= Solution
\b[Covered by the official notes:] definitions of four-force and four-velocity and the vanishing antisymmetric contraction (PDF p. 15; handwritten p. 10).<Symmetry and Relativity 2011 Exam Solution Notes>[PDF p. 15; handwritten p. 10: definitions of four-force and four-velocity and the vanishing antisymmetric contraction]{ref}
To make the necessity statement precise, mass preservation must hold \b[for every allowed timelike four-velocity], with the same field <tensor> and nonzero charge. It is not necessary for one prescribed trajectory alone. Use <metric signature> $(-,+,+,+)$, so $U_aF^a=-c^2dm/d\tau$. Put $H^{ab}=(\mathcal F^{ab}+\mathcal F^{ba})/2$. The mass-preservation condition requires the <quadratic form> $H^{ab}U_aU_b=0$ for every timelike $U$. Every timelike vector can be rescaled to a physical <four-velocity>; by homogeneity the quadratic form vanishes on the open timelike cone. A polynomial vanishing on an open set is identically zero, so $H^{ab}=0$. Thus the <electromagnetic field tensor> must be antisymmetric under this universal quantifier. For a single velocity, the scalar contraction can vanish even for a nonzero symmetric part, which is why the quantifier is essential.
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