Physics 171
Problem Set #2
Fall 2014
DUE: TUESDAY OCTOBER 28, 2014
1. Consider an inertial frame K that is moving with velocity v along the x axis
with respect to an inertial frame K. A photon of angular frequency is omitted from
a source located at the o
Physics 171
Solutions to Problem Set #4
Fall 2014
1. In class, I proved that for any covariant vector V ,
( )V = R V .
(1)
(a) Show that for a contravariant vector W ,
( )W = R W .
(2)
Noting that g = 0, it follows that ( )g = 0 after applying Leibnizs
ru
Physics 171
Solutions to Problem Set #3
Fall 2014
1. Consider the connection coecients prior to imposing the condition that is
symmetric under the interchange of its lower two indices. The torsion tensor, T , is dened
as
T .
(a) In class, I derived the tr
Physics 171
Solutions to Problem Set #2
Fall 2014
1. Consider an inertial frame K that is moving with velocity v along the x axis with respect
to an inertial frame K. A photon of angular frequency is omitted from a source located at
the origin of K and is
Physics 171
Solutions to Problem Set #1
Fall 2014
1. Suppose that a and b are proper Euclidean three-vectors, whereas c and d are improper
three-vectors (also called pseudovectors). Show that
Given that a and b are proper Euclidean three-vectors, whereas
Physics 171
Midterm Exam Solutions
Fall 2014
1. [10] The metric tensor in curved spacetime is denoted by g . The Levi-Civita tensor is
denoted by . One may be tempted to dene the dual metric tensor (analogous to the
dual electromagnetic eld strength tenso
Physics 171
Final Exam Solutions
Fall 2014
1. Assume that f (x) is a well behaved scalar function of the spacetime coordinate x .
Evaluate:
( )f (x) .
First, we consider f (x) f (x) . Noting that f (x) is a covariant vector, in
order to compute its covari
Physics 171
MIDTERM EXAM
Fall 2014
INSTRUCTIONS: This is a take-home exam. The point value of each problem is indicated in brackets. The exam should be completed and returned to me or to my ISB
mailbox at the end of the day on Friday November 14.
While wo
Physics 171
FINAL EXAM
Fall 2014
You have three hours to complete this exam. The point value of each of the three problems
is indicated in brackets, so use your time wisely. In answering the questions, you may quote
results that have been derived in the t
Physics 171
Problem Set #5
Fall 2014
DUE: FRIDAY DECEMBER 12, 2014
FINAL EXAM ALERT: The nal exam will be an in-class exam, and will take place in
Nat. Sci. Annex 103 from 123 pm on Monday December 15, 2003. The exam will cover
the entire course material.
Physics 171
Problem Set #4
Fall 2014
DUE: TUESDAY DECEMBER 2, 2014
1. In class, I proved that for any covariant vector V ,
( )V = R V .
(1)
(a) Show that for a contravariant vector W ,
( )W = R W .
(2)
(b) Given a contravariant second rank tensor T and a
Physics 171
Problem Set #3
Fall 2014
MIDTERM ALERT: The midterm exam will be a take-home exam. The exam will be
handed out in class on Thursday November 13. Completed exams should be returned
to me by the end of the day on Friday November 14. The exam wil
Physics 171
Problem Set #1
Fall 2014
DUE: TUESDAY OCTOBER 14, 2014
1. Suppose that a and b are proper Euclidean three-vectors, whereas c and d are improper
three-vectors (also called pseudovectors). Show that
(a) a b and c d are both scalar quantities, wh
Physics 171
Solutions to Problem Set #5
Fall 2014
1. Once across the event horizon of a black hole, what is the longest proper time the observer
can spend before reaching the singularity of the black hole?
A curve of extremal proper time is a geodesic. Fo