ASSIGNMENT 3
Note: every time you are asked to prove the tensor properties of some object, you should demonstrate
it by presenting the appropriate transformation rules.
1. One has a 4-th rank tensor R . What kind of tensor R is ? Prove it.
Since we are gi
ASSIGNMENT 4
1. Show by explicit computation that for the 2D metric that describes the surface of a
sphere (of unit radius)
ds2 = d2 + sin2 d2
(1)
in coordinates x = (, ) , = 1, 2, the following three vector elds are Killing vector
elds
=
, = (cos , cot
ASSIGNMENT 6, due date Dec 5th
Poisson brackets A particle moves in some external eld. It is angular momentum (with respect to the
arbitrary chosen origin) is L = r p.
1. Calculate all Poisson brackets of the components of the angular momentum cfw_Lx , Ly
University of Alberta
Department of physics
Jose Luis A. Nandez, Dmitri Pogosian
November 25, 2012
Problem 1 Is the 2D space-time, described by the following time-dependent metric
ds2 = dt2 + t2 dx2 ,
(1)
curved or at ? Prove your statement by necessary c
ASSIGNMENT 5, Due date Nov 28th
1. Let us derive the second order equation for the Killing vector eld. We have showm
that Killing eld satises
; + ; = 0
We can covariantly dierentiate this relation once with respect to x to obtain
; + ; = 0
Obviously, we c
University of Alberta
Department of physics
Jose Luis A. Nandez, Dmitri Pogosian
November 7, 2011
Problem 1 In three-dimensional Euclidean space a coordinate system x a is related to
the Cartesian coordinates xa by
x1 = x 1 + x 2 ,
x2 = x 1 x 2 ,
x3 = 2x
Worked Solutions to Hobson, Efstathiou, and Lasenbys General
Relativity: An Introduction for Physicists
M. Haddad
May 25, 2016
2
Contents
1 The
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
Spacetime of Special Relativity
Deriving
Math 209 (2012)
Midterm Examination
Multiple Choice Questions.
1. A company produces cylindrical pop cans with internal base radius r = 5cm and height h = 15cm. They
wish to reduce r by 0.1cm. In order to keep the volume of pop the same, we find, using di
Math 209- Midterm Exam Practice Questions
1. Find and sketch
p the domain of the function:
f (x, y) = y + 25 x2 y 2
2. Evaluate the limit or show that it does not exist:
(a)
xy
p
lim
(x,y)(0,0)
x2 + y 2
(b)
lim
(x,y)(0,0) x2
2xy
+ 2y 2
(c)
x3 + y 3
(x,y)(
MATH 209Final Examination
Date: December 20, 2012
Time: 2 hours
Surname:
Given name(s):
(Please, print!)
ID#:
Signature:
Please, check your section/instructor!
Section Instructor X
EA1
EB1
EC1
ED1
EE1
EF1
EG1
EH1
Multiple Choice
Maximum
Mark
25
Long Answe
MATH 209
FINAL EXAMINATION
Date: December 15, 2007.
Time: 2 hours
GIVEN NAME(S):
SURNAME:
(PLEASE PRINT)
Signature:
INSTRUCTIONS:
1. There are 6 multiple choice questions and 3 long response questions. For
the long response questions you must show all you
Solutions to ASSIGNMENT 1
1. It is always best to start with a plot of the space-time diagram. Since
that we will be asked in point b) anyway, we shall choose the origin
of all coordinate systems to be at the event of the departure of a twin
from earth.
F
Solutions to ASSIGNMENT 2
Problem 5.4
The four-velocity is u = (c, V ), where three-velocity V = V i =
To compute the four acceleration a = du we need
d
is the usual three-acceleration a = dV /dt. Thus
d
dt
dxi
.
dt
a
= 3 Vc2 , where a
dt du
d d
V a 2 V a
Solutions to ASSIGNMENT 1
Problem 2. In the rest frame of the rocket the light has velocity c so
it takes t = L/c time to reach to nose of the rocket. This event
has coordinates (L/c, L) in rocket rest-frame. The earth moves with
V = 2/3c, transformation
Some formulae of tensor calculus and dierential geometry
1
General notation
We use Greek letters , . . . = 0, 1, 2, 3 for components of 4-vectors,tensors, etc and Roman letters
i, j, k . . . = 1, 2, 3 for their spatial components. Ordinary partial derivat
Part I:
1. Which of the intervals in Fig. 1 are time-like, space-like or null ?
ct
(a)
(b)
(c)
x
Figure 1: Diagram of Minkowski space-time. Light cone of the event at origin is shown in dashed
line (scales of the ct and x axes are the same)
a) - space-lik
FORMULA SHEET FOR ASSIGNMENT 2
1
Tensor transformation rules
Tensors are dened by their transformation properties under coordinate change. One distinguishes convariant and contravariant indexes. Number of indexes is tensors rank, scalar and vector quantit
ASSIGNMENT 6 solutions
I show the solution using Shwartzschild coordinates (which gives the right answer although the coordinates formally fail at r = rg ). One could also solve the problem in Lemaitre
coordinates.
The frequency of light, propagating with
ASSIGNMENT 3
Note: every time you are asked to prove the tensor properties of some object, you should demonstrate
it by presenting the appropriate transformation rules.
1. One has a 4-th rank tensor R . What kind of tensor R is ? Prove it.
Since we are gi
MATH 209
Final ExaminationSolutions
1
Multiple Choice Questions
White exams
1. Let S denote the surface consisting of the part of the plane x + y + z = 1 that lies inside the
the cylinder x2 + y 2 = 1, and let C denote its boundary. If
~ (x, y, z) = cos10