**M04Dec and R07Dec**: this week I sketch how Einstein upgraded Newtonian gravity to a relativistic tensor theory and derive Einstein's equations from an action principle.**M27Nov and R30Nov**: this week I give a brief introduction to cosmology.**M20Nov and R23Nov**: this week I describe gravitational waves and the recent discoveries announced by the LIGO team.**M13Nov and R16Nov**: this week I describe several experimental successes of GR.**R02Nov**: Today I discussed the Kerr black hole, frame dragging, and the Penrose process enabling energy extraction from a rotating black hole at the price of reducing its rotation.**M30Oct**: Today I finished analyzing the geodesics of Schwarzschild, and described other coordinate systems for the Schwarzschild black hole and its causal structure.**R26Oct**: Today I first discussed the TOV equation for a star, after introducing the energy-momentum tensor for a perfect fluid. Next, I discussed the Pound-Rebka experiment measuring gravitational redshift, one of the famous experiments shown to be consistent with GR's predictions. Then I began analyzing of the geodesics of Schwarzschild, focusing on first integrals arising from symmetries and the four-velocity constraint for massless/massive particles.**M23Oct**: Today I derived the Schwarzschild solution for a static, spherically symmetric system, and Birkhoff's Theorem. I highlighted the event horizon at the Schwarzschild radius and the curvature singularity at the centre.**R19Oct**: Today I discussed Killing vectors arising when a spacetime has symmetries, and how they lead to conservation laws for particles moving on geodesics. I also touched on maximally symmetric spacetimes, which have the most symmetry available: flat Minkowski spacetime for zero cosmological constant, Anti de Sitter spacetime (AdS) for negative cosmological constant, and de Sitter (deS) spacetime for positive cosmological constant.
**M16Oct**: Today I finished my discussion of tidal forces in General Relativity -- and how the Newtonian description of tidal forces is recovered in the limit of weak gravity and slow speeds. Then I briefly mentioned a more sophisticated treatment of geodesic deviation by studying geodesic congruences, resulting in the Raychaudhuri equation describing whether geodesics focus. Then I discussed the Lie derivative along a vector field, which can be defined even without having an affine connection.**R12Oct**: Today I gave a simple example of geodesic deviation. Then I started describing tidal forces, first reviewing how they occur in Newtonian gravity. Then I began introducing how we find Newtonian gravity as a limit of Einstein gravity, in relation to tidal forces.**R05Oct**: Today I introduced Riemann normal coordinates, and showed how they help us derive identities obeyed by the Riemann tensor. In particular, I showed which symmetries the all-downstairs version of Riemann obeys under exchange of various combinations of its indices. Knowing these symmetries is a crucially important way to cut down on work when computing components of Riemann. I also counted the number of independent components in Riemann as a function of spacetime dimension. Then I introduced the geodesic deviation equation and showed how Riemann plays a central role in it.**M02Oct**: Today I introduced the Riemann tensor, derived an expression for its components in terms of Christoffels, and worked an example of computing the Riemann tensor, the Ricci tensor, and the Ricci scalar.**R29Sep**: Today I worked a very explicit example of computing Christoffel symbols for a simple example spacetime in great detail.**M25Sep**: Today I showed why partial derivatives acting on tensors generically do not produce another bona fide tensor, when the transformation between two different sets of coordinates depends on spacetime position. I then outlined the story of the metric-compatible Levi-Civita connection, whose components are the Christoffel symbols $\Gamma^\mu_{\ \nu\sigma}$, which are uniquely determined by first derivatives of the metric tensor. What the connection does physically is to knit together the various infinitesimal neighbourhoods around various points into a coherent curved spacetime, in such a way that we can define a covariant derivative that*is*a true tensor. I showed details of how the covariant derivative $\nabla_\mu$ acting on tensors is built from a linear combination of the partial derivative $\partial_\mu$ and the Christoffels $\Gamma^\mu_{\ \nu\sigma}$. Then I showed how to take covariant derivative of an arbitrary tensor. Next, I introduced the idea of the directional covariant derivative along a curve in spacetime, and the concept of parallel transport. Then I defined geodesics as curves that parallel transport their own tangent vector, and wrote down the geodesic equation. I also showed how to obtain the geodesic equation from a variational principle starting from the action for the massive relativistic point particle. Lastly, I explained why an observer in freefall will always measure the*most*elapsed proper time: anyone who accelerates will measure less.**R21Sep**: Today I introduced tensors in curved spacetime. I started by mentioning the Einstein Equivalence Principle, which says that you cannot do a local experiment to tell the difference between acceleration due to rockets and acceleration due to gravity. I introduced spacetime as a curved manifold, which at every point has the structure of flat spacetime in an infinitesimal neighbourhood. The interesting part is how all these neighbourhoods are sewn together to make the fabric of curved spacetime. I introduced the coordinate basis, $dx^\mu$ and $\partial_\mu$, and the metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. I emphasized that the key property of a tensor in curved spacetime is the same as it was in flat spacetime: it transforms in a well-defined way under coordinate transformations. The main difference is that the Jacobian between the original and transformed coordinates is not a bunch of constants: it depends on spacetime position. I also discussed two important rules of index gymnastics for tensors: be very careful about the upstairs/downstairs position of any given index, and also its left/right position. The metric tensor is symmetric under a switch of its two indices, but generic tensors are not.**M18Sep**: Today I began by finishing the discussion of the action principle for the relativistic point particle that I started last time. Then I moved to introducing some important new four-vectors. First, I showed how the partial derivative four-vector naturally arises as a covariant vector with a downstairs index. Then I introduced the four-vector gauge potential of electromagnetism and showed how the electric and magnetic fields are embedded in the gauge field strength tensor, which naturally arises with two downstairs indices and is antisymmetric. I also introduced the 4-vector current and showed how Maxwell's equations can be written in much more succinct form in relativistic notation. (You will flesh out the details of how this works in HW1 problem 2.) Then I described how to understand the case of constant relativistic acceleration in special relativity, and mentioned that you can use the formulae I derived to solve the Twin Paradox. In brief: the physics resolution of theparadox

is that yes, the accelerating twin does age less quickly than their homebody twin! (You will flesh out the details of this in HW1 problem 1.)**R14Sep**: Today, I started with the invariant interval and the definition of the light cone. Then I introduced four-vectors and 4D Minkowski spacetime. In particular, I introduced the Minkowski metric tensor, which is used to raise and lower indices, or to take the inner product between two 4-vectors. I noted that in this course I will be using the mostly minus signature convention for the Minkowski metric. Then I introduced the general definition for a tensor: something that transforms in a particular well-defined way under coordinate transformations. The general utility of tensors in physics is that they are the objects we use to write down dynamical equations, like generalizations of Newton's 2nd Law. I gave examples of the momentum 4-vector, and the 4-velocity and the 4-acceleration vectors for massive particles. In any tensor equation we might write, the number and type of indices must match on the left and right hand side, like in the equation relating the 4-momentum of a point particle to its mass (a relativistic invariant) and its 4-velocity. I finished up by discussing action principles for point particles.**M11Sep**: Today, I introduced the concepts of contravariant and covariant vectors in 3D Euclidean space, which you can think of as like column vectors and row vectors respectively. I explained that rotations have the important property of leaving the norm of a 3-vector invariant, and I showed how rotations affect the components of 3-vectors. Next, I introduced index notation and the Einstein summation convention, and showed that they provide a simpler and more compact way of representing multiplication of matrices and vectors. Then I introduced the Kronecker delta tensor, which is used to raise and lower vector indices or to take the inner (dot) product between two vectors in Euclidean 3-space. I also introduced the permutation tensor, which can be used in 3D to take the outer (cross) product of two vectors. Then I showed how Lorentz transformations look a lot more natural in the rapidity parametrization.**R07Sep**: Most of what I discussed today was organizational matters -- the syllabus, when homeworks and the midterm will be, textbooks, prerequisites, deadline and accommodation policies, etc. Please read through all the pages on this website to familiarize yourself with their contents. Also, office hours for the prof. were settled to be on Tuesdays from 16:15-18:00.