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 jA j (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 rαA = @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.