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General Relativity.
- Format:
- Book
- Author/Creator:
- Arias Abad, Camilo.
- Language:
- Spanish
- Physical Description:
- 1 online resource (425 pages)
- Edition:
- 1st ed.
- Place of Publication:
- Bogotá : Universidad Nacional de Colombia, Unibiblos, 2025.
- Summary:
- Aimed at readers with a mathematical background, this text offers an introduction to the general theory of relativity. The book is structured in four parts: Geometry, Electromagnetism and Special Relativity, Gravity and Curvature, and Solutions to Einstein's Equations. Each part is designed to build progressively on the previous ones, and attempts to combine mathematical rigor and physical insight.This text is suitable for advanced undergraduates, graduate students, and researchers who are looking for a well-rounded and rigorous exploration of general relativity. Whether used as a course textbook or for independent study, it provides a pathway into one of the most profound theories in physics.
- Contents:
- Intro
- Introduction
- I Geometry
- Differentiable manifolds
- Manifolds
- The tangent space and the derivative
- Vector bundles
- The tangent bundle and vector fields
- Vector fields and flows
- The cotangent bundle and tensor fields
- Differential forms and integration
- Differential forms
- Classical vector calculus
- Manifolds with boundary
- Oriented manifolds
- Integration of forms
- Stokes' theorem
- The classical integral theorems
- An application: conservation of mass
- The metric determines the geometry
- The metric tensor
- Length of a curve
- Isometries and Killing vector fields
- Connections, parallel transport and geodesics
- Connections
- The Levi-Civita connection
- The pullback of bundles and connections
- Parallel transport
- Geodesics
- Curvature
- The Riemann curvature tensor
- The Ricci tensor and scalar curvature
- Sectional curvature
- Curvature and parallel transport
- Geodesic deviation and Jacobi fields
- Gauss' lemma and curvature
- II Electromagnetism and Special Relativity
- Electricity and Magnetism
- Coulomb's and Lorentz force laws
- Electrostatics: charges at rest
- Electrodynamics: moving charges
- The Law of Biot-Savart
- There are no magnetic monopoles
- Magnetostatics
- Varying electric fields
- Faraday's law of induction
- Conservation of energy
- Conservation of linear momentum
- Maxwell's equations and waves
- Galilean transformations and the speed of light
- Special Relativity
- The Michelson-Morley experiment
- Lorentz tranformations
- Minkowski spacetime
- Motion of particles and observers in Minkowski Spacetime
- Twins
- Time travel and causality
- Length contraction
- Velocities under Lorentz Transformations
- Bell's spaceship paradox
- The Doppler effect
- Aberration of Light
- Muon Decay: An experimental test for Special Relativity.
- Energy, Momentum and Mass
- Electromagnetism and Special Relativity
- III Gravity and Curvature
- From Minkowski to curved spacetimes
- Light cones and causality
- Proper time, velocity and momentum
- Geodesic motion and Fermi coordinates
- Acceleration and Fermi-Walker coordinates
- An observer moving with constant acceleration
- A Journey to Kepler 22-b
- Redshift and blueshift
- Fermi-Walker transport for circular motion
- The physical meaning of coordinates
- The equivalence principle and tidal forces
- Tidal forces: newtonian analysis
- Time dilation due to acceleration
- Gravitational time dilation and redshift
- Comparing the two situations
- The Energy-Momentum tensor
- The equation of continuity
- Euler's equation
- The momentum flux
- Perfect fluid energy-momentum tensor
- Conservation of energy-momentum
- Electromagnetic energy-momentum tensor
- The symmetry of T
- The Field equation
- Newton's law and the Poisson equation
- Units and dimensions
- The Einstein tensor
- Newtonian limit and the value of
- Estimate of g00
- The cosmological constant
- The geometric meaning of Einstein's equation
- The astonishing analogy: geodesic deviation and tidal forces
- Einstein's equation and variational principles
- Predictions and tests
- IV Solutions to Einstein's Equation
- The Schwarzschild solution
- Gravitational potential of a point mass
- Spherical symmetry and Birkhoff's theorem
- The Schwarzschild metric
- Planetary motion in Newtonian gravity
- Timelike geodesics in Schwarzschild spacetime
- Precession of Mercury's Perihelion
- Lightlike geodesics in Schwarzschild spacetime
- Gravitational bending of light
- Conformal maps and Carter-Penrose diagrams
- Incoming Eddington-Finkelstein and black holes
- Outgoing Eddington-Finkelstein and white holes
- Kruskal-Szekeres coordinates.
- Interior of a non rotating star
- Interior of a Uniformly Dense Star
- Geometry Inside a Spherical Empty Cavity
- Time Machines
- The FLRW metric and Cosmology
- The FLRW metric
- Lightlike godesics
- Conformal flatness and Penrose diagram
- Cosmological red shift and Hubble's law
- Age and diameter of the observable universe
- V Appendices
- Linear algebra and tensors
- Linear algebra and matrices
- Tensor products
- Tensors
- Change of basis
- Contraction of tensors.
- Topology and analysis
- The Hodge star operator
- The Picard-Lindelöf theorem
- Lie groups
- The linking number
- The future is open
- Other geometric results
- The theorem of Hopf-Rinow
- The theorem of Ambrose
- Constant curvature and Cartan-Hadamard
- Constant curvature and Killing-Hopf
- Bibliography
- Index.
- Notes:
- Description based on publisher supplied metadata and other sources.
- ISBN:
- 958-50-5888-X
- OCLC:
- 1517038451
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