Introduction to General Relativity and Cosmology

Cover Page
Title
Copyright
Dedication
Preface
About the Author
Contents
1. Differential Geometry
	1.1 The Concept of a Vector
		1.1.1 Vector operations
		1.1.2 Projections and basis vectors
		1.1.3 Towards tangent space and all that
		1.1.4 Index notation
	1.2 Manifolds and Tensors
		1.2.1 Tangent space and vector fields
		1.2.2 Tensors
		1.2.3 Manifolds and metric
		1.2.4 Examples of metrics
		1.2.5 Geodesics
		1.2.6 Covariant derivative
		1.2.7 Parallel transport and geodesics
	1.3 Curvature
		1.3.1 Infinitesimal parallelogram
		1.3.2 Riemann, Ricci and Weyl tensors
		1.3.3 Geodesic deviation equation
	1.4 Euler–Lagrange Equations
	1.5 Further Reading
	1.6 Exercises
2. Einstein Field Equations
	2.1 Some Physics Background
		2.1.1 Newton’s theory of gravity
		2.1.2 Special relativity
		2.1.3 Maxwell equations
		2.1.4 Matter tensors
	2.2 Geometry and Gravity
		2.2.1 Geodesics and Newton’s law
		2.2.2 Curvature and the Poisson equation
		2.2.3 Field equations of General Relativity
		2.2.4 The principle of minimal gravitational coupling
	2.3 Weak Gravity
		2.3.1 Linearised Riemann and Ricci tensors
		2.3.2 Gauge transformations
		2.3.3 Linearised Einstein field equations
		2.3.4 Gravitational waves
	2.4 Variational Approach to General Relativity
	2.5 Further Reading
	2.6 Exercises
3. Schwarzschild Solutions
	3.1 Spherical Symmetry and Birkhoff’s Theorem
	3.2 The Schwarzschild Solution
	3.3 The Schwarzschild Interior Solution
	3.4 Geodesics in Schwarzschild Spacetime
	3.5 Testing General Relativity — The Classical Tests
		3.5.1 Perihelion precession of Mercury
		3.5.2 Light deflection by the Sun
		3.5.3 Gravitational redshift of light
		3.5.4 Radar echo or gravitational time delay
	3.6 The Schwarzschild Radius
		3.6.1 Radial null geodesics
		3.6.2 Eddington–Finkelstein coordinates
		3.6.3 Kruskal–Szekeres coordinates
		3.6.4 Black holes
	3.7 Further Reading
	3.8 Exercises
4. Cosmology
	4.1 Classical and Modern Cosmology
		4.1.1 Cosmological principle
		4.1.2 Geometry of constant time hypersurfaces
		4.1.3 Friedmann–Lemaître–Robertson–Walker metric
		4.1.4 Particle horizons
		4.1.5 Field equations
	4.2 Cosmological Solutions
		4.2.1 Matter-dominated universe
		4.2.2 Radiation-dominated universe
		4.2.3 The Einstein static universe
		4.2.4 De Sitter universe
	4.3 Physical Cosmology
		4.3.1 Cosmological parameters
		4.3.2 Redshift
		4.3.3 Distances in cosmology
		4.3.4 Distance redshift relationships
		4.3.5 The universe today
		4.3.6 Shortcomings in cosmology
	4.4 Inflation
		4.4.1 Accelerated expansion
		4.4.2 Scalar fields in cosmology
		4.4.3 Slow-roll inflation
	4.5 Further Reading
	4.6 Exercises
5. Solutions to Exercises
	5.1 Solutions: Differential Geometry
		The concept of a vector
		Manifolds and tensors
		Curvature
	5.2 Solutions: Einstein Field Equations
		Some physics background
		Geometry and gravity
		Weak gravity
		Variational approach to General Relativity
	5.3 Solutions: Schwarzschild Solutions
		The Schwarzschild solution
		The Schwarzschild interior solution
		Geodesics in Schwarzschild spacetime
		Testing General Relativity — the classical tests
		The Schwarzschild radius
	5.4 Solutions: Cosmology
		Classical and Modern Cosmology
		Physical cosmology
		Inflation
Bibliography
Index