URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=34GTR&collegejaar=2026&taal=enDescription
Manifolds and coordinates- The concept of a manifold
- Coordinates; Curves and surfaces
- Coordinate transformations
- Summation convention
- Geometry of manifolds
- Riemannian geometry
- Intrinsic and extrinsic geometry
- Examples of non-Euclidean geometry
- Lengths, areas and volumes
- Local Cartesian coordinates
- Tangent spaces to manifolds
- Pseudo-Riemannian manifolds
- Integration over general submanifolds
- Topology of manifolds
Vector calculus on manifolds
- Scalar fields on manifolds
- Vector fields on manifolds
- Tangent vector to a curve
- Basis vectors
- Raising and lowering vector indices
- Basis vectors and coordinate transformations
- Coordinate-independent properties of vectors
- Derivatives of basis vectors and the affine connection
- Transformation properties of the affine connection
- Relationship of the connection and the metric
- Local geodesic and Cartesian coordinates
- Covariant derivative of a vector
- Vector operators in component form
- Intrinsic derivative of a vector along a curve
- Parallel transport
- Null curves, non-null curves and affine parameters
- Geodesics
- Stationary property of non-null geodesics
- Lagrangian procedure for geodesics
- Alternative form of the geodesic equations
Tensor calculus on manifolds
- Tensor fields on manifolds
- Components of tensors
- Symmetries of tensors
- The metric tensor
- Raising and lowering tensor indices
- Mapping tensors into tensors
- Elementary operations with tensors
- Tensors as geometrical objects
- Tensors and coordinate transformations
- Tensor equations
- The quotient theorem
- Covariant derivative of a tensor
- Intrinsic derivative of a tensor along a curve
The equivalence principle and spacetime curvature
- Newtonian gravity
- The equivalence principle
- Gravity as spacetime curvature
- Local inertial coordinates
- Observers in a curved spacetime
- Weak gravitational fields and the Newtonian limit
- Intrinsic curvature of a manifold
- The curvature tensor
- Properties of the curvature tensor
- The Ricci tensor and curvature scalar
- Curvature and parallel transport
- Curvature and geodesic deviation
- Tidal forces in a curved spacetime
The gravitational field equations
- The energy–momentum tensor
- The energy–momentum tensor of a perfect fluid
- Conservation of energy and momentum for a perfect fluid;
- The Einstein equations
- The Einstein equations in empty space
- The weak-field limit of the Einstein equations
- The cosmological-constant term
- Geodesic motion from the Einstein equations
The Schwarzschild geometry
- The general static isotropic metric
- Solution of the empty-space field equations
- Birkhoff’s theorem
- Gravitational redshift for a fixed emitter and receiver
- Geodesics in the Schwarzschild geometry
- Trajectories of massive particles
- Radial motion of massive particles
- Circular motion of massive particles
- Stability of massive particle orbits
- Trajectories of photons
- Radial motion of photons
- Circular motion of photons
- Stability of photon orbits
Experimental tests of general relativity
- Precession of planetary orbits
- The bending of light
- Radar echoes
Schwarzschild black holes
- The characterisation of coordinates
- Singularities in the Schwarzschild metric
- Radial photon worldlines in Schwarzschild coordinates
- Radial particle worldlines in Schwarzschild coordinates
- Eddington–Finkelstein coordinates
- Gravitational collapse and black-hole formation
- Spherically symmetric collapse of dust
- Tidal forces near a black hole
Linearised general relativity
- The weak-field metric
- The linearised gravitational field equations
- Linearised gravity in the Lorenz gauge
- General properties of the linearised field equations
- Solution of the linearised field equations in vacuo
Gravitational waves
- Plane gravitational waves and polarisation states
- Transforming to the transverse-traceless gauge
- The detection of gravitational waves
Objectives
After the end of this course, you (the student) will be able to:- Being able to do calculus with four-vectors and tensors in a four-dimensional pseudo-Riemannian space.
- Being able to do absolute and covariant differentiations.
- Being able to evaluate the metric properties of a non-Euclidean space, which is defined by a given metric tensor, by calculating the affine connections and the curvature tensors.
- Being able to determine the geodesics of such a space for mass and light particles.
- Being able to manipulate with the Schwarzschild metric and from this metric to evaluate time dilatations and general particle trajectories.
- Being able to assess the concept of a black hole.
- Starting from this being able to calculate general particle trajectories.