Books 1 General Relativity 1.1 Classical • Gravitation and Cosmology S. Weinberg , Wiley (1972) • Gravitation Charles W. Misner, Kip S. Thorne and John A. Wheeler, Freeman (1973) • Problem Book in Relativity and Gravitation A.P. Lightman, W.H. Press, R.H. Price and S.A.Teukolky, Princeton (1975) 1.2 Textbooks • A First Course in General Relativity B.F.Schutz, Cambridge (1986) • Gravitation and Spacetime Hans Ohanian and Remo Ruffini, W.W.Norton (1994) • GRAVITY : an introduction to Einstein’s General Relativity J.B. Hartle, Addison-Wesley (2003) • Relativity: An Introduction to Special and General Relativity Hans Stefani, Cambridge (2004) (also in German) • An Introduction to General Relativity: SPACETIME and GEOMETRY S.M. Carroll, Addisson- Wesley (2004) • Relativity, Gravitation and Cosmology : A Basic Introduction Ta-Pei Cheng, Oxford (2005) • Relativity : Special, General and Cosmological W.Rindler, Oxford (2006) • General Relativity : An Introduction for Physicists M.P. Hobson, G. Efstathiou and A.N. Lasenby, Cambridge (2006) 2 Neutron Stars, Relativistic Astrophysics • Compact Stars: Nuclear Physics, Particle Physics and General Relativity Norman K. Glenden- ning, Springer (2000) • Black Holes, White Dwarfs and Neutron Stars Stuart L. Shapiro and Saul A. Teukolsky, Willey (1983) • NEUTRON STARS I : Equations of State and Structure P. Haensel, A.Y. Potekhin, D.G. Yakovlev, Springer (2007) 1 Tensor Algebra A Short Introduction to Tensors Kostas Kokkotas May 2, 2007 Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra 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 sys tem 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 neighb ouring point A  with coordinates x µ + dx µ . The components of such a vector are the differentials dx µ . Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra 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 sys tem 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 neighb ouring point A  with coordinates x µ + dx µ . The components of such a vector are the differentials dx µ . Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra 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 (1) 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 ∂f µ ∂x ν dx ν for ν = 1, , n (2) ˜v µ = d ˜x µ dλ =  ν ∂˜x µ ∂x ν dx ν dλ =  ν ∂˜x µ ∂x ν v ν (3) Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra 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 ˜a µ in ˜x µ by a relation of the form ˜a µ =  ν ∂˜x µ ∂x ν a ν (4) 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  µ b µ a µ =  µ ˜ b µ ˜a µ for any x µ → ˜x µ [Scalar Product] (5) The covariant vector will transform as: ˜ b µ =  ν ∂x ν ∂˜x µ b ν (6) Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra 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 ˜a λ ˜ b µ ≡ 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 µ ν & ˜ T αβ = ∂x µ ∂˜x α ∂x ν ∂˜x β T µν The Kronecker symbol δ λ µ is a mixed tensor having fram e independent values for its components. Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra Tensors: tensor algebra Tensor multiplication : The product of two vectors is a tensor of order 2, because ˜a α ˜ b β = ∂˜x α ∂x µ ∂˜x β ∂x ν a µ 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 Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra Tensors: Differentiation The simplest tensor field is a scalar field φ = φ(x α ) and its derivatives are the c omponents of a covariant tensor! ∂φ ∂˜x λ = ∂x α ∂˜x λ ∂φ ∂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 ≡ ∂A µ ∂x α = ∂ ∂x α  ∂x µ ∂˜x ν ˜ A ν  = ∂˜x ρ ∂x α ∂ ∂˜x ρ  ∂x µ ∂˜x ν ˜ A ν  = ∂ 2 x µ ∂˜x ν ∂˜x ρ ∂˜x ρ ∂x α ˜ A ν + ∂x µ ∂˜x ν ∂˜x ρ ∂x α ∂ ˜ A ν ∂˜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. Kostas Kokkotas A Short Introduction to Tensors Tensor Algebra Tensors: Connections The transformation (x µ → ˜x µ ) of the derivative of a vector is: A µ ,α = ∂x µ ∂˜x ν ∂˜x ρ ∂x α ( ˜ A ν ,ρ + ∂ 2 x κ ∂˜x σ ∂˜x ρ ∂x ν ∂˜x κ    ˜ Γ ν σρ ˜ A σ ) (13) in another coordinate (x µ → x µ ) we get again A µ ,α = ∂x µ ∂x ν ∂x ρ ∂x α  A ν ,ρ + Γ ν σρ A σ  . (14) Suggesting that the transformation (˜x µ → x µ ) will be: ˜ A µ ,α + ˜ Γ µ αλ ˜ A λ = ∂˜x µ ∂x ν ∂x ρ ∂˜x α  A ν ,ρ + Γ ν σρ A σ  (15) The necessary and sufficie nt condition for A µ ,α to be a tensor is: Γ λ ρν = ∂ 2 x µ ∂x ν ∂x ρ ∂x λ ∂x µ + ∂x κ ∂x ρ ∂x σ ∂x ν ∂x λ ∂x µ Γ µ κσ . (16) Γ λ ρν is the called the connection of the space. Kostas Kokkotas A Short Introduction to Tensors . and A.N. Lasenby, Cambridge (2006) 2 Neutron Stars, Relativistic Astrophysics • Compact Stars: Nuclear Physics, Particle Physics and General Relativity Norman K. Glenden- ning, Springer (2000) •