Course: Relativity & Cosmology

This online course is closely linked to the textbook A College Course on Relativity and Cosmology by Ta-Pei Cheng (Oxford 2015). It is divided into 8 PARTS, presented in 30 lectures (each typically in 3 video segments). The relevant book chapters and lecture segments (in clickable links) are listed in the table below. View this course on YouTube here.

There is no need to follow the whole lecture course; one can use the table of contents to choose whatever topics one is interested in. [Suggestion: To avoid excessive disruptions of YouTube ads, instead of clicking on individual video segments, it will be a good idea to start with PLAY ALL at each Part, then pick on the segments. The same strategy also works for the table below: start with the Part # playlist on YouTube rather than the individual segments. Viewers only interested in the basics can skip those details in the video segments marked by a star *.]

The viewer is encouraged to do exercises disbursed throughout the book (can compare to the solutions given below after the video list in each Part) and study the Review Questions (and their keys) at the end of each chapter.

PART-1: Special Relativity
(Book Ch 1, Ch 2, and Ch 3, and video lectures #1 to #6)

Description: We present special relativity as first introduced by Einstein and then study its geometric formulation in Minkowski spacetime.

Learning objective: Understanding the meaning of relativity (as a coordinate symmetry) and the key role played by Einstein’s new conception of time in its formulation. Understanding why the proper physics arena is the 4D spacetime and the sense that the flat spacetime metric, the diagonal matrix of (-1,1,1,1), embodies all special relativity. Learning to use 4D tensors (such as 4- position, 4-derivative, 4-momentum, and electromagnetic field tensor and energy-momentum-stress tensor, etc.) to construct relativistic equations.

See (in conjunction with Ch 3) Supplementary notes on the math of tensors

View the Part 1 playlist on YouTube or click on the individual videos below.

PART-2: Equivalence Principle
(Book Ch 4, and video lectures #07 to #09) 

Description: Historically, Einstein used the idea of the equivalence between gravitation and inertia to proceed from special to general relativity. Some GR results can be obtained in this physically more accessible approach. 

Learning objective: Understanding how Einstein used the Equivalence Principle (EP) to extract some GR results (gravitational redshift, time dilation, and light deflection) and in turn how such physics led him to the idea that gravitational field is simply a curved spacetime.

View the Part 2 playlist on YouTube or click on the individual videos below.

Part 3: General Relativity
(Book Ch 5 & Ch 6, and video lectures #10 to #15) 

Description: Einstein’s theory of general relativity posits that the gravitational field is a curved 4D spacetime. We first learn how to describe a warped manifold using the metric tensor. From this, we are motivated immediately to write down the GR equation of motion. The more difficult topic of the GR field equation is covered in the second half of this Part (Chapter 6). Using the spherically symmetric solution of the Einstein equation, we study the trajectories of light and a planet in the solar system.  

Learning objective: Learning how a curved space can be described by length measurements. The metric tensor connects length measurements to the chosen Gaussian coordinate and can determine the geometry of the space, e.g., the equation that fixes a geodesic line. Understanding how the EP physics (particularly gravitational time dilation) has led Einstein to the idea that GR is a field theory of gravity with the field being the curved spacetime. In this field theoretical framework, the geodesic equation is the equation of motion, and the field equation relates geometry to the matter/energy source, with the geometric side being the curvature (tidal gravity) and the source side the energy-momentum-stress tensor. The spacetime outside a spherical source is given by the Schwarzschild solution, leading to the geodesic equations for light (as in light deflection by the sun) and for a matter test particle (such as the planet Mercury).

View the Part 3 playlist on YouTube or click on the individual videos below.

Part 4: Tensor Formalism for GR
(Book Ch 11, and video lectures #16 to #17)

 Description: The more difficult topic of deriving Riemann curvature tensor is presented here. In this way, the Einstein field equation is justified with the proper mathematics. Chapter 11 may be viewed as the math appendix of the book. Viewers less interested in the mathematical aspect of GR can skip over and go directly to Parts 5 - 8 (black holes & cosmology).

Learning objective: Viewers should enjoy seeing some of the basic features of differential geometry (covariant differentiation and parallel transport, etc.) for a proper formulation of general relativity. One can then derive the Riemann curvature tensor by parallel transporting a vector around a closed path or through the geodesic deviation equation. From the Principle of General Covariance, one can then derive the GR equation of motion and the Einstein field equation (with the help of Bianchi identity). With this math background, a viewer should also be able to understand the Action-Principle approach to the GR field equation.

View the Part 4 playlist on YouTube or click on the individual videos below.

Part 5: Black Holes
(Book Ch 7, and video lectures #18 to #20) 

Description: The gravity of a black hole is so strong, and the spacetime so warped, that the roles of space and time are interchanged across its event horizon. For black holes to be relevant for realistic astrophysical situations one must first demonstrate that ordinary stars over a certain mass limit eventually undergo gravitational collapse into black holes. The gravitational binding energy of matter around a black hole is immense, giving rise to many energetic events observed in the universe. While no signal can be transmitted from the interior of a black hole in classical physics, quantum fluctuation around a BH can bring about Hawking radiation -- the thermal emission of particles and light from a black hole.

Learning objective: Understanding the spacetime structure of black holes, showing that the lightcones tip over when crossing the event horizon so that no signal can leave the BH. Such classical consideration leads one to conclude that nothing can ever come out of a black hole, but in the context of quantum field theory in curved spacetime, there will be thermal emission known as Hawking radiation.

View the Part 5 playlist on YouTube or click on the individual videos below.

Part 6: GR & Cosmology
(Book Ch 8, and video lectures #21 to #24) 

Description: The feature that the space is dynamic in GR naturally leads to the observed expanding universe. Based on the Cosmological Principle (supported by direct observation), which implies an expanding universe according to Hubble’s law (first written down by Lemaitre). The spacetime geometry is described by the Robertson-Walker metric. With an ideal cosmic fluid as the source, the resultant Einstein equation is in the form of Friedmann equations. This is the FLRW cosmology. Einstein also discovered that the GR field equation naturally allows for the presence of vacuum energy, called the cosmological constant, which gives rise to a gravitational repulsion that increases with distance.

Learning objective: Understanding why general relativity is the natural framework for studying our universe as a physical system.

View the Part 6 playlist on YouTube or click on the individual videos below.

Part 7: The Big Bang Cosmology
(Book Ch 9, and video lectures #25 to #27)

Description: Extrapolating back in an expanding universe suggests a big hot beginning. The system must pass through thermal equilibria with various constituent particles as the universe expands and cools. This picture is strongly supported by the presence of these Big Bang thermal relics in the present era, for instance, the light nuclear elements of helium, deuterium, etc., with just the correct relative abundance. Most importantly, the decoupled photons became the cosmic microwave background (CMB) observed today; the statistical feature of its anisotropy contains much information about the geometry and energy contents of the universe.

Learning objective: Understanding how the observed thermal relics of cosmic microwave radiation and the abundance of light nuclear elements, etc supports the big bang theory.

View the Part 7 playlist on YouTube or click on the individual videos below.

Part 8: Cosmology of Inflation & Accelerating Universe 
(Book Ch 10, and video lectures #28 to #30) 

Description: While Parts 6 & 7 describe the basic FLRW cosmology, in Part 8, to this model added the cosmological constants, resulting in the current standard model of ΛCDM cosmology. The presence of such a cosmological constant is the basis of the inflation theory of the Big Bang, which can leave just the right conditions for the start of FLRW evolution. Furthermore, astronomical observations are all consistent with a cosmological expansion accelerating in the present epoch due to another cosmological constant (dark energy). Thus, our universe appears to be dominated by dark energy (Λ) and cold dark matter (CDM), leading to the current standard model of ΛCDM cosmology. 

Learning objective: Understanding how GR can accommodate a cosmological repulsion that manifests itself as the primordial ‘inflation’ of the Big Bang and as the ‘dark energy’ that propels an accelerated expansion.

View the Part 8 playlist on YouTube or click on the individual videos below.