Newman–Penrose formalism


Newman–Penrose formalism

The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose[1] for General Relativity. Their notation is an effort to treat General Relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR.

The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the space-time, leading to simplified expressions for physical observables.

In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime.

The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor. In particular, it can be shown that one of these scalars--Ψ4 in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.[2]

Contents

Notation

The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric. At each point, a tetrad (set of four vectors) is introduced. The first two vectors, nμ and lμ are just a pair of standard (real) null vectors such that nμlμ = 1. For example, we can think in terms of spherical coordinates, and take nμ to be the outgoing null vector, and lμ to be the ingoing null vector. A complex null vector is then constructed by combining a pair of real, orthogonal unit space-like vectors. In the case of spherical coordinates, the standard choice is

m^\mu = \frac{1}{\sqrt{2}}\left( \hat{\theta} + i \hat{\phi} \right)^\mu\ .

The complex conjugate of this vector then forms the fourth element of the tetrad. The orthogonality relations between these vectors are then:

l_\mu l^\mu = n_\mu n^\mu = m_\mu m^\mu = \bar{m}_\mu \bar{m}^\mu = 0\ ,
l_\mu n^\mu = -m_\mu \bar{m}^\mu = -1\ ,
l_\mu m^\mu = l_\mu \bar{m}^\mu = n_\mu m^\mu = n_\mu \bar{m}^\mu = 0\ ,

if we assume the usual − + + + sign convention for the metric.

Newman and Penrose then introduce some functions using this tetrad:

  • twelve complex spin coefficients which describe the change in the tetrad from point to point: \kappa, \rho, \sigma, \tau, \epsilon, \alpha, \beta, \gamma, \pi, \lambda, \mu, \nu.
  • five complex functions encoding various pieces of the Weyl tensor in the tetrad basis: \Psi_0, \ldots, \Psi_4.
  • ten complex functions encoding pieces of the Ricci tensor in the tetrad basis: Φ000102101112202122.

In many situations—especially algebraically special spacetimes or vacuum spacetimes—the Newman–Penrose formalism simplifies dramatically, as many of the functions go to zero. This simplification allows for various theorems to be proven more easily than using the standard form of Einstein's equations.

Radiation field

The Weyl scalar Ψ4 was defined by Newman & Penrose as

\Psi_4 = -C_{\alpha\beta\gamma\delta} n^\alpha \bar{m}^\beta n^\gamma \bar{m}^\delta\

(note, however, that the overall sign is arbitrary, and that Newman & Penrose worked with a "timelike" metric signature of ( + , − , − , − )). In empty space, the Einstein Field Equations reduce to Rαβ = 0. From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, Cαβγδ = Rαβγδ. We can make the standard choice for the tetrad at infinity:

\vec{l} = \frac{1}{\sqrt{2}} \left( \hat{t} + \hat{r} \right)\ ,
\vec{n} = \frac{1}{\sqrt{2}} \left( \hat{t} - \hat{r} \right)\ ,
\vec{m} = \frac{1}{\sqrt{2}} \left( \hat{\theta} + i\hat{\phi} \right)\ .

In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as

 \frac{1}{4}\left( \ddot{h}_{\hat{\theta}\hat{\theta}} - \ddot{h}_{\hat{\phi}\hat{\phi}} \right) = -R_{\hat{t}\hat{\theta}\hat{t}\hat{\theta}} = -R_{\hat{t}\hat{\phi}\hat{r}\hat{\phi}} = -R_{\hat{r}\hat{\theta}\hat{r}\hat{\theta}} = R_{\hat{t}\hat{\phi}\hat{t}\hat{\phi}} = R_{\hat{t}\hat{\theta}\hat{r}\hat{\theta}} = R_{\hat{r}\hat{\phi}\hat{r}\hat{\phi}}\ ,
 \frac{1}{2} \ddot{h}_{\hat{\theta}\hat{\phi}} = -R_{\hat{t}\hat{\theta}\hat{t}\hat{\phi}} = -R_{\hat{r}\hat{\theta}\hat{r}\hat{\phi}} = R_{\hat{t}\hat{\theta}\hat{r}\hat{\phi}} = R_{\hat{r}\hat{\theta}\hat{t}\hat{\phi}}\ ,

assuming propagation in the \hat{r} direction. Combining these, and using the definition of Ψ4 above, we can write

 \Psi_4 = \frac{1}{2}\left( \ddot{h}_{\hat{\theta} \hat{\theta}} - \ddot{h}_{\hat{\phi} \hat{\phi}} \right) + i \ddot{h}_{\hat{\theta}\hat{\phi}} = -\ddot{h}_+ + i \ddot{h}_\times\ .

Far from a source, in nearly flat space, the fields h + and h_\times encode everything about gravitational radiation propagating in a given direction. Thus, we see that Ψ4 encodes in a single complex field everything about (outgoing) gravitational waves.

Radiation from a finite source

Using the wave-generation formalism summarised by Thorne,[3] we can write the radiation field quite compactly in terms of the mass multipole, current multipole, and spin-weighted spherical harmonics:

\Psi_4(t,r,\theta,\phi) = - \frac{1}{r\sqrt{2}} \sum_{l=2}^{\infty} \sum_{m=-l}^l \left[ {}^{(l+2)}I^{lm}(t-r) -i\ {}^{(l+2)}S^{lm}(t-r) \right] {}_{-2}Y_{lm}(\theta,\phi)\ .

Here, prefixed superscripts indicate time derivatives. That is, we define

{}^{(l)}G(t) = \left( \frac{d}{dt} \right)^l G(t)\ .

The components Ilm and Slm are the mass and current multipoles, respectively. − 2Ylm is the spin-weight -2 spherical harmonic.

See also

References

  1. ^ Ezra T. Newman and Roger Penrose (1962). "An Approach to Gravitational Radiation by a Method of Spin Coefficients". Journal of Mathematical Physics 3 (3): 566–768. Bibcode 1962JMP.....3..566N. doi:10.1063/1.1724257.  The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.
  2. ^ Saul Teukolsky (1973). "Perturbations of a rotating black hole". Astrophysical Journal 185: 635–647. Bibcode 1973ApJ...185..635T. doi:10.1086/152444. 
  3. ^ Thorne, Kip S. (April 1980). "Multipole expansions of gravitational radiation". Rev. Mod. Phys. 52 (2): 299–339. Bibcode 1980RvMP...52..299T. doi:10.1103/RevModPhys.52.299.  A broad summary of the mathematical formalism used in the literature on gravitational radiation.
  • Wald, Robert (1984). General Relativity. University of Chicago Press. ISBN 0-226-87033-2.  Wald treats the more succinct version of the Newman–Penrose formalism in terms of more modern spinor notation.
  • S. W. Hawking and G. F. R. Ellis (1973). The large scale structure of space-time. Cambridge University Press. ISBN 0-226-87033-2.  Hawking and Ellis use the formalism in their discussion of the final state of a collapsing star.

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