Practice Set

February 18, 2023

1. Manifold is a space that looks like Rn locally.

There do exist manifolds that have no smooth struc-
ture. Smooth structures can be used to construct various
functions giving an algebraic picture of the manifold. A
manifold is differentiable if it is possible to define a scalar
field at each point of the manifold that can be differenti-
ated everywhere. In a non differentiable manifold every
chart in its atlas is continuous but nowhere differentiable.
Please explain whether following sets are manifold or not.
If they are manifold then think about its differentiability,
its dimensionality and number of coordinate charts in its
atlas.
(a). A single point R0 = {0}.
(b). Rn
(c). A zero sphere, S0
(d). A n-sphere Sn
(e). n-torus , Tn
(f ). A set of all continuous transformations (For exam-
ple, rotation)
(g). A line segment
(h). A one dimensional line running into a 2D plane.
(i). A single cone
(j). 2 cones stuck together at their vertices.
(k). The phase space of N particles kept in a cubical

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box at rest on the Earth.
(l). The sphere Sn with antipodal points identified, RPn

2. The important quantities in a manifold are points

and geometrical or topological relationship between them
, but not the coordinate system. Let f : M → N
be a map from an m-dimensional manifold M to an n-
dimensional manifold N. Show that the differentiability
of f is independent of coordinate charts in N.
3. Consider the following coordinate transformation from
rectangular (x, y) to a new set,
x = µν ; y = (µ2 − ν 2 )/2
(a). Sketch the curves of constant µ and constant ν in
(x, y) plane.
(b). Transform the line element ds2 = dx2 + dy 2 into
(µ, ν) coordinate.
(c). Do the curves of constant µ and constant ν intersect
at right angles ?
(d). Find the equation of a circle of radius r centered at
the origin in terms of µ and ν.
(e). Calculate the ratio of circumference to diameter of
a circle using (µ, ν) coordinates. Do you get the correct
answer?
4. An equal area map projection is one for which there
is a constant proportionality between areas on the map
and areas on the surface of the globe. Given x = Lφ/2π
what function y(λ) would make an equal area map?
5. A curve is defined by {x = f (λ), y = g(λ), 0 ≤ λ ≤ 1}.
Show that the tangent vector space (dx/dλ, dy/dλ) does
actually lie tangent to the curve.
6. Consider the 2D space with line element
dr2
ds2 = 1−2µ/r + r2 dφ2
Show that this geometry can be embeded in a 3 dimen-

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sional Euclidean space. Please also find the equation for
the corresponding 2D plane.
7. Let M be a manifold and f : M → R be a smooth
function such that df = 0 at some point p ∈ M . Let xµ
be a coordinate chart defined in a neighbourhood of p.
2
Define Fµν = ∂x∂ µf∂ ν . By considering the transformation
law for components show that Fµν defines a (0, 2) tensor.
(This is called the Hessian of f at p.) Construct also a
coordinate free definition and demonstrate its tensorial
property.
8. Show that given a conserved stress energy tensor T αβ
such that ∇α T αβ = 0 , and a Killing vector field Kα ,
the current defined by Jα = Kβ T αβ is conserved i.e, di-
vergence free.
9. Using the tensor transformation law applied to stress
tensor ,F µν , show how the electric and magnetic 3 vec-
tors transform under
(a). rotation about z-axis
(b). a boost along the y-axis.
10. A 3-sphere can be parameterized by Euler angles
(θ, φ, ψ) where 0 < θ < π, 0 < φ < 2π, 0 < ψ < 4π.
Define the following 1 forms:
σ1 = −sinψdθ +cosψsinθdφ ; σ2 = cosψdθ +sinψsinθdφ
; σ3 = dψ + cosθdφ
Show that dσ1 = σ2 ∧ σ3 with analogous results for dσ2
and dσ3 .
11. Consider a change of basis ẽµ = (A−1 )νµ eν . Find
relation of the components of Levi-Civita connection in
new basis with old basis. Also show that difference of
two connections transform as a tensor.
12. How many independent components does the Rie-
mann tensor tensor (of the Levi-Civita connection) have
in 1,2, 3 and 4 dimensions? Show that in 2D

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 1
Rµνρσ = R(gµρ gνσ − gµσ gνρ ) (1)
2
Discuss the implications for GR in 2 spacetime dimen-
sions. Can there be a gravitational field in empty space
in dimensions less than 4?
13(a). Show that the metric ensor, the Levi-Civita
tensor and the Kroneckar delta tensor for flat 3+1 di-
mensional Minkowski space-time remains invariant un-
der Lorentz transformation Λµν .
(b). Obtain the corresponding Levi-Civita products for
3+1 dimensional Minkowski space and determine all pos-
sible contractions.
14. Show explicitly that s2 = −c2 t2 + d~r2 is an invariant
under a boost with velocity ~v in +x direction.
15. Show that 1D electromagnetic wave equation is
not Galilean covariant but 1D Schrodinger equation is
Galilean covariant subjected to a phase transformation
of the wave function.
16(a). Find the equation for the geodesic in a non Rie-
mannian metric:
dsn = gµ1....µn dxµ1 .....dxµn (2)
(b). What kind of geometries (instead of the Minkowskian
one ) are there, if gµ1....µn has only constant (coordinate
independent) components? In the case of Minkowskian
signature there is a light cone , which allows one to spec-
ify which events are causally connected. What does one
have instead of that in the case of constant metrics with
more indices?
(c). What kind of geometry (instead of the Minkowskian
one) is there , if gµν = Diag(1,1,-1,-1)? What is there in-
stead of the light cone and causality?
17. Consider a 2-sphere with coordinates (θ, φ) and met-

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ric

ds2 = dθ2 + sin2 θdφ2 (3)

(a). Show that lines of constant longitudes (φ = constant)
are geodesics, and that the only line of constant lati-
tude (θ = constant) that is a geodesic is the equator
(θ = π/2).
(b). Calculate the 3 Killing vectors corresponding to
this sphere. (c). Further, show that the commutators of
these Killing vectors are cyclic

[A, B] = C, [B, C] = A, [C, A] = B (4)

18(a). Show that if photon has a tiny mass m, then

we will have an extra term m2 Aµ Aµ the the electromag-
netic Lagrangian and the velocity of the electromagnetic
waves will not be universal but will depend on frequency.
Suppose there are no massless particles at all in nature.
How will one formulate special relativity and interpret
the universal speed c? Is there any experiment which
shows the mass of photon to be exactly 0?
(b). Is it possible somewhere in the unverse where light
itself crosses the speed limit c ? ( Note that it is lo-
cal speed which is asked, not like superluminal velocity
which is an apparent phenomenon)
(c). Think about the behaviour of electromagnetism in
dimensions less than 1+3 using Maxwell’s equations.
19. Show that if a fluid is at rest in a static metric then
√
its temperature satisfies the relation T −g00 = constant.
20. Consider the conformal transformation gab → Ω2 (x)gab , g ab →
Ω−2 (x)g ab . Show that the free field electromagnetic ac-
tion remain invariant under these transformations. What
does it imply for the solutions of Maxwell’s equations in

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a spacetime with a metric of the form gµν = f 2 (x)ηµν ?

21. Calculate the curvature scalar for the metric of

2D Euclidean space written in polar coordinates.
22. Suppose v α = dxα /dλ? obeys the geodesic equation
in the form Dv α /dλ? = κ(λ? )v α . Clearly λ? is not a
affine parrameter. Show that uα = dxα /dλ obeys the
geodesic equation
R in the form Duα /dλ = 0 provided that
dλ/dλ? = exp[ κ(λ? )dλ? ] .
23. A particle with electric charge e moves with 4 veloc-
ity uα in a spacetime with metric gαβ in the presence of
a vector potential Aµ . The spacetime admits a Killing
vector field ξ α such that Lξ~gαβ =0 , Lξ~Aα =0. Show
that the quantity (uα + eAα )ξ α is a constant along the
worldline of the particle.
26. In an intertial frame O, calculate components of the
stress energy tensor of the following systems:
(a). A group of particles all moving with the same 3-
velocity ~v = β~ex as seen in O. Let the rest-mass density
of these particles be ρ0 , as measured in their own rest
frame. Assume a sufficiently high density of particles to
enable treating them as a continuum.

(b) A ring of N similar particles of rest mass m rotat-

ing counter-clockwise in the xy plane about the origin of
O, at a radius a from this point, with an angular velocity
ω. The ring is a torus of circular cross-section δa << a,
within which the particles are uniformly distributed with
a high enough density for the continuum approximation
to apply. Do not include the stress-energy of whatever
forces keep them in orbit. Part of this calculation will
relate 0 of part (a) to N, a, δ a, and ω.

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 (c). Two such rings of particles, one rotating clock-
wise and the other counter-clockwise, at the same radius
a. The particles do not collide or otherwise interact in
any way.
27. A satellite is in circular orbit of radius r around the
earth (radius R, mass M) . A standard clock C on the
satellite is compared with an identical clock C0 at the
south pole of the Earth. Show that the ratio of the rate
of the orbitting clock to that of clock on the Earth is
approximately 1 + GM GM
Rc2 − 3 2rc2 . Note that the orbiting
clock is faster only if r > 3R/2 i.e, r − R > 3184 km.
28. Show that the line element ds2 = y 2 dx2 + x2 dy 2
represents the Euclidean plane , but the line element
ds2 = ydx2 + xdy 2 represents a curved 2D manifold.
29. Show that any Killing vector ξ i satisfies the equation
ξ i + Rki ξ k = 0.
(5)

30. If T µν is the energy momentum tensor and i

R ξi kis a
timelike Killing vector, show that the integral Tk ξ dΣ
over the whole spacelike hypersurface is independent of
the choice of the hypersurface.