NOTE
PHZ 6607 Lecture Notes
1. Lecture 2
1.1. Definitions
Books:
(i) Tensor Analysis on Manifolds
(ii) The mathematical theory of black holes
(iii) Carroll
(iv) Schutz
Vector:
(i) In an N-Dimensional space, a vector is defined as a linear combination of tangential
directions within a basis of that space; denotes direction.
(ii) Vectors are not free to slide from point to point outside the context of that spaces.
A vector is attached to a single point. A=Aµ êµ
(iii) We define a vector which relies only on infinitesimal neighbourhoods.
d
du
=
dxi d
du dxi
Tensors:
(i) Within a vector space V, a tensor is a scalar-valued multilinear function with
variables all in either V or V*.
(ii) A family of tangent vector spaces over the curve of a manifold.
(iii) A tensor T of type (k,l) is a multilinear map from a collection of dual-vectors and
vectors to R.
(iv) A tensor of type (N,N) at P is defined to be a linear function which takes a
arguements N one-forms and N’ vectors and whose value is a real number.
Manifold:
(i) A topological space in which some neighbourhood of each point admits a coordinate
system consisting of real coordinate functions on points of the neighbourhood which
determine the position of points and the topology of that neighbourhood: aka it is
locally cartesian.
(ii) When a Eucledian space is stripped of its vector space only its differentiable
structure is retained; by piercing together domains of the Eucledian space in a
smooth manner we obtain a manifold.
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(iii) A manifold is a space that may be curved and have complicated topology but
in local regions look just like Rn . The local regions of same dimension patched
together create the manifold.
(iv) A set of points M is defined to be a manifold if each point of M has an open
neighbourhood which has a continuous 1-1 map onto an open set of Rn for some n.
Tangent vector:
(i) Directional derivative on an n-dimensional manifold.
(ii) The set of numbers
dxl
dλ
on the curve xl (λ)
(iii) The tangent vector V (λ) has components V µ =
dxµ
dλ
(iv) For a parametrized curve f (x0 (λ)) the tangent vector is
d
dλ
=
dxi d
dλ dxi
one-form:
(i) A linear mapping of the tangent space onto the reals.
(ii) The dual mapping of vectors onto a real number space.
(iii) One forms are elements of the dual vector space T ∗p
W = Wµ θ̂µ
(iv) A one-form is a linear real valued function of vectors. A one form W̃ at P associates
with a vector V~ at P a real number which we call W̃ (V~ )
Notes:
Answer 1.3—–Two points P1 and P2 are (x1 , y1 ) (x2 , y2 ) in old coordinate system
and are (u1 , v1 ) (u2 , v2 ) in new coordinate system
A
2B
A
∆l2 = ∆x2 + ∆y 2 = ∆u2 +
∆u∆v + ∆v 2
D
D
D
if u,v are orthogonal ⇒ ∆u∆v coefficient = 0
1.2. METRIC
ds2 = gab (dxa )(dxb )
gab is called metric
Example, For spherical polar coordinates θ, phi, R = a
ds2 = a2 (dθ2 + Sin2 θ dφ2 )
Looking close to top of sphere(on a tangent plane),
sin θ ≈ θ ⇒
ds2 = a2 (dθ2 + θ2 dφ2 )
but this corresponds to coordinate transformation x = aθ cos φ and y = aθ sin φ which
implies that metric for sphere becomes metric for cartesian coordinates under the assumption of small θ. This is an example of going to local cartesian coordinates
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Relation between ea and ea ::
~, B
~ , C
~
Let basis vectors for ea space be A
a
~ , E
~ , F~
Let basis vectors for e space be D
~D
~ is not necessarily 0,1 but can be
Since these are two different vector spaces, hence A.
any number α and similarly with their other combinations. Using these basis vectors
~ 0 .D
~ = 1. Similarly , B
~ 0 .E
~ = 1 and
we can construct another set of vectors such that A
0
~ .F~ = 1.
C
In Q1.3, u=ax+by and v=cx+dy.
How to find eu and eu ?
∂
.
Example of ex is ∂x
∂y
∂
∂x
⇒ eu = ∂u = ∂u ex + ∂u
ey
x
Example of e is dx.
⇒ eu = du = ∂u
ex + ∂u
ey
∂x
∂y
Note that î is ex
eu eu = 1
eu ev = 0
Parametrized curve is defined as:
X = X 1 (λ)
Y = X 2 (λ)
Z = X 3 (λ)
At each point we can define a vector which is tangent to curve
i
V i (X) = dX
dλ
i dxj
Scalar created by tangent vector : gij dx
dλ dλ
1.3. Relationship between upper and lower index basis vectors
u = ax + by
v = cx + dy
for x,y space we have ex , ey , ex , ey . What is eu , eu ?
A possible basis for ordinary vectors in x,y space is given by the differential
∂
∂
operators ∂x
and ∂y
. We can write
∂
∂x ∂
∂y ∂
=
+
∂u
∂u ∂x ∂u ∂y
We can generalize to
∂x
∂y
eu =
ex +
ey
∂u
∂u
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and
∂u x ∂u y
e +
e
∂x
∂y
Orthogonality between upper and lower indices is by decree.
eu =
1.4. Define a tangent vector
In space xi , λ is the arbitrary parameterization of the curve.
The tangent vector on the curve is
dxi
v =
dλ
i
then
1
L = mv 2
2
1
dxi dxj
= mgij
2
dλ dλ
find the variation of action with respect to different paths
Z
Z
dxi dxj
1
S = Ldλ =
mgij
dλ
2
dλ dλ
First path: xi (λ) Second path: xi (λ) + δxi (λ)
For the second path
Z
Z
d(xi + δxi ) d(xj + δxj )
1
S = Ldλ =
mgij (xk + δxk )
dλ
2
dλ
dλ
To first order
Z 1
∂gij k dxi dxj
dδxi dxj
dxi dδxj
δS = m
δx
+ gij
+ gij
dλ
2
∂xk
dλ dλ
dλ dλ
dλ dλ
We are at liberty to change labels within the summation
Z ∂gij k dxi dxj
dδxk dxj
dxi dδxk
1
δx
+ gkj
+ gik
dλ
δS = m
2
∂xk
dλ dλ
dλ dλ
dλ dλ
Using
d
dλ
gkj δx
k dx
j
dλ
=
dgkj k dxj
d2 xj
dδxk dxj
δx
+ gkj
+ gkj δxk 2
dλ
dλ
dλ dλ
dλ
we get
m
δS =
2
d2 xj
∂gjk dxi dxj
d2 xi ∂gik dxj dxi ∂gij dxi dxj
−gjk 2 −
− gik 2 −
+ k
dλ
∂xi dλ dλ
dλ
∂xj dλ dλ
∂x dλ dλ
Z
j
i
m
d
dx
dx k
+
gkj δxk
+ gik
δx dλ
2
dλ
dλ
dλ
Z δxk dλ
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Z d2 xj
1 ∂gik ∂gjk ∂gij dxi dxj
−gjk 2 −
+
−
δxk dλ
δS = m
dλ
2 ∂xj
∂xi
∂xk dλ dλ
Z
j
m
d
dxi k
k dx
+
gkj δx
+ gik
δx dλ
2
dλ
dλ
dλ
then
d2 xj
1
−gjk 2 −
dλ
2
∂gik ∂gjk ∂gij
+
−
∂xj
∂xi
∂xk
dxi dxj
=0
dλ dλ
multiply by (−g kl )
d2 xl 1 lk
+ g
dλ2
2
∂gik ∂gjk ∂gij
+
−
∂xj
∂xi
∂xk
dxi dxj
=0
dλ dλ
the boundary term goes away if we keep the end points fixed