Mathematical predictions of General Relativity

Explain the invariant in a centrally symmetric gravitational field in general relativity, described in spherical coordinates with a function of r and time, preserving symmetry while including angular terms carefully.

Transform coordinates to set A zero and K minus R squared, then derive the metric tensor from the invariant and compute Christoffel symbols and Ricci tensor with Matlab.

Learn to compute Christoffel symbols from a diagonal metric by deriving the inverse metric and partial derivatives, and verify results with Matlab.

Compute Christoffel symbols and the Ricci and Einstein tensors in Matlab using the symbolic toolbox. Set up the metric, define coordinates, derive derivatives, and verify results against the Christoffel expressions.

Rewrite the mixed Einstein tensor terms, analyze the extensor G AlphaBeta and its matrix components, and define sigma terms, then relate the vacuum setting to the energy-momentum tensor.

this lecture derives the stress-energy tensor for a macroscopic body using the four-velocity, defines energy density and pressure, and shows the rest-frame diagonal form.

The lecture derives exact field equations for a centrally symmetric vacuum gravitational field, setting stress-energy components to zero and analyzing lambda and its derivatives to show no extra information arises.

Derives the vacuum field equations for a centrally symmetric gravitational field, yielding the Schwarzschild metric and Schwarzschild radius, and shows alignment with Newton's law at large distances.

Explore how the Schwarzschild radius shapes the metric and time dilation near massive bodies, reveals curvature effects on distances, and connects to Mercury’s perihelion shift, light bending, and gravitational waves.

Derive the relativistic correction to planetary orbits using the Schwarzschild metric and conserved quantities, and compute the Mercury perihelion advance from energy and angular momentum.

Explain how general relativity predicts light deflection by gravity, deriving the sun’s deflection angle using the Schwarzschild metric and Fermat’s principle, yielding alpha = 4 GM/(B c^2).

Relate the Schwarzschild radius to energy balance, derive r_s = 2GM/c^2, and discuss relativistic corrections affecting stable orbits and light's different critical radius near 3GM/c^2.

Derive the relativistic Hamilton-Jacobi equation from the action and Hamiltonian for a massive object, bridging special and general relativity with covariant derivatives and metric for light bending and perihelion shift.

Show how the action stays invariant under canonical transformations, use generating functions to link coordinates and momenta, and relate constants like energy and angular momentum to planetary trajectories.

Derive the relativistic correction to Mercury's orbit in a central Schwarzschild field by using conserved energy and angular momentum, the metric, and an orbital integral to obtain the trajectory.

Compute the relativistic corrections to Mercury’s orbit by expanding the trajectory in powers of RG over R, compare earth’s and sun’s cases, and outline Matlab implementation for higher-order terms.

Compute a Maclaurin series of a function in Matlab with symbolic variables and a third-order truncation. Substitute sigma one to obtain the series used to derive Mercury's perihelion advance.

Derives the relativistic correction to Mercury's perihelion by expanding the radial action and isolating the 1 over r^2 term, yielding the advance delta phi per revolution.

Calculate the delta phi correction for the advance of Earth and Mercury perihelion using eccentricity and the semi-major axis, and express results in arc seconds per century.

Explore how light bends in a center-symmetric gravitational field by applying the Hamilton-Jacobi framework for massless photons, deriving the light-ray trajectory and the leading relativistic corrections.

Compute relativistic corrections to the bending of light by reformulating the integral and extracting the deflection angle, showing delta phi relates to the gravitational parameter R_G for large distances.

Derive gravitational waves from the field equations in a weak vacuum, showing metric perturbations propagate as waves at the speed of light.