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| Calculations in the [[Newman–Penrose formalism|Newman–Penrose (NP) formalism]] of [[general relativity]] normally begin with the '''construction of a complex null tetrad''' <math>\{l^a,n^a,m^a,\bar{m}^a\}</math>, where <math>\{l^a,n^a\}</math> is a pair of ''real'' null vectors and <math>\{m^a,\bar{m}^a\}</math> is a pair of ''complex'' null vectors. These tetrad [[Vector field|vectors]] respect the following normalization and metric conditions assuming the spacetime signature <math>(-,+,+,+):</math>
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| *<math>l_a l^a=n_a n^a=m_a m^a=\bar{m}_a \bar{m}^a=0\,;</math>
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| *<math>l_a m^a=l_a \bar{m}^a=n_a m^a=n_a \bar{m}^a=0\,;</math>
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| *<math>l_a n^a=l^a n_a=-1\,,\;\; m_a \bar{m}^a=m^a \bar{m}_a=1\,;</math>
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| *<math>g_{ab}=-l_a n_b - n_a l_b +m_a \bar{m}_b +\bar{m}_a m_b\,, \;\; g^{ab}=-l^a n^b - n^a l^b +m^a \bar{m}^b +\bar{m}^a m^b\,.</math>
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| Only after the tetrad <math>\{l^a,n^a,m^a,\bar{m}^a\}</math> gets constructed can one move forward to compute the [[Newman–Penrose_formalism#Four_directional_derivatives|directional derivatives]], [[Newman–Penrose_formalism#Twelve_spin_coefficients|spin coefficients]], [[Newman–Penrose_formalism#Commutators|commutators]], [[Weyl scalar|Weyl-NP scalars]] <math>\Psi_i</math>, [[Ricci scalars (Newman-Penrose formalism)|Ricci-NP scalars]] <math>\Phi_{ij}</math> and [[Newman–Penrose_formalism#Maxwell-NP_scalars.2C_Maxwell_equations_in_NP_formalism|Maxwell-NP scalars]] <math>\phi_i</math> and other quantities in NP formalism. There are three most commonly used methods to construct a complex null tetrad:
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| # All four tetrad vectors are [[Holonomic basis|nonholonomic]] combinations of [[Tetrad formalism|orthonormal holonomic tetrads]];<ref name=demystified>David McMahon. ''Relativity Demystified - A Self-Teaching Guide''. Chapter 9: ''Null Tetrads and the Petrov Classification''. New York: McGraw-Hill, 2006.</ref>
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| # <math>l^a</math> (or <math>n^a</math>) are aligned with the outgoing (or ingoing) tangent vector field of [[Null vector|null]] radial [[Geodesic (general relativity)|geodesics]], while <math>m^a</math> and <math>\bar{m}^a</math> are constructed via the nonholonomic method;<ref name=chandra>Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Section ξ20, Section ξ21, Section ξ41, Section ξ56, Section ξ63(b). Chicago: University of Chikago Press, 1983.</ref>
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| # A tetrad which is adapted to the spacetime structure from a 3+1 perspective, with its general form being assumed and tetrad functions therein to be solved.
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| In the context below, it will be shown how these three methods work.
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| Note: In addition to the convention <math>\{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}</math> employed in this article, the other one in use is <math>\{(+,-,-,-); l^a n_a=1\,,m^a \bar{m}_a=-1\}</math>.
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| ==Nonholonomic tetrad==
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| The primary method to construct a complex null tetrad is via combinations of orthonormal bases.<ref name="demystified" /> For a spacetime <math>g_{ab}</math> with an orthonormal tetrad <math>\{\omega_0\,,\omega_1\,,\omega_2\,,\omega_3 \}</math>,
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| <math>g_{ab}=-\omega_0\omega_0+\omega_1\omega_1+\omega_2\omega_2+\omega_3\omega_3\,,</math>
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| the covectors <math>\{l_a\,,n_a\,,m_a\,,\bar{m}_a\}</math> of the ''nonholonomic'' complex null tetrad can be constructed by
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| <math>l_adx^a=\frac{\omega_0+\omega_1}{\sqrt{2}}\,,\quad n_adx^a=\frac{\omega_0-\omega_1}{\sqrt{2}}\,,</math><br />
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| <math>m_adx^a=\frac{\omega_2+i\omega_3}{\sqrt{2}}\,,\quad \bar{m}_adx^a=\frac{\omega_2-i\omega_3}{\sqrt{2}}\,,</math>
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| and the tetrad vectors <math>\{l^a\,,n^a\,,m^a\,,\bar{m}^a\}</math> can be obtained by raising the indices of <math>\{l_a\,,n_a\,,m_a\,,\bar{m}_a\}</math> via the inverse metric <math>g^{ab}</math>.
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| Remark: The nonholonomic construction is actually in accordance with the local [[light cone]] structure.<ref name="demystified" />
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| <div style="clear:both;width:65%;" class="NavFrame collapsed">
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| <div class="NavHead" style="background-color:#FFFFFF; text-align:left; font-size:larger;">Example: A nonholonomic tetrad</div>
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| <div class="NavContent" style="text-align:left;">
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| Given a spacetime metric of the form (in signature(-,+,+,+))
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| :<math>g_{ab}=-g_{tt}dt^2+g_{rr}dr^2+g_{\theta\theta}d\theta^2+g_{\phi\phi}d\phi^2\,,</math>
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| the nonholonomic orthonormal covectors are therefore
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| :<math>\omega_t=\sqrt{g_{tt}}dt\,,\;\;\omega_r=\sqrt{g_{rr}}dr\,,\;\;\omega_\theta=\sqrt{g_{\theta\theta}}d\theta\,,\;\;\omega_\phi=\sqrt{g_{\phi\phi}}d\phi\,,</math>
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| and the nonholonomic null covectors are therefore
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| :<math>l_adx^a=\frac{1}{\sqrt{2}}(\sqrt{g_{tt}}dt+\sqrt{g_{rr}}dr)\,,</math> <math> n_adx^a=\frac{1}{\sqrt{2}}(\sqrt{g_{tt}}dt-\sqrt{g_{rr}}dr)\,,</math>
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| :<math>m_adx^a=\frac{1}{\sqrt{2}}(\sqrt{g_{\theta\theta}}d\theta+i\sqrt{g_{\phi\phi}}d\phi)\,,</math> <math> \bar{m}_adx^a=\frac{1}{\sqrt{2}}(\sqrt{g_{\theta\theta}}d\theta-i\sqrt{g_{\phi\phi}}d\phi)\,.</math>
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| </div>
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| </div>
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| ==l<sup>a</sup> (n<sup>a</sup>) aligned with null radial geodesics==
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| In [[Minkowski spacetime]], the nonholonomically constructed null vectors <math>\{l^a\,,n^a\}</math> respectively match the outgoing and ingoing ''null radial'' rays. As an extension of this idea in generic curved spacetimes, <math>\{l^a\,,n^a\}</math> can still be aligned with the tangent vector field of null radial [[Congruence (general relativity)|congruence]].<ref name="chandra" /> However, this types of adaption only work for <math>\{t,r,\theta,\phi\}</math>, <math>\{u,r,\theta,\phi\}</math> or <math>\{v,r,\theta,\phi\}</math> coordinates where the ''radial'' behaviors can be well described, with <math>u</math> and <math>v</math> denote the outgoing (retarded) and ingoing (advanced) null coordinate respectively.
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| <div style="clear:both;width:65%;" class="NavFrame collapsed">
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| <div class="NavHead" style="background-color:#FFFFFF; text-align:left; font-size:larger;">Example: Null tetrad for Schwarzschild metric in Eddington-Finkestein coordinates </div>
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| <div class="NavContent" style="text-align:left;">
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| The Schwarzschild metric in Eddington-Finkestein coordinates reads
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| <math>ds^2=-Fdv^2+2dvdr+r^2(d\theta^2+\sin^2\!\theta\,d\phi^2)\,,\;\;\text{with } F\,:=\,\Big(1-\frac{M}{r} \Big)^2\,,</math>
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| so the Lagrangian for null radial [[Geodesics in general relativity|geodesics]] of the Schwarzschild spacetime is
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| <math>L=-F\dot{v}^2+2\dot{v}\dot{r}\,,</math>
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| which has an ''ingoing'' solution <math>\dot{v}=0</math> and an outgoing solution <math>\dot{r}=\frac{F}{2}\dot{v}</math>. Now, we can construct a complex null tetrad which is adapted to the ingoing null radial geodesics:
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| <math>l^a=(1,\frac{F}{2},0,0)\,,\quad n^a=(0,-1,0,0)\,,\quad m^a=\frac{1}{\sqrt{2}\,r}(0,0,1,i\,\csc\theta)\,,</math>
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| and the dual basis covectors are therefore
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| <math>l_a=(-\frac{F}{2},1,0,0)\,,\quad n_a=(-1,0,0,0)\,,\quad m_a=\frac{r}{\sqrt{2}}(0,0,1,\sin\theta)\,.</math>
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| Here we utilized the cross-normalization condition <math>l^an_a=n^al_a=-1</math> as well as the requirement that <math>g_{ab}+l_an_b+n_al_b</math> should span the induced metric <math>h_{AB}</math> for cross-sections of {v=constant, r=constant}, where it is important to recall that <math>dv</math> and <math>dr</math> are not mutually orthogonal. Also, the remaining two tetrad (co)vectors is constructed nonholonomically. With the tetrad defined, we are now able to respectively find out the spin coefficients, Weyl-Np scalars and Ricci-NP scalars that
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| <math>\kappa=\sigma=\tau=0\,,\quad \nu=\lambda=\pi=0\,,\quad \gamma=0 </math><br />
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| <math>\rho=\frac{-r+2M}{2r^2}\,,\quad \mu=-\frac{1}{r}\,,\quad \alpha=-\beta=\frac{-\sqrt{2}\cot\theta}{4r}\,,\quad \varepsilon=\frac{M}{2r^2}\,;</math>
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| <math>\Psi_0=\Psi_1=\Psi_3=\Psi_4=0\,,\quad \Psi_2=-\frac{M}{r^3}\,,</math>
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| <math>\Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{11}=\Phi_{12}=\Phi_{22}=\Lambda=0 \,.</math>
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| </div>
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| </div>
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| <div style="clear:both;width:65%;" class="NavFrame collapsed">
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| <div class="NavHead" style="background-color:#FFFFFF; text-align:left; font-size:larger;">Example: Null tetrad for extremal Reissner-Nordström metric in Eddington-Finkestein coordinates</div>
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| <div class="NavContent" style="text-align:left;">
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| The Reissner-Nordström metric in ingoing Eddington-Finkestein coordinates reads
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| :<math>ds^2=- G dv^2+2dvdr+r^2 d\theta^2+r^2\sin^2\!\theta\,d\phi^2\,,\;\;\text{with } G\,:=\,\Big(1-\frac{M}{r} \Big)^2\,,</math>
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| so the Lagrangian is
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| :<math>2L=- G \dot v^2+2\dot v \dot r+r^2 ({\dot\theta}^2+r^2\sin^2\!\theta\,\dot\phi^2\,.</math>
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| For null radial geodesics with <math>\{L=0\,,\dot\theta=0\,,\dot\phi=0\}</math>, we have two solutions
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| :<math>\dot v=0</math> (ingoing) and <math>\dot r=2F\dot v</math> (outgoing),
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| and therefore the tetrad for an ingoing observer can be set up as
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| :<math>l^a\partial_a\,=\, \Big(1\,,\frac{F}{2}\,,0\,,0 \Big)\,,\quad n^a\partial_a\,=\,\Big(0\,,-1\,,0\,,0 \Big)\,, </math>
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| :<math>l_adx^a\,=\, \Big(-\frac{F}{2}\,,1\,,0,0 \Big)\,,\quad n_adx^a\,=\,\Big(-1\,,0\,,0\,,0 \Big)\,,</math>
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| :<math>m^a\partial_a\,=\,\frac{1}{\sqrt{2}}\, \Big(0\,,0\,,\frac{1}{r}\,,\frac{i}{r\sin\theta} \Big) \,,\quad m_a dx^a\,=\,\frac{1}{\sqrt{2}}\,\Big(0\,,0\,,r\,,i\sin\theta \Big)\,.</math>
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| With the tetrad defined, we are now able to work out the spin coefficients, Weyl-NP scalars and Ricci-NP scalars that
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| <math>\kappa=\sigma=\tau=0\,,\quad \nu=\lambda=\pi=0\,,\quad \gamma=0 </math><br />
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| <math>\rho=\frac{(r-M)^2}{2r^3}\,,\quad \mu=-\frac{1}{r}\,,\quad \alpha=-\beta=\frac{-\sqrt{2}\cot\theta}{4r}\,,\quad \varepsilon=\frac{M(r-M)}{2r^3}\,;</math>
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| <math>\Psi_0=\Psi_1=\Psi_3=\Psi_4=0\,,\quad \Psi_2=-\frac{(Mr-M)}{r^4}\,,</math>
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| <math>\Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{12}=\Phi_{22}=\Lambda=0 \,,\quad \Phi_{11}=-\frac{M^2}{2r^4} \,.</math>
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| </div>
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| </div>
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| ==Tetrads adapted to the spacetime structure==
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| At some typical boundary regions such as [[Null vector|null]] infinity, [[Timelike Infinity|timelike infinity]], [[Spacelike vector|spacelike]] infinity, [[black hole]] horizons and [[cosmological horizon]]s, null tetrads adapted to spacetime structures are usually employed to achieve the most succinct [[Newman-Penrose formalism|Newman-Penrose]] descriptions.
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| ===Newman-Unti tetrad for null infinity===
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| For null infinity, the classic Newman-Unti (NU) tetrad<ref>Ezra T Newman, Theodore W J Unti. ''Behavior of asymptotically flat empty spaces''. Journal of Mathematical Physics, 1962, '''3'''(5): 891-901.</ref><ref>Ezra T Newman, Roger Penrose. ''An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Section IV. Journal of Mathematical Physics, 1962, '''3'''(3): 566-768.</ref><ref name=AppendixB>E T Newman, K P Tod. ''Asymptotically Flat Spacetimes'', Appendix B. In A Held (Editor): ''General relativity and gravitation: one hundred years after the birth of Albert Einstein''. Vol(2), page 1-34. New York and London: Plenum Press, 1980.</ref> is employed to study [[asymptotic behavior]]s at ''null infinity'',
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| <math>l^a\partial_a=\partial_r:=D\,,</math><br />
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| <math>n^a\partial_a=\partial_u +U\partial_r +X\partial_\varsigma+\bar{X} \partial_{\bar \varsigma}:=\Delta\,,</math><br />
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| <math>m^a\partial_a=\omega\partial_r+\xi^3\partial_\varsigma +\xi^4\partial_{\bar \varsigma}:=\delta\,,</math><br />
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| <math>\bar{m}^a\partial_a=\bar{\omega}\partial_r+\bar{\xi}^3\partial_{\bar\varsigma} +\bar{\xi}^4\partial_{ \varsigma}:=\bar\delta\,,</math>
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| where <math>\{U, X, \omega, \xi^3, \xi^4\}</math> are tetrad functions to be solved. For the NU tetrad, the foliation leaves are parameterized by the ''outgoing'' (advanced) null coordinate <math>u</math> with <math>l_a=du</math>, and <math>r</math> is the normalized [[Affine parameter|affine]] coordinate along <math>l^a</math> <math>(Dr=l^a\partial_ar=1)</math>; the ingoing null vector <math>n^a</math> acts as the null generator at null infinity with <math>\Delta u=n^a\partial_a u=1</math>. The coordinates <math>\{u,r,\varsigma, \bar{\varsigma}\}</math> comprise two real affine coordinates <math>\{u,r\}</math> and two complex [[stereographic]] coordinates <math>\{\varsigma:= e^{i\phi}\cot\frac{\theta}{2}, \bar{\varsigma}=e^{-i\phi}\cot\frac{\theta}{2}\}</math>, where <math>\{\theta,\phi\}</math> are the usual spherical coordinates on the cross-section <math>\hat\Delta_u=S^2_u</math> (as shown in ref.,<ref name="AppendixB" /> ''complex stereographic'' rather than ''real [[Isothermal coordinates|isothermal]]'' coordinates are used just for the convenience of completely solving NP equations).
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| Also, for the NU tetrad, the basic gauge conditions are
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| <math>\kappa=\pi=\varepsilon=0\,,\quad \rho=\bar\rho\,,\quad \tau=\bar\alpha+\beta\,.</math>
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| === Adapted tetrad for exteriors and near-horizon vicinity of isolated horizons ===
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| For a more comprehensive view of black holes in quasilocal definitions, adapted tetrads which can be smoothly transited from the exterior to the [[Near-horizon metric|near-horizon vicinity]] and to the horizons are required. For example, for [[isolated horizons]] describing black holes in equilibrium with their exteriors, such a tetrad and the related coordinates can be constructed this way.<ref>Xiaoning Wu, Sijie Gao. ''Tunneling effect near weakly isolated horizon''. Physical Review D, 2007, '''75'''(4): 044027. [http://arxiv.org/abs/gr-qc/0702033 arXiv:gr-qc/0702033v1]</ref><ref>Xiaoning Wu, Chao-Guang Huang, Jia-Rui Sun. ''On gravitational anomaly and Hawking radiation near weakly isolated horizon''. Physical Review D, 2008, '''77'''(12): 124023. [http://arxiv.org/abs/0801.1347 arXiv:0801.1347v1(gr-qc)]</ref><ref>Yu-Huei Wu, Chih-Hung Wang. ''Gravitational radiation of generic isolated horizons''. [http://arxiv.org/abs/0807.2649 arXiv:0807.2649v1(gr-qc)]</ref><ref>Xiao-Ning Wu, Yu Tian. ''Extremal isolated horizon/CFT correspondence''. Physical Review D, 2009, '''80'''(2): 024014. [http://arxiv.org/abs/0904.1554v3 arXiv: 0904.1554(hep-th)]</ref><ref>Yu-Huei Wu, Chih-Hung Wang. ''Gravitational radiations of generic isolated horizons and non-rotating dynamical horizons from asymptotic expansions''. Physical Review D, 2009, '''80'''(6): 063002. [http://arxiv.org/abs/0906.1551 arXiv:0906.1551v1(gr-qc)]</ref><ref>Badri Krishnan. ''The spacetime in the neighborhood of a general isolated black hole''. [http://arxiv.org/abs/1204.4345 arXiv:1204.4345v1 (gr-qc)]</ref> Choose the first real null covector <math>n_a</math> as the gradient of foliation leaves
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| <math>
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| n_a\,=-dv \,,
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| </math><br />
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| where <math>v</math> is the ''ingoing'' (retarded) [[Eddington-Finkelstein coordinates|Eddington-Finkelstein-type]] null coordinate, which labels the foliation cross-sections and acts as an affine parameter with regard to the outgoing null vector field <math>l^a\partial_a</math>, i.e.
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| <math>
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| Dv=1 \,,\quad \Delta v=\delta v=\bar\delta v=0\,.
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| </math><br />
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| Introduce the second coordinate <math>r</math> as an affine parameter along the ingoing null vector field <math>n^a</math>, which obeys the normalization
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| <math>
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| n^a\partial_a r \,=\,-1 \; \Leftrightarrow\; n^a\partial_a \,=\, -\partial_r\,.
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| </math>
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| Now, the first real null tetrad vector <math>n^a</math> is fixed. To determine the remaining tetrad vectors <math>\{l^a,m^a,\bar m^a\}</math> and their covectors, besides the basic cross-normalization conditions, it is also required that: (i) the outgoing null normal field <math>l^a</math> acts as the null generators; (ii) the null frame (covectors) <math>\{l_a, n_a, m_a, \bar m_a\}</math> are parallelly propagated along <math>n^a\partial_a</math>; (iii) <math>\{m^a,\bar m^a\}</math> spans the {t=constant, r=constant} cross-sections which are labeled by ''real'' [[isothermal coordinates]] <math>\{y,z\}</math>.
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| Tetrads satisfying the above restrictions can be expressed in the general form that
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| <math>l^a\partial_a=\partial_v +U\partial_r +X^3\partial_y+X^4 \partial_{ z }\, := \,D \,,</math><br />
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| <math>n^a\partial_a=-\partial_r\, := \,\Delta \,,</math><br />
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| <math>m^a\partial_a=\Omega\partial_r+\xi^3\partial_y +\xi^4\partial_{ z } \, := \,\delta \,,</math><br />
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| <math>\bar{m}^a\partial_a=\bar{\Omega}\partial_r +\bar{\xi}^3\partial_{ y}+\bar{\xi}^4\partial_{ z } \, := \,\bar\delta \,.</math>
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| The gauge conditions in this tetrad are
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| <math>\nu=\tau=\gamma=0\,,\quad \mu=\bar\mu\,,\quad \pi=\alpha+\bar\beta\,,</math>
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| Remark: Unlike [[Schwarzschild coordinates|Schwarzschild-type coordinates]], here r=0 represents the [[Event horizon|horizon]], while r>0 (r<0) corresponds to the exterior (interior) of an isolated horizon. People often [[Taylor expansion|Taylor]] expand a scalar <math>Q</math> function with respect to the horizon r=0,
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| <math>
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| Q=\sum_{i=0} Q^{(i)}r^i=Q^{(0)}+Q^{(1)}r+\cdots +Q^{(n)}r^n+\ldots
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| </math>
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| where <math>Q^{(0)}</math> refers to its on-horizon value. The very coordinates used in the adapted tetrad above are actually the [[Gaussian null coordinates]] employed in studying near-horizon geometry and mechanics of black holes.
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| ==See also==
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| *[[Newman-Penrose formalism]]
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| ==References==
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| {{reflist}}
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| [[Category:General relativity]]
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| [[Category:Mathematical methods in general relativity]]
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