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
![{\displaystyle l_{a}l^{a}=n_{a}n^{a}=m_{a}m^{a}={\bar {m}}_{a}{\bar {m}}^{a}=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d75c6aaccf0b3d78bc2b1188cedd16c13e2664c)
![{\displaystyle l_{a}m^{a}=l_{a}{\bar {m}}^{a}=n_{a}m^{a}=n_{a}{\bar {m}}^{a}=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb411132d94c19dd18b55c279e9feda517c7eda)
![{\displaystyle l_{a}n^{a}=l^{a}n_{a}=-1\,,\;\;m_{a}{\bar {m}}^{a}=m^{a}{\bar {m}}_{a}=1\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb5fb357f82354a2f96ba74250e1f1b9122d692)
![{\displaystyle 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}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f4d4dfdce6ebd6cc0a3ef09354b362dea8628d)
Only after the tetrad
gets constructed can one move forward to compute the
directional derivatives,
spin coefficients,
commutators,
Weyl-NP scalars
,
Ricci-NP scalars
and
Maxwell-NP scalars
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 tetrads;
[1]
(or
) are aligned with the outgoing (or ingoing) tangent vector field of
null radial
geodesics, while
and
are constructed via the nonholonomic method;
[2]
- 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.
[1] For a spacetime
with an orthonormal tetrad
,
the covectors
of the nonholonomic complex null tetrad can be constructed by
![{\displaystyle l_{a}dx^{a}={\frac {\omega _{0}+\omega _{1}}{\sqrt {2}}}\,,\quad n_{a}dx^{a}={\frac {\omega _{0}-\omega _{1}}{\sqrt {2}}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/008b4726344739782d6f1ade6f916c00d9046e02)
and the tetrad vectors
can be obtained by raising the indices of
via the inverse metric
.
Remark: The nonholonomic construction is actually in accordance with the local
light cone structure.
[1]
Example: A nonholonomic tetrad
Given a spacetime metric of the form (in signature(-,+,+,+))
![{\displaystyle g_{ab}=-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }d\phi ^{2}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc809208faecdca68d56725f465aac7ae3110210)
the nonholonomic orthonormal covectors are therefore
![{\displaystyle \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 \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d37656f240593d006c6d7962261b1b0f835bdcef)
and the nonholonomic null covectors are therefore
![{\displaystyle n_{a}dx^{a}={\frac {1}{\sqrt {2}}}({\sqrt {g_{tt}}}dt-{\sqrt {g_{rr}}}dr)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3564075d7af928288d83400dbde53e08899bbcec)
![{\displaystyle {\bar {m}}_{a}dx^{a}={\frac {1}{\sqrt {2}}}({\sqrt {g_{\theta \theta }}}d\theta -i{\sqrt {g_{\phi \phi }}}d\phi )\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1a4e15ce90ce1a294b44fbe7a7458f0073dccd)
la (na) aligned with null radial geodesics
In
Minkowski spacetime, the nonholonomically constructed null vectors
respectively match the outgoing and ingoing null radial rays. As an extension of this idea in generic curved spacetimes,
can still be aligned with the tangent vector field of null radial
congruence.
[2] However, this type of adaption only works for
,
or
coordinates where the radial behaviors can be well described, with
and
denote the outgoing (retarded) and ingoing (advanced) null coordinate, respectively.
Example: Null tetrad for Schwarzschild metric in Eddington-Finkelstein coordinates reads
so the Lagrangian for null radial
geodesics of the Schwarzschild spacetime is
which has an ingoing solution
and an outgoing solution
. Now, one can construct a complex null tetrad which is adapted to the ingoing null radial geodesics:
and the dual basis covectors are therefore
Here we utilized the cross-normalization condition
as well as the requirement that
should span the induced metric
for cross-sections of {v=constant, r=constant}, where
and
are not mutually orthogonal. Also, the remaining two tetrad (co)vectors are constructed nonholonomically. With the tetrad defined, one is now able to respectively find out the spin coefficients, Weyl-Np scalars and Ricci-NP scalars that
![{\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \gamma =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65846ceef3d53764c64ff0e462fe9e4d5ab078b)
Example: Null tetrad for extremal Reissner–Nordström metric in Eddington-Finkelstein coordinates reads
![{\displaystyle ds^{2}=-Gdv^{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}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56faf37a05ce7b8ac307bdfda01a20c29cbbee36)
so the Lagrangian is
![{\displaystyle 2L=-G{\dot {v}}^{2}+2{\dot {v}}{\dot {r}}+r^{2}({\dot {\theta }}^{2}+\sin ^{2}\!\theta \,{\dot {\phi }}^{2})\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf457eee9424da5c1a870c936621ccd59618803)
For null radial geodesics with
, there are two solutions
(ingoing) and
(outgoing),
and therefore the tetrad for an ingoing observer can be set up as
![{\displaystyle l^{a}\partial _{a}\,=\,{\Big (}1\,,{\frac {F}{2}}\,,0\,,0{\Big )}\,,\quad n^{a}\partial _{a}\,=\,{\Big (}0\,,-1\,,0\,,0{\Big )}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8812ad755a96b863aac9fb9f5e7016213fc3fcf2)
![{\displaystyle l_{a}dx^{a}\,=\,{\Big (}-{\frac {F}{2}}\,,1\,,0,0{\Big )}\,,\quad n_{a}dx^{a}\,=\,{\Big (}-1\,,0\,,0\,,0{\Big )}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94d2f2e4462faa18d631e944f25943e0c6574022)
![{\displaystyle 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 {r}{\sqrt {2}}}\,{\Big (}0\,,0\,,1\,,i\sin \theta {\Big )}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/614219d22568665e044b552439e17d9791964559)
With the tetrad defined, we are now able to work out the spin coefficients, Weyl-NP scalars and Ricci-NP scalars that
![{\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \gamma =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65846ceef3d53764c64ff0e462fe9e4d5ab078b)
Tetrads adapted to the spacetime structure
At some typical boundary regions such as
null infinity,
timelike infinity,
spacelike infinity,
black hole horizons and
cosmological horizons, null tetrads adapted to spacetime structures are usually employed to achieve the most succinct
Newman–Penrose descriptions.
Newman-Unti tetrad for null infinity
For null infinity, the classic Newman-Unti (NU) tetrad
[3]
[4]
[5] is employed to study
asymptotic behaviors at null infinity,
![{\displaystyle l^{a}\partial _{a}=\partial _{r}:=D\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d975f0643395a56911761f3dcc2df47d9ddb16e)
![{\displaystyle n^{a}\partial _{a}=\partial _{u}+U\partial _{r}+X\partial _{\varsigma }+{\bar {X}}\partial _{\bar {\varsigma }}:=\Delta \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51fe7f77131dd55b1f56a55850ac4731936b0efe)
![{\displaystyle m^{a}\partial _{a}=\omega \partial _{r}+\xi ^{3}\partial _{\varsigma }+\xi ^{4}\partial _{\bar {\varsigma }}:=\delta \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7b741e7bc679e1fbd9d117c4362c4c457f9b1e)
where
are tetrad functions to be solved. For the NU tetrad, the foliation leaves are parameterized by the outgoing (advanced) null coordinate
with
, and
is the normalized
affine coordinate along
; the ingoing null vector
acts as the null generator at null infinity with
. The coordinates
comprise two real affine coordinates
and two complex
stereographic coordinates
, where
are the usual spherical coordinates on the cross-section
(as shown in ref.,
[5] complex stereographic rather than real
isothermal coordinates are used just for the convenience of completely solving NP equations).
Also, for the NU tetrad, the basic gauge conditions are
Adapted tetrad for exteriors and near-horizon vicinity of isolated horizons
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 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.
[6]
[7]
[8]
[9]
[10]
[11] Choose the first real null covector
as the gradient of foliation leaves
![{\displaystyle n_{a}\,=-dv\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7b905b18398083f7347d3e3fe5b8a5128b0808)
where
is the ingoing (retarded)
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
, i.e.
![{\displaystyle Dv=1\,,\quad \Delta v=\delta v={\bar {\delta }}v=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d7881e4a0c542b677bf90ba31fa335f1accbf9)
Introduce the second coordinate
as an affine parameter along the ingoing null vector field
, which obeys the normalization
Now, the first real null tetrad vector
is fixed. To determine the remaining tetrad vectors
and their covectors, besides the basic cross-normalization conditions, it is also required that: (i) the outgoing null normal field
acts as the null generators; (ii) the null frame (covectors)
are parallelly propagated along
; (iii)
spans the {t=constant, r=constant} cross-sections which are labeled by real
isothermal coordinates
.
Tetrads satisfying the above restrictions can be expressed in the general form that
![{\displaystyle l^{a}\partial _{a}=\partial _{v}+U\partial _{r}+X^{3}\partial _{y}+X^{4}\partial _{z}\,:=\,D\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb03086f977dbbcb49f6cab6f4438617a5290f)
![{\displaystyle n^{a}\partial _{a}=-\partial _{r}\,:=\,\Delta \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f85eebdbec49529175fead0246d1e770cc7c5033)
![{\displaystyle m^{a}\partial _{a}=\Omega \partial _{r}+\xi ^{3}\partial _{y}+\xi ^{4}\partial _{z}\,:=\,\delta \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3cf7b1e31d2b201eee7cbf0514a7ca74ec8f1e4)
The gauge conditions in this tetrad are
Remark: Unlike
Schwarzschild-type coordinates, here r=0 represents the
horizon, while r>0 (r<0) corresponds to the exterior (interior) of an isolated horizon. People often
Taylor expand a scalar
function with respect to the horizon r=0,
where
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.
See also
References
- ^
a
b
c David McMahon. Relativity Demystified - A Self-Teaching Guide. Chapter 9: Null Tetrads and the Petrov Classification. New York: McGraw-Hill, 2006.
- ^
a
b 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.
-
^ Ezra T Newman, Theodore W J Unti. Behavior of asymptotically flat empty spaces. Journal of Mathematical Physics, 1962, 3(5): 891-901.
-
^ 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.
- ^
a
b 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.
-
^ Xiaoning Wu, Sijie Gao. Tunneling effect near weakly isolated horizon. Physical Review D, 2007, 75(4): 044027.
arXiv:gr-qc/0702033v1
-
^ 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.
arXiv:0801.1347v1(gr-qc)
-
^ Yu-Huei Wu, Chih-Hung Wang. Gravitational radiation of generic isolated horizons.
arXiv:0807.2649v1(gr-qc)
-
^ Xiao-Ning Wu, Yu Tian. Extremal isolated horizon/CFT correspondence. Physical Review D, 2009, 80(2): 024014.
arXiv: 0904.1554(hep-th)
-
^ 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.
arXiv:0906.1551v1(gr-qc)
-
^ Badri Krishnan. The spacetime in the neighborhood of a general isolated black hole.
arXiv:1204.4345v1 (gr-qc)