Book contents
- Frontmatter
- Contents
- Preface
- Suggestions for using this book
- 1 General relativity preliminaries
- 2 The 3 + 1 decompostion of Einstein's equations
- 3 Constructing initial data
- 4 Choosing coordinates: the lapse and shift
- 5 Matter sources
- 6 Numerical methods
- 7 Locating black hole horizons
- 8 Spherically symmetric spacetimes
- 9 Gravitational waves
- 10 Collapse of collisionless clusters in axisymmetry
- 11 Recasting the evolution equations
- 12 Binary black hole initial data
- 13 Binary black hole evolution
- 14 Rotating stars
- 15 Binary neutron star initial data
- 16 Binary neutron star evolution
- 17 Binary black hole–neutron stars: initial data and evolution
- 18 Epilogue
- A Lie derivatives, Killing vectors, and tensor densities
- B Solving the vector Laplacian
- C The surface element on the apparent horizon
- D Scalar, vector and tensor spherical harmonics
- E Post-Newtonian results
- F Collisionless matter evolution in axisymmetry: basic equations
- G Rotating equilibria: gravitational field equations
- H Moving puncture representions of Schwarzschild: analytical results
- I Binary black hole puncture simulations as test problems
- References
- Index
1 - General relativity preliminaries
Published online by Cambridge University Press: 05 March 2013
- Frontmatter
- Contents
- Preface
- Suggestions for using this book
- 1 General relativity preliminaries
- 2 The 3 + 1 decompostion of Einstein's equations
- 3 Constructing initial data
- 4 Choosing coordinates: the lapse and shift
- 5 Matter sources
- 6 Numerical methods
- 7 Locating black hole horizons
- 8 Spherically symmetric spacetimes
- 9 Gravitational waves
- 10 Collapse of collisionless clusters in axisymmetry
- 11 Recasting the evolution equations
- 12 Binary black hole initial data
- 13 Binary black hole evolution
- 14 Rotating stars
- 15 Binary neutron star initial data
- 16 Binary neutron star evolution
- 17 Binary black hole–neutron stars: initial data and evolution
- 18 Epilogue
- A Lie derivatives, Killing vectors, and tensor densities
- B Solving the vector Laplacian
- C The surface element on the apparent horizon
- D Scalar, vector and tensor spherical harmonics
- E Post-Newtonian results
- F Collisionless matter evolution in axisymmetry: basic equations
- G Rotating equilibria: gravitational field equations
- H Moving puncture representions of Schwarzschild: analytical results
- I Binary black hole puncture simulations as test problems
- References
- Index
Summary
In this chapter we assemble some of the elements of Einstein's theory of general relativity that we will be working with in later chapters. We assume that the geometric objects and equations that we list, as well as their interpretation, are already very familiar to readers. The discussion below should serve simply as a checklist of a few of the basics that we need to pack with us before embarking on our voyage into numerical spacetime.
Throughout this book we adopt the (−+++) metric signature together with all the sign conventions of Misner et al. (1973). Following that book, but in this chapter only, we will display a tensor in spacetime by a symbol in boldface when emphasizing its coordinatefree character, or by its components when the tensor has been expanded in a particular set of basis tensors. However, unlike that book, we will use Latin indices a, b, … instead of Greek letters to denote the spacetime indices of the tensor components, with the values of the indices running from 0 to 3. This choice anticipates a switch we will make to abstract index notation in all subsequent chapters of this book. We will introduce this switch in Section 2.1. We adopt the usual Einstein convention of summing over repeated indices. Finally, here and throughout we will use geometrized units in which both the gravitational constant and the speed of light are assigned the values of one, G = c = 1.
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- Numerical RelativitySolving Einstein's Equations on the Computer, pp. 1 - 22Publisher: Cambridge University PressPrint publication year: 2010