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Construction of a complex null tetrad ：ウィキペディア英語版

Construction of a complex null tetrad
Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad

\

, where

\

is a pair of ''real'' null vectors and

\

is a pair of ''complex'' null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature

(-,+,+,+):

l_a l^a=n_a n^a=m_a m^a=\bar_a \bar^a=0\,;

l_a m^a=l_a \bar^a=n_a m^a=n_a \bar^a=0\,;

l_a n^a=l^a n_a=-1\,,\;\; m_a \bar^a=m^a \bar_a=1\,;

g_=-l_a  n_b - n_a  l_b +m_a  \bar_b +\bar_a  m_b\,,  \;\; g^=-l^a  n^b - n^a  l^b +m^a  \bar^b +\bar^a  m^b\,.

Only after the tetrad

\

gets constructed can one move forward to compute the directional derivatives, spin coefficients, commutators, Weyl-NP scalars

\Psi_i

, Ricci-NP scalars

\Phi_

and Maxwell-NP scalars

\phi_i

and other quantities in NP formalism. There are three most commonly used methods to construct a complex null tetrad:
# All four tetrad vectors are nonholonomic combinations of orthonormal holonomic tetrads;〔David McMahon. ''Relativity Demystified - A Self-Teaching Guide''. Chapter 9: ''Null Tetrads and the Petrov Classification''. New York: McGraw-Hill, 2006.〕
#

l^a

(or

n^a

) are aligned with the outgoing (or ingoing) tangent vector field of null radial geodesics, while

m^a

and

\bar^a

are constructed via the nonholonomic method;〔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.〕
# 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.
In the context below, it will be shown how these three methods work.
Note: In addition to the convention

\

employed in this article, the other one in use is

\

.
==Nonholonomic tetrad==

The primary method to construct a complex null tetrad is via combinations of orthonormal bases.〔 For a spacetime

g_

with an orthonormal tetrad

\

g_=-\omega_0\omega_0+\omega_1\omega_1+\omega_2\omega_2+\omega_3\omega_3\,,

the covectors

\

of the ''nonholonomic'' complex null tetrad can be constructed by

l_adx^a=\frac_adx^a=\frac^a\}

can be obtained by raising the indices of

\

via the inverse metric

g^

.
Remark: The nonholonomic construction is actually in accordance with the local light cone structure.〔

Example: A nonholonomic tetrad

Given a spacetime metric of the form (in signature(-,+,+,+))
:

g_=-g_dt^2+g_dr^2+g_d\theta^2+g_d\phi^2\,,

the nonholonomic orthonormal covectors are therefore
:

\omega_t=\sqrt}dr\,,\;\;\omega_\theta=\sqrt}d\phi\,,

and the nonholonomic null covectors are therefore
:

l_adx^a=\frac}dt+\sqrt}dt-\sqrt}d\theta+i\sqrt_adx^a=\frac}d\theta-i\sqrt{g_{\phi\phi}}d\phi)\,.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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