Astronomy 345, General Relativity and Gravitation 1

Lecture 1, 2022 September 21, 15:00Introduction, covering part 1 of the notesIntroductory remarks, some thought experiments as motivation, and remarks about reading (mp3, transcripts: html, txt, vtt).

Lecture 2, 2022 September 28, 15:00An introduction to vectors, one-forms and tensorsAt this point we take a holiday from the physics, in favour of mathematical preliminaries. This part is concerned with defining vectors, tensors and functions reasonably carefully, and showing how they are linked with the notion of coordinate systems. This will take us to the point where, in part 3, we can talk about doing calculus with these objects. You may well be familiar with many of the mathematical concepts in this part – functions, vector spaces, vector bases, and basis transformations – but I will (re)introduce them below with a slightly more sophisticated mathematical notation that will allow us to make use of them later. The exception to that is tensors, which may have seemed slightly gratuitous, if you have encountered them at all before; they are vital in relativity. (mp3, transcripts: html, txt, vtt).

Lecture 3, 2022 September 30, 16:00Vector, one-form and tensor componentsGiven a set of basis vectors, and corresponding basis one-forms, we can identify the components of tensors with respect to those bases. (mp3, transcripts: html, txt, vtt).

Lecture 4, 2022 October 05, 15:00Changing basesThere is no preferred basis, so we have to be able to change from one set of basis vectors to another. We see some further examples of some important vector spaces. This lecture covers the material from Sect.2.2.7 to the end. (mp3, transcripts: html, txt, vtt).

Lecture 5, 2022 October 12, 15:00The tangent planeAn introduction to manifolds, tangent vectors, and the tangent plane, covering section 3.1 of the notes. (mp3, transcripts: html, txt, vtt).

Lecture 6, 2022 October 19, 15:00The Christoffel symbols, and covariant differentiationHow do we describe how basis vectors differ at different places in the space? We learn about how the Christoffel symbols describe this, and how that in turn lets us describe how a vector field changes, in a coordinate-independent way. So far, we're differentiating vector fields in a flat space, only; next time we'll learn how to do the same in a curved space. (mp3, transcripts: html, txt, vtt).

Lecture 7, 2022 October 26, 15:00Covariant differentiation in a curved spaceLast time, we learned how to define differentiation in a flat space; this time we learn how we can use that work to define differentiation in a curved space. It turns out to be more direct that we might guess. We define the 'local inertial frame', in which the metric is flat in a region around a point, and define 'parallel transport' as a way of moving a vector from one tangent space to another. Between them, these allow us to define differentiation in a curved space in terms of the covariant derivative in the locally flat space of free fall. (mp3, transcripts: html, txt, vtt).

Whole-class supervision, 2022 October 28, 14:00The first whole-class supervision sessionSeeded by some good questions in the padlet, we talked over how we define tensors in terms of vectors and one-forms, and a recap of what the 'outer product' means and how we can build up higher-rank tensors from the outer product of lower-rank ones. That in turn led to a conversation about components, and about the transformation of components between frames. (mp3, transcripts: html, txt, vtt).

Lecture 8, 2022 November 02, 15:00Curvature and the Riemann tensorWhen we move a vector along a closed path in a space, it picks up information about the curvature of that space. We work through the details of this for a vector moving through an arbitrary space, and discover the Riemann tensor. (mp3, transcripts: html, txt, vtt).

Lecture 9, 2022 November 09, 15:00Dust and the energy-momentum temsorFirst, we make plausible the definition of the flux vector, and from that the energy-momentum tensor, which describes the flux of energy-momentum across spacelike surfaces (ie, matter and energy flow) and across timelike ones (ie, flow into the future). Then we move on to another version of the Equivalence Principle, which allows us to make the link between the physics we understand, namely the behaviour of objects in free-fall, to the physics we don't yet, namely the behaviuor of objects in a curved spacetime. (mp3, transcripts: html, txt, vtt).

Lecture 10, 2022 November 16, 15:00The Equivalence principle and Einstein's equationA further form of the Equivalence Principle, which we first saw in lecture 1, lets us predict how objects move in a curved spacetime: they move along timelike geodesics. We can then go on to ask 'what governs the shape of spacetime?', and by a heuristic argument, follow Einstein in guessing a constraint for the shape of spacetime, in the form of Einstein's equations. Together, these form the mathematical basis of the famous summary 'Spacetime tells matter how to move; matter tells spacetime how to curve.' (mp3, transcripts: html, txt, vtt).

Supervision 2, 2022 November 18, 14:00The second whole-class supervision sessionAgain seeded by good padlet questions, we talked over timelike geodesics, how we get the Bianchi identities and what the comma-goes-to-semicolon rule means, and some (admittedly rather handwaving) thoughts about the physical intuition behind the Ricci tensor and the curvature scalar. (mp3, transcripts: html, txt, vtt).

Lecture 11, 2022 November 23, 15:00The weak-field solutionI cannot leave you without showing at least one solution of Einstein's equations. We take a summary look at the solution of the equations in the limit of a small central mass, and discover that we recover something corresponding to Newton's expression for the gravitational field, with geodesics which correspond to the Newtonian prediction for movement in that field. (mp3, transcripts: html, txt, vtt).

Tutorial, 2022 December 02, 16:00GRG1 (whole-class) tutorialThough labelled differently on the timetable, this followed the same format as the earlier whole-class supervisions. We mostly talked about the various sections within part 4: the ideas behind creating a tensor description of energy-momentum, the guesswork behind Einstein's equations, and the general approach to, and difficulties of, solving the equations to get geodesics. (mp3, transcripts: html, txt, vtt).