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- 95

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## Homework Statement

Show that the Lorentz Force Law, [itex]\frac{dp^{\nu}}{d \tau} = -q U_{\mu} F^{\mu \nu}[/itex], is consistent with [itex]P^\mu P_\mu= -m^2.[/itex] Here U is the 4-velocity, F is the Electromagnetic Tensor, and p is the 4-momentum. (Minkowski Space)

## Homework Equations

As stated above.

## The Attempt at a Solution

I initially tried differentiating [itex]p^\nu p_\nu = -m^2[/itex] w.r.t. tau. This gave me, through the product rule,

[tex]p_\nu \frac{d p^\nu}{d \tau} + p^\nu \frac{d p_\nu}{d \tau} = 0 [/tex]. Since the only difference between the covariant and contravariant forms of the Lorentz Force Law is whether the [itex]\nu[/itex] on the F is raised of lowered. Substituting the Lorentz Force Law into the above, we get:

[tex] -q ( U_\mu F^{\mu \nu} p_\nu + U_\mu F^\mu_\nu p^\nu) = 0 [/tex]

-q cancels, leaving

[tex] ( U_\mu F^{\mu \nu} p_\nu + U_\mu F^\mu_\nu p^\nu) = 0 [/tex]

This is where I'm stuck. Both the left hand term and the right hand term are constants, and if I showed that one was the negative of the other I'd probably be set, but I'm not sure where to go from here.

Thanks!