Newman–Penrose formalism

The Newman–Penrose (NP) formalism[1][2] is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). 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,[3] 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 spacetime, 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 Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.[4]

Newman and Penrose introduced the following functions as primary quantities using this tetrad:[1][2]

  • Twelve complex spin coefficients (in three groups) which describe the change in the tetrad from point to point: .
  • Five complex functions encoding Weyl tensors in the tetrad basis: .
  • Ten functions encoding Ricci tensors in the tetrad basis: (real); (complex).

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.

In this article, we will only employ the tensorial rather than spinorial version of NP formalism, because the former is easier to understand and more popular in relevant papers. One can refer to ref.[5] for a unified formulation of these two versions.

Null tetrad and sign conventionEdit

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,   and   are just a pair of standard (real) null vectors such that  . For example, we can think in terms of spherical coordinates, and take   to be the outgoing null vector, and   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

 

The complex conjugate of this vector then forms the fourth element of the tetrad.

Two sets of signature and normalization conventions are in use for NP formalism:   and  . The former is the original one that was adopted when NP formalism was developed[1][2] and has been widely used[6][7] in black-hole physics, gravitational waves and various other areas in general relativity. However, it is the latter convention that is usually employed in contemporary study of black holes from quasilocal perspectives[8] (such as isolated horizons[9] and dynamical horizons[10][11]). In this article, we will utilize   for a systematic review of the NP formalism (see also refs.[12][13][14]).

It's important to note that, when switching from   to  , definitions of the spin coefficients, Weyl-NP scalars   and Ricci-NP scalars   need to change their signs; this way, the Einstein-Maxwell equations can be left unchanged.

In NP formalism, the complex null tetrad contains two real null (co)vectors   and two complex null (co)vectors  . Being null (co)vectors, self-normalization of   naturally vanishes,


 ,

so the following two pairs of cross-normalization are adopted


 

while contractions between the two pairs are also vanishing,


 .

Here the indices can be raised and lowered by the global metric   which in turn can be obtained via


 

NP quantities and tetrad equationsEdit

Four covariant derivative operatorsEdit

In keeping with the formalism's practice of using distinct unindexed symbols for each component of an object, the covariant derivative operator   is expressed using four separate symbols ( ) which name a directional covariant derivative operator for each tetrad direction. Given a linear combination of tetrad vectors,  , the covariant derivative operator in the   direction is  .

The operators are defined as
 
 

which reduce to   when acting on scalar functions.

Twelve spin coefficientsEdit

In NP formalism, instead of using index notations as in orthogonal tetrads, each Ricci rotation coefficient   in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex spin coefficients (in three groups),


 
 

 
 

 
 
 
 

Spin coefficients are the primary quantities in NP formalism, with which all other NP quantities (as defined below) could be calculated indirectly using the NP field equations. Thus, NP formalism is sometimes referred to as spin-coefficient formalism as well.

Transportation equations: covariant derivatives of tetrad vectorsEdit

The sixteen directional covariant derivatives of tetrad vectors are sometimes called the transportation/propagation equations,[citation needed] perhaps because the derivatives are zero when the tetrad vector is parallel propagated or transported in the direction of the derivative operator.

These results in this exact notation are given by ODonnell:[5]:57–58(3.220)
 
 
 
 

 
 
 
 

 
 
 
 

 
 
 
 

Interpretation of   from   and  Edit

The two equations for the covariant derivative of a real null tetrad vector in its own direction indicate whether or not the vector is tangent to a geodesic and if so, whether the geodesic has an affine parameter.

A null tangent vector   is tangent to an affinely parameterized null geodesic if  , which is to say if the vector is unchanged by parallel propagation or transportation in its own direction.[15]:41(3.3.1)

  shows that   is tangent to a geodesic if and only if  , and is tangent to an affinely parameterized geodesic if in addition  . Similarly,   shows that   is geodesic if and only if  , and has affine parameterization when  .

(The complex null tetrad vectors   and   would have to be separated into the spacelike basis vectors   and   before asking if either or both of those are tangent to spacelike geodesics.)

CommutatorsEdit

The metric-compatibility or torsion-freeness of the covariant derivative is recast into the commutators of the directional derivatives,


 
 
 
 

which imply that


 
 
 
 

Note: (i) The above equations can be regarded either as implications of the commutators or combinations of the transportation equations; (ii) In these implied equations, the vectors   can be replaced by the covectors and the equations still hold.

Weyl–NP and Ricci–NP scalarsEdit

The 10 independent components of the Weyl tensor can be encoded into 5 complex Weyl-NP scalars,


         

The 10 independent components of the Ricci tensor are encoded into 4 real scalars  ,  ,  ,   and 3 complex scalars   (with their complex conjugates),


 

 
 
 

In these definitions,   could be replaced by its trace-free part  [13] or by the Einstein tensor   because of the normalization relations. Also,   is reduced to   for electrovacuum ( ).

Einstein–Maxwell–NP equationsEdit

NP field equationsEdit

In a complex null tetrad, Ricci identities give rise to the following NP field equations connecting spin coefficients, Weyl-NP and Ricci-NP scalars (recall that in an orthogonal tetrad, Ricci rotation coefficients would respect Cartan's first and second structure equations),[5][13]


These equations in various notations can be found in several texts.[3]:46–47(310(a)-(r))[13]:671–672(E.12) The notation in Frolov and Novikov [13] is identical and the typesetting matches pixel by pixel. (Springer seems to use a substantially similar LaTex package).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Also, the Weyl-NP scalars   and the Ricci-NP scalars   can be calculated indirectly from the above NP field equations after obtaining the spin coefficients rather than directly using their definitions.

Maxwell–NP scalars, Maxwell equations in NP formalismEdit

The six independent components of the Faraday-Maxwell 2-form (i.e. the electromagnetic field strength tensor)   can be encoded into three complex Maxwell-NP scalars[12]


 

and therefore the eight real Maxwell equations   and   (as  ) can be transformed into four complex equations,


 
 
 
 


with the Ricci-NP scalars   related to Maxwell scalars by[12]


 

It is worthwhile to point out that, the supplementary equation   is only valid for electromagnetic fields; for example, in the case of Yang-Mills fields there will be   where   are Yang-Mills-NP scalars.[16]

To sum up, the aforementioned transportation equations, NP field equations and Maxwell-NP equations together constitute the Einstein-Maxwell equations in Newman–Penrose formalism.

Applications of NP formalism to gravitational radiation fieldEdit

The Weyl scalar   was defined by Newman & Penrose as

 

(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  . From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor,  . We can make the standard choice for the tetrad at infinity:

 
 
 

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

 
 

assuming propagation in the   direction. Combining these, and using the definition of   above, we can write

 

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

Radiation from a finite sourceEdit

Using the wave-generation formalism summarised by Thorne,[17] we can write the radiation field quite compactly in terms of the mass multipole, current multipole, and spin-weighted spherical harmonics:

 

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

 

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

See alsoEdit

NotesEdit

ReferencesEdit

  1. ^ a b c 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. ^ a b c Ezra T Newman, Roger Penrose. Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients. Journal of Mathematical Physics, 1963, 4(7): 998.
  3. ^ a b Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Oxford Classics Series ed.). Oxford University Press. p. 40. ISBN 0-19850370-9. Retrieved 31 May 2019. The Newman–Penrose formalism is a tetrad formalism with a special choice of the basis vectors.
  4. ^ Saul Teukolsky (1973). "Perturbations of a rotating black hole". Astrophysical Journal. 185: 635–647. Bibcode:1973ApJ...185..635T. doi:10.1086/152444.
  5. ^ a b c Peter O'Donnell. Introduction to 2-Spinors in General Relativity. Singapore: World Scientific, 2003.
  6. ^ Subrahmanyan Chandrasekhar. The Mathematical Theory of Black Holes. Chicago: University of Chikago Press, 1983.
  7. ^ J B Griffiths. Colliding Plane Waves in General Relativity. Oxford: Oxford University Press, 1991.
  8. ^ Ivan Booth. Black hole boundaries. Canadian Journal of Physics, 2005, 83(11): 1073-1099. [arxiv.org/abs/gr-qc/0508107 arXiv:gr-qc/0508107v2]
  9. ^ Abhay Ashtekar, Christopher Beetle, Jerzy Lewandowski. Geometry of generic isolated horizons. Classical and Quantum Gravity, 2002, 19(6): 1195-1225. arXiv:gr-qc/0111067v2
  10. ^ Abhay Ashtekar, Badri Krishnan. Dynamical horizons: energy, angular momentum, fluxes and balance laws. Physical Review Letters, 2002, 89(26): 261101. [arxiv.org/abs/gr-qc/0207080 arXiv:gr-qc/0207080v3]
  11. ^ Abhay Ashtekar, Badri Krishnan. Dynamical horizons and their properties. Physical Review D, 2003, 68(10): 104030. [arxiv.org/abs/gr-qc/0308033 arXiv:gr-qc/0308033v4]
  12. ^ a b c Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Chapter 2.
  13. ^ a b c d e Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998. Appendix E.
  14. ^ Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. Isolated horizons: Hamiltonian evolution and the first law. Physical Review D, 2000, 62(10): 104025. Appendix B. gr-qc/0005083
  15. ^ Robert M. Wald (1984). General Relativity.
  16. ^ E T Newman, K P Tod. Asymptotically Flat Spacetimes, Appendix A.2. In A Held (Editor): General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein. Vol(2), page 27. New York and London: Plenum Press, 1980.
  17. ^ Thorne, Kip S. (April 1980). "Multipole expansions of gravitational radiation" (PDF). 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.

External linksEdit