PH5011 General Relativity - Astronomy Group

    PH5011
General Relativity

Notes from Martinmas 2011
[email protected]

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

coordinate line given by for all

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 depend on

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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

construction: Kronecker‐delta
orthogonality
normalization for
for
 orthogonal basis for
orthonormal basis
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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)
as matrix symmetry:

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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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With the reciprocal basis 1 Curvilinear coordinates

1.3 Metric

,

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 (→ 1.6)
covariant components

“raising/lowering indices”

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1.5 Tensor fields 1 Curvilinear coordinates

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

rank of tensor tensor of rank 0 scalar
tensor of rank 1 vector
tensor of rank 2 square matrix
tensor of rank 3 cube
              ........

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basis vectors 1 Curvilinear coordinates
⤿
1.5 Tensor fields

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 derivatives
differentials
} {contravariant
components transform like coordinate

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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Proof: 1 Curvilinear coordinates
⤿
1.7 Affine connection
(II) + (III) ‐ (I) :
⤿ (I)
(II)
(III)

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2 Tensor analysis and the basis vectors

2.1 Covariant derivative

vector field
    both the vector components
     depend on the coordinates

derivative:

define covariant derivative of a contravariant vector component
as so that

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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 eachupper {index , add
lower

where takes place of in or

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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

antisymmetri[[c 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: is a scalar
The scalar product of two vectors
⤿

On the other hand

⤿
(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)

⤿ matches required conditions

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