Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 8

Covered by the official notes: definitions of four-force and four-velocity and the vanishing antisymmetric contraction (PDF p. 15; handwritten p. 10).^[ref]

To make the necessity statement precise, mass preservation must hold 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 $U_a{F}^a=-c^{2}dm/d\tau$. Put $H^{ab}=({\mathcal{F}}^{ab}+{\mathcal{F}}^{ba})/2$. The mass-preservation condition requires the quadratic form $H^{ab}{U}_a{U}_b=0$ 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.

The official notes state only ${\mathcal{F}}^{\prime }=\Lambda \mathcal{F}{\Lambda }^T$ (PDF p. 15, handwritten p. 10); they do not evaluate the requested coefficients. For a boost along positive $x$, use ${\Lambda }^0{}_0={\Lambda }^1{}_1=\gamma$, ${\Lambda }^0{}_1={\Lambda }^1{}_0=-\gamma v/c$, with the transverse diagonal entries equal to one. Multiplying the specified electromagnetic field tensor givesThe corresponding Lorentz transformation of electromagnetic fields isConsequentlyThese coefficients depend on the frame speed, but are constant for the fixed boost.

Covered by the official notes: explicit field contraction and Lorentz invariance as a tensor scalar (PDF pp. 15–16; handwritten pp. 10–11).^[ref]

Covered by the official notes: rocket-frame electric and magnetic fields, proper three-force and proper acceleration (PDF p. 16; handwritten p. 11).^[ref]