R2REL

1. From a boost in the [math]\displaystyle{ x }[/math] direction, determine the form of a boost in a direction [math]\displaystyle{ \overrightarrow n=(\cos\theta, sin\theta, 0) }[/math]. Compare with the general form given in lecture.
2. Consider boosts [math]\displaystyle{ B_x(v) }[/math] and [math]\displaystyle{ B_y(v) }[/math] respectively in the [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] directions with the same value of the boost velocity [math]\displaystyle{ v }[/math]. Show that the product [math]\displaystyle{ B_x(v)B_y(v) }[/math] is equivalent to a rotation of an angle [math]\displaystyle{ \phi }[/math] around the [math]\displaystyle{ z }[/math] axis, followed by a boost in a direction [math]\displaystyle{ \overrightarrow n=(\cos\theta, sin\theta, 0) }[/math] of velocity [math]\displaystyle{ v_\theta }[/math]. Determine [math]\displaystyle{ \phi }[/math], [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ v_\theta }[/math].
3. Show that the two boosts of question 2 do not commute. Is the commutator [math]\displaystyle{ [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]\displaystyle{ x(t) }[/math] of a particle with a constant proper acceleration [math]\displaystyle{ \alpha }[/math] in the [math]\displaystyle{ x }[/math] direction, starting from rest at [math]\displaystyle{ 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.