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 / 7

The official notes give only “Part 1: as in textbooks” (PDF p. 13, handwritten p. 8). The requested derivation is supplied here.

Put a light source at the rod midpoint in its rest frame $S$ and emit two pulses simultaneously toward its ends, each a distance $\ell$ away. In $S$, the reflection events are $(t,x)=(\ell /c,\pm \ell )$. Invariance of light speed, the Einstein's principle of relativity and linearity give a Lorentz transformation $x^{\prime }=A(x-vt)$, $t^{\prime }=A(t-vx/c^2)$. Applying the inverse transformation with speed $-v$ gives $A^2(1-v^2/c^2)=1$, hence $A=\gamma$.

The transformed reflection events haveThese events are not simultaneous in $S^{\prime }$. Both rod ends move at $-v$ there, so extrapolating each endpoint to $t^{\prime }=0$ gives $x_{\pm }^{\prime }(0)=x_{\pm }^{\prime }+v{t}_{\pm }^{\prime }=\pm \ell /\gamma$. Measuring simultaneous endpoints therefore yields length contraction:

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]

The orientation-dependent results in the official notes show that the measured spring constant is not a Lorentz scalar. A Lorentz transformation changes both the simultaneous extension and the measured three-force; their ratio depends on the orientation. This does not make the spring intrinsically anisotropic in its own rest frame, or require a preferred inertial frame. It is a quasistatic effective stiffness comparison between frames; an extended real spring also has finite internal signal speeds.

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]