Manifolds & charts A manifold M of dimension n is a topological space that near each n point resembles n-dimensional Euclidean space R . That is, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n n Let M , and R . U ✓ V ✓ A homeomorphism φ : , φ(u)=(x (u),...,x (u)) U ! V 1 n is a coordinate system on , and coordinates x 1 ,...,x n . The pair ( , φ ) is called a coordinateU chart U Equivalence principle: choice of charts is arbitrary 15 • Space-time is a Lorentzian Manifold smooth manifold: locally similar enough to Euclidean space to do calculus Pseudo-Riemannian manifold: equipped with an inner product on tangent space. Family of inner products is called Riemannian metric (tensor). Metric need not to be positive definite Lorentzian manifold: 4 dimensional, (-,+,+,+) tangent vectors can be classified into time-like, null, space-like 16 Some basic rules • In curved space-times we cannot use Cartesian coordinates • Expressions have to be coordinate invariant • coordinate transformation for tensors @xµ @x⌫ @~xµ0 @~x⌫0 ~ ~µ0⌫0 µ⌫ gµ0⌫0 = ~ ~ gµ⌫ g = g @xµ0 @x⌫0 µ ⌫ or @x @x “covariant index” ↵ δ “contravariant index” ! S γ ··· = ··· ··· µ⌫ µ µ⌫ • g g ⌫ = δ σ , i.e., g is the inverse of gµ⌫ 17 µ ⌫ ~ @x @x gµ0⌫0 = ~ ~ gµ⌫ The metric @xµ0 @x⌫0 • In GR the most important tensor is the metric gµ⌫ which is the generalisation of ⌘µ⌫ 2 µ ⌫ • line element ds = gµ⌫ dx dx • there exists a single point on the manifold such that @ gµ⌫ ⌘µ⌫ and even g µ ⌫ 0 (local inertial frame, LIF) ! @x ! • g µ ⌫ is determined by Einstein’s equations • scalar product g aµb⌫ = a b⌫ (a b) µ⌫ ⌫ ⌘ · 18 Derivatives • In Euclidean space: - d A i of vector A i is also a vector space-independent - @ k A i form a tensor • This is generally not the case in curvilinear coordinates: ⌫ ⌫ ⌫ µ @x0 x0 = f 0 (x ) - covariant vector Aµ = A0 @xµ ⌫ ⌫ 2 ⌫ @x0 @ x0 - its differential dAµ = dA0 + A0 dx @xµ ⌫ ⌫ @xµ@x • Must generalise partial derivative to covariant derivative 19 The covariant derivative rµ • Cannot be “too different” from the partial derivative. Locally free falling coordinate system (LIF): @ rµ ! µ V ⌫ = @ V ⌫ + Γ⌫ V λ rµ µ µλ ⌫ • Γ µ λ is the connection coefficient; result of parallel transport Locally free falling coordinate system (LIF): Γ⌫ 0 µλ ! LIF Must also hold in any ! • @ g (P ) g =0 µ ↵ ⌘rµ ↵ coordinate system! 1 Γ↵ = g↵ (@ g + @ g @ g ) µ⌫ 2 ⌫ βµ µ ⌫ − β µ⌫ ⌫ ⌫ Christoffel symbol (Γµλ = Γ⌫ ) 20 Covariant derivatives II • ↵ Γ µ ⌫ are constructed to be non-tensorial: Instead, v ↵ transforms as a tensor rµ • linearity (T + S)= T + S r r r • product (T S)=( T ) S + T ( S) r ⌦ r ⌦ ⌦ r • On scalars we simply have µΦ = @µΦ r ↵ ↵ ↵ ⌫ • On vectors µv = @µv + Γ v r µ⌫ v = @ v Γ⌫ v rµ ↵ µ ↵ − ↵µ ⌫ 21.