Books on vehicle attitude and motion often use tensors in their analyses, and I have discussed the reasons for that in a previous chapter. But tensors also carry an esotericism arising from being used to quantify the curved spacetime of general relativity. And so I end the book by telling the inquisitive reader how tensors ‘work’ more generally, and how this more advanced topic makes quick work of calculating the gradient, divergence, laplacian, and curl of vector calculus. I end with a discussion of parallel transport, which has found its way into the exotic ‘wander azimuth’ axes used in some navigation systems.

The principle of relativity requires that no interaction can propagate instantly. Gravitational waves (GW) must exist, propagating with the same speed as light. The specific characters of GW are predicted by Einstein’s general relativity (GR). After decades of efforts to develop detectors, on 11 February 2016, the LIGO and Virgo Collaboration published the discovery of a GW.

The elements of GR relevant for GW production, propagation and detection. How the GR field, which is the dimensionless metric tensor, differs from the other fundamental fields, which have physical dimensions. The instruments and the discovery. After the first observation, dozens of gravitational signals have been detected, the vast majority from merging black holes and one, on 17 August 2017, from the merger of neutron stars. In this case, electromagnetic signals are expected, and have been detected, providing unique information to astrophysics and to fundamental physics as well. The measurement of the speed of the GW and the establishment of a bound on the mass of the graviton.

The metric tensor and the (pseudo-)Riemannian manifolds are defined. The results of the earlier chapters are specialised to this case, in particular the affine connection coefficients are shown to reduce to the Christoffel symbols. The signature of a metric, the timelike, null and spacelike vectors are defined and the notion of a light cone is introduced. It is shown that in two dimensions the notion of curvature agrees with intuition. It is also shown that geodesic lines extremise the interval (i.e. the ‘distance’). Mappings between Riemann spaces are discussed. Conformal curvature (= the Weyl tensor) is defined and it is shown that zero conformal curvature on a manifold of dimension >=4 implies that the metric is proportional to the flat one. Conformal flatness in three dimensions and the Cotton–York tensor are discussed. Embeddings of Riemannian manifolds in Riemannian manifolds of higher dimension are discussed and the Gauss–Codazzi equations derived. The Petrov classification of conformal curvature tensors in four dimensions with signature (+ - - -) is introduced at an elementary level.

We equip differentiable manifolds with a metric and introduce differential geometry, which provides the mathematical formalism underlying the theory of general relativity and many other applications in different areas of physics, science, and engineering.