General Relativity and Cosmology
Curved spacetime, black holes and the expanding universe: the cracks in Newtonian gravity and Einstein's road to the 1915 field equations, with geometrized units and the compactness that measures when relativity matters; special relativity in geometric form, with the interval, light cones, proper time, four-vectors, the Minkowski metric, invariant mass, rapidity and accelerated observers; the weak and Einstein equivalence principles, tests from Eötvös to MICROSCOPE, gravitational time dilation and redshift, the Pound–Rebka experiment and the relativistic corrections to GPS clocks; metrics, locally inertial coordinates, Christoffel symbols and the geodesic equation derived from stationary proper time, with Newton's law recovered from the warping of time; tensors, parallel transport, the Riemann, Ricci and Einstein tensors, geodesic deviation as tidal acceleration, the stress-energy tensor and the Einstein field equation with its Newtonian limit; the Schwarzschild solution, Birkhoff's theorem, Flamm's paraboloid, conserved energy and angular momentum, the effective potential, the innermost stable circular orbit, the photon sphere and radial infall; the classic tests of perihelion precession, light bending, gravitational lensing, the Shapiro delay and gravitational redshift, and the parametrized post-Newtonian framework; black-hole horizons, Eddington–Finkelstein coordinates, the Kerr solution, the ergosphere and frame dragging, the area theorem, Hawking radiation and black-hole entropy; linearized gravity, gravitational-wave polarizations, the quadrupole formula and chirp mass, the Hulse–Taylor pulsar, LIGO, Virgo and KAGRA, LISA and pulsar timing arrays; the FLRW metric, Hubble's law, redshift, the Friedmann equations, density parameters, cosmic ages and distances and the supernova evidence for acceleration; the hot Big Bang, neutron freeze-out, Big Bang nucleosynthesis, recombination and the cosmic microwave background with its acoustic peaks; and a capstone on dark matter, dark energy, inflation, the Hubble tension and quantum gravity, with runnable Python examples throughout.
PHYS-450 Year 4 3 credits 12 lessons · 4 h 41 min ▶ Video lessons coming soon
Offered by HSM University · Physics
This is the Canadian version of this course. See the American version.
What you'll learn
- Explain why Newtonian gravity is incomplete and describe the central idea of general relativity, using the compactness \(GM/Rc^2\) and geometrized units to judge when relativistic effects matter.
- Use spacetime diagrams, the Minkowski interval, proper time, four-vectors and Lorentz transformations to solve problems in special relativity, including invariant mass and accelerated motion.
- State the weak and Einstein equivalence principles, derive gravitational time dilation and redshift from them, and compute the relativistic corrections to GPS clocks.
- Work with metrics and Christoffel symbols, derive the geodesic equation from the principle of stationary proper time, and recover Newton's law of gravity in the weak-field limit.
- Explain the Riemann, Ricci and Einstein tensors and the stress-energy tensor, interpret curvature as tidal acceleration through geodesic deviation, and state the Einstein field equation and its Newtonian limit.
- Analyse the Schwarzschild solution, including its conserved quantities, effective potential, innermost stable circular orbit, photon sphere and radial infall.
- Derive and evaluate the perihelion advance, light deflection, Shapiro delay and gravitational redshift, and interpret solar-system tests through the parametrized post-Newtonian parameters.
- Describe black-hole horizons, the Kerr solution, the ergosphere and frame dragging, the area theorem, and the Hawking temperature and entropy of black holes.
- Derive the wave equation of linearized gravity, describe gravitational-wave polarizations and sources, use the chirp-mass formulas, and explain how waves are detected across frequency bands.
- Derive the Friedmann equations from the FLRW metric, relate redshift to the scale factor, compute ages and distances in model universes, and explain the evidence for cosmic acceleration.
- Explain the thermal history of the early universe, estimate the primordial helium abundance and the temperature of recombination, and interpret the spectrum and acoustic peaks of the cosmic microwave background.
- Evaluate the evidence for dark matter and dark energy, explain how inflation addresses the horizon and flatness problems, and assess the Hubble tension and the prospects for resolving it.
Course content
12 lessons · 4 h 41 min total · final exam
- 1 Why general relativity: gravity as geometry 24 min · quiz
- 2 Special relativity review: four-vectors, the metric and spacetime diagrams 22 min · quiz
- 3 The equivalence principle and gravitational time dilation 23 min · quiz
- 4 Curved spacetime: metrics, geodesics and the geodesic equation 25 min · quiz
- 5 Tensors, curvature and the Einstein field equations 25 min · quiz
- 6 The Schwarzschild solution and orbits around a mass 23 min · quiz
- 7 Classic tests: perihelion precession, light bending, Shapiro delay and redshift 23 min · quiz
- 8 Black holes: horizons, Kerr spin and Hawking radiation 24 min · quiz
- 9 Gravitational waves: linearized gravity, sources and detection 23 min · quiz
- 10 Cosmology I: the FLRW metric, Hubble's law and the Friedmann equations 22 min · quiz
- 11 Cosmology II: the hot Big Bang, nucleosynthesis and the cosmic microwave background 22 min · quiz
- 12 Capstone: open questions — dark matter, dark energy, inflation and the Hubble tension 25 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Curved spacetime, black holes and the expanding universe: the cracks in Newtonian gravity and Einstein's road to the 1915 field equations, with geometrized units and the compactness that measures when relativity matters; special relativity in geometric form, with the interval, light cones, proper time, four-vectors, the Minkowski metric, invariant mass, rapidity and accelerated observers; the weak and Einstein equivalence principles, tests from Eötvös to MICROSCOPE, gravitational time dilation and redshift, the Pound–Rebka experiment and the relativistic corrections to GPS clocks; metrics, locally inertial coordinates, Christoffel symbols and the geodesic equation derived from stationary proper time, with Newton's law recovered from the warping of time; tensors, parallel transport, the Riemann, Ricci and Einstein tensors, geodesic deviation as tidal acceleration, the stress-energy tensor and the Einstein field equation with its Newtonian limit; the Schwarzschild solution, Birkhoff's theorem, Flamm's paraboloid, conserved energy and angular momentum, the effective potential, the innermost stable circular orbit, the photon sphere and radial infall; the classic tests of perihelion precession, light bending, gravitational lensing, the Shapiro delay and gravitational redshift, and the parametrized post-Newtonian framework; black-hole horizons, Eddington–Finkelstein coordinates, the Kerr solution, the ergosphere and frame dragging, the area theorem, Hawking radiation and black-hole entropy; linearized gravity, gravitational-wave polarizations, the quadrupole formula and chirp mass, the Hulse–Taylor pulsar, LIGO, Virgo and KAGRA, LISA and pulsar timing arrays; the FLRW metric, Hubble's law, redshift, the Friedmann equations, density parameters, cosmic ages and distances and the supernova evidence for acceleration; the hot Big Bang, neutron freeze-out, Big Bang nucleosynthesis, recombination and the cosmic microwave background with its acoustic peaks; and a capstone on dark matter, dark energy, inflation, the Hubble tension and quantum gravity, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 4.