PART I THE PHYSICS OF RELATIVISTIC HYDRODYNAMICS
1 A Brief Review of General Relativity ....................... 3
1.1 Why this chapter? .......................................... 3
1.2 The concept of spacetime ................................... 4
1.3 Spacetime as a manifold .................................... 5
1.3.1 Coordinates ......................................... 6
1.3.2 Curves and paths .................................... 9
1.3.3 Tangent vectors ..................................... 9
1.3.4 Gradients of a function ............................ 12
1.3.5 A geometrical view of vectors and covectors ........ 13
1.3.6 Tensors ............................................ 15
1.3.7 Tensor algebra ..................................... 17
1.3.8 The most important tensor: the metric .............. 20
1.3.9 Splitting a tensor through a vector ................ 23
1.4 Rat spacetime: special relativity ......................... 24
1.5 Curved spacetimes: general relativity ..................... 29
1.5.1 Lie derivative ..................................... 31
1.5.2 Covariant derivative and Christoffel symbols ....... 34
1.5.3 Symmetries and Killing vector fields ............... 37
1.5.4 Geodesic equation .................................. 38
1.5.5 The Riemann tensor ................................. 41
1.6 Einstein equations ........................................ 46
1.7 Spacetimes of astrophysical relevance ..................... 48
1.7.1 Non-rotating black holes: the Schwarzschild
solution ........................................... 49
1.7.2 Rotating black holes: the Kerr solution ............ 54
1.7.3 The Friedmann-Robertson-Walker metric .............. 59
1.8 Gravitational radiation ................................... 61
1.9 Further reading ........................................... 66
1.10 Problems .................................................. 67
2 A Kinetic-Theory Description of Fluids .................... 68
2.1 On the fluid approximation ................................ 68
2.2 Newtonian kinetic theory .................................. 70
2.2.1 The Boltzmann equation ............................. 70
2.2.2 The H-theorem ...................................... 75
2.2.3 The moment equations ............................... 78
2.2.4 The Maxwell-Boltzmann equilibrium distribution ..... 81
2.2.5 The zero-order approximation: perfect fluids ....... 84
2.2.6 The first-order approximation: non-perfect fluids .. 87
2.3 Relativistic kinetic theory ............................... 89
2.3.1 The relativistic Boltzmann equation ................ 90
2.3.2 Relativistic transport fluxes ...................... 91
2.3.3 The relativistic H-theorem ......................... 93
2.3.4 The relativistic moment equations .................. 95
2.3.5 The general-relativistic hydrodynamic equations .... 96
2.3.6 Relativistic equilibrium distributions ............. 97
2.3.7 The laws of thermodynamics ........................ 101
2.4 Equations of state ....................................... 103
2.4.1 Degenerate relativistic fluid ..................... 109
2.4.2 Non-degenerate relativistic fluid ................. 110
2.4.3 Non-degenerate non-relativistic fluid ............. 111
2.4.4 Ultrarelativistic fluid ........................... 112
2.4.5 Degenerate Fermi fluid ............................ 114
2.4.6 Ideal fluid ....................................... 115
2.4.7 Polytropic fluid .................................. 118
2.4.8 Radiation fluid ................................... 123
2.4.9 Dark-energy fluid ................................. 126
2.4.10 Newtonian and relativistic barotropic fluids ...... 128
2.5 Further reading .......................................... 131
2.6 Problems ................................................. 132
3 Relativistic Perfect Fluids .............................. 133
3.1 Kinematic properties of fluids ........................... 133
3.1.1 Kinematic shear, expansion and vorticity .......... 133
3.1.2 Evolution laws of the kinematic quantities ........ 137
3.2 Mass current and energy-momentum of perfect fluids ....... 138
3.3 Hydrodynamics equations of perfect fluids ................ 143
3.4 Perfect fluids and symmetries ............................ 145
3.5 The Newtonian limit of the hydrodynamic equations ........ 147
3.6 Stationary flows ......................................... 150
3.6.1 Bernoulli's theorem ............................... 150
3.6.2 Relativistic Bernoulli theorem .................... 152
3.7 Irrotational flows ....................................... 152
3.7.1 Newtonian irrotational flows ...................... 152
3.7.2 Kelvin-Helmholtz theorem .......................... 154
3.7.3 Relativistic vorticity ............................ 155
3.7.4 Relativistic irrotational flows ................... 157
3.7.5 Relativistic Kelvin-Helmholtz theorem ............. 159
3.8 Isentropic flows ......................................... 162
3.9 A velocity-potential approach to relativistic
hydrodynamics ............................................ 164
3.10 A variational principle for relativistic hydrodynamics ... 168
3.11 Perfect multifluids ...................................... 175
3.11.1 Coupled multifluids ............................... 175
3.11.2 Interacting multifluids ........................... 179
3.12 Further reading .......................................... 187
3.13 Problems ................................................. 188
4 Linear and Nonlinear Hydrodynamic Waves .................. 190
4.1 Hyperbolic systems of partial differential equations ..... 190
4.1.1 Quasi-linear formulation .......................... 190
4.1.2 Conservative formulation .......................... 195
4.2 Linear and nonlinear behaviour ........................... 198
4.2.1 Characteristic equations for linear systems ....... 198
4.2.2 Riemann invariants ................................ 200
4.2.3 Characteristic curves and caustics ................ 203
4.2.4 Domain of determinacy and region of influence ..... 206
4.3 Linear hydrodynamic waves ................................ 208
4.3.1 Sound waves ....................................... 208
4.4 Nonlinear hydrodynamic waves ............................. 209
4.4.1 Simple waves and discontinuous waves .............. 209
4.4.2 Rarefaction waves ................................. 211
4.4.3 Shockwaves ........................................ 214
4.4.4 Contact discontinuities ........................... 222
4.5 The Riemann problem ...................................... 223
4.6 Solution of the one-dimensional Riemann problem .......... 227
4.6.1 Limiting relative velocities ...................... 228
4.7 Solution of the multidimensional Riemann problem ......... 233
4.7.1 Jumps across a shock wave ......................... 235
4.7.2 Jumps across a rarefaction wave ................... 236
4.7.3 Limiting relative velocities ...................... 237
4.7.4 Relativistic effects in multidimensional Riemann
problems .......................................... 239
4.7.5 Shock-detection techniques ........................ 243
4.8 Stability of shock waves ................................. 245
4.9 General-relativistic discontinuities ..................... 249
4.10 Further reading .......................................... 255
4.11 Problems ................................................. 256
5 Reaction Fronts: Detonations and Deflagrations ........... 258
5.1 Basic properties of reaction fronts ...................... 258
5.2 Reaction adiabat ......................................... 259
5.3 Relativistic detonations ................................. 262
5.4 Relativistic deflagrations ............................... 266
5.5 Stability of reaction fronts ............................. 268
5.5.1 Stability of detonations .......................... 271
5.5.2 Stability of deflagrations ........................ 281
5.6 Further reading .......................................... 283
5.7 Problems ................................................. 284
6 Relativistic Non-Perfect Fluids .......................... 285
6.1 On the four-velocity of a non-perfect fluid .............. 285
6.2 The energy-momentum tensor of non-perfect fluids ......... 287
6.3 Hydrodynamic equations of non-perfect fluids ............. 290
6.3.1 The general form of the momentum and energy
equations ......................................... 290
6.3.2 The equilibrium state ............................. 291
6.4 Classical Irreversible Thermodynamics (first-order
theories) ................................................ 291
6.4.1 The constitutive equations ........................ 292
6.4.2 The Newtonian limit: Navier-Stokes and heat
conduction ........................................ 294
6.5 The importance of a causal theory ........................ 296
6.5.1 Parabolic versus hyperbolic ....................... 296
6.5.2 Non-causality of Classical Irreversible
Thermodynamics .................................... 298
6.6 Extended Irreversible Thermodynamics (second-order
theories) ................................................ 299
6.6.1 The Israel-Stewart formulation .................... 300
6.6.2 Characteristic speeds of the Israel-Stewart
formulation ....................................... 303
6.6.3 Divergence-type theories .......................... 306
6.7 Concluding remarks ....................................... 312
6.8 Further reading .......................................... 314
6.9 Problems ................................................. 315
PART II NUMERICAL RELATIVISTIC HYDRODYNAMICS
7 Formulations of the Einstein-Euler Equations ............. 319
7.1 The 3 + 1 decomposition of spacetime ..................... 319
7.2 Formulations of the Einstein equations ................... 324
7.2.1 Spherically symmetric Lagrangian formulations ..... 325
7.2.2 The ADM formulation ............................... 329
7.2.3 Conformal traceless formulations .................. 335
7.2.4 Gauge conditions in 3+1 formulations .............. 343
7.2.5 The generalised harmonic formulation .............. 346
7.2.6 Constraint equations, initial data and
constrained evolution ............................. 350
7.3 Formulations of the hydrodynamic equations ............... 360
7.3.1 The Wilson formulation ............................ 360
7.3.2 The importance of conservative formulations ....... 362
7.3.3 The 3 + 1 "Valencia" formulation .................. 364
7.3.4 The covariant formulation ......................... 374
7.3.5 The light-cone formulation ........................ 375
7.3.6 The discontinuous Galerkin formulation ............ 377
7.4 Further reading .......................................... 383
7.5 Problems ................................................. 384
8 Numerical Relativistic Hydrodynamics: Finite-Difference
Methods .................................................. 386
8.1 The discretisation process ............................... 387
8.2 Numerical errors ......................................... 390
8.2.1 Consistency, convergence and stability ............ 394
8.3 Finite-difference methods ................................ 396
8.3.1 Analysis of the numerical stability ............... 396
8.3.2 The upwind scheme ................................. 399
8.3.3 The FTCS scheme ................................... 401
8.3.4 The Lax-Friedrichs scheme ......................... 402
8.3.5 The leapfrog scheme ............................... 404
8.3.6 The Lax-Wendroff scheme ........................... 405
8.3.7 Kreiss-Oliger dissipation ......................... 407
8.4 Artificial-viscosity approaches .......................... 409
8.5 Further reading .......................................... 412
8.6 Problems ................................................. 413
9 Numerical Relativistic Hydrodynamics: HRSC Methods ....... 414
9.1 Conservative schemes ..................................... 414
9.1.1 Rankine-Hugoniot conditions ....................... 414
9.1.2 Finite-volume conservative numerical schemes ...... 416
9.1.3 Finite-difference conservative numerical schemes .. 418
9.2 Upwind methods ........................................... 420
9.2.1 Monotone methods .................................. 420
9.2.2 Total variation diminishing methods ............... 421
9.2.3 Godunov methods ................................... 423
9.3 Reconstruction techniques ................................ 427
9.3.1 Slope-limiter methods ............................. 428
9.3.2 The piecewise-parabolic method .................... 430
9.3.3 Reconstruction in characteristic variables ........ 434
9.4 Approximate Riemann solvers .............................. 436
9.4.1 Incomplete Riemann solvers ........................ 436
9.4.2 Complete Riemann solvers .......................... 439
9.5 The method of lines ...................................... 447
9.5.1 Explicit Runge-Kutta methods ...................... 448
9.5.2 Implicit-explicit Runge-Kutta methods ............. 449
9.6 Central numerical schemes ................................ 452
9.6.1 Staggered central schemes ......................... 453
9.6.2 Non-staggered central schemes ..................... 455
9.7 Further reading .......................................... 457
9.8 Problems ................................................. 458
10 Numerical Relativistic Hydrodynamics: High-Order
Methods .................................................. 459
10.1 Why high-order numerical methods? ........................ 459
10.2 ENO and WENO methods for hyperbolic conservation laws .... 460
10.2.1 Finite-volume ENO schemes ......................... 461
10.2.2 Finite-volume WENO schemes ........................ 466
10.2.3 Finite-difference ENO schemes ..................... 468
10.2.4 Finite-difference WENO schemes .................... 472
10.3 Discontinuous Galerkin methods ........................... 472
10.3.1 The essence of DG methods ......................... 472
10.3.2 An example: a RKDG scheme in spherical symmetry ... 475
10.4 The ADER approach ........................................ 479
10.4.1 The original formulation .......................... 479
10.4.2 The local spacetime DG scheme ..................... 482
10.5 Extension to multidimensional problems ................... 485
10.5.1 Finite-difference multidimensional schemes ........ 486
10.5.2 Finite-volume multidimensional schemes ............ 487
10.6 Further reading .......................................... 489
10.7 Problems ................................................. 490
PART III APPLICATIONS OF RELATIVISTIC HYDRODYNAMICS
11 Relativistic Hydrodynamics of Non-Selfgravitating
Fluids ................................................... 493
11.1 Similar and self-similar flows ........................... 494
11.1.1 One-dimensional self-similar flows ................ 495
11.1.2 Self-similar hydrodynamics of a bubble ............ 501
11.1.3 Self-similar hydrodynamics of a drop .............. 504
11.2 Relativistic blast waves ................................. 507
11.3 Spherical flows onto and out of a compact object ......... 513
11.4 Spherical accretion onto a black hole .................... 516
11.5 Non-spherical accretion onto a moving black hole ......... 524
11.5.1 Accreting potential flows ......................... 525
11.5.2 Bondi-Hoyle-Lyttleton flows ....................... 529
11.6 Fluids in circular motion around a black hole ............ 537
11.6.1 von Zeipel cylinders .............................. 537
11.7 Geometrically thick tori ................................. 541
11.7.1 The "runaway" instability ......................... 548
11.7.2 On the sound speed in polytropic tori ............. 549
11.7.3 Thick tori in Schwarzschild-de Sitter spacetimes .. 551
11.8 Relativistic accreting discs ............................. 553
11.8.1 Rest-mass conservation ............................ 558
11.8.2 Radial momentum conservation ...................... 559
11.8.3 Angular momentum conservation ..................... 560
11.8.4 Hydrostatic vertical equilibrium .................. 562
11.8.5 Energy conservation ............................... 563
11.9 Relativistic jets ........................................ 565
11.9.1 Apparently superluminal jets ...................... 566
11.9.2 Hydrodynamic acceleration mechanisms .............. 569
11.9.3 Numerical modelling of relativistic jets .......... 577
11.10 Relativistic heavy-ion collisions ....................... 580
11.10.1 Basic concepts ................................... 580
11.10.2 One-dimensional Bjorken flow ..................... 584
11.10.3 Cylindrically symmetric flows .................... 585
11.11 Further reading ......................................... 589
11.12 Problems ................................................ 590
12 Relativistic Hydrodynamics of Selfgravitating Fluids ..... 593
12.1 Spherical stars .......................................... 593
12.2 Gravastars ............................................... 599
12.3 Rotating stars ........................................... 604
12.3.1 Uniformly rotating stars ...........................604
12.3.2 Differentially rotating stars ..................... 608
12.4 Collapse of a compact star to a black hole ............... 612
12.4.1 Dust collapse: the Oppenheimer-Snyder solution .... 612
12.4.2 Fluid collapse .................................... 617
12.5 Dynamics of binary neutron stars ......................... 623
12.5.1 Broadbrush picture ................................ 623
12.5.2 Dynamics of equal-mass binaries ................... 627
12.5.3 Dynamics of unequal-mass binaries ................. 637
12.6 Dynamics of black-hole-neutron-star binaries ............. 647
12.6.1 Broadbrush picture ................................ 648
12.7 Further reading .......................................... 657
12.8 Problems ................................................. 658
Appendix A Geometrised System of Units ....................... 659
Appendix В Notable Thermodynamic Expressions ................. 661
B.l Thermodynamic quantities and potentials .................. 661
B.2 Maxwell relations ........................................ 663
Appendix С Notable Tensors ................................... 665
C.1 Relativistic expressions ................................. 665
C.2 Newtonian expressions .................................... 667
Appendix D Common Practices in Numerical Relativistic
Hydrodynamics ................................................. 668
D.1 Conversion from conserved to primitive variables ......... 668
D.l.l Analytic equations of state ........................ 668
D.1.2 Tabulated equations of state ....................... 671
D.2 Treatment of atmospheres ................................. 673
D.3 Guaranteeing the positivity of pressure .................. 674
D.4 Domain excision .......................................... 675
Appendix E Numerical Building Blocks ......................... 678
E.l TVD slope limiters ....................................... 678
E.2 Basic Riemann solvers .................................... 679
E.3 Reference one-dimensional pseudo-code .................... 680
References ............................................... 682
Index ......................................................... 721
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