Special and General Relativity Lecture Notes:
Day 2 (08/29/08)
Haridis Pal, Benjamin Hall
Contents
1 Vectors
1.1 Vector Types . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 1-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Tensors
2.1 Tensor Types .
2.2 Transformations
2.3 Other examples:
2.4 Other types: . .
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3 Homework from Lecture 1
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4 Next Class
4.1 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Topics for next class . . . . . . . . . . . . . . . . . . . . . . .
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1
1
Vectors
1.1
Vector Types
• as displacement
• as operators
• by components
• basis vectors
• as geometrical objects
Note that, for vectors:
1. index location matters
2. Einstein index summation applies
There are 2 types of vectors distinguished by placing the indices as a
superscript or a subscript:
dxa (λ)
dλ
∂φ
=
∂xa
Va =
(1)
Va
(2)
where φ and λ are scalars.
Also, they are differentiated by the way they transform:
Let xa → X A (xa ), then:
VA ≡
∂φ
∂xa ∂φ
∂xa
≡
=
Ua
∂X A
∂X A ∂xa
∂xA
(3)
Note: In this sense, the position vector is not a vector, but change in position
is.
2
Although we have 2 types of vectors, our definition is slightly different
from a mathematician’s definition:
V a e~a ' V ' e~a Va
(4)
Above e~a are basis vectors, not necessarily normalized
• Basis vectors exist.
• Come in two types.
• Conventional relation:
ea eb = δab
Now, V → v a ∂x∂ a . Here
1.2
∂
∂xa
(5)
behaves as e~a , i.e., as basis vectors.
1-form
∂φ a
dx
∂xa
V 2 = v a va
dφ =
3
(6)
(7)
2
Tensors
2.1
Tensor Types
If v a , wa are vectors, (i.e. first rank tensors), then we can form new quantities
called tensors:
y ab = v a wb , v a zb , va zb
(8)
All these are second rank tensors of different type.
2.2
Transformations
Transformation → y AB =
2.3
∂xA ∂xB ab
y
∂xa ∂xb
Other examples:
• Mba Ndb = Pda
• gab g bc = δac
v a → called contravariant; va → called covariant.
Let’s see how much we can proceed further without using a metric.
Let Aa ,Ba → matrices such that Aa Bb = Mba
M |n×n = εabc...n Msa Mrb Mtc ...Mba εsrt...n
∂φ
Henceforth we will write ∂x
a as ∂a φ.
The covariant way to write this is ∇a φ.
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(9)
2.4
Other types:
• ∂i Bj − ∂j Bi : exterior derivative
• vj = Ai ∂i Bj + Bi ∂j Ai : lieder, transform as a vector
• y a = Mba V b ; xa Mba = Zb
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Homework from Lecture 1
Example of scalars without using the metric:
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R
∇a φdxa , εabc Aa Bb Cc
Next Class
4.1
Reading
Handout #3 and #5
4.2
Exercises
Problems 1 - 3 on http://www.phys.ufl.edu/courses/phz6607/fall08/Questions.pdf
4.3
Topics for next class
• Parameterized curves
• Metric
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