A Short Introduction to Tensor Analysis Kostas Kokkotas a March 14, 2019 Eberhard Karls Univerity of Tubingen¨ 1 a This chapter based strongly on “Lectures of General Relativity” by A. Papapetrou, D. Reidel publishing company, (1974) Scalars and Vectors A 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 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 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,...) 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˜ ˜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... 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 ˜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 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 algebra ii α α • Trace: of the mixed tensor T β is called the scalar T = T α. • Symmetric Tensor : 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 connection of the space and it is not tensor. 11 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 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 Parallel Transport of a vectori Let aµ be some covariant vector field.