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A Short Introduction to Analysis

Kostas Kokkotas a March 14, 2019

Eberhard Karls Univerity of Tubingen¨

1 a This chapter based strongly on “Lectures of ” by A. Papapetrou, D. Reidel publishing company, (1974) Scalars and Vectors

A n-dim is a space M on every point of which we can assign n numbers( x 1,x 2,...,x n) - the coordinates - in such a way that there will be an one to one correspondence between the points and the n numbers. Every point of the manifold has its own neighborhood which can be mapped to a n-dim Euclidean space. The manifold cannot be always covered by a single system of coordinates and there is not a preferable one either. The coordinates of the point P are connected by relations of the form: 0 0 x µ = x µ x 1, x 2, ..., x n for µ0 = 1, ..., n and their inverse  0 0 0  x µ = x µ x 1 , x 2 , ..., x n for µ = 1, ..., n. If there exist µ0 ν µ0 ∂x ν ∂x µ0 A = and A 0 = ⇒ det |A | (1) ν ∂x ν µ ∂x µ0 ν then the manifold is called differential.

2 • Vector field (contravariant): an example is the infinitesimal displacement vector, leading from a point A with coordinates x µ to a neighbouring point A0 with coordinates x µ + dx µ. The components of such a vector are the differentials dx µ.

Scalars and Vectors

Any physical quantity, e.g. the velocity of a particle, is determined by a set of numerical values - its components - which depend on the . Studying the way in which these values change with the coordinate system leads to the concept of tensor.

With the help of this concept we can express the physical laws by tensor equations, which have the same form in every coordinate system.

• Scalar field : is any physical quantity determined by a single numerical value i.e. just one component which is independent of the coordinate system (mass, charge,...)

3 Scalars and Vectors

Any physical quantity, e.g. the velocity of a particle, is determined by a set of numerical values - its components - which depend on the coordinate system. Tensors Studying the way in which these values change with the coordinate system leads to the concept of tensor.

With the help of this concept we can express the physical laws by tensor equations, which have the same form in every coordinate system.

• Scalar field : is any physical quantity determined by a single numerical value i.e. just one component which is independent of the coordinate system (mass, charge,...) • Vector field (contravariant): an example is the infinitesimal displacement vector, leading from a point A with coordinates x µ to a neighbouring point A0 with coordinates x µ + dx µ. The components of such a vector are the differentials dx µ.

3 Vector Transformations

From the infinitesimal vector AA~ 0 with components dx µ we can construct a finite vector v µ defined at A. This will be the tangent vector to the curve x µ = f µ(λ) passing from the points A and A0 corresponding to the values λ and λ + dλ of the parameter. Then dx µ v µ = . (2) dλ Any transformation from x µ to˜x µ (x µ → x˜µ) will be determined by n equations of the form:˜x µ = f µ(x ν ) where µ , ν = 1, 2, ..., n. This means that : X ∂f µ X ∂x˜µ dx˜µ = dx ν = dx ν for ν = 1, ..., n (3) ∂x ν ∂x ν ν ν and dx˜µ X ∂x˜µ dx ν X ∂x˜µ v˜µ = = = v ν (4) dλ ∂x ν dλ ∂x ν ν ν

4 Contravariant and Covariant Vectors

Contravariant Vector: is a quantity with n components depending on the coordinate system in such a way that the components aµ in the coordinate system x µ are related to the components˜aµ in˜x µ by a relation of the form X ∂x˜µ a˜µ = aν (5) ∂x ν ν

Covariant Vector: eg. bµ, is an object with n components which depend on the coordinate system on such a way that if aµ is any contravariant vector, the following sums are scalars

X µ X µ µ µ bµa = b˜µa˜ = φ for any x → x˜ [ Scalar Product] (6) µ µ

The covariant vector will transform as (why?): X ∂x ν X ∂x˜ν b˜ = b or b = b˜ (7) µ ∂x˜µ ν µ ∂x µ ν ν ν What is Einstein’s summation convention?

5 Tensors: at last

A contravariant tensor of order 2 is a quantity having n2 components T µν µ µ which transforms (x → x˜ ) in such a way that, if aµ and bµ are arbitrary covariant vectors the following sums are scalars:

λµ λµ µ µ T aµbλ = T˜ a˜λb˜µ ≡ φ for any x → x˜ (8)

Then the transformation formulae for the components of the tensors of order 2 are (why?):

∂x˜α ∂x˜β ∂x˜α ∂x ν ∂x µ ∂x ν T˜ αβ = T µν , T˜ α = T µ & T˜ = T ∂x µ ∂x ν β ∂x µ ∂x˜β ν αβ ∂x˜α ∂x˜β µν The Kronecker symbol ( λ 0 if λ 6= µ , δ µ = 1 if λ = µ .

is a having frame independent values for its components.

αβγ... ? Tensors of higher order: T µνλ...

6 Tensor algebrai

• Tensor addition : Tensors of the same order( p, q) can be added, their sum being again a tensor of the same order. For example: ∂x˜ν a˜ν + b˜ν = (aµ + bµ) (9) ∂x µ

• Tensor multiplication : The product of two vectors is a tensor of order 2, because ∂x˜α ∂x˜β a˜αb˜β = aµbν (10) ∂x µ ∂x ν in general:

µν µ ν µ µ T = A B or T ν = A Bν or Tµν = AµBν (11)

• Contraction: for any mixed tensor of order( p, q) leads to a tensor of order( p − 1, q − 1)(prove it!)

λµν µν T λα = T α (12)

7 Tensor ii

α α • Trace: of the mixed tensor T β is called the scalar T = T α.

• : Tλµ = Tµλ orT(λµ) ,

Tνλµ = Tνµλ or Tν(λµ)

• Antisymmetric : Tλµ = −Tµλ or T[λµ],

Tνλµ = −Tνµλ or Tν[λµ]

Number of independent components : Symmetric: n(n + 1)/2, Antisymmetric: n(n − 1)/2

8 Tensors: Differentiation & Connectionsi

We consider a region V of the space in which some tensor, e.g. a covariant α vector aλ, is given at each point P(x ) i.e.

α aλ = aλ(x )

We say then that we are given a tensor field in V and we assume that the components of the tensor are continuous and differentiable functions od x α.

Question: Is it possible to construct a new tensor field by differentiating the given one?

The simplest tensor field is a scalar field φ = φ(x α) and its derivatives are the components of a covariant tensor! ∂φ ∂x α ∂φ ∂φ = we will use: = φ ≡ ∂ φ (13) ∂x˜λ ∂x˜λ ∂x α ∂x α ,α α

i.e. φ,α is the gradient of the scalar field φ.

9 Tensors: Differentiation & Connections ii The derivative of a contravariant vector field Aµ is : ∂Aµ ∂  ∂x µ  ∂x˜ρ ∂  ∂x µ  ≡ Aµ = A˜ν = A˜ν ∂x α ,α ∂x α ∂x˜ν ∂x α ∂x˜ρ ∂x˜ν ∂2x µ ∂x˜ρ ∂x µ ∂x˜ρ ∂A˜ν = A˜ν + (14) ∂x˜ν ∂x˜ρ ∂x α ∂x˜ν ∂x α ∂x˜ρ Without the first term (red) in the right hand side this equation would be the transformation formula for a mixed tensor of order 2. The transformation (x µ → x˜µ) of the derivative of a vector is:

∂2x µ ∂x˜ν ∂x˜σ ∂x µ ∂x˜ρ Aµ − Aκ = A˜ν (15) ,α ∂x˜ν ∂x˜σ ∂x κ ∂x α ∂x˜ν ∂x α ,ρ

| {zµ } Γακ

in another coordinate (x 0µ → x˜µ) we get again: ∂2x 0µ ∂x˜ν ∂x˜σ ∂x 0µ ∂x˜ρ A0µ − A0κ = A˜ν (16) ,α ∂x˜ν ∂x˜σ ∂x 0κ ∂x α ∂x˜ν ∂x 0α ,ρ | {z } 0µ Γ ακ

10 Tensors: Differentiation & Connections iii

Suggesting that the transformation (x µ → x 0µ) will be:

∂x µ ∂x 0ρ Aµ + Γµ Aκ = A0ν + Γ0ν A0σ (17) ,α ακ ∂x 0ν ∂x α ,ρ σρ

µ µ κ The necessary and sufficient condition for A ,α + Γ ακA to be a tensor is:

∂2x µ ∂x 0λ ∂x κ ∂x σ ∂x 0λ Γ0λ = + Γµ . (18) ρν ∂x 0ν ∂x 0ρ ∂x µ ∂x 0ρ ∂x 0ν ∂x µ κσ

λ The objectΓ ρν is the called the of the space and it is not tensor.

11

According to the previous assumptions, the following quantity transforms as a tensor of order 2

µ µ µ λ µ µ µ λ A ;α = A ,α + Γ αλA or ∇αA = ∂αA + Γ αλA (19)

and is called absolute or covariant derivative of the contravariant vector Aµ. In similar way we get (how?):

φ;λ = φ,λ (20) ρ Aλ;µ = Aλ,µ − ΓµλAρ (21) λµ λµ λ αµ µ λα T ;ν = T ,ν + Γαν T + Γαν T (22) λ λ λ α α λ T µ;ν = T µ,ν + Γαν T µ − Γµν T α (23) α α Tλµ;ν = Tλµ,ν − Γλν Tµα − Γµν Tλα (24) λµ··· λµ··· T νρ··· ;σ = T νρ··· ,σ λ αµ··· µ λα··· + ΓασT νρ··· + ΓασT νρ··· + ··· α λµ··· α λµ··· − ΓνσT αρ··· − ΓρσT να··· − · · · (25) 12 of a vectori

Let aµ be some covariant vector field. Consider two neighbouring points P and P0, and the displacement PP0 having components dx µ.

0 At the point P we may define two vectors associated to aµ(P): 0 a) The vector aµ(P ) defined as:

0 ν aµ(P )= aµ(P) + aµ,ν (P)dx = aµ(P) +∆ aµ(P) (26) 0 ν The difference aµ(P ) − aµ(P) = aµ,ν (P)dx ≡ ∆aµ(P) is not a tensor.

0 0 b) The vector Aµ(P ) after its “parallel transport” from P to P

0 Aµ(P )= aµ(P) + δaµ(P). (27)

δaµ(P) is not yet defined.

13 Parallel Transport of a vector ii

λ The connectionΓ µν allows us to define a covariant vector aλ;µ which is a λ tensor. In other words with the help ofΓ µν we can define a vector at a point 0 P , which has to be considered as “equivalent” to the vector aµ defined at P. The difference of the two vectors

0 0 aµ(P ) − Aµ(P ) = aµ + ∆aµ − (aµ + δaµ) = ∆aµ − δaµ | {z } | {z } | {z } | {z } vector at point P at point P vector ν λ  ν λ ν = aµ,ν dx − δaµ = aµ,ν − Cµν aλ dx i.e. δaµ = Cµν aλdx | {z } vector

λ λ Which leads to the obvious suggestion that Cµν ≡ Γ µν .

14 Parallel Transport of a vector iii

Parallel Transport λ The connectionΓ µν allows to define the transport of a vector aλ from a point P to a neighbouring point P0 (Parallel Transport).

λ ν δaµ = Γ µν aλdx for covariant vectors (28) µ µ λ ν δa = −Γ λν a dx for contravariant vectors (29)

•The parallel transport of a scalar field is zero! δφ = 0(why?)

15 Review of the 1st Lecture

Tensor Transformations X ∂x ν X ∂x˜µ b˜ = b and˜ aµ = aν (30) µ ∂x˜µ ν ∂x ν ν ν

∂x˜α ∂x˜β ∂x˜α ∂x ν ∂x µ ∂x ν T˜ αβ = T µν , T˜ α = T µ & T˜ = T ∂x µ ∂x ν β ∂x µ ∂x˜β ν αβ ∂x˜α ∂x˜β µν Covariant Derivative

φ;λ = φ,λ (31) ρ Aλ;µ = Aλ,µ − ΓµλAρ (32) λµ λµ λ αµ µ λα T ;ν = T ,ν + Γαν T + Γαν T (33)

Parallel Transport

λ ν δaµ = Γ µν aλdx for covariant vectors (34) µ µ λ ν δa = −Γ λν a dx for contravariant vectors (35)

16 Curvature Tensori

The“expedition” of a vector parallel transported along a closed path

The parallel transport of the vector aλ from the point P to A leads to a change λ µ ν of the vector by a quantityΓ µν (P)a dx and the new vector is:

λ λ λ µ ν a (A) = a (P) − Γµν (P)a (P)dx (36)

A further parallel transport to the point B will lead the vector

λ λ λ ρ σ a (B) = a (A) − Γ ρσ(A)a (A)δx λ λ µ ν λ  ρ ρ β ν  σ = a (P) − Γ µν (P)a (P)dx − Γ ρσ(A) a (P) − Γ βν (P)a dx δx

17 Curvature Tensor ii

Since both dx ν and δx ν assumed to be small we can use the following expression λ λ λ µ Γ ρσ(A) ≈ Γ ρσ(P) + Γ ρσ,µ(P)dx . (37) Thus we have estimated the total change of the vector aλ from the point P to B via A (all terms are defined at the point P).

λ λ λ µ ν λ ρ σ λ ρ β ν β λ ρ τ σ a (B)= a − Γµν a dx − Γρσa δx + Γ ρσΓβν a dx δx + Γρσ,τ a dx δx λ ρ β τ ν σ − Γ ρσ,τ Γ βν a dx dx δx

If we follow the path P → C → B we get:

λ λ λ µ ν λ ρ σ λ ρ β ν β λ ρ τ σ a (B)= a −Γ µν a δx −Γ ρσa dx +Γ ρσΓ βν a δx dx +Γ ρσ,τ a δx dx

and the accumulated effect on the vector will be

λ λ λ β ν σ σ ν λ ρ λ  δa ≡ a (B) − a (B) = a (dx δx − dx δx ) Γ ρσΓβν + Γ βσ,ν (38)

18 Curvature Tensor iii By exchanging the indices ν → σ and σ → ν we construct a similar relation

λ β σ ν ν σ λ ρ λ  δa = a (dx δx − dx δx ) Γ ρν Γ βσ + Γ βν,σ (39)

and the total change will be given by the following relation: 1 δaλ = − aβ Rλ (dx σδx ν − dx ν δx σ) (40) 2 βνσ

where λ λ λ µ λ µ λ R βνσ = −Γ βν,σ + Γ βσ,ν − Γ βν Γ µσ + Γ βσΓ µν (41) is the curvature tensor.

Figure 1: Measuring the curvature for the space.

19

• For a vector uλ at point P we apply the parallel transport along a curve on an n-dimensional space which will be given by n equations of the form: x µ = f µ(λ); µ = 1, 2, ..., n

µ dxµ • If u = dλ is the tangent vector at P, the parallel transport of this vector will determine at another point of the curve a vector which will not be in general tangent to the curve. • If the transported vector is tangent to any point of the curve then this curve is a curve of this space and is given by the equation : duρ + Γρ uµuν = 0 . (42) dλ µν • Geodesic curves are the shortest curves connecting two points on a curved space.

20

A space is called a metric space if a prescription is given attributing a scalar distance to each pair of neighbouring points The distance ds of two points P(x µ) and P0(x µ + dx µ) is given by 2 2 2 ds2 = dx 1 + dx 2 + dx 3 (43) In another coordinate system,˜x µ, we will get ∂x ν dx ν = dx˜α (44) ∂x˜α which leads to: 2 µ ν α β ds =g ˜µν dx˜ dx˜ = gαβ dx dx . (45) This gives the following transformation relation (why?): ∂x α ∂x β g˜ = g (46) µν ∂x˜µ ∂x˜ν αβ suggesting that the quantity gµν is a symmetric tensor, the so called metric tensor. The relation (45) characterises a Riemannian space: This is a metric space in which the distance between neighbouring points is given by (45). 21 • If at some point P there are given 2 infinitesimal displacements d (1)x α and d (2)x α , the metric tensor allows to construct the scalar

(1) α (2) β gαβ d x d x

which shall call scalar product of the two vectors. • Properties:

α λα λ µ µα gµαA = Aµ, gµαT = T µ, gµν T α = Tνα, gµν gασT = Tνσ

• Metric element for Minkowski spacetime

ds2 = −dt2 + dx 2 + dy 2 + dz2 (47) ds2 = −dt2 + dr 2 + r 2dθ2 + r 2 sin2 θdφ2 (48)

• For a sphere with radius R :

ds2 = R2 dθ2 + sin2 θdφ2 (49)

• The metric element of a torus with radii a and b

ds2 = a2dφ2 + (b + a sin φ)2 dθ2 (50)

22 • The contravariant form of the metric tensor:

αβ β αβ 1 αβ gµαg = δ µ where g = G ← minor determinant (51) det |gµν | With g µν we can now raise lower indices of tensors

µ µν µν µρ ν µρ νσ A = g Aν , T = g T ρ = g g Tρσ (52)

• The angle, ψ, between two infinitesimal vectors d (1)x α and d (2)x α is 1:

g d (1)x α d (2)x β cos(ψ) = αβ . (53) p (1) ρ (1) σp (2) µ (2) ν gρσ d x d x gµν d x d x

1Remember the Euclidean relation: A~ · B~ = ||A|| · ||B|| cos(ψ)

23 The Determinant of gµν

The quantity g ≡ det |gµν | is the determinant of the metric tensor. The determinant transforms as :  ∂x α ∂x β   ∂x α   ∂x β  g˜ = detg ˜ = det g = det g · det · det µν αβ ∂x˜µ ∂x˜ν αβ ∂x˜µ ∂x˜ν  ∂x α 2 = det g = J 2g (54) ∂x˜µ where J is the Jacobian of the transformation. This relation can be written also as: p|g˜| = J p|g| (55) i.e. the quantity p|g| is a scalar density of weight 1. • The quantity p|g|δV ≡ p|g|dx 1dx 2 ... dx n (56) is the invariant volume element of the Riemannian space. • If the determinant vanishes at a point P the invariant volume is zero and this point will be called a singular point.

24 Christoffel Symbols

In Riemannian space there is a special connection derived directly from the metric tensor. This is based on a suggestion originating from that is : “ If a vector aλ is given at some point P, its length must remain unchanged under parallel transport to neighboring points P0”.

2 2 µ ν 0 µ 0 ν 0 |~a|P = |~a|P0 or gµν (P)a (P)a (P) = gµν (P )a (P )a (P ) (57) Since the distance between P and P0 is |dx ρ| we can get

0 ρ gµν (P ) ≈ gµν (P) + gµν,ρ(P)dx (58) µ 0 µ µ σ ρ a (P ) ≈ a (P) − Γσρ(P)a (P)dx (59) 25 By substituting these two relation into equation (57) we get (how?)

σ σ µ ν ρ (gµν,ρ − gµσΓ νρ − gσν Γ µρ) a a dx = 0 (60)

This relations must by valid for any vector aν and any displacement dx ν which leads to the conclusion the relation in the parenthesis is zero. Closer observation shows that this is the covariant derivative of the metric tensor !

σ σ gµν;ρ = gµν,ρ − gµσΓ νρ − gσν Γ µρ =0 . (61) i.e. gµν is covariantly constant. This leads to a unique determination of the connections of the space (Riemannian space) which will have the form (why?)

1 Γα = g αν (g + g − g ) (62) µρ 2 µν,ρ νρ,µ ρµ,ν and will be called Christoffel Symbols .

α α It is obvious thatΓ µρ = Γ ρµ .

26 Geodesics in a Riemann Space

The geodesics of a Riemannian space have the following important property. If a geodesic is connecting two points A and B is distinguished from the neighboring lines connecting these points as the line of minimum or maximum length. The length of a curve, x µ(s), connecting A and B is:

Z B Z B µ ν 1/2 h α dx dx i S = ds = gµν (x ) ds A A ds ds

A neighboring curve˜x µ(s) connecting the same points will be described by the equation: x˜µ(s) = x µ(s) +  ξµ(s) (63) where ξµ(A) = ξµ(B) = 0. The length of the new curve will be: Z B h dx˜µ dx˜ν i1/2 S˜ = g (˜x α) ds (64) µν ds ds A 27 For simplicity we set: α α α µ ν 1/2 α α α µ ν 1/2 f (x , u ) = [gµν (x )u u ] and f˜(˜x , u˜ ) = [gµν (˜x )˜u u˜ ]

uµ =x ˙ µ = dx µ/ds and uµ = x˜˙ µ = dx˜µ/ds =x ˙ + ξ˙µ (65)

Then we can create the difference δS = S˜ − S 2 Z B Z B δS = δf ds = f˜ − f  ds A A Z B Z B h ∂f d  ∂f i α d  ∂f α =  α − α ξ ds +  α ξ ds A ∂x ds ∂u A ds ∂u The last term does not contribute and the condition for the length of S to be an extremum will be expressed by the relation:

2 We make use of the following relations:

α α α α α α α α  α ∂f α ∂f  2 f (˜x , u˜ )= f (x + ξ , u + ξ˙ ) = f (x , u ) +  ξ + ξ˙ + O( ) ∂xα ∂uα

d  ∂f α ∂f α d  ∂f  α ξ = ξ˙ + ξ ds ∂uα ∂uα ds ∂uα

28 Z B h ∂f d  ∂f i α δS =  α − α ξ ds = 0 (66) A ∂x ds ∂u Since ξα is arbitrary, we must have for each point of the curve: d  ∂f  ∂f − = 0 (67) ds ∂uµ ∂x µ Notice that the Langrangian of a freely moving particle with mass m = 2, is: µ ν 2 L = gµν u u ≡ f this leads to the following relations ∂f 1 ∂L ∂f 1 ∂L = L−1/2 and = L−1/2 (68) ∂uα 2 ∂uα ∂x α 2 ∂x α and by substitution in (67) we come to the condition

d  ∂L  ∂L − = 0 . (69) ds ∂uµ ∂x µ

29 µ ν Since L = gµν u u we get: ∂L ∂uµ ∂uν = g uν + g uµ ∂uα µν ∂uα µν ∂uα µ ν µ ν µ = gµν δ αu + gµν u δ α =2 gµαu (70) ∂L = g uµuν (71) ∂x α µν,α thus d dg duµ duµ (2g uµ)=2 µα uµ + 2g = 2g uν uµ + 2g ds µα ds µα ds µα,ν µα ds duµ = g uν uµ + g uµuν + 2g (72) µα,ν να,µ µα ds and by substitution (71) and (72) in (69) we get duµ 1 g + [g + g − g ] uµuν = 0 µα ds 2 µα,ν αµ,ν µν,α if we multiply with g ρα the geodesic equations duρ + Γρ uµuν = 0 , or uρ uν = 0 (73) ds µν ;ν ρ ρ µ because du /ds = u ,µu .

30 Euler-Lagrange Eqns vs Geodesic Eqns

µ ν The Lagrangian for a freely moving particle is: L = gµν u u and the Euler-Lagrange equations: d  ∂L  ∂L − = 0 ds ∂uµ ∂x µ are equivalent to the geodesic equation

duρ d 2x ρ dx µ dx ν + Γρ uµuν = 0 or + Γρ = 0 ds µν ds2 µν ds ds Notice that if the metric tensor does not depend from a specific coordinate e.g. x κ then d  ∂L  = 0 ds ∂x˙ κ which means that the quantity ∂L/∂x˙ κ is constant along the geodesic. Then ∂L µ eq (70) implies that ∂x˙ κ = gµκu that is the κ component of the generalized µ momentum pκ = gµκu remains constant along the geodesic.

31 Tensors : Geodesics

If we know the tangent vector uρ at a given point of a known space we can determine the geodesic curve. Which will be characterized as:

• timelike if |~u|2 > 0 • null if |~u|2 = 0 • spacelike if |~u|2 < 0

2 µ ν where |~u| = gµν u u

If gµν 6= ηµν then the light cone is affected by the curvature of the spacetime. For example, in a space with metric ds2 = −f (t, x)dt2 + g(t, x)dx 2 the light cone will be drawn from the relation dt/dx = ±pg/f which leads to STR results for f , g → 1.

32 Null Geodesics

For null geodesics ds = 0 and the proper length s cannot be used to parametrize the geodesic curves. Instead we will use another parameter λ and the equations will be written as:

d 2x κ dx µ dx ν + Γκ = 0 (74) dλ2 µν dλ dλ and obiously: dx µ dx ν g = 0 . (75) µν dλ dλ

33 Geodesic Eqns & Affine Parameter

In deriving the geodesic equations we have chosen to parametrize the curve via the proper length s. This choice simplifies the form of the equation but it is not a unique choice. If we chose a new parameter, σ then the geodesic equations will be written:

d 2x µ dx α dx β d 2σ/ds2 dx µ + Γµ = − (76) dσ2 αβ dσ dσ (dσ/ds)2 dσ where we have used dx µ dx µ dσ d 2x µ d 2x µ  dσ 2 dx µ d 2σ = and = + (77) ds dσ ds ds2 dσ2 ds dσ ds2 The new geodesic equation (76), reduces to the original equation(74) when the right hand side is zero. This is possible if

d 2σ = 0 (78) ds2 which leads to a linear relation between s and σ i.e. σ = αs + β where α and β are arbitrary constants. σ is called affine parameter.

34 Riemann - Ricci & Einstein Tensorsi

When in a space we define a metric then is called metric space or Riemann space. For such a space the curvature tensor

λ λ λ σ λ σ λ R βνµ = −Γβν,µ + Γβµ,ν − Γβν Γσµ + ΓβµΓσν (79)

is called Riemann Tensor and can be also written as: 1 R = g Rλ = (g + g − g − g ) κβνµ κλ βνµ 2 κµ,βν βν,κµ κν,βµ βµ,κν α ρ α ρ  + gαρ ΓκµΓβν − Γκν Γβµ

• Properties of the Riemann Tensor:

Rκβνµ = −Rκβµν , Rκβνµ = −Rβκνµ , Rκβνµ = Rνµκβ , Rκ[βµν] = 0

Thus in an n-dim space the number of independent components is (how?):

n2(n2 − 1)/12 (80)

35 Riemann - Ricci & Einstein Tensors ii

Thus in a 4-dimensional space the Riemann tensor has only 20 independent components. • The contraction of the Riemann tensor leads to Ricci Tensor

λ λµ Rαβ = R αλβ = g Rλαµβ µ µ µ ν µ ν =Γ αβ,µ − Γαµ,β + Γαβ Γνµ − Γαν Γβµ (81)

which is symmetric i.e. Rαβ = Rβα. • Further contraction leads to the Ricci or Curvature Scalar

α αβ αβ µν R = R α = g Rαβ = g g Rµανβ . (82)

• The following combination of Riemann and Ricci tensors is called 1 G = R − g R (83) µν µν 2 µν

36 Riemann - Ricci & Einstein Tensors iii

with the very important property:

µ  µ 1 µ  G ν;µ = R ν − δ ν R = 0 . (84) 2 ;µ

This results from the Bianchi Identity (how?)

λ R µ[νρ;σ] = 0 (85)

37 Flat & Empty Spacetimes

• When Rαβµν = 0 the spacetime is flat

• When Rµν = 0 the spacetime is empty

Prove that : λ λ λ κ a ;µ;ν − a ;ν;µ = −R κµν a

38 Weyl Tensori

Important relations can be obtained when we try to express the Riemann or Ricci tensor in terms of trace-free quantities. 1 S = R − g R ⇒ S = Sµ = g µν S = 0 . (86) µν µν 4 µν µ µν

or the Cλµνρ: 1 R = C + (g S + g S − g S − g S ) λµνρ λµνρ 2 λρ µν µν λρ λν µρ µρ λν 1 + R (g g − g g ) (87) 12 λρ µν λν µρ 1 C = R − (g R + g R − g R − g R ) λµνρ λµνρ 2 λρ µν µν λρ λν µρ µρ λν 1 − R (g g − g g ) . (88) 12 λρ µν λν µρ and we can prove (how?) that :

λρ g Cλµρν = 0 (89)

39 Weyl Tensor ii • The Weyl tensor is the trace-free part of the Riemann tensor. In the absence of sources, the trace part of the Riemann tensor will vanish due to the Einstein equations, but the Weyl tensor can still be non-zero. This is the case for gravitational waves propagating in vacuum.

• The Weyl tensor expresses the tidal force (as the does) that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force.

40 Weyl Tensor iii • The Weyl tensor is called also conformal curvature tensor because it has the following property : If we consider besides the Riemannian space M with

metric gµν a second Riemannian space M˜ with metric

2A g˜µν = e gµν

where A is a function of the coordinates. The space M˜ is said to be conformal to M. One can prove that

α ˜ α α ˜ α Rβγδ 6= Rβγδ while Cβγδ = Cβγδ (90)

i.e. a “conformal transformation” does not change the Weyl tensor.

41 Weyl Tensor iv

• It can be verified at once from equation (88) that the Weyl tensor has the same symmetries as the Riemann tensor. Thus it should have 20 independent components, but because it is traceless [condition (89)] there are 10 more conditions for the components therefore the Weyl tensor has 10 independent components. In general, the number of independent components is given by 1 n(n + 1)(n + 2)(n − 3) (91) 12

• In 2D and 3D spacetimes the Weyl curvature tensor vanishes identically. Only for ≥ 4, the Weyl curvature is generally nonzero. • If in a spacetime with ≥ 4 dimensions the Weyl tensor vanishes then the metric is locally conformally flat, i.e. there exists a local coordinate system in which the metric tensor is proportional to a constant tensor.

• On the symmetries of the Weyl tensor is based the Petrov classification of the space times (any volunteer?).

42 Tensors : An example for parallel transport

θ φ A vector A~ = A ~eθ + A ~eφ is parallel transported along a closed line on the surface of a sphere with metric ds2 = dθ2 + sin2θdφ2 and Christoffel symbols 1 θ 2 φ Γ22 = Γφφ = − sin θ cos θ andΓ 12 = Γθφ = cot θ . α α µ ν The eqns δA = −Γµν A dx for parallel transport will be written as: ∂A1 ∂Aθ = −Γ1 A2 ⇒ = sin θ cos θAφ ∂x 2 22 ∂φ ∂A2 ∂Aφ = −Γ2 A1 ⇒ = − cot θAθ ∂x 2 12 ∂φ

43 • i.e. different components but the measure is still the same ~ 2 µ ν θ2 2 φ2 |A| = gµν A A = A + sin θ A sin2(2π cos θ) = cos2(2π cos θ) + sin2 θ = 1 sin2 θ

The solutions will be: ∂2Aθ = − cos2 θAθ ⇒ Aθ = α cos(φ cos θ) + β sin(φ cos θ) ∂φ2 ⇒ Aφ = − [α sin(φ cos θ) − β cos(φ cos θ)] sin−1 θ

θ φ and for an initial unit vector( A , A ) = (1, 0) at( θ, φ) = (θ0, 0) the integration constants will be α = 1 and β = 0.

• The solution is: sin(2π cos θ) A~ = Aθ~e + Aφ~e = cos(2π cos θ)~e − ~e θ φ θ sin θ φ

Question : What is the condition for the path followed by the vector to be a geodesic? 44 The solutions will be: ∂2Aθ = − cos2 θAθ ⇒ Aθ = α cos(φ cos θ) + β sin(φ cos θ) ∂φ2 ⇒ Aφ = − [α sin(φ cos θ) − β cos(φ cos θ)] sin−1 θ

θ φ and for an initial unit vector( A , A ) = (1, 0) at( θ, φ) = (θ0, 0) the integration constants will be α = 1 and β = 0.

• The solution is: sin(2π cos θ) A~ = Aθ~e + Aφ~e = cos(2π cos θ)~e − ~e θ φ θ sin θ φ

• i.e. different components but the measure is still the same ~ 2 µ ν θ2 2 φ2 |A| = gµν A A = A + sin θ A sin2(2π cos θ) = cos2(2π cos θ) + sin2 θ = 1 sin2 θ

Question : What is the condition for the path followed by the vector to be a geodesic? 44 Extension 1-forms (*)i

• A 1-form can be defined as a linear, real valued function of vectors. 3 • A 1-form˜ω at a point P associates with a vector v at P a real number, which maybe called˜ω(v). We may say that˜ω is a function of vectors while the linearity of this function means:

• ω˜(av + bw) = aω˜(v) + bω˜(w) a, b ∈ IR • (aω˜)(v) = a[˜ω(v)] • (˜ω +σ ˜)(v) =ω ˜(v) +σ ˜(v)

• Thus 1-forms at a point P satisfy the axioms of a , which is ∗ called the dual space to TP , and is denoted by TP . • The linearity also allows to consider the vector as function of 1-forms: Thus vectors and 1-forms are thus said to be dual to each other.

45 1-forms (*) ii

• Their value on one another can be represented as follows:

ω˜(v) ≡ v(˜ω) ≡< ω,˜ v > . (92)

α • We may introduce a set of 1-forms˜ω dual to the basis vectors eα. • An arbitrary 1-form B˜ can be expanded in its covariant components according to α B˜ = Bαω˜ . (93)

• The scalar product of two 1-forms A˜ and B˜ is

α β  αβ αβ α β A˜ · B˜ = (Aαω˜ ) · Bβ ω˜ = g AαBβ where g =ω ˜ · ω˜ . (94)

• A basis of 1-forms dual to the basis eα always satisfies

α α ω˜ · eβ = δ β . (95)

46 1-forms (*) iii

• The scalar product of a vector with a 1-form does not involve the metric, but only a summation over an index

α β  α β α A · B˜ = (A eα) · Bβ ω˜ = A δα Bβ = A Bα . (96)

A vector A carries the same information as the corresponding 1- form A˜, and we often make no distinction between them, recall that :

β α αβ Aα = gαβ A and A = g Aβ . (97)

α A coordinate basis of 1-forms may be written˜ωα = d˜x , remember that α eα = ∂/∂x ≡ ∂α. α Geometrically the basis form d˜x may be thought of as surfaces of constant x α. • An orthonormal basis˜ωαˆ satisfies the relation

ˆ ˆ ω˜αˆ · ω˜β = ηαˆβ . (98)

47 1-forms (*) iv

• The gradient, d˜f , of an arbitrary scalar function f is particularly useful 1-form.

α • In a coordinate basis, it may be expanded according to d˜f = ∂αf dx˜ (its components are ordinary partial derivatives). • The scalar product between an arbitrary vector v and the 1-form d˜f gives the of f along v

α ˜β  α v · d˜f = (v eα) · ∂β f dx = v ∂αf . (99)

3See B. F. Schutz “Geometrical methods of mathematical ” Cambridge, 1980.

48 1-forms or co-vectors (*)i

Based on the previous, any 4-vector A can be expanded in contravariant components 4 α A = A eα . (100)

Here the four basis vectors eα span the vector spaceT P M tangent to the spacetime manifold M

49 1-forms or co-vectors (*) ii

and

gαβ = eα · eβ . (101)

In a coordinate basis, the basis vectors are tangent vectors to coordinate lines α and can be written as eα = ∂/∂x ≡ ∂α. The coordinate vectors commute. We may also set an orthonormal basis vectors at a point (orthonormal tetrad) for which

eαˆ · eβˆ = ηαˆβˆ . (102) • In general, orthonormal basis vectors do not form a coordinate basis and do not commute. • In a flat spacetime it is always possible to transform to coordinates which

are everywhere orthonormal i.e. ηαβ = diag(−1, 1, 1, 1). • For a genaral spacetime this is not possible, but one may select an event in the spacetime to be the origin of a local inertial coordinate frame, where

50 1-forms or co-vectors (*) iii

gαβ = ηαβ at this point and in addition the 1st derivatives of the metric tensor

at that point vanish, i.e. ∂γ gαβ = 0. • An observer in such a coordinate frame is called a local inertial and can use a coordinate basis that forms that forms a local orthonormal tetrad to make measurements in SR. • Such n observer will find that all the (non-gravitational) laws of physics in this frame are the same as in . • The scalar product of two 4-vectors A and B is

α β  α β A · B = (A eα) · B eβ = gαβ A B . (103)

4Baumgarte-Shapiro ”Numerical Relativity”, Cambridge 2010

51 Lie Derivatives, Isometries & Killing Vectorsi

Up to now we consider coordinate transformations x µ → x˜µ with the following meaning: to the point P with initial coordinate values x µ we assign new coordinates˜x µ determined from x µ via the n functions

x˜µ = f µ(x α) , µ = 1...n (104)

We shall give to the transformation x µ → x˜µ the following meaning: To the point P having the coordinate values x µ we correspond another point Q of the same space having the coordinate values˜x µ in the same coordinate system. This operations is a mapping of the space into itself. In the infinitesimal mapping

x˜µ = x µ + ξµ (105)

where ξµ = ξµ(x α) is a given vector field and  an infinitesimal parameter.

52 Lie Derivatives, Isometries & Killing Vectors ii The meaning is: in the point P with coordinate values x µ we correspond the point Q with coordinate values x µ + ξµ (in the same coordinate system). Thus we can define the difference between two tensors defined at the point Q this will be the Lie derivative. The first tensor is the one defined by the field at Q while the second will be the one defined if we ”take over” the value of the field at P and map it via (105) into the point Q.

53 Lie Derivatives, Isometries & Killing Vectors iii The Lie derivative of a Scalar

µ Lets assume a scalar field at a point P i.e. φP = φ(x ), since it is a scalar we postulate that the mapping P → Q will leave it unchanged. Thus we have at Q two scalars

µ µ φQ = φ(˜x ) + φ,µξ and φP→Q = φP (106)

The Lie derivative of the scalar φ with respect to ξµ is defined as follows:

φQ − φP→Q Lξφ = lim (107) →0  therefore due to (106) we get:

µ Lξφ = φ,µξ (108)

54 Lie Derivatives, Isometries & Killing Vectors iv

The Lie derivative of a Contravariant

Let’s assume that the contravariant vector kµ = kµ(x α) is the tangent vector to a curve passing from the point at P. If this curve is described by a parameter λ we have:

dx µ kµ = (109) dλ where dx µ are the components of the vector PP’. If the points Q and Q0 correspond to P and P0 via the mapping (105) then the components of the vector QQ0 will be

µ µ µ µ µ µ ν δx = dx + ξP0 − ξP = dx + ξ ,ν dx (110)

55 Lie Derivatives, Isometries & Killing Vectorsv

The transport of the vector kµ from P to Q by the mapping (105) will be:

δx µ kµ = = kµ + ξµ kν (111) P→Q dλ P ,ν and the definition of the Lie derivative of the vector kµ is: µ µ µ kQ − kP→Q Lξk = lim (112) →0  and since µ µ α µ µ ν kQ = k (˜x ) = kP + k ,ν ξ

56 Lie Derivatives, Isometries & Killing Vectors vi we finally get µ µ ν µ ν Lξk = k ,ν ξ − ξ ,ν k . (113)

... The Lie derivative of any other tensor T... will be defined as

...... (T... )Q − (T... )P→Q LξT... = lim (114) →0 

The detailed formulae will be derived from the formulae that we have already derived for the scalar and the contravariant vector.

57 Lie Derivatives, Isometries & Killing Vectors vii The Lie derivative of a Covariant Vector Field

µ For a covariant vector field pµ with the help of a contravariant vector k we get

µ µ µ Lξ (pµk ) = Lξpµk + pµLξk (115)

For the left hand side due to (108) we get

µ µ ν µ µ ν Lξ (pµk ) = (pµk ),ν ξ = (pµ,ν k + pµk ,ν ) ξ (116) Then by introducing in the last term of (115) the expression (113) we get:

µ  ν ν  k Lξpµ − pµ,ν ξ − pν ξ ,µ = 0 (117)

and since kµ is arbitrary we get:

ν ν Lξpµ = pµ,ν ξ + pν ξ ,µ (118)

In a similar way we can find the formulae for the Lie derivative of any tensor

ρ ρ ρ LξTµν = Tµν,ρξ + Tρν ξ ,µ + Tµρξ ,ν (119)

58 Lie Derivatives, Isometries & Killing Vectors viii

The Lie derivative of the metric tensor

If we apply the formula (119) to the metric tensor we get:

ρ ρ ρ Lξgµν = gµν,ρξ + gρν ξ ,µ + gµρξ ,ν (120)

α α and if we use the relation gλµ;ν = gλµ,ν − gαµΓλν − gλαΓµν = 0 we get

ρ ρ Lξgµν = gρν ξ ;µ + gµρξ ;ν (121) ρ ρ but since gρν ξ ;µ = (gρν ξ );µ = ξν;µ we get

Lξgµν = ξµ;ν + ξν;µ (122)

59 Isometries & Killing Vectorsi

QUESTION: If the neighboring points P and P0 go over, under the mapping (105), to the points Q and Q0 what is the condition ensuring that

2 2 dsPP0 = dsQQ0 (123)

for any pair of points P and P0. If such a mapping exist, it will be called isometric mapping. We have 2 µ ν dsPP0 = (gµν )P dx dx (124) But then according to the second of (106):

2 µ ν µ ν dsPP0 = (gµν dx dx )P→Q = (gµν )P→Q δx δx (125)

where dx µ are the components of PP0 and δx µ the components of QQ0. On the other side 2 µ ν dsQQ0 = (gµν )Q δx δx (126)

60 Isometries & Killing Vectors ii

Therefore the condition for the validity of (123) is

(gµν )P→Q = (gµν )Q .

and according to the definition (114) of the Lie derivative:

Lξgµν = 0 (127)

Therefore we found that the condition for the existence of isometric mappings is the existence of solutions ξµ of the equation

ξµ;ν + ξν;µ = 0 (128)

This is the Killing equation and the vectors ξµ which satisfy it are called Killing vectors.

The existence of a Killing vector, i.e. of an isometric mapping of the space onto itself, it is an expression of a certain intrinsic symmetry property of the space.

61 Isometries & Killing Vectors iii

According to Noether’s theorem, to every continuous symmetry of a physical system corresponds a conservation law.

• Symmetry under spatial translations → conservation of momentum • Symmetry under time translations → conservation of energy • Symmetry under rotations → conservation of

PROVE: If ξµ is a Killing vector, then, for a particle moving along a geodesic, the scalar product of this killing vector and the momentum Pµ = m dx µ/dτ of the particle is a constant.

62