PH5011 General Relativity - ASTRONOMY GROUPstar-hz4/gr/GRlec1+2+3.pdf · General Relativity Notes...

    PH5011

General Relativity

Notes from Martinmas 2011

md35@st-andrews.ac.uk

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

2

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

3

depend on

1 Curvilinear coordinates 1.1 Basis and coordinates

Example A: Cartesian coordinates (I)

4

1 Curvilinear coordinates 1.1 Basis and coordinates

Example B: Constant, non‐orthogonal system (I)

5

1 Curvilinear coordinates

1.2 Reciprocal basis

Kronecker‐delta

construction:

orthogonality

normalization

for

for

 orthogonal basis

orthonormal basis

6

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)

⤿

9

1 Curvilinear coordinates

1.3 Metric

⤿

coefficients of metric tensor (→ 1.5)

symmetry:

as matrix

10

1 Curvilinear coordinates 1.3 Metric

Examples A+B: Cartesian & non‐orthogonal constant basis (III)

⤿

⤿

11

1 Curvilinear coordinates 1.3 Metric

length of curve given by

parametric representation of curve

12

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”

15

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

⤿

21

1 Curvilinear coordinates 1.6 Coordinate transformations

Proof: are covariant components of a tensor

⤿

22

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]

⤿

⤿

⤿

25

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