General Relativity – APM 426/MAT1700 – Volker Schlue
Problem Set 2
Due:
Tue 27 Jan
Problem 1 [Receding observers in Minkowski spacetime ]:
Consider a set of observers in uniform motion receding from a center. In other words
consider a family of observers whose histories are straight lines in Minkowski space intersecting in a single point. These are rays contained in the future of the origin:
3
X
+
0
1
2
3
1+3
0 2
i 2
0
I0 = (x , x , x , x ) ∈ R
: −(x ) +
(x ) < 0 and x > 0 .
(1)
i=1
(i) Depict the family of receding observers and the set I0+ in a diagram.
(ii) At each p ∈ I0+ construct the hyperplane of simultaneous events Σp to p as judged
by the observer passing through p.
(iii) Do there exist global hypersurfaces of simultaneity in I0+ ?
In other words, is the distribution of planes
∆ := {Σp : p ∈ I0+ }
(2)
integrable?
Hint: Consider the hyperboloids Hτ+ given by
3
X
+
0
1
2
3
+
0 2
i 2
2
Hτ := (x , x , x , x ) ∈ I0 : (x ) −
(x ) = τ
(3)
i=1
and show that for a given event p ∈ Hτ+ the tangent plane at p to Hτ+ coincides
with Σp .
(iv) Consider the hyperboloid H1+ . Assign to each event p ∈ H1+ the polar normal
coordinates (χ, θ, φ), where θ and φ are the polar and azimuthal angle of the standard
coordinates on the unit sphere, and χ is the geodesic distance on H1+ from (1, 0, 0, 0)
to the event p. Verify that the induced metric on H1+ is given by
dσ 2 = dχ2 + sinh2 χ(dθ2 + sin2 θ dφ2 ).
(4)
Show that the Gauss curvature of H1+ is equal to −1.
Hints: Consider a χ-θ-section, in fact choose θ = π2 . Recall from the tutorial that
the Gauss curvature K of a 2-dimensional Riemannian metric in geodesic normal
coordinates,
dσ 2 = dr2 + R2 (r)dφ2
(5)
dk(r)
− k 2 (r),
dr
(6)
is given by
K=−
where
k(r) =
1 dR(r)
.
R(r) dr
(7)
(v) Show that the Minkowski metric can be written in I0+ as
ds2 = −dτ 2 + τ 2 dσ 2 .
(8)
That is to say I0+ can be foliated by the hyperboloids Hτ+ and the induced metric
on Hτ+ is precisely τ 2 dσ 2 . Also show that the Gauss curvature of Hτ+ is equal to
K = −τ −2 .
(9)
(vi) Consider two receding observers. Suppose a light signal is sent from one to the
other. Let
v = tanh χ
(10)
be the velocity of the observer emitting the light as measured in the rest frame of
the observer receiving the signal. Show that a redshift is observed with
ωe
= eχ ,
ωr
(11)
where ωe and ωr are the frequencies of the emitted and the received light respectively.
Hints: Consider the situation in a reference frame in which the observer receiving
the light is at rest. Then consider the situation in the x0 -x3 -plane (x1 = x2 = 0),
where
x0 = τ cosh χ,
x3 = τ sinh χ
and χ coincides with the χ-coordinate of the moving observer.
(12)