where $\iota_u F$ denotes the interior product of the vector $u$ with the 2-form $F$, and $(\iota_u F)_\nu$ are the components of the resulting 1-form. (Note that if you contract one index of a $p$-form you will have $p-1$ lower antisymmetric indices left, and that's why the interior product takes $p$-forms to $p-1$-forms.)

Using the interior product, we can now very easily write down the force 1-form $f$ as the interior product of the 4-velocity $u$ with the electromagnetic 2-form $F$, times the electric charge $q$:

$$f = q \, \iota_u F,$$

or in terms of components,

$$f_\nu = q \, u^\mu F_{\mu\nu}.$$

Note: Sometimes you will see the interior product written as $\iota_u F = u \lrcorner F$. In this notation, the Lorentz force is $f = q \, u \lrcorner F$.

Further reading: Geometry, Topology And Physics, 2nd Edition, by Nakahara.


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