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R2REL

From IFPA wiki
  1. From a boost in the [math]x[/math] direction, determine the form of a boost in a direction [math]\overrightarrow n=(\cos\theta, sin\theta, 0)[/math]. Compare with the general form given in lecture.
  2. Consider boosts [math]B_x(v)[/math] and [math]B_y(v)[/math] respectively in the [math]x[/math] and [math]y[/math] directions with the same value of the boost velocity [math]v[/math]. Show that the product [math]B_x(v)B_y(v)[/math] is equivalent to a rotation of an angle [math]\phi[/math] around the [math]z[/math] axis, followed by a boost in a direction [math]\overrightarrow n=(\cos\theta, sin\theta, 0)[/math] of velocity [math]v_\theta[/math]. Determine [math]\phi[/math], [math]\theta[/math] and [math]v_\theta[/math].
  3. Show that the two boosts of question 2 do not commute. Is the commutator [math][B_x(v),B_y(v)]=B_x(v)B_y(v)-B_y(v)B_x(v)[/math] a rotation?
  4. What is the worldline [math]x(t)[/math] of a particle with a constant proper acceleration [math]\alpha[/math] in the [math]x[/math] direction, starting from rest at [math]t=0[/math]?
  5. Calculate the value of the mass shift of a hydrogen atom in its ground state. Compare it with the sum of the masses of the electron and of the proton. Same question for deuterium (bound state of a neutron and of a proton), and compare this mass shift to the sum of the masses of the proton and of the neutron.

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