1
Problems on coordinate changes and metrics
Problem 1: Consider 3-d Cartesian coordinates
ds2 = dx2 + dy 2 + dz 2
(1)
If we have spherical symmetry in a physical problem, then one coordinate we would like to
use i the radial distance r given by
r2 = x2 + y 2 + z 2
(2)
We need two other coordinates to complete the definition of spherical polar coordinates.
We define a coordinate θ which breaks up the sum in (2) as
z 2 = r2 cos2 θ,
x2 + y 2 = r2 sin2 θ
(3)
Thus we have
0≤θ≤π
z = r cos θ,
(4)
where the coordinate range for θ is seen to allow all the possible values for z. We now split
the sum in x2 + y 2 = (r sin θ)2 in a similar way by introducing another coordinate φ
x2 = (r2 sin2 θ) cos2 φ,
y 2 = (r2 sin2 θ)2 sin2 φ
(5)
Thus we get
x = r sin θ cos φ,
y = r sin θ sin φ,
0 ≤ φ < 2π
(6)
where the coordinate range for φ given above is seen to cover all allowed values of x, y.
(a) Find the points where the spherical polar coordinates break down.
At the points θ = 0 and θ = π we find that all values 0 ≤ φ < 2π correspond to the
same values of x, y, z. Thus the φ coordinate changes while none of the x, y, z coordinates
move. This fact can be seen from the vanishing of the determinant
 ∂x ∂x ∂x 


cos(φ) sin(θ) r cos(θ) cos(φ) −r sin(θ) sin(φ)
∂r
∂θ
∂φ

∂y
∂y 
det  ∂y
= det  sin(θ) sin(φ) r cos(θ) sin(φ) r cos(φ) sin(θ)  = r2 sin θ
∂r
∂θ
∂φ 
∂z
∂z
∂z
cos(θ)
−r sin(θ)
0
∂r
∂θ
∂φ
(7)
which vanishes at θ = 0, π.
(b) Find the metric in spherical polar coordinates.
1
We have
dx = dr sin θ cos φ + r cos θ cos φdθ − r sin θ sin φdφ
dy = dr sin θ cos φ + r cos θ sin φdθ + r sin θ cos φdφ
dz = dr cos θ − r sin θdθ
ds2 = dx2 + dy 2 + dz 2
= (dr sin θ cos φ + r cos θ cos φdθ − r sin θ sin φdφ)2
+(dr sin θ cos φ + r cos θ sin φdθ + r sin θ cos φdφ)2
+(dr cos θ − r sin θdθ)2
= dr2 + r2 dθ2 + r2 sin2 θdφ2
(8)
Problem 2: A hypersurface with Euclidean signature
Start with the space with metric
ds2 = −dx21 + dx22 + dx23 + dx24 + dx25
(9)
−x21 + x22 + x23 + x24 + x25 = −a2
(10)
Consider the hypersurface
Find the metric on this hypersurface, by breaking up the sum above as
x21 = (acoshα)2
x22 + x23 + x24 + x25 = (asinhα)2
(11)
The coordinates x2 , x3 , x4 , x5 make a S 3 . We write
x1 = acoshα
x2 = asinhα cos θ
x3 = asinhα sin θ cos φ
x4 = asinhα sin θ sin φ cos χ
x5 = asinhα sin θ sin φ sin χ
(12)
ds2 = −dx21 + dx22 + dx23 + dx24 + dx25
i
h
= a2 dα2 + sinh2 α(dθ2 + sin2 θdφ2 + sin2 θ sin2 φdχ2 )
h
i
= a2 dα2 + sinh2 α dΩ23
(13)
We get
2
where dΩ23 is the metric of a uniform 3-sphere.
Problem 3: De Sitter spacetime in open slicing
Start with the space with metric
ds2 = −dx21 + dx22 + dx23 + dx24 + dx25
(14)
−x21 + x22 + x23 + x24 + x25 = a2
(15)
Consider the hypersurface
Find the metric on this hypersurface, by breaking up the sum above as
x22 = (acoshα)2
−x21 + x23 + x24 + x25 = (asinhα)2
(16)
We write
x2 = acoshα
x1 = asinhαcoshβ
x3 = asinhαsinhβ cos φ
x4 = asinhαsinhβ sin φ cos χ
x5 = asinhαsinhβ sin φ sin χ
(17)
ds2 = −dx21 + dx22 + dx23 + dx24 + dx25
h
i
= a2 − dα2 + sinh2 α dβ 2 + sinh2 β(dφ2 + sin2 φdχ2 )
h
i
= a2 − dα2 + sinh2 α dH32
(18)
dH32 = dβ 2 + sinh2 βdΩ22
(19)
We get
where
is the metric of a hyperbolic 3-dimensional Euclidean space.
Problem 4: Consider the way we put spherical polar coordinates on a S 2 of radius a.
We start with
x2 + y 2 + z 2 = a2
(20)
and break this sum into two parts
z 2 = a2 cos2 θ,
x2 + y 2 = a2 sin2 θ
3
(21)
Now suppose the problem has ellipsoidal symmetry; i.e., instead of the spherical surfaces
we get for constant r, we want ellipsoidal surfaces. An ellipsoid with axial symmetry around
the z axis has the form
z 2 x2 + y 2
+
=1
(22)
a2
b2
Thus we write
z 2 = a2 cos θ, x2 + y 2 = b2 sin2 θ
(23)
Followed by the further break up of the second sum
x = b sin θ cos φ,
y = b sin θ sin φ
(24)
We still have to choose a, b. Typically, as we go far from the origin, we recover spherical
symmetry, so we would like ab → 1 at large distances. Suppose we want prolate spheroids;
p
i.e., z > x2 + y 2 . We choose a constant A, and write
a = Acoshα,
b = Asinhα
(25)
Thus α governs the overall scale of the ellipsoidal surface, and so is like a radial coordinate;
further we see that limα→∞ ab = 1. Thus we have
x = Asinhα sin θ cos φ
y = Asinhα sin θ sin φ
z = Acoshα cos θ
p
Suppose we want oblate spheroids; i.e. z < x2 + y 2 . Then we write
a = Asinhα,
b = Acoshα
(26)
(27)
and get
x = Acoshα sin θ cos φ
y = Acoshα sin θ sin φ
z = Asinhα cos θ
(28)
(a) Find the metric in prolate spheroidal coordinates.
(b) Find the metric in oblate spheroidal coordinates.
(a) We have
h
i
ds2 = A2 (sinh2 α cos2 θ + cosh2 α sin2 θ)(dα2 + dθ2 ) + sinh2 α sin2 θdφ2
(29)
(b) We have
h
i
ds2 = A2 (cosh2 α cos2 θ + sinh2 α sin2 θ)(dα2 + dθ2 ) + cosh2 α sin2 θdφ2
4
(30)