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0 General issues
0.1 Summation convention
dimension of coordinate space
pairwise indices imply sum
0.2 Indices
Apart from a few exceptions, upper and lowerindices
are to be distinguished thoroughly
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1 Curvilinear coordinates
1.1 Basis and coordinates
location described by set of coordinates
for all coordinate line given by
tangent vector at
≡ basis vector related to coordinate
⤿ set of basis vectors spans tangent space at
infinitesimal displacement in space on variation of coordinate
given by line element
in general, the basis vectors
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depend on
1 Curvilinear coordinates 1.1 Basis and coordinates
Example A: Cartesian coordinates (I)
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1 Curvilinear coordinates 1.1 Basis and coordinates
Example B: Constant, non‐orthogonal system (I)
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1 Curvilinear coordinates
1.2 Reciprocal basis
Kronecker‐delta
construction:
orthogonality
normalization
for
for
orthogonal basis
orthonormal basis
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for
1 Curvilinear coordinates 1.2 Reciprocal basis
Special case: 3 dimensions
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1 Curvilinear coordinates 1.2 Reciprocal basis
Example A: Cartesian coordinates (II)
⤿
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1 Curvilinear coordinates 1.2 Reciprocal basis
Example B: Constant, non‐orthogonal system (II)
⤿
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1 Curvilinear coordinates
1.3 Metric
⤿
coefficients of metric tensor (→ 1.5)
symmetry:
as matrix
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1 Curvilinear coordinates 1.3 Metric
Examples A+B: Cartesian & non‐orthogonal constant basis (III)
⤿
⤿
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1 Curvilinear coordinates 1.3 Metric
length of curve given by
parametric representation of curve
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1 Curvilinear coordinates 1.3 Metric
Example: Length of equator in spherical coordinates
use parameter along the azimuth
⤿
⤿ in
one only needs to consider
one full turn for and
:
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1 Curvilinear coordinates 1.3 Metric
With the reciprocal basis ,
one defines reciprocal components of the metric tensor
which fulfill ,
equivalent to the condition for the inverse matrix
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1 Curvilinear coordinates 1.3 Metric
metric tensor
orthonormality condition
⤿
“lowers index”
“raises index”
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1 Curvilinear coordinates
1.4 Vector fields
mathematics: vector field
physics: vector (field)
vector components defined by means of basis vectors
contravariant components
covariant components (→ 1.6)
“raising/lowering indices”
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1 Curvilinear coordinates
1.5 Tensor fields
mathematics: tensor field
physics: tensor (field)
tensor is multi‐dimensional generalization of vector
product of vector spaces
behaves like a vector with respect to each of the vector spaces
tensor of rank 0
tensor of rank 1
tensor of rank 2
tensor of rank 3
........
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rank of tensor
scalar
vector
square matrix
cube
1 Curvilinear coordinates 1.5 Tensor fields
basis vectors
⤿
apply to each of the vector spaces
contravariant components
covariant components
mixed components
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1 Curvilinear coordinates 1.5 Tensor fields
Example: Rank‐2 tensor
⤿
Coincidentally, with the matrix product
For Cartesian coordinates:
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1 Curvilinear coordinates
1.6 Coordinate transformations
consider different set of coordinates
(chain rule)
different coordinate systems describe same locations
⤿
⤿
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1 Curvilinear coordinates 1.6 Coordinate transformations
vector fields
⤿
covariant contravariant }
components transform like coordinate {
derivatives
differentials
tensor fields
⤿
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1 Curvilinear coordinates 1.6 Coordinate transformations
Proof: are covariant components of a tensor
⤿
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1 Curvilinear coordinates
1.7 Affine connection
in general, basis vectors depend on the coordinates
derivative of basis vector written in basis
affine connection (Christoffel symbol)
derivative of reciprocal basis vector:
⤿
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1 Curvilinear coordinates 1.7 Affine connection
Example C: Spherical coordinates (IV)
⤿
⤿
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1 Curvilinear coordinates 1.7 Affine connection
Example C: Spherical coordinates (IV) [continued]
⤿
⤿
⤿
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1 Curvilinear coordinates 1.7 Affine connection
given that
the Christoffel symbolscan be expressed by means
of the components of the metric tensorand their derivatives
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1 Curvilinear coordinates 1.7 Affine connection
Proof:
⤿ (I)
(II)
(III)
(II) + (III) ‐ (I) :
⤿
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2 Tensor analysis
2.1 Covariant derivative
vector field
both the vector components
depend on the coordinates
derivative:
and the basis vectors
define covariant derivative of a contravariant vector component
as
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so that
2 Tensor analysis 2.1 Covariant derivative
derivatives transform as
⤿ can be considered the covariant components
of the vector
(gradient)
covariant components of a vector
form components of a tensor, not
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2 Tensor analysis 2.1 Covariant derivative
contravariant components covariant components
covariant derivatives of tensor components
for each { }
upper
lower index
in
, add {
or where takes place of
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2 Tensor analysis 2.1 Covariant derivative
Covariant derivative of 2nd‐rank tensor
⤿
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2 Tensor analysis
2.2 Riemann tensor
order of 2nd covariant derivatives of vector
is not commutative, but
with the Riemann (curvature) tensor
(not intended to be memorized)
with
⤿
and m
Rilkj = gim R lkj
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2 Tensor analysis 2.2 Riemann tensor
Riemann tensor
[[
has two pairs of indices and is
antisymmetric in the indices of each pair
symmetric in exchanging the pairs
Moreover,
(1st Bianchi identity)
(2nd Bianchi identity)
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2 Tensor analysis 2.2 Riemann tensor
Proof:
The scalar product of two vectors
⤿
On the other hand
is a scalar
⤿
(Riemann curvature tensor is antisymmetric in first two indices)
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2 Tensor analysis
2.3 Einstein tensor
2nd‐rank curvature tensor fulfilling
must relate to Riemann tensor
only a single non‐vanishing contraction (up to a sign)
(Ricci tensor)
with next‐level contraction
(Ricci scalar)
⤿
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matches required conditions
