So the Lorentz force on a massive particle is given by $f^{\mu} = qg^{\mu\alpha}F_{\alpha\beta}\hat{v}^{\beta}$, where $\hat{v}^{\beta}$ is the four vector of the particle and $F_{\alpha\beta} = \partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$ is the EM field strength tensor, with EM potential $A_{\alpha}$.
I want to show that this force will always be spacelike. And since the four-velocity of a massive particle will always be time-like, then I can prove $f^{\mu}$ is always spacelike if: $$g_{\mu\nu}\hat{v}^{\mu}f^{\nu}=0$$
I feel like what I am doing is correct, but I am unable to show its zero. I think part of the problem is that I am quite new to abstract index notation, and am unsure how to simplify expressions. Thus far, I have worked out:
$$g_{\mu\nu}\hat{v}^{\mu}f^{\nu} = g_{\mu\nu}\hat{v}^{\mu}qg^{\nu\alpha}F_{\alpha\beta}\hat{v}^{\beta}=q\delta^{\alpha}_{\mu}F_{\alpha\beta}\hat v^{\beta}\hat v^{\mu} $$
Assuming this approach is valid, can anyone hint at where one can go from here? Also, what are some general guidelines or rules for manipulating/simplifying tensor expressions such as these?