A Short Introduction to Tensor Analysis

Kostas Kokkotas

May 5, 2009

Kostas Kokkotas A Short Introduction to Tensor Analysis Scalars and Vectors

An n-dim manifold 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 a one to one correspondence between the points and the n numbers. 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. Kostas Kokkotas A Short Introduction to Tensor Analysis I 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.

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. 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.

I 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,...)

Kostas Kokkotas A Short Introduction to Tensor Analysis 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. 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.

I 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,...)

I 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.

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 of the curve x µ = f µ(λ) where the points A and A0 correspond 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 ∂x˜µ X ∂f 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

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 ˜aµ = 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˜µ˜a = φ for any x → x˜ [ Scalar Product] µ µ (6) The covariant vector will transform as: 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? Kostas Kokkotas A Short Introduction to Tensor Analysis Tensors: at last

A conravariant tensor of order 2 is a quantity having n2 components µν µ µ T 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˜ ˜aλb˜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 mixed tensor having frame independent values for its components. αβγ... Tensors of higher order: T µνλ...

Kostas Kokkotas A Short Introduction to Tensor Analysis Tensor algebra

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 ˜aν + 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 ˜aαb˜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)

Kostas Kokkotas A Short Introduction to Tensor Analysis Tensor algebra

I 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)

I Symmetric Tensor : Tλµ = Tµλ orT(λµ) , Tνλµ = Tνµλ or Tν(λµ)

I 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

Kostas Kokkotas A Short Introduction to Tensor Analysis Tensors: Differentiation

We consider a region V of the space in which some tensor, e.g. a α covariant 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 . 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. The derivative of a contravariant vector field Aµ is : ∂Aµ ∂ ∂x µ  ∂x˜ρ ∂ ∂x µ  Aµ ≡ = A˜ν = A˜ν ,α ∂x 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 in the right hand side this equation would be the transformation formula for a covariant tensor of order 2. Kostas Kokkotas A Short Introduction to Tensor Analysis Tensors: Connections

The transformation (x µ → x˜µ) of the derivative of a vector is:

∂x µ ∂x˜ρ ∂2x κ ∂x˜ν Aµ = [A˜ν + A˜σ] (15) ,α ∂x˜ν ∂x α ,ρ ∂x˜σ∂x˜ρ ∂x κ | {z } ˜ν Γσρ in another coordinate (x µ → x 0µ) we get again: ∂x µ ∂x 0ρ Aµ = [A0ν + Γ0ν A0σ] . (16) ,α ∂x 0ν ∂x α ,ρ σρ Suggesting that the transformation (˜x µ → x 0µ) will be: ∂x˜µ ∂x 0ρ A˜µ + Γ˜µ A˜λ = (A0ν + Γ0ν A0σ) (17) ,α αλ ∂x 0ν ∂x˜α ,ρ σρ µ The necessary and sufficient condition for A ,α 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 Γ ρν is the called the connection of the space and it is not tensor.

Kostas Kokkotas A Short Introduction to Tensor Analysis Covariant Derivative

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 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)

Kostas Kokkotas A Short Introduction to Tensor Analysis Parallel Transport of a vector

λ The connectionΓ µν helps is determining a vector Aµ = aµ + δaµ, at a point P0(x ν + δx ν ) , which can be considered as “equivalent” to the ν vector aµ given at P(x )

0 ν ν ∆aµ = aµ(P ) − aµ(P) = aµ(P) + aµ,ν dx − aµ(P) = aµ,ν dx

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 Kostas Kokkotas A Short Introduction to Tensor Analysis λ ν δaµ = Γ µν aλdx for covariant vectors (26) µ µ λ ν δa = −Γ λν a dx for contravariant vectors (27)

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

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

Kostas Kokkotas A Short Introduction to Tensor Analysis Curvature Tensor

The ”trip” of a parallel transported vector along a closed path

The parallel transport of the vector aλ from the point P to A leads to a λ µ ν change 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 dx (28) 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 dx − Γ ρσ(A) a (P) − Γ βν (P)a dx δx

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 . (29) 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 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 (30)

Kostas Kokkotas A Short Introduction to Tensor Analysis By exchanging the indices ν → σ and σ → ν we construct a similar relations

λ β σ ν ν σ λ ρ λ
δa = a (dx δx − dx δx ) Γ ρν Γ βσ + Γ βν,σ (31) and the total change will be given by the following relation: 1 δaλ = − aβRλ (dx σδx ν − dx ν δx σ) (32) 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 (33)

is the curvature tensor.

Kostas Kokkotas A Short Introduction to Tensor Analysis Geodesics

λ I For a vector u 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 I 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.

I If the transported vector is tangent to any point of the curve then this curve is a geodesic curve of this space and is given by the equation : duρ + Γρ uµuν = 0 . (34) dλ µν

I Geodesic curves are the shortest curves connecting two points on a curved space.

Kostas Kokkotas A Short Introduction to Tensor Analysis Metric Tensor

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
(35) In another coordinate system,˜x µ, we will get ∂x ν dx ν = dx˜α (36) ∂x˜α which leads to: 2 µ ν α β ds =g ˜µν dx˜ dx˜ = gαβdx dx . (37) This gives the following transformation relation (why?): ∂x α ∂x β g˜ = g (38) µν ∂x˜µ ∂x˜ν αβ

suggesting that the quantity gµν is a symmetric tensor, the so called metric tensor. The relation (37) characterises a Riemannian space: This is a metric space in which the distance between neighbouring points is given by (37).

Kostas Kokkotas A Short Introduction to Tensor Analysis • If at some point P we 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:

µ µα α µ µα gµν A = Aν , gµν T = T ν , gµν T α = Tαν , gµν gασT = Tνσ

I Metric element for Minkowski spacetime ds2 = −dt2 + dx 2 + dy 2 + dz2 (39) ds2 = −dt2 + dr 2 + r 2dθ2 + r 2 sin2 θdφ2 (40)

I For a sphere with radius R : ds2 = R2 dθ2 + sin2 θdφ2
(41)

I The metric element of a torus with radii a and b ds2 = a2dφ2 + (b + a sin φ)2 dθ2 (42)

Kostas Kokkotas A Short Introduction to Tensor Analysis • The contravariant form of the metric tensor:

αβ β αβ 1 αβ gµαg = δµ where g = G ← minor determinant det |gµν | (43) 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 (44)

• The angle, ψ, between two infinitesimal vectors d (1)x α and d (2)x α is: g d (1)x α d (2)x β cos(ψ) = αβ . (45) p (1) ρ (1) σp (2) µ (2) ν gρσ d x d x gµν d x d x

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 (46) ∂x˜µ where J is the Jacobian of the transformation. This relation can be written also as: p|g˜| = J p|g| (47) 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 (48) 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.

Kostas Kokkotas A Short Introduction to Tensor Analysis Christoffel Symbols

In Riemannian space there is a special connection derived directly from the metric tensor. This is based on a suggestion originating from Euclidean geometry 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 ) (49) Since the distance between P and P0 is dx ρ we can get 0 ρ gµν (P ) ≈ gµν (P) + gµν,ρ(P)dx (50) µ 0 µ µ σ ρ a (P ) ≈ a (P) − Γσρ(P)a (P)dx (51)

Kostas Kokkotas A Short Introduction to Tensor Analysis By substituting these two relation into equation (49) we get (how?)

σ σ
µ ν ρ gµν,ρ − gµσΓνρ − gσν Γµρ a a dx = 0 (52)

This relations must by valid for any vector aν and any displacement dx ν which leads to the conclusion the 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 − gµσΓνρ − gσν Γµρ =0 . (53) 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 ) (54) µρ 2 µν,ρ νρ,µ ρµ,ν and will be called Christoffel Symbols . α α It is obvious thatΓ µρ = Γ ρµ .

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 α dx dx 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) (55) where ξµ(A) = ξµ(B) = 0. The length of the new curve will be: Z B  µ ν 1/2 α dx˜ dx˜ S˜ = gµν (˜x ) ds (56) A ds ds Kostas Kokkotas A Short Introduction to Tensor Analysis 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˜ ]

µ uµ =x ˙ µ = dx µ/ds and uµ = x˜˙ = dx˜µ/ds =x ˙ + ξ˙µ (57)

Then we can create the difference δS = S˜ − S 1

Z B Z B   δS = δf ds = f˜ − f ds A A Z B    Z B   ∂f d ∂f 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:

1 We make use of the following relations:

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

d „ ∂f « ∂f d „ ∂f « ξα = ξ˙α + ξα ds ∂uα ∂uα ds ∂uα

Kostas Kokkotas A Short Introduction to Tensor Analysis Z B    ∂f d ∂f α δS =  α − α ξ ds = 0 (58) A ∂x ds ∂u Since ξα is arbitrary, we must have for each point of S:

d  ∂f  ∂f − = 0 (59) ds ∂uµ ∂x µ

Notice that the Langrangian of a freely moving particle with mass m = 2, µ ν 2 is: L = gµν u u ≡ f this leads to the following relations ∂f 1 ∂L ∂f 1 ∂L = L−1/2 and = L−1/2 (60) ∂uα 2 ∂uα ∂x α 2 ∂x α and by substitution in (59) we come to the condition

d  ∂L  ∂L − = 0 . (61) ds ∂uµ ∂x µ

Kostas Kokkotas A Short Introduction to Tensor Analysis µ ν Since L = gµν u u we get:

µ ν ∂L ∂u ν µ ∂u µ ν µ ν µ = gµν u + gµν u = gµν δ u + gµν u δ = 2gµαu (62) ∂uα ∂uα ∂uα α α ∂L µ ν = gµν,αu u (63) ∂xα 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 (64) µα,ν να,µ µα ds and by substitution in (61) 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 we the geodesic equations duρ + Γρ uµuν = 0 , or uρ uν = 0 (65) ds µν ;ν

ρ ρ µ because du /ds = u ,µu .

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 k then d  ∂L  = 0 ds ∂x˙ κ which means that the quantity ∂L/∂x˙ κ is constant along the geodesic. ∂L µ Then eq (62) implies that ∂x˙ κ = gµκu that is the κ component of the µ generalized momentum pκ = gµκu remains constant along the geodesic.

Kostas Kokkotas A Short Introduction to Tensor Analysis 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:

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

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 (66) dλ2 µν dλ dλ and obiously: dx µ dx ν g = 0 . (67) µν dλ dλ

Kostas Kokkotas A Short Introduction to Tensor Analysis 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 = − (68) 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 = + (69) ds dσ ds ds2 dσ2 ds dσ ds2 The new geodesic equation (68), reduces to the original equation(66) when the right hand side is zero. This is possible if d 2σ = 0 (70) 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.

Kostas Kokkotas A Short Introduction to Tensor Analysis Riemann Tensor

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 (71)

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:

n2(n2 − 1)/12 (72)

For a 4-dimensional space only 20 independent components

Kostas Kokkotas A Short Introduction to Tensor Analysis The Ricci and Einstein Tensors

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 + ΓαβΓνµ − Γαν Γβµ (73)

which is symmetric Rαβ = Rβα. Further contraction leads to the Ricci or Curvature Scalar α αβ αβ µν R = R α = g Rαβ = g g Rµανβ . (74) The following combination of Riemann and Ricci tensors is called Einstein Tensor 1 G = R − g R (75) µν µν 2 µν with the very important property:   µ µ 1 µ G ν;µ = R ν − δ ν R = 0 . (76) 2 ;µ This results from the following important identity (Bianchi Identity)

λ R µ[νρ;σ] = 0 (77)

Kostas Kokkotas A Short Introduction to Tensor Analysis Flat & Empty Spacetimes

I When Rαβµν = 0 the spacetime is flat

I When Rµν = 0 the spacetime is empty

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

Kostas Kokkotas A Short Introduction to Tensor Analysis Weyl Tensor

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 = S µ = g µν S = 0 . (78) µν µν 4 µν µ µν

or the Weyl tensor 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 ) (79) 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 ) . (80) 6 λρ µν λν µρ and we can prove (how?) that :

λρ g Cλµνρ = 0 (81)

Kostas Kokkotas A Short Introduction to Tensor Analysis The Weyl tensor is called also conformal curvature tensor because it has the following property : • 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 (82) i.e. a “conformal transformation” does not change the Weyl tensor. • It can be verified at once from equation (80) 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 (81)] there are 10 more conditions for the components therefore the Weyl tensor has 10 independent components.

Kostas Kokkotas A Short Introduction to Tensor Analysis Tensors : An example for parallel transport

~ 1 2 A vector A = A ~eθ + A ~eφ be parallel transported along a closed line on the surface of a sphere with metric ds2 = dθ2 + sin2dφ2 and Christoffel 1 2 symbolsΓ 22 = − sin θ cos θ andΓ 12 = cot θ . α α µ ν The eqns δA = −Γµν A dx for parallel transport will be written as: ∂A1 ∂A1 = −Γ1 A2 ⇒ = sin θ cos θA2 ∂x 2 22 ∂φ ∂A2 ∂A2 = −Γ2 A1 ⇒ = − cot θA1 ∂x 2 12 ∂φ

Kostas Kokkotas A Short Introduction to Tensor Analysis The solutions will be: ∂2A1 = − cos2 θA1 ⇒ A1 = α cos(φ cos θ) + β sin(φ cos θ) ∂φ2 ⇒ A2 = − [α sin(φ cos θ) − β cos(φ cos θ)] sin−1 θ

1 2 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~ = A1~e + A2~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 1
2 2 2
2 |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?

Kostas Kokkotas A Short Introduction to Tensor Analysis