General Relativity – APM 426/MAT1700 – Volker Schlue
Problem Set 1
Due:
Jan 15
Problem 1 [Lorentz transformations ]: Let e0 be a unit time-like vector at a point p in
1 + 1-dimensional Minkowski space, and e1 a unit spacelike vector at p orthogonal to e0 .
Moreover let v ∈ R, |v| < c, c being the speed of light.
(i) Show that
1
v/c
Ov : e0 7→ p
e0 + p
e1
(1)
1 − v 2 /c2
1 − v 2 /c2
maps e0 to unit future-directed vectors. Sketch the curve v 7→ Ov e0 in a Cartesian
coordinate system based on (e0 , e1 ).
(ii) Define Ov e1 explicitly – similarly to Ov e0 in (1) – such that Ov is a Lorentz transformation. Sketch (Ov e0 , Ov e1 ) in a Cartesian coordinate system based on (e0 , e1 ).
(iii) Fix v ∈ R, and denote by e00 = Ov e0 , and e01 = Ov e1 . Let q be an event with
Cartesian coordinates (ct, x) relative to the unprimed basis, and coordinates (ct0 , x0 )
relative to the primed basis, i.e.
q = cte0 + xe1 = ct0 e00 + x0 e01 .
(2)
Express (ct0 , x0 ) in terms of (ct, x).
(iv) In the relations found in (iii) take c → ∞. These should be the Galilean transformation laws from one inertial system to another.
Problem 2 [Special Relativity ]: In 2 + 1-dimensional Minkowski space, consider one
system of reference S0 in uniform motion relative to another S. Suppose an observer at
rest in S0 carries with her a round disk, (in the sense we can think of the boundary as
traced out by a continuously turning pointer attached at the centre).
(i) If an observer at rest relative to S were to draw a closed curve C that encloses
the disc as it passes by at an instance of time as judged by him, what would the
geometrical shape of C be?
Hint: Recall that light emitted from a point propagates on cocentric spheres relative to every system of reference.
(ii) Suppose the speed of the observer in S0 relative to the one in S is v, and the
diameter of the disc as measured in S0 is D. Can you calculate the diameter of C
in the direction of the motion in terms of v, and D?
Hint: Reduce for simplicity to a 1 + 1-dimensional problem, and use the considerations of Problem 1.