This course starts from the incompatibility between Galileo’s principle and Maxwell’s equations, and expands on that in order to consistently formulate Special Relativity and later on, in the second part of the course, General Relativity. The other main purpose is to stimulate students to develop the mathematical intuition required to fully grasp and appreciate the contents of these subjects. Therefore, EVERY equation in this course will be motivated. Besides, other key concepts such as: Lagrangian mechanics (i.e. the Action Principle, Lagrange equations), tensors, will be fully covered in the course. The main prerequisites to the course are Calculus and Multivariable Calculus, especially: the divergence theorem, vectors, dot and cross products, matrix multiplication, determinants. Some (basic) knowledge of Classical physics is recommended, such as: scalar potential, Newton laws, Kinetic energy, Energy conservation, Wave equation (and I mean just the mathematical form of the equation).
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In the first part of the course Lorentz transformations are derived in two different ways. The mathematics to be able to follow this part can be more easily digested than the mathematics required to follow the part on General Relativity. For General Relativity, it is recommended to follow along with a piece of paper and pencil and derive the equations. Please make sure that you meet the prerequisite requirements.
Introduction
introduction: disagreement between electromagnetism and classical mechanics
Galileo’s transformations
Lorentz Transformations
Lorentz trasformation derivation part 1
Lorentz trasformation derivation part 2 – Simultaneity
Lorentz trasformation derivation part 3
Lorentz transformation derivation part 4
Lorentz transformation derivation part 5
Lorentz transformation derivation part 6
Lorentz transformation derivation part 7
Lorentz invariant quantity
A different derivation of Lorentz transformations part 1
A different derivation of Lorentz transformations part 2: rotation matrices
A different derivation of Lorentz transformations part 3
A different derivation of Lorentz transformations part 4
A different derivation of Lorentz transformations part 5
Length contraction, Time dilation, Proper time
Length contraction and time dilation
Non inertial frames and proper time
Lagrangian mechanics
Introduction to Lagrangian mechanics
Lagrangian mechanics part 1
Lagrangian mechanics part 2
Lagrangian mechanics part 3
Lagrangian mechanics part 4
Derivation of the Hamiltonian
Definition of momentum
Energy and Momentum in Special Relativity
Lagrangian in Special Relativity
Derivation of Momentum in Special Relativity
Derivation of E=mc^2
Relation between Energy and momentum
Momentum of a Photon
Transition from Special to General Relativity
Introduction to General Relativity: Einstein’s “happiest” thought
Highlighting the need for Differential Geometry
Invariant in tensor notation
Tensors
Tensor transformations part 1
Tensor transformations part 2
Higher rank tensors from lower rank tensors
Lower rank tensors from higher rank tensors
Transformation of Euclidean derivatives
Covariant Derivative
Some properties of the metric tensor
Christoffel symbol in terms of the metric tensor part 1
Christoffel symbol in terms of the metric tensor part 2
Covariant derivative of the metric tensor
Covariant derivative of a contravariant vector part 1
Covariant derivative of a contravariant vector part 2
Proof that the covariant derivative of the metric tensor is zero
Core of General Relativity
Equation of a geodesic
Geodesic and parallel transport
Riemann tensor part 1
Riemann tensor part 2
Some properties of : Riemann_tensor, Ricci tensor, Ricci scalar
Action in General Relativity
Invariant 4-volume element in the action
Determinant of the metric tensor
Variation of the action of gravity part 1
Variation of the action of gravity part 2
Variation of the action of gravity part 3
Einstein field equations part 1
Einstein field equations part 2: another property of the Riemann tensor
Einstein field equations part 3: energy momentum tensor
Field equations in classical physics
Reducing General Relativity to Newtonian laws
Final form of the field equations
Appendix
Rigorous proof that the variation of Christoffel symbol is a tensor
A more rigorous derivation of the Riemann tensor
