Problem sheet 2

Lecture: Introduction to general relativity, 2023-2024

Deadline: Nov 17th, 2023 (3 weeks)

Topic: special relativity, tensor algebra

Please send the solutions via email to korzynski@cft.edu.pl. I prefer PDF's (for example LaTeX-generated) or common graphic formats (for example scans of your hand-written notes).

Problem 1.

Recall the Faraday tensor from Lecture 3 and the derivation of the two of four Maxwell's equations during the lecture. Derive the two remaining Maxwell's equations from the equation

F_{\mu\nu,\alpha} + F_{\nu\alpha,\mu} + F_{\alpha\mu,\nu} = 0.

Hint:

Show that if the value of any two or three indices repeats in the formula above (for example $\alpha=\mu=0$ ) then it is trivially satisfied, i.e. it yields no new equations.
Consider the formula above for all possible triples of different index values, i.e. $(\alpha, \mu, \nu) = (0,1,2), (0, 1, 3), (0, 2, 3), \dots$ . Substitute these values and show that you obtain the rest of the Maxwell's equations component by component.

Problem 2. Relativistic beaming paradox

This is a more of a conceptual problem than a calculational one. Don't worry if you find it too difficult, it's not obligatory, but it can be fun to think about.

Consider a massive point source $S$ emitting isotropically light in all directions as measured in its inertial rest frame $P$ . In a boosted inertial frame $Q$ the body has velocity $\vec v$ and its radiation appears anisotropic, i.e. the energy is collimated in the direction of propagation of the body. Therefore the electromagnetic radiation emitted by $S$ has a non-vanishing net momentum in the direction of $\vec v$ in $Q$ .

This in turn means that there must be a net force in direction $-\vec v$ acting on $S$ due to the momentum conservation. It follows that $S$ should experience a drag-like force due to its light emission and gradually slow down its motion.

But this conclusion is obviously incompatible with special relativity or the principle of relativity: the direction and the magnitude of this drag and of the body's deceleration depends on the velocity $\vec v$ , and thus on the moving frame $Q$ we pick. More specifically, the appearance of this drag is incompatible with all inertial frames being physically equivalent and describing the same physics: the body $S$ should remain intertial according the an observer in $P$ , decelerate according to an observer in $Q$ etc.

How would you resolve this paradox? Is this drag-like force real?

Click here if yon need a hint.