A Short Introduction to Tensors

Transcript

1 May 2, 2007

2 Tensors: 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. 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 Tensors: 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. 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 A with coordinates x µ + dx µ. The components of such a vector are the differentials dx µ.

4 Tensors: Vector Transformations From the infinitesimal vector AA 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 A correspond to the values λ and λ + dλ of the parameter. Then v µ = dx µ 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 : d x µ = ν x µ x ν dx ν = ν (1) f µ x ν dx ν for ν = 1,..., n (2) ṽ µ = d x µ dλ = ν x µ dx ν x ν dλ = ν x µ x ν v ν (3)

5 Tensors: Contravariant and Covariant Vectors Contravariant Vector: is a quantity with n components depending on the coordinate system in such a way that the components a µ in the coordinate system x µ are related to the components ã µ in x µ by a relation of the form ã µ = x µ x ν aν (4) ν Covariant Vector: eg. b µ, is an object with n components which depend on the coordinate system on such a way that if a µ is any contravariant vector, the following sums are scalars b µ a µ = b µ ã µ for any x µ x µ [Scalar Product] (5) µ µ The covariant vector will transform as: b µ = x ν x µ b ν (6) ν

6 Tensors: at last A conravariant tensor of order 2 is a quantity having n 2 components T µν which transforms (x µ x µ ) in such a way that, if a µ and b µ are arbitrary covariant vectors the following sums are scalars: T λµ a µ b λ = T λµ ã λ bµ φ (7) Then the transformation formulae for the components of the tensors of order 2 are (why?): T αβ = x α x µ x β x ν T µν, T α β = x α x µ x ν x β T ν µ & T αβ = x µ x ν x α x β T µν The Kronecker symbol δ λ µ is a mixed tensor having frame independent values for its components.

7 Tensors: tensor algebra Tensor multiplication : The product of two vectors is a tensor of order 2, because ã α b β = x α x β x µ x ν aµ b ν (8) in general: T µν = A µ B ν or T µ ν = A µ B ν or T µν = A µ B ν (9) Contraction: for any mixed tensor of order (p, q) leads to a tensor of order (p 1, q 1) T λµν λα = T µν α (10) Symmetric Tensor : T λµ = T µλ ort (λµ), T νλµ = T νµλ or T ν(λµ) Antisymmetric : T λµ = T µλ or T [λµ], T νλµ = T νµλ or T ν[λµ] No of independent components : Symmetric : n(n + 1)/2, Antisymmetric : n(n 1)/2

8 Tensors: Differentiation The simplest tensor field is a scalar field φ = φ(x α ) and its derivatives are the components of a covariant tensor! φ x λ = x α φ x λ x α we will use: φ x α = φ,α (11) i.e. φ,α is the gradient of the scalar field φ. The derivative of a contravariant vector field A µ is : A µ,α Aµ x α = ( ) x µ x α x ν Ãν = x ρ ( x µ x α x ρ = x ν Ãν 2 x µ x ρ x ν x ρ x α Ãν + x µ x ρ Ãν x ν x α x ρ (12) Without thefirst term in the right hand side this equation would be the transformation formula for a covariant tensor of order 2. )

9 Tensors: Connections The transformation (x µ x µ ) of the derivative of a vector is: A µ,α = x µ x ρ x ν x α (Ãν,ρ + 2 x κ x ν } x σ x {{ ρ x κ Ã σ ) (13) } Γ ν σρ in another coordinate (x µ x µ ) we get again A µ,α = x µ x ρ ( A ν x ν x α,ρ + Γ ν σρa σ). (14) Suggesting that the transformation ( x µ x µ ) will be: Ã µ,α + Γ µ αλãλ = x µ x ρ ( A ν x ν x α,ρ + Γ ν σρa σ) (15) The necessary and sufficient condition for A µ,α to be a tensor is: Γ λ ρν = 2 x µ x λ x ν x ρ x µ + x κ x σ x λ x ρ x ν x µ Γµ κσ. (16) Γ λ ρν is the called the connection of the space.

10 Tensors: Covariant Derivative According to the previous assumptions, the following quantity transforms as a tensor of order 2 A µ ;α = A µ,α + Γ µ αλa λ (17) and is called covariant derivative of the contravariant vector A µ. In similar way we get (how?) : φ ;λ = φ,λ (18) A λ;µ = A λ,µ Γ ρ µλ a ρ (19) T λµ ;ν = T λµ,ν + Γ λ ανt αµ + Γ µ ανt λα (20) T λ µ;ν = T λ µ,ν + Γ λ ανt α µ Γ α µνt λ α (21) T λµ;ν = T λµ,ν Γ α λν T µα Γ α µνt λα (22) T λµ νρ ;σ = T λµ νρ,σ + Γ λ ασt αµ νρ + Γ µ ασt λα νρ + Γ α νσt λµ αρ Γ α ρσt λµ να (23)

11 Tensors: Parallel Transport Γ λ µν helps to determine a vector A λ, at a point P, which has to be considered as equivalent to the vector a λ given at P. δ (1) a µ = a µ (P ) a µ (P) = a µ (P) + a µ,ν dx ν a µ (P) = a µ,ν dx ν a µ (P ) A µ (P ) }{{} vector = a µ,ν dx ν δa µ = }{{} vector = a µ + δ (1) a µ (a µ + δa µ ) = δ (1) a µ δa µ }{{}}{{}}{{} at point P at point P vector ) (a µ,ν Cµνa λ λ dx ν i.e. δa µ = Cµνa λ λ dx ν δa µ = Γ λ µνa λ dx ν for covariant vectors (24) δa µ = Γ µ λν aλ dx ν for contravariant vectors (25) The connection Γ λ µν allows to define the transport of a vector a λ from a point P to a neighbouring point P (Parallel Transport). The parallel transport of a scalar field is zero! δφ = 0 (why?)

12 Tensors: Curvature Tensor The trip of a parallel transported vector along a closed path leads to the following total change: δa λ = 1 2 aβ Rβνσ λ (dx σ δx ν dx ν δx σ ) (26) where Rβνσ λ = Γλ βν,σ + Γλ βσ,ν Γµ βν Γλ µσ + Γ µ βσ Γλ µν (27) is the curvature tensor.

13 Tensors: Geodesics For a vector u λ 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 If u µ = dxµ 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 geodesic curve of this space and is given by the equation : du ρ dλ + Γρ µνu µ u ν = 0. (28) Geodesic curves are the shortest curves connecting two points of a curved space.

14 Tensors: Metric Tensor The distance ds of two points P(x µ ) and P (x µ + dx µ ) is given by ds 2 = ( dx 1) 2 + ( dx 2 ) 2 + ( dx 3 ) 2 (29) In another coordinate system, x µ, we will get dx ν = x ν x α d x α (30) α which leads to: ds 2 = g µν d x µ d x ν = g αβ dx α dx β. which gives the following transformation relation: g µν = x µ x ν x α x β g αβ (31) suggesting that the quantity g µν is a symmetric tensor, the so called metric tensor. Properties: g µν A µ = A ν, g µν T µα = T α ν, g µν T µ α = T αν, g µν g ασ T µα = T νσ

15 Tensors: Metric Tensor - Christoffel Symbols Metric element for Minkowski spacetime ds 2 = dt 2 + dx 2 + dy 2 + dz 2 (32) ds 2 = dt 2 + dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (33) For a sphere with radius R : ds 2 = R 2 ( dθ 2 + sin 2 θdφ 2) The metric element of a torus with radii a and b ds 2 = a 2 dφ 2 + (b + a sin φ) 2 dθ 2 (34) The contravariant form of the metric tensor: g µα g αβ = δµ β where g αβ 1 = det g µν G αβ (35) The angle between two infinitesimal vectors d (1) x α and d (1) x α is: cos(ψ) = g αβ d (1) x α d (2) x β g ρσ d (1) x ρ d (2) x σ g µν d (1) x µ d (2) x ν. (36)

16 Tensors: Christoffel Symbols The covariant derivative of the metric tensor is g µν;ρ = g µν,ρ g µσ Γ σ νρ g σν Γ σ µρ = 0. (37) i.e. g µν is covariantly constant. This leads to a unique determination of the connections of the space (Riemannian space) Γ α µρ = 1 2 g αν (g µν,ρ + g νρ,µ g ρµ,ν ). (38) which are called Christoffel Symbols. It is obvious that Γ α µρ = Γ α ρµ.

17 Tensors: 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 R λ βνµ = Γ λ βν,µ + Γλ βµ,ν Γσ βν Γλ σµ + Γ σ βµ Γλ σν (39) is called Riemann Tensor. R κβνµ = g κλ R λ βνµ = 1 2 (g κµ,βν + g βν,κµ g κν,βµ g βµ,κν ) ) + g αρ (Γ α κµγ ρ βν Γα κνγ ρ βµ Properties of the Riemann Tensor: R κβνµ = R κβµν, R κβνµ = R βκνµ, R κβνµ = R νµκβ, R κ[βµν] = 0 Thus in an n-dimensional space the number of independent components is: n 2 (n 2 1)/12 (40) For a 4-dimensional space only 20 independent components

18 Tensors : Ricci and Einstein The contraction of the Riemann tensor leads to Ricci Tensor R αβ = R λ αλβ = g λµ R λαµβ = Γ µ αβ,µ Γµ αµ,β + Γµ αβ Γν νµ Γ µ ανγ ν βµ (41) which is symmetric R αβ = R βα. Further contraction leads to the Ricci Scalar R = R α α = g αβ R αβ = g αβ g µν R µανβ. (42) The following combination of Riemann and Ricci tensors is called Einstein Tensor G µν = R µν 1 2 g µνr (43) with the very important property: ( G µ ν;µ = R µ ν 1 ) 2 δµ νr ;µ = 0. (44)

19 Tensors : Flat and Empty Spacetimes R αβµν = 0 flat spacetime R µν = 0 empty spacetime Prove that : a λ ;µ;ν a λ ;ν;µ = R λ κµνa κ

20 Tensors : Euler-Lagrange Eqns vs Geodesic Eqns The Lagrangian for a freely moving particle is: L = g µν u µ u ν and the Euler-Lagrange equations: ( ) d L ds u µ L x µ = 0 are equivalent to the geodesic equation du ρ ds + Γρ µνu µ u ν d 2 x ρ = 0 or ds 2 + dx µ dx ν Γρ µν ds ds = 0 If we know the tangent vector u ρ at a given point of a known space we can determine the geodesic curve. Which will be called: u 2 < 0 timelike u 2 = 0 null u 2 > 0 spacelike where u 2 = g µν u µ u ν

21 Tensors : An example for parallel transport I A vector A = A 1 e θ + A 2 e φ be parallel transported along a closed line on the surface of a sphere with metric ds 2 = dθ 2 + sin 2 dφ 2 and Christoffel symbols Γ 1 22 = sin θ cos θ and Γ2 12 = cot θ. The eqns δa α = Γ α µνa µ dx ν for parallel transport will be written as: A 1 x 2 = Γ 1 22A 2 A1 φ A 2 x 2 = Γ 2 12A 1 A2 φ = sin θ cos θa2 = cot θa1

22 Tensors : An example for parallel transport II The solutions will be: 2 A 1 φ 2 = cos2 θa 1 A 1 = α cos(φ cos θ) + β sin(φ cos θ) A 2 = [α sin(φ cos θ) β cos(φ cos θ)] sin 1 θ and for a unit vector (A 1, A 2 ) = (1, 0) at (θ, φ) = (θ 0, 0) the intebration constants will be α = 1 and β = 0. The solution is: A = A 1 e θ + A 2 e φ = cos(2π cos θ) e θ sin(2π cos θ) e φ sin θ i.e. different components but the measure is still the same A 2 = g µν A µ A ν = ( A 1) 2 + sin 2 θ ( A 2) 2 = cos 2 (2π cos θ) + sin 2 θ sin2 (2π cos θ) sin 2 θ = 1