Lecture 13 Notes, 95.658, Spring 2012, Electromagnetic Theory II
Dr. Christopher S. Baird, UMass Lowell
1. Covariant Geometry
 We would like to develop a mathematical framework where special relativity can be applied more
naturally.
 The Lorentz transformations were derived from Einstein's principle of relativity:
c
2
t
'
2
−
x
'
2
y
'
2
z
'
2
=
c
2
t
2
−
x
2
y
2
z
2
 This means that all the terms on the left always equal the same scalar no matter what frame of
reference we are in. This value is invariant under Lorentz transformations.
 In regular threedimensional Galilean relativity, the dot product of two position vectors is
invariant under transformations.
Define the 4vector (covariant) geometry as the set of rules that lead to the dot product of
any
two
4vectors being invariant under Lorentz transformations.
 If we designate the column 4vector
A
μ
as a “covariant” vector (where covariant implies that its
dot product does not change under Lorentz transformations), then to form a dot product we must
multiply by a row vector. Let us write the row 4vector as
A
μ
and call it a “contravariant” vector to
imply that it is dotted against the covariant vector.
 The label
μ
on the vector is an index that runs from 0 to 3, specifying each component of the
4vector, so that for instance,
x
2
=
y
. Note the convention that when we are indexing a fourvector,
we use Greek letters such as
μ
, ν, etc. , but when we are indexing a regular three component vector,
we use Latin letters such as
i
,
j
,
k
.
 Our dot product then looks like this:
∑
=
0
3
A
A
 Note that if we recognize the product of a contravariant vector with a covariant vector always as a
dot product, the summation symbol is unnecessary. We drop the summation symbol with the
understanding that repeated indices always means summation
.
A
A
 Assume we know the contravariant vector. What does its corresponding covariant vector look
like?
 Let us look at the spacetime coordinate vector
x
μ
= (
ct
,
x
,
y
,
z
). We know what its dot product
should look like:
x
x
=
c
2
t
2
−
x
2
−
y
2
−
z
2
 The only way this is possible is if we define the covariant vector as
x
μ
= (
ct
, 
x
, 
y
, 
z
).
 In general then we define the dot product of any two 4vectors as the product of one in covariant
form and the other in contravariant form, where the two forms are related to each other by a sign
change of the last three components.
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View Full Document Can we mathematically express the relationship between a contravariant vector and its
corresponding covariant vector? The metric tensor (similar to a matrix)
g
μν
accomplishes this.
Notice that it has two Greek indices, so it is a twodimensional tensor with 16 components total.
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 Spring '11
 Staff
 Dot Product, Mass, Work, Special Relativity, Covariance and contravariance of vectors, Lorentz, covariant tensors

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