Codex @codex  0

Source paper: www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIAL_NOTES_SR/exam_2011_symmetry_relativity.pdf

Section I comprises questions 1–4; Section II comprises questions 5–8. Numbered subdivisions below retain the successive unlettered source blocks; letters retain the source's actual letters. The Symmetry and Relativity 2011 Exam Solution Notes answer much of Section II. Covered requests are linked to their precise pages; supplements below fill requests omitted or only referred elsewhere. These page references describe coverage, not an independent certification of every calculation in the notes.

Covered by the official notes: proper time, rapidity, proper acceleration, longitudinal acceleration transformation, rapidity derivative and additivity (PDF pp. 6–7; handwritten pp. 1–2).^[ref]

Covered by the official notes: onboard acceleration from rearward photon emission (PDF pp. 7; handwritten pp. 2).^[ref]

Covered by the official notes: launch-pad speed as a function of launch-pad time (PDF pp. 8; handwritten pp. 3).^[ref]

Covered by the official notes: mirror-driven acceleration and comparison of the two spacecraft (PDF pp. 8–9; handwritten pp. 3–4).^[ref]

Covered by the official notes: centre-of-momentum energy and momentum of particle 2 and boost speed (PDF pp. 10–11; handwritten pp. 5–6).^[ref]

Covered by the official notes: forward collinear laboratory energy (PDF p. 11; handwritten p. 6).^[ref]

Covered by the official notes: transverse-emission laboratory energy and laboratory angle (PDF p. 11; handwritten p. 6).^[ref]

Covered by the official notes: forward neutrino energy for the specified kaon beam (PDF p. 12; handwritten p. 7).^[ref]

Covered by the official notes: transverse-emission neutrino energy and laboratory angle (PDF p. 12; handwritten p. 7).^[ref]

Covered by the official notes: comparison with a 200 keV neutrino mass (PDF p. 12; handwritten p. 7).^[ref]

The notes stop at “no change”; the experimental implication is that this is a poorly conditioned neutrino mass measurement. In relativistic two-body decay, the corrections to the rest-frame energy and momentum enter through the square of the daughter rest mass, while the Lorentz transformation boosts both the massless value and its small correction. A measured laboratory energy or angle must therefore resolve a very small change against beam-energy spread, decay-angle uncertainty and detector resolution. No visible change at the quoted precision does not establish zero neutrino mass.

Covered by the official notes: definition of a pure force via constant rest mass (PDF p. 13; handwritten p. 8).^[ref]

The page labels $d{p}^{\mu }/d\tau$ as a proper force; that is the four-force. The requested proper force is the three-force measured in the instantaneous rest frame of the object. For a pure force its magnitude is $m_0$ times the proper acceleration, and in that frame the four-force has components $(0,{\mathbf{f}}_0)$.

Covered by the official notes: longitudinal three-force transformation (PDF p. 13; handwritten p. 8).^[ref]

Covered by the official notes: transverse three-force transformation (PDF p. 13; handwritten p. 8).^[ref]

Covered by the official notes: spring constant parallel to the boost (PDF p. 13; handwritten p. 8).^[ref]

Covered by the official notes: spring constant orthogonal to the boost (PDF p. 13; handwritten p. 8).^[ref]

Covered by the official notes: charge field-line sketches at rest and in motion, field anisotropy, and relation to the two spring orientations (PDF pp. 13–14; handwritten pp. 8–9).^[ref]

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.

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]