All questions may be attempted but only marks obtained on the best four solutions will

count.
The use of an electronic calculator is not permitted in this examination.

Christoffel symbol:
1
Γcab = g cd (gda,b + gbd,a − gab,d ). (1)
2
Geodesics parameterised by λ:

d2 X a b
a dX dX
c
+ Γ bc = 0. (2)
dλ2 dλ dλ
Riemann curvature tensor:

Rmnr s = Γsmr,n − Γsnr,m + Γamr Γsan − Γanr Γsam , (3)

Rabcd = −Rbacd , Rabcd = −Rabdc , Rabcd = Rcdab ,
Rdabc + Rdcab + Rdbca = 0. (4)

Ricci tensor and Ricci scalar:

Rmr = Rmnr n , R = g mr Rmr . (5)

Schwarzschild-de Sitter line element:

   −1
2 2m Λ 2 2 2m Λ 2
ds = − 1 − − r dt + 1 − − r dr2 + r2 (dθ2 + sin2θdφ2 ). (6)
r 3 r 3

MATH3305 PLEASE TURN OVER

1
 1. (a) Describe how a contravariant vector V a transforms under coordinate transfor-
mations between coordinates X a and X 0a .
(b) Describe how a covariant vector Wa transforms under coordinate transforma-
tions between coordinates X a and X 0a .
(c) Let M a bc be a rank 3 tensor. Show that M b ab transforms like a covariant vector
under coordinate transformations.
(d) Let M be a N -dimensional manifold and let gab be a metric tensor. Compute
gab g bc and gab g ab .
(e) Let ∇a be a covariant derivative operator on a manifold M. Let Fab be an
anti-symmetric tensor. Express ∇a Fbc + ∇b Fca + ∇c Fab in terms of partial
derivatives only.

2. Let Rabcd be the Riemann curvature tensor defined by

∇a ∇b Wc − ∇b ∇a Wc = Rabc d Wd . (∗)

(a) Show that this implies

Rabcd + Rcabd + Rbcad = 0.

[Hint: Start by writing out (∗) three times with permuted indices.]
(b) Show that the Ricci tensor is symmetric.

Let Lv be the so-called Lie derivative with respect to the vector v a . For f ∈ C ∞ (M)
it is defined by Lv f := v a ∇a f . Moreover,

Lv ua := v b ∇b ua + ub ∇a v b ,

and it satisfies the Leibniz rule.

(c) Show that Lv wa = v b ∇b wa − wb ∇b v a .

MATH3305 CONTINUED

2
 3. Consider a 2-dimensional manifold with line element

ds2 = (1 + y 2 )dy 2 + 2ydydz + dz 2 .

(a) Compute the Christoffel symbol components Γ211 and Γ122 using Eq. (1).
(b) Obtain the geodesic equations parametrised by λ, say, using the variational
approach.
(c) Solve the geodesic equations and suggest an improved coordinate system. What
is the metric in the new coordinates? What lines describe the geodesics geo-
metrically?
(d) What can you say about the Riemann curvature tensor, the Ricci tensor and
the Ricci scalar of this manifold?
(e) Consider the Einstein field equations in a 2-dimensional setting. Show that
(2d)
they imply Tab = 0.

4. Consider the static and spherically symmetric metric given by

ds2 = −e2ν(χ) dt2 + dχ2 + R2 (χ)(dθ2 + sin2θdφ2 ).

The vacuum Einstein field equations with cosmological term are

1 − R02 − 2RR00
− Λ = 0,
R2
R02 − 1 + 2RR0 ν 0
+ Λ = 0,
R2
ν 0 R0 + R(ν 02 + ν 00 ) + R00
+ Λ = 0,
R
where the prime denotes differentiation with respect to χ.

(a) Solve the vacuum field equations under the assumption that R(χ) = R0 =
const.
(b) For which Λ does this solution exist?
(c) Identify the values of χ for which the metric is well defined.
√ √
(d) Let r be a new coordinate defined by r = (1/ Λ) sin( Λχ + C) where C is
some constant. Show that the metric found in (a) takes the form

−1 2 1
ds2 = − 1 − r2 /α2 dt2 + 1 − r2 /α2 dr + (dθ2 + sin2θdφ2 ) ,


Λ
where α2 = 1/Λ.

MATH3305 PLEASE TURN OVER

3
 5. Light deflection by the Sun is described by the deflection angle ∆φ = 2φ+ − π. The
geodesic equation for massless particles reduces to
 2
`2
dr E 2 − 1 − 2M r r2
= 2 4
.
dφ ` /r

(a) Explain the meaning of the quantities E and `.

(b) Rewrite the differential equation using the impact parameter or distance of
closest approach. This new equation should be independent of E and `.
(c) Find an integral which determines the value of φ+ .
(d) You are given that

∂φ+ 2
= .
∂M M =0 r0

Show that light deflection by the Sun is given by 4GM /(c2 R ). Discuss any
approximation used.
(e) Briefly discuss light deflection by a massive object in Newtonian physics.

MATH3305 END OF PAPER

4
All questions may be attempted but only marks obtained on the best four solutions will
count.
The use of an electronic calculator is not permitted in this examination.

Christoffel symbol:
1
Γcab = g cd (gda,b + gbd,a − gab,d ). (1)
2
Geodesics parameterised by λ:

d2 X a b
a dX dX
c
+ Γ bc = 0. (2)
dλ2 dλ dλ
Riemann curvature tensor:

Rmnr s = Γsmr,n − Γsnr,m + Γamr Γsan − Γanr Γsam , (3)

Rabcd = −Rbacd , Rabcd = −Rabdc , Rabcd = Rcdab ,
Rdabc + Rdcab + Rdbca = 0. (4)

Ricci tensor and Ricci scalar:

Rmr = Rmnr n , R = g mr Rmr . (5)

Schwarzschild-de Sitter line element:

   −1
2 2m Λ 2 2 2m Λ 2
ds = − 1 − − r dt + 1 − − r dr2 + r2 (dθ2 + sin2θdφ2 ). (6)
r 3 r 3

MATH3305 PLEASE TURN OVER

1
 1. (a) Describe how a contravariant vector V a transforms under coordinate transfor-
mations between coordinates X a and X 0a .
(b) Describe how a covariant vector Wa transforms under coordinate transforma-
tions between coordinates X a and X 0a .
(c) Let ∇a be a covariant derivative operator on a manifold M. Let Fab be an
anti-symmetric tensor. Express ∇a Fbc + ∇b Fca + ∇c Fab in terms of partial
derivatives only.
(d) Consider an arbitrary curve C given by X i = X i (λ). Find the transformation
properties of

d2 X i
dλ2
under arbitrary coordinate transformations.

2. Let g denote the determinant of the metric tensor g = det(gij ) on a Riemannian

manifold. You are given that
∂g ∂gmn
k
= gg mn .
∂X ∂X k
(a) Prove the identity:
√
∂ log g
Γiki = .
∂X k

(b) Next, prove the identity:

√
1 ∂( gg ik )
g mn Γimn = −√ .
g ∂X k

˜ satisfying ∇
(c) Consider a covariant derivative ∇ ˜ a gbc = Qa gbc 6= 0. Show that
˜ a g = −Qa g , where g is the usual inverse metric.
this implies ∇ bc bc bc

MATH3305 CONTINUED

2
 3. Consider a 2-dimensional manifold with line element
dz 2
ds2 = dy 2 + .
cosh4 z
(a) Calculate the Christoffel symbol components using Eq. (1).
(b) Obtain the geodesic equations parametrised by λ, say, using the variational
approach.
(c) Solve the geodesic equations and suggest an improved coordinate system. What
is the metric in the new coordinates? What lines describe the geodesics geo-
metrically?
(d) In 2 dimensions the Riemann curvature tensor has 1 independent component
only. Show that this implies the vanishing of the Einstein tensor in 2 dimen-
sions.

4. Consider the static and spherically symmetric metric given by

ds2 = −e2ν(χ) dt2 + dχ2 + R2 (χ)(dθ2 + sin2θdφ2 ).

The vacuum Einstein field equations with cosmological term are

1 − R02 − 2RR00
− Λ = 0,
R2
R02 − 1 + 2RR0 ν 0
+ Λ = 0,
R2
ν 0 R0 + R(ν 02 + ν 00 ) + R00
+ Λ = 0,
R
where the prime denotes differentiation with respect to χ.

(a) Solve the vacuum field equations under the assumption that R(χ) = R0 =
const.
(b) For which Λ does this solution exist?
(c) Identify the values of χ for which the metric is well defined.
√ √
(d) Let r be a new coordinate defined by r = (1/ Λ) sin( Λχ + C) where C is
some constant. Show that the metric found in (a) takes the form

−1 2 1
ds2 = − 1 − r2 /α2 dt2 + 1 − r2 /α2 dr + (dθ2 + sin2θdφ2 ) ,


Λ
where α2 = 1/Λ.

MATH3305 PLEASE TURN OVER

3
 5. Consider geodesics in the Schwarzschild spacetime. You are given the geodesic
equation in the form
  2 
1 2 1 2M ` 1 2
ṙ + 1− − L = E .
2 2 r r2 2

(a) Explain the meaning of the quantities E, ` and L.

(b) State the effective potential and discuss the meaning of the 4 terms that appear.
(c) Determine whether a photon can have an exactly circular orbit and if so, what
the radius of this orbit would be. Discuss the stability of this orbit.
(d) The weak-field Einstein field equation reads −2h̄ab = 2κTab where h̄ab is the
trace reversed metric perturbation. Briefly discuss the implications of this
equation in the context of gravitational waves. Describe the two possible po-
larisations of gravitational waves.

MATH3305 END OF PAPER

4
All questions may be attempted but only marks obtained on the best four solutions will
count.
The use of an electronic calculator is not permitted in this examination.

Christoffel symbol:
1
Γcab = g cd (gda,b + gbd,a − gab,d ). (1)
2
Geodesics parameterised by λ:

d2 X a b
a dX dX
c
+ Γ bc = 0. (2)
dλ2 dλ dλ
Riemann curvature tensor:

Rmnr s = Γsmr,n − Γsnr,m + Γamr Γsan − Γanr Γsam , (3)

Rabcd = −Rbacd , Rabcd = −Rabdc , Rabcd = Rcdab ,
Rdabc + Rdcab + Rdbca = 0. (4)

Ricci tensor and Ricci scalar:

Rmr = Rmnr n , R = g mr Rmr . (5)

Schwarzschild-de Sitter line element:

   −1
2 2m Λ 2 2 2m Λ 2
ds = − 1 − − r dt + 1 − − r dr2 + r2 (dθ2 + sin2θdφ2 ). (6)
r 3 r 3

MATH0025 PLEASE TURN OVER

1
 1. (a) Describe how a contravariant vector V a transforms under coordinate transfor-
mations between coordinates X a and X 0a .
(b) Describe how a covariant vector Wa transforms under coordinate transforma-
tions between coordinates X a and X 0a .
(c) Let Qi j be a rank two tensor satisfying Qi j Qj k = δki . Show that this tensor
satisfies the equation

Qj k ∇s Qi j = −Qi j ∇s Qj k .

(d) The setting of this part is three-dimensional flat space R3 with Cartesian co-
ordinates X i = {x, y, z}. The matrix curl of some matrix R is defined by

(Curl R)ij = εjmn ∂m Rin ,

where we sum over repeated indices. Compute Curl R for the matrix
 
cos φ(x, y) − sin φ(x, y) 0
R =  sin φ(x, y) cos φ(x, y) 0 ,
0 0 1

where φ is a function of the first two coordinates only.

Hint: 7 of the 9 components of (Curl R)ij are zero.

2. Let ∇ be the covariant derivative on a manifold (metric compatible and torsion-

free). You are given the identity ∇a ∇d U b − ∇d ∇a U b = −Radi b U i for some vector
U b where Radi b is the Riemann curvature tensor.

(a) Define the rank 2 tensor W ij = U i V j where V j is another arbitrary vector.

Show that

∇a ∇d W ij − ∇d ∇a W ij = −Rads i W sj − Rads j W is .

(b) Let F mn be a skew-symmetric tensor, this means F mn = −F nm . Show that

∇i Fjk + ∇j Fki + ∇k Fij = ∂i Fjk + ∂j Fki + ∂k Fij .

(c) Next define the object J i = ∇n F in (for skew-symmetric F mn ) and prove that

∇s J s = 0 .

MATH0025 CONTINUED

2
 3. The setting of this question is Minkowski space with signature (+, −, −, −). This
means we work with the line element ds2 = dt2 − dx2 − dy 2 − dz 2 .

(a) Consider the curve defined by X i (u) = uv i where u is an affine parameter and
v i = (1, 0, 0, 1). Show that X i is a null curve by showing that gij Ẋ i Ẋ j = 0 for
all u.
(b) Show that X i (u) satisfies the geodesic equations and hence is a null geodesic.
(c) Let k i be another null vector gij k i k j = 0 which is normalised relative to v i
such that vi k i = 1. k i will have two independent components. Write k i =
(k 0 , α, β, k 3 ). Find explicit expressions for k 0 and k 3 in terms of α, β.
(d) Introduce a new coordinate system X i = uv i + rk i and show the explicit
expressions
r
t = u + (α2 + β 2 + 1) ,
2
r
z = u + (α2 + β 2 − 1) ,
2
x = rα , y = rβ .

(e) Finally show that the Minkowski line element in the new coordinates {u, r, α, β}
becomes

ds2 = 2du dr − r2 (dα2 + dβ 2 ) .

4. Consider a 2-dimensional manifold with line element

ds2 = −e−2τ dτ 2 + e2z dz 2 .

(a) Calculate the Christoffel symbol components using Eq. (1).

(b) Obtain the geodesic equations parametrised by λ, say, using the variational
approach.
(c) Solve the geodesic equations and suggest an improved coordinate system. What
is the metric in the new coordinates? What curves describe the geodesics
geometrically?
(d) Find coordinates u and v in terms of τ and z such that the line element take
the form ds2 = dudv.

MATH0025 PLEASE TURN OVER

3
 5. The most general static and spherically symmetric line element is given by

ds2 = −a(r)dt2 + b(r)dr2 + r2 (dθ2 + sin2θdφ2 ),

and results in the following two Einstein tensor components

b0 (r) 1 1
Gtt = − 2
+ 2 − 2,
rb(r) r b(r) r
0
a (r) 1 1
Grr = + 2 − 2,
ra(r)b(r) r b(r) r

where the prime denotes differentiation with respect to r.

(a) Solve the static and spherically symmetric vacuum field equations.
(b) Discuss the importance of this Schwarzschild solution by considering the limit
r → ∞ and r → rc where rc is a critical radius you should interpret physically.
(c) Consider the so called Robertson expansion

m2 m3
 
m
a(r) = 1 − 2 + 2(β − γ) 2 + O ,
r r r3
 2
m m
b(r) = 1 + 2 γ + O .
r r2

What are the values of β and γ for the Schwarzschild metric?

(d) Show the the quantity u given by

u = t − r − 2m log |r − 2m| ,

is constant for outgoing radial null geodesics.

MATH0025 END OF PAPER

4
UNIVERSITY COLLEGE LONDON

EXAMINATION FOR INTERNAL STUDENTS

MODULE CODE : MATH0025

ASSESSMENT : MATH0025A6UB
PATTERN

MODULE NAME : Mathematics For General Relativity

LEVEL: : Undergraduate

DATE : 04-June-2020

TIME : 15:00

This paper is suitable for candidates who attended classes for this
module in the following academic year(s):

Year
2019/20
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE

Hall Instructions
Standard Calculators
Non-Standard
Calculators

TURN OVER
All questions should be attempted, and marks obtained on all solutions will count.
The examiners will consider separately your performance in the more straightforward (A)
and more challenging (B) parts of questions, when determining your final grade.

Christoffel symbol:
1
Γcab = g cd (gda,b + gbd,a − gab,d ). (1)
2
Geodesics parameterised by λ:

d2 X a b
a dX dX
c
+ Γ bc = 0. (2)
dλ2 dλ dλ
Riemann curvature tensor:

Rmnr s = ∂n Γsmr − ∂m Γsnr + Γamr Γsan − Γanr Γsam , (3)

Rabcd = −Rbacd , Rabcd = −Rabdc , Rabcd = Rcdab ,
Rdabc + Rdcab + Rdbca = 0. (4)

Ricci tensor and Ricci scalar:

Rmr = Rmnr n , R = g mr Rmr . (5)

Geodesic deviation equation:

D2 N a
2
= (Rijc a T j T c )N i . (6)
Dλ
Schwarzschild-de Sitter line element:
   −1
2 2m Λ 2 2 2m Λ 2
ds = − 1 − − r dt + 1 − − r dr2 + r2 (dθ2 + sin2θdφ2 ). (7)
r 3 r 3

MATH0025 PLEASE TURN OVER

1
1.A (a) Describe how a contravariant vector V a transforms under coordinate transfor-
mations between coordinates X a and X 0a . Derive this result from first princi-
ples.
(b) Describe how a covariant vector Wa transforms under coordinate transforma-
tions between coordinates X a and X 0a . Derive this result from first principles.
(c) You are given a covariant derivative operator ∇a which satisfies ∇a gbc =
−Qa gbc 6= 0. Show that ∇a g bc = Qa g bc .

1.B (d) You are given a torsion-free covariant derivative operator ∇a which satisfies
∇a gbc = −Qa gbc 6= 0. Show that the Christoffel symbol components in this
case are given by

bs = 1 g sb (∂a gbc − ∂b gca + ∂c gab ) + 1 g sb (Qa gbc − Qb gca + Qc gab ) ,

Γac
2 2
and interpret your result.
(e) You are given another covariant derivative operator ∇a which satisfies ∇a gbc =
es 6= Γ
−Qabc 6= 0 and with Γ es . Find the explicit expression for the Christoffel
ac ca
symbol components in this case.

MATH0025 CONTINUED

2
2.A You are given the Euclidean space R3 with Cartesian coordinates X i = {x, y, z} and
standard line element ds2 = dx2 + dy 2 + dz 2 . The ring torus can be parametrised
as follows: x = (A + B cos θ) cos φ, y = (A + B cos θ) sin φ and z = B sin θ, here A
and B are two constant parameters.

(a) Show that the induced metric of the ring torus in coordinates Y i = {θ, φ} is

ds2 = B 2 dθ2 + (A + B cos θ)2 dφ2 .

(b) Using the Lagrangian approach, find the non-vanishing Christoffel symbols and
state the geodesic equations.
(c) Using the explicit formula [see Eq. (1)], find the non-vanishing Christoffel sym-
bols and state the geodesic equations.

2.B (d) Find all non-vanishing Riemann tensor components.

(e) Show that φ̇(R + r cos θ)2 = C is a constant.
(f) Derive the equation

C2
θ̇2 = D2 −
B 2 (A + B cos θ)2

where D is another constant.

(g) Give a qualitative description of the geodesics.

MATH0025 PLEASE TURN OVER

3
3.A Consider the line element ds2 = −(1 + 2φ)dt2 + (1 − 2φ)[dx2 + dy 2 + dz 2 ] with
coordinates X i = {t, x, y, z}, i = 0, 1, 2, 3, where X α = {x, y, z}, α = 1, 2, 3 are
Cartesian coordinates and φ(x, y, z)  1 is the static (and weak) gravitational
potential. Let ui be a test particle’s 4-velocity where we also assume non-relativistic
(slow) motions.

(a) Show that the Schwarzschild solution can be written in the above form, state
any assumptions used.
(b) Under the above assumptions which relations are implied on |u0 | and |uα |?
What is the interpretation of uα under the above assumptions?
(c) Consider the geodesic equations u̇i + Γijk uj uk = 0. Solve the i = 0 equation
and interpret the result.
(d) What is the interpretation of the three i = α equations?

p
3.B (e) Show that m −gij ui uj , using the above assumptions, is the difference between
potential and kinetic energy of a particle in a gravitational field. Constant
terms may be neglected.
(f) Why is the difference between potential and kinetic energy relevant? Write no
more than 2 paragraphs.
(g) Consider a light ray travelling along the x-axis. Show that the corresponding
geodesic equations satisfy
s
dx 1 + 2φ
=± .
dt 1 − 2φ

MATH0025 CONTINUED

4
4.A You are given a family of curves given by X i (t, s) where t is assumed to be Newtonian
time. Define the vector N i = ∂X i /∂s.

(a) Use Newton’s force law to derive the equation

∂ 2N i ∂ 2Φ
= − Nk
∂t2 ∂X k ∂Xi
where Φ is the Newtonian gravitational potential.
(b) Discuss the existence and key properties of gravitational wave solutions in the
context of the linearised Einstein field equations when the metric is assumed to
be of the form gij = ηij + hij with |hij |  1. Write no more than 2 paragraphs.

4.B (c) Show that the geodesic deviation equation [see Eq. (6)] for the Newtonian
potential implies

∂ 2Φ
Ri0k0 =
∂X i ∂X k
and that all other components of the Riemann tensor vanish.
(d) Find the non-vanishing Ricci tensor components.
(e) Find the linearised Einstein field equations and derive the Newtonian vacuum
equations.

MATH0025 PLEASE TURN OVER

5
Submit only four solutions. If more solutions are submitted only the first four will be
marked.

Christoffel symbol:
1
Γcab = g cd (gda,b + gbd,a − gab,d ). (1)
2
Geodesics parameterised by λ:

d2 X a b
a dX dX
c
+ Γ bc = 0. (2)
dλ2 dλ dλ
Riemann curvature tensor:

Rmnr s = ∂n Γsmr − ∂m Γsnr + Γamr Γsan − Γanr Γsam , (3)

Rabcd = −Rbacd , Rabcd = −Rabdc , Rabcd = Rcdab ,
Rdabc + Rdcab + Rdbca = 0. (4)

Ricci tensor and Ricci scalar:

Rmr = Rmnr n , R = g mr Rmr . (5)

Geodesic deviation equation:

D2 N a
= (Rijc a T j T c )N i . (6)
Dλ2
Schwarzschild-de Sitter line element:
   −1
2 2m Λ 2 2 2m Λ 2
ds = − 1 − − r dt + 1 − − r dr2 + r2 (dθ2 + sin2θdφ2 ). (7)
r 3 r 3

MATH0025 PLEASE TURN OVER

1
1.A (a) Let Aj be a covariant vector. Show explicitly how ∂i Aj transforms under
coordinate transformations.
(b) Now discuss the transformation properties of ∂i Aj − ∂j Ai under coordinate
transformations.
(c) Derive the identity

∇a ∇b W cd − ∇b ∇a W cd + Rabs c W sd + Rabs d W cs = 0 ,

using the definition of the Riemann curvature tensor.

1.B (d) You are given a covariant derivative operator ∇a and the quantity ∇a Γijk .
Discuss whether this object is a tensor.
(e) Consider the conformal transformation g̃ab = Ω2 (X i ) gab where Ω is an arbitrary
function of the coordinates. Compute the Christoffel symbol Γ̃ijk and write it as
the sum of Γijk and a new term. Show explicitly that the new term is symmetric
in the lower indices.

MATH0025 CONTINUED

2
2.A You are given the Euclidean space E3 with Cartesian coordinates X i = {x, y, z} and
standard line element ds2 = dx2 +dy 2 +dz 2 . You are given the following surface: x =
(1 + (v/2) cos(u/2)) cos(u), y = (1 + (v/2) cos(u/2)) sin(u) and z = (v/2) sin(u/2).

(a) Compute the induced metric in coordinates Y i = {u, v}, i = 1, 2. It is of the

form
1
ds2 = f (u, v) du2 + dv 2 .
4
State the function f explicitly.
(b) Find Γ111 , Γ112 and Γ211 . The other Christoffel symbol components vanish.

2.B (c) Show that all non-vanishing Riemann tensor components Rijkl are of the form
β/f . Determine the constant β.
(d) Using cylindrical coordinates {ρ, ϕ, z} show that the surface can be written as
ρ − 1 = z/ tan(ϕ/2).

MATH0025 PLEASE TURN OVER

3
 / := M −I tr M/3 is trace-free,
3.A (a) Let M be a symmetric 3×3 matrix. Show that M
here I stands for the identity matrix.
(b) Now consider an n-dimensional manifold with metric tensor gij and Ricci tensor
Rij . Let Zij := Rij − gij R/α where α is a constant. Determine the value of α
if Zij is assumed to be trace-free.
(c) Write the 4-dimensional Einstein field equations with cosmological constant
without matter (Tij = 0) in two separate equations: a trace-free equation and
a trace equation. You should use the tensor Zij for the trace-free equation.

3.B Your are given a vector va that satisfies ∇a vb + ∇b va = 0.

(d) Show that ∇a ∇b vc = Rcba s vs .

(e) Show that v a ∇a R = 0.

MATH0025 CONTINUED

4
4.A The charged generalisation of the Schwarzschild solution is known as the Reissner-
Nordström solution. It is given by
 2M Q2   2M Q2 −1 2
ds2 = − 1 − + 2 dt2 + 1 − + 2 dr + r2 dΩ2 ,
r r r r
where M is the mass parameter and Q is the charge. We assume r > 0.

(a) This line element has up to two possible coordinate singularities r+ and r− ,
we will call these horizons. Find explicit expressions for r± in terms of M and
Q. State the necessary condition for the existence of two distinct horizons, one
horizon and no horizon.
(b) Assuming θ = π/2 throughout, state the remaining geodesic equations and
interpret the two conserved quantities which will appear.

4.B (c) Consider the trajectory of a photon (massless particle) and find possible radii
such that this photon has an exactly circular orbit. This is known as the photon
sphere.
(d) Discuss the condition under which such orbits can exist. What is the maximum
possible number of such orbits for the Reissner-Nordström solutions?
(e) Assuming that there are two horizons r+ and r− , how many photon spheres are
present in the Reissner-Nordström manifold? Carefully justify your answer.

MATH0025 END OF PAPER

5
 MATH0025

Answer all questions.

1. The setting of this question is an n-dimensional manifold M equipped with

metric tensor gij and covariant derivative ∇a . We work in a local coordinate system
X i . Let C be a curve written as X i (λ) and let its tangent vector be T i .
(a) Consider a vector V a evaluated along the curve C and define
∆V a := V a (X i (λ + δλ)) − V a (X i (λ))
where δλ  1. Write ∆V a in terms of the curve’s tangent vector, δλ and
the derivatives of V a with respect to the local coordinates.
(b) Introduce the object
D̄ a
V := T i ∂i V a .
D̄λ
Write this quantity in terms of ∆V a and δλ.
(c) When discussing the geodesic deviation equation we introduced the nota-
tion
D a
V := T i ∇i V a .
Dλ
Write the equation for parallel transport using this notation and also write
the geodesic (or autoparallel) equation in this notation.
(d) Motivate the introduction of the connection Γijk using the object ∆V a .
(e) Let U i be a vector with length ` = Us U s , (` 6= 0) which is parallelly
transported along the curve C. Find a condition on gij such that ` is
preserved under parallel transport.
(25 marks)

2. The setting of this question is the 2-dimensional manifold M with line element
 1 
ds2 = gij dX i dX j = cosh(y) dx2 + x2 dy 2 + x sinh(y) dx dy ,
4
where X i = {x, y} and i = 1, 2.
(a) Using any suitable method, find the non-vanishing Christoffel symbol com-
ponents.
(b) Show that the geodesic equations can be brought into the form
1 2
ẍ + xẏ 2 = 0 , ÿ + ẋẏ = 0 .
4 x
(c) Show that x2 ẏ is a conserved quantity along the geodesics.
(d) By solving the geodesic equations, or otherwise, find an improved coordi-
nate system to show that this is a flat 2-dimensional manifold.
(25 marks)

MATH0025 Page 1 of 3
3. The setting of this question is a 4-dimensional Lorentzian manifold M as it is
used in General Relativity. Consider the perfect fluid energy-momentum tensor
given by
T ij = (ρ + p)ui uj + pg ij ,
where ρ is the energy density, p is the isotropic pressure and ui is the 4-velocity
which satisfies ui ui = −1.
(a) Define the so-called projection hij := gij + ui uj . Show that
hij hjk = hik and hij uj = 0 ,
and also
T ij = ρui uj + phij .
Justify why hij is called a projection.
(b) Show that ui ∇j ui = 0.
(c) Show that uj ∇i T ij = 0 is equivalent to
ui ∇i ρ + (ρ + p)∇i ui = 0 . (*)
(d) Next, show that hkj ∇i T ij = 0 is equivalent to
hik ∇i p + (ρ + p)(hkj ui ∇i uj ) = 0 .
(e) Assuming weak gravitational fields and small velocities (compared to the
speed of light) rewrite equation (∗) and interpret your result.
(25 marks)

MATH0025 Page 2 of 3
4. The line element of a Schwarzschild black hole in an external magnetic field is
given by
ds2 = −A2 (1 − 2m/r)dt2 + A2 (1 − 2m/r)−1 dr2 + A2 r2 dθ2 + A−2 r2 sin2θdφ2 ,
where the function A = A(r, θ) is given by
1
A = 1 + B02 r2 sin2θ .
4
The magnetic field is denoted by B0 . When discussing geodesics θ = π/2 may be
assumed without further justifications.
(a) Determine the possible location(s) of singularities, coordinate singularities
and horizons and compare the results with the usual Schwarzschild solution.
(b) Show that the geodesic equations give rise to two conserved quantities.
State the relevant equations and give a physical interpretation.
(c) Show that the geodesic equation for radial null geodesics can be written as
dr E
=± .
dλ a(r)2
Explain the meaning of the two signs and determine the function a(r).
(d) Integrate the geodesic equation for radial null geodesics and show that r =
2m can be reached for a finite parameter value λ? . (An explicit expression
for λ? does not have to be derived.)
(e) Show that the geodesic equations can be written in the form of an ‘energy’
equation
 2m  E2
ṙ2 + 1 − f (r) = .
r a(r)4
Find the explicit form of the function f (r).
(25 marks)

MATH0025 Page 3 of 3