General Relativity. The Theoretical Minimum

GENERAL
	RELATIVITY
	Contents
	Preface
	Lecture 1: Equivalence Principle and Tensor Analysis
	Introduction
	Equivalence Principle
	Accelerated Reference Frames
	Curvilinear Coordinate Transformations
	Effect of Gravity on Light
	Tidal Forces
	Non-Euclidean Geometry
	Riemannian Geometry
	page furled
	Metric Tensor
	Mathematical Interlude: Dummy Variables
	Mathematical Interlude: Einstein Summation Convention
	Y,VvUv
	First Tensor Rule: Contravariant Components of Vectors
	[ ATX(P), X2(P), ... ,XN(P)]
	[y^p), y2(P),... ,yN(p)]
	dXm
	Mathematical Interlude: Vectors and Tensors
	9Xp
	= w, —
	Second Tensor Rule: Covariant Components of Vectors
	Covariant and Contravariant Components of Vectors and Tensors
	Lecture 2: Tensor Mathematics
	Introduction
	Flat Space
	Metric Tensor
	Scalar, Vector, and Tensor Fields
	s\'(r) = s(X)	(2)
	Geometric Interpretation of Contravariant and Covariant Components of a Vector
	v = V1^ + V2e2 + V3e3
	Mathematical Interlude: Dot Product of Two Vectors
	Vi = V • a
	V = VTnem
	Vn = V-en
	Tensor Mathematics
	Tensor Algebra
	(v” H\'\"’\' = ar^ (l\'\"‘ H\'h, (24)
	More on the Metric Tensor
	Mathematical Interlude: The Metric is a Symmetric Tensor
	The Matrix Inverse of the Metric
	Lecture 3: Flatness and Curvature
	Introduction
	General Relativity in Modern Physics
	Riemannian Geometry
	Gaussian Normal Coordinates
	Covariant Derivatives
	- r*m vt	(io)
	Christoffel Symbols
	rf = rf	(id
	Curvature Tensor
	D9Drvn = ds [drvn - r‘nyf]
	R,^ = drr‘n - asr‘n + r?nrL. - nnr* (23)
	Lecture 4: Geodesics and Gravity
	Introduction
	Parallel Transport
	Tangent Vectors and Geodesics
	Example of Calculations with Christoffel Symbols
	More on Geodesics
	Space-Time
	t 4
	Q
	p.
	At2 = At2 - AX2
	►
	X
	Special Relativity
	Uniform Acceleration
	Uniform Gravitational Field
	Motion of a Particle
	Lecture 5:
	Metric for a Gravitational Field
	Time-like, Space-like, and Light-like Intervals and Light Cones
	Geodesics and Euler—Lagrange Equations
	shortest curve
	A = / £(X, X) dt
	Schwarzschild Metric
	Black Holes
	Event Horizon of a Black Hole
	Motion of a Light Ray
	Lecture 6: Black Holes
	Schwarzschild Metric
	Schwarzschild Radius or Black Hole Event Horizon
	Light Ray Orbiting a Black Hole
	horizon
	orbiting particle
	light-ray
	O black hole
	Photon Sphere
	black hole
	Hyperbolic Coordinates Revisited
	Interchange of Space and Time Dimensions at the Horizon
	Black Hole Singularity
	Alice
	No Escaping from a Black Hole
	Lecture 7: Falling into a Black Hole
	Introduction
	Schwarzschild Metric, Event Horizon, and Singularity
	horizon
	r = 0 r = 1	r
	Fundamental Diagram of Space-Time near a Black Hole
	Notes on the Fundamental Diagram
	History of Black Holes
	Falling into a Black Hole
	Questions/Answers Session
	Lecture 8: Formation of a Black Hole
	Introduction
	Kruskal—Szekeres Coordinates
	Penrose Diagrams
	T=0
	T=-l
	T=-2
	T = 2
	T=1
	r=o
	T=-l
	T=-2
	Wormholes
	Formation of a Black Hole and Newton’s Shell Theorem
	inside the shell
	Discussion of the Time Variable
	Lecture 9: Einstein Field Equations
	Introduction
	Newtonian Gravitational Field
	Continuity Equation
	Energy-Momentum Tensor
	Ricci Tensor and Curvature Scalar
	ft = ft£ =
	Einstein Tensor and Einstein Field Equations
	Questions/Answers Session
	Lecture 10: Gravitational Waves
	Introduction
	r = h-^	(6)
	Gravitational Waves
	x = xf + /(x\', y\')
	(14) y = y\' + 5(x\', y\')
	Questions/Answers Session
	Einstein—Hilbert Action for General Relativity
	Steps to Derive the Field Equations from the Action
	Index