Let and . a Homeomorphism Is a Coordinate System on , And

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 : , (u)=(x (u),...,x (u)) U ! V 1 n is a coordinate system on , and coordinates x 1 ,...,x n . The pair ( , ) 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” ↵ “contravariant index” ! S ··· = ··· ···

µ⌫ µ µ⌫ • g g ⌫ = , 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 ⌫ + ⌫ V rµ µ µ ⌫ • µ is the connection coefﬁcient; result of parallel transport Locally free falling coordinate system (LIF): ⌫ 0 µ ! LIF Must also hold in any ! • @ g (P ) g =0 µ ↵ ⌘rµ ↵ coordinate system!

1 ↵ = g↵ (@ g + @ g @ g ) µ⌫ 2 ⌫ µ µ ⌫ µ⌫ ⌫ ⌫ Christoffel symbol (µ = ⌫ )

20 Covariant derivatives II

• ↵ 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 µ = @µ r ↵ ↵ ↵ ⌫ • On vectors µv = @µv + v r µ⌫ v = @ v ⌫ v rµ ↵ µ ↵ ↵µ ⌫