*Miguel Alcubierre*

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Gravitational waves are one of the most important physical phenomena associated with the presence of strong and dynamic gravitational fields, and as such they are of great interest in numerical ...
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Gravitational waves are one of the most important physical phenomena associated with the presence of strong and dynamic gravitational fields, and as such they are of great interest in numerical relativity. There are two main approaches to the extraction of gravitational wave information from a numerical simulation. Traditional approach has been based on the theory of perturbations of a Schwarzschild spacetime developed originally by Regge and Wheeler, Zerilli, and a number of other authors, and later recast as a gauge invariant framework by Moncrief. In recent years, however, it has become increasingly common in numerical relativity to extract gravitational wave information in terms of the components of the Weyl curvature tensor with respect to a frame of null vectors, using what is known as the Newman-Penrose formalism. This chapter presents a brief introduction to both these approaches, and describes how to calculate the energy and momentum radiated by gravitational waves in each case.Less

Gravitational waves are one of the most important physical phenomena associated with the presence of strong and dynamic gravitational fields, and as such they are of great interest in numerical relativity. There are two main approaches to the extraction of gravitational wave information from a numerical simulation. Traditional approach has been based on the theory of perturbations of a Schwarzschild spacetime developed originally by Regge and Wheeler, Zerilli, and a number of other authors, and later recast as a gauge invariant framework by Moncrief. In recent years, however, it has become increasingly common in numerical relativity to extract gravitational wave information in terms of the components of the Weyl curvature tensor with respect to a frame of null vectors, using what is known as the Newman-Penrose formalism. This chapter presents a brief introduction to both these approaches, and describes how to calculate the energy and momentum radiated by gravitational waves in each case.

*Spyros Alexakis*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter fleshes out the strategy of iteratively decomposing any P(g) = unconverted formula 1 for which ∫P(g)dVsubscript g is a global conformal invariant. It makes precise the notions of better ...
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This chapter fleshes out the strategy of iteratively decomposing any P(g) = unconverted formula 1 for which ∫P(g)dVsubscript g is a global conformal invariant. It makes precise the notions of better and worse complete contractions in P(g) and then spells out (1.17), via Propositions 2.7, 2.8. In particular, using the well-known decomposition of the curvature tensor into its trace-free part (the Weyl tensor) and its trace part (the Schouten tensor), it reexpresses P(g) as a linear combination of complete contractions involving differentiated Weyl tensors and differentiated Schouten tensors, as in (2.47). The chapter also proves (1.17) when the worst terms involve at least one differentiated Schouten tensor.Less

This chapter fleshes out the strategy of iteratively decomposing any *P*(*g*) = unconverted formula 1 for which ∫*P*(*g*)*dV*subscript *g* is a global conformal invariant. It makes precise the notions of better and worse complete contractions in *P*(*g*) and then spells out (1.17), via Propositions 2.7, 2.8. In particular, using the well-known decomposition of the curvature tensor into its trace-free part (the Weyl tensor) and its trace part (the Schouten tensor), it reexpresses *P*(*g*) as a linear combination of complete contractions involving differentiated Weyl tensors and differentiated Schouten tensors, as in (2.47). The chapter also proves (1.17) when the worst terms involve at least one differentiated Schouten tensor.

*Spyros Alexakis*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter proves (1.17) when the worst terms in P(g) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need ...
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This chapter proves (1.17) when the worst terms in P(g) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need both a local conformal invariant W(g) and a divergence divᵢTⁱ(g) to prove (1.17). One obvious difficulty is how, upon inspection of P(g)subscript worst-piece, to separate the piece that must be cancelled out by a local conformal invariant from the piece that is cancelled out by a divergence. In a first step, we prove that we can first explicitly construct a local conformal invariant and a divergence and subtract them from P(g)subscript worst-piece, to be left with a new worst piece, which has some additional algebraic properties. In a second step, we show that this new worst piece can be cancelled out by subtracting a divergence.Less

This chapter proves (1.17) when the worst terms in *P*(*g*) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need both a local conformal invariant *W*(*g*) and a divergence divᵢ*T*ⁱ(*g*) to prove (1.17). One obvious difficulty is how, upon inspection of *P*(*g*)subscript worst-piece, to separate the piece that must be cancelled out by a local conformal invariant from the piece that is cancelled out by a divergence. In a first step, we prove that we can first explicitly construct a local conformal invariant and a divergence and subtract them from *P*(*g*)subscript worst-piece, to be left with a new worst piece, which has some additional algebraic properties. In a second step, we show that this new worst piece can be cancelled out by subtracting a divergence.