This book discusses analogies between relativistic cosmology and various physical systems or phenomena, mostly in the earth sciences, that are described formally by the same equations. Of the two independent equations describing the universe as a whole, one (the Friedmann equation) has the form of an energy conservation equation for one-dimensional motion. The second equation is fairly easy to satisfy (although not automatic): as a result, cosmology lends itself to analogies with several systems. Given that a variety of universes are mathematically possible, several analogies exist. Analogies discussed in this book include equilibrium beach profiles, glacial valleys, the shapes of glaciers, heating/cooling models, freezing bodies of water, capillary fluids, Omori's law for earthquake aftershocks, lava flows, and a few mathematical analogies (Fibonacci's sequence, logistic equation, geodesics of various spaces, and classic variational problems). A century of research in cosmology can solve problems on the other side of an analogy, which in turn can suggest ideas in gravity. Finding a cosmic analogy solves the inverse variational problem of finding a Lagrangian and a Hamiltonian for that system, when nobody thought one exists. Often, the symmetries of the cosmological equations translate in new symmetries of the analogous system. The book surprises the reader with analogies between natural systems and exotic systems such as possible universes.
Author(s): Valerio Faraoni
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 286
City: London
Contents
Preface
List of Notations
Acknowledgments
List of Figures
1. Relativistic cosmology: a primer
1.1 General relativity
1.1.1 Equivalence Principle
1.1.2 Differential geometry
1.1.2.1 Manifolds and coordinates
1.1.2.2 Tensors
1.1.2.3 Tensor symmetries
1.1.2.4 Tensor fields
1.1.3 Metric tensor
1.1.4 Levi–Civita symbol and tensor densities
1.1.5 Covariant derivative
1.1.6 Connection, geodesics, and curvature
1.1.7 Energy–momentum tensors
1.1.7.1 Perfect fluids
1.1.7.2 Scalar field
1.1.7.3 Energy conditions
1.1.8 Einstein equations
1.2 Cosmology
1.2.1 Spatial homogeneity and isotropy
1.2.2 Friedmann–Lemaître–Robertson–Walker spacetimes
1.2.3 Geometry of three–spaces of constant curvature
1.2.3.1 Three–sphere (closed universe)
1.2.3.2 Flat space
1.2.3.3 Hyperbolic space (open universe)
1.3 Einstein–Friedmann equations
1.3.1 General case
1.3.1.1 Open universe (κ = –1)
1.3.1.2 Closed universe (κ = +1)
1.3.2 Energy conservation
1.3.2.1 Evolution of the Hubble function
1.3.3 Integration of the energy equation for constant equation of state
1.3.4 Dominance of a perfect fluid
1.4 Perfect fluid solutions
1.4.1 Spatially flat, Λ = 0, FLRW universes
1.4.1.1 Dust
1.4.1.2 Radiation
1.4.2 Cosmological constant
1.4.2.1 Solutions for spatially curved universes with Λ = 0
1.4.3 FLRW Lagrangian and Hamiltonian
1.5 Symmetries of the Einstein–Friedmann equations for spatially flat universes
1.6 The accelerating universe and dark energy
2. Cosmic analogies in mathematics
2.1 Inverse variational problems
2.2 When the same equation describes many phenomena
2.2.1 Stability of the equilibrium solutions y = yc
2.2.2 Symmetry for power–law f(y)
2.2.3 Lagrangian and Hamiltonian formulations
2.2.4 Mechanical analogy
2.3 Exponential growth and decay
2.4 de Sitter universe and Fibonacci’s rabbits
2.4.1 The cosmic analogy
2.4.2 Lagrangian, Hamiltonian, and a new invariant
2.5 Cosmic analogues of the logistic equation
2.5.1 Comoving time analogy I
2.5.2 Comoving time analogy II
2.5.3 Conformal time analogy I
2.5.4 Conformal time analogy II
2.6 Geodesics of the Euclidean plane
2.7 Geodesics of the Poincaré half–plane
2.8 Cosmology and the catenary problem
2.9 Cosmic analogue of the minimal surface of revolution
2.9.1 Dependent variable x = x(y)
2.9.2 Dependent variable y = y(x)
2.10 Strategies to find analogies
3. Heating, cooling, and their cosmic analogues
3.1 Physics of heating and cooling
3.2 Cool cosmic analogies
3.2.1 Newton cooling
3.2.2 Dulong–Petit cooling
3.2.3 Simplified Newton–Stefan cooling
3.2.4 Other cooling models
3.3 Cooling of lava flows
3.4 Freezing of lakes in winter
3.4.1 The analogous radiation–dominated universe
3.4.2 Symmetries
4. Cosmic analogies in gravity
4.1 Falling raindrops, debris flows, and avalanches
4.1.1 Falling raindrop
4.1.2 Landslides, debris flows, and avalanches
4.2 Brachistochrone problem
4.3 Gravity tunnels
4.4 Terrestrial brachistochrone
4.5 Newtonian analogy for FLRW cosmology
4.6 Turning Newtonian cosmology into a relativistic problem
4.6.1 Test particle in radial motion above a ball
4.6.1.1 Massive test particle
4.6.1.2 Massless test particle
4.6.2 Generalization to any static spherical geometry
4.6.3 An exact expanding relativistic sphere
4.6.3.1 Special case of the Smoller–Temple shock wave solution
4.6.3.2 Vaidya geometries
4.6.3.3 Quasilocal energy, Ricci and Weyl tensors
4.6.3.4 Symmetries of the Newtonian problem
5. Cosmology on the beach
5.1 Equilibrium beach profiles
5.2 Point–particle mechanical analogy
5.3 Cosmic analogue of an equilibrium beach profile
5.4 Beach profile solutions via the analogous cosmology
5.4.1 Case n = 1/3
5.4.2 n = –1 (linearly expanding universe)
5.4.3 n = –1=3 (dust–dominated cosmos)
5.4.4 General n and cosmic fluid
5.4.5 Deep water approximation
5.4.6 Beach profiles are roulettes
5.5 Maximum or minimum energy dissipation?
6. Cosmology on a glacier
6.1 Introduction
6.2 Growth of grains in glacier firn
6.3 The universe in a glacial valley
6.3.1 Mechanical analogy for glacial valleys
6.3.1.1 Parameter range λ > 1
6.3.1.2 Parameter λ = 1
6.3.1.3 Parameter range 0 < λ < 1
6.3.1.4 Negative parameter λ
6.3.1.5 Analogue of the unconstrained variational problem
6.3.2 Cosmological analogy
6.3.3 Maximum or minimum friction?
6.4 The universe in a glacier slope
6.4.1 Lagrangian formulation of the Vialov equation
6.4.2 Point–particle analogy
6.4.3 Cosmological analogy
6.4.4 Nye profile as a generating function
7. Earthquakes and their cosmic cousins
7.1 Omori’s law
7.2 Mechanical analogy and Lagrangian formulation
7.3 Analogy with the Big Rip
7.4 Generalizations
8. Miscellaneous cosmic analogies
8.1 Introduction
8.2 Charge buildup in a thunderstorm cloud
8.3 de Sitter space in a meniscus
8.4 Other fluids ruled by surface tension
8.4.1 Dynamical bubbles
8.4.2 Capillary flow
8.5 Analogies with negative density fluids
8.5.1 Plasma dispersion relation
8.5.2 Fronts of lava flows
8.5.3 River floods
8.6 Conclusions
Appendix A
A.1 Einstein equations for a spatially flat FLRW universe
Appendix B
B.1 Analogue of the terrestrial brachistochrone
Appendix C
C.1 Radial geodesics of static spherical geometries
C.1.1 Reissner–Nordström black hole
C.1.2 (Anti–)de Sitter universe
C.1.3 Schwarzschild–de Sitter/Kottler spacetime
C.1.4 Kiselev black hole
C.1.5 Barriola–Vilenkin global monopole
C.1.6 Bardeen regular black hole
C.1.7 Morris–Thorne wormhole
C.1.8 Wyman’s “other” solution
Bibliography
Index
