Construction of a complex null tetrad

The primary method to construct a complex null tetrad is via combinations of orthonormal bases.[1] For a spacetime ${\displaystyle g_{ab}}$ with an orthonormal tetrad ${\displaystyle \{\omega _{0}\,,\omega _{1}\,,\omega _{2}\,,\omega _{3}\}}$,

${\displaystyle g_{ab}=-\omega _{0}\omega _{0}+\omega _{1}\omega _{1}+\omega _{2}\omega _{2}+\omega _{3}\omega _{3}\,,}$

the covectors ${\displaystyle \{l_{a}\,,n_{a}\,,m_{a}\,,{\bar {m}}_{a}\}}$ of the nonholonomic complex null tetrad can be constructed by

${\displaystyle l_{a}dx^{a}={\frac {\omega _{0}+\omega _{1}}{\sqrt {2}}}\,,\quad n_{a}dx^{a}={\frac {\omega _{0}-\omega _{1}}{\sqrt {2}}}\,,}$
${\displaystyle m_{a}dx^{a}={\frac {\omega _{2}+i\omega _{3}}{\sqrt {2}}}\,,\quad {\bar {m}}_{a}dx^{a}={\frac {\omega _{2}-i\omega _{3}}{\sqrt {2}}}\,,}$

and the tetrad vectors ${\displaystyle \{l^{a}\,,n^{a}\,,m^{a}\,,{\bar {m}}^{a}\}}$ can be obtained by raising the indices of ${\displaystyle \{l_{a}\,,n_{a}\,,m_{a}\,,{\bar {m}}_{a}\}}$ via the inverse metric ${\displaystyle g^{ab}}$.

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(-,+,+,+))

${\displaystyle g_{ab}=-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }d\phi ^{2}\,,}$

the nonholonomic orthonormal covectors are therefore

${\displaystyle \omega _{t}={\sqrt {g_{tt}}}dt\,,\;\;\omega _{r}={\sqrt {g_{rr}}}dr\,,\;\;\omega _{\theta }={\sqrt {g_{\theta \theta }}}d\theta \,,\;\;\omega _{\phi }={\sqrt {g_{\phi \phi }}}d\phi \,,}$

and the nonholonomic null covectors are therefore

${\displaystyle l_{a}dx^{a}={\frac {1}{\sqrt {2}}}({\sqrt {g_{tt}}}dt+{\sqrt {g_{rr}}}dr)\,,}$ ${\displaystyle n_{a}dx^{a}={\frac {1}{\sqrt {2}}}({\sqrt {g_{tt}}}dt-{\sqrt {g_{rr}}}dr)\,,}$

${\displaystyle m_{a}dx^{a}={\frac {1}{\sqrt {2}}}({\sqrt {g_{\theta \theta }}}d\theta +i{\sqrt {g_{\phi \phi }}}d\phi )\,,}$ ${\displaystyle {\bar {m}}_{a}dx^{a}={\frac {1}{\sqrt {2}}}({\sqrt {g_{\theta \theta }}}d\theta -i{\sqrt {g_{\phi \phi }}}d\phi )\,.}$

In Minkowski spacetime, the nonholonomically constructed null vectors ${\displaystyle \{l^{a}\,,n^{a}\}}$ respectively match the outgoing and ingoing null radial rays. As an extension of this idea in generic curved spacetimes, ${\displaystyle \{l^{a}\,,n^{a}\}}$ can still be aligned with the tangent vector field of null radial congruence.[2] However, this type of adaption only works for ${\displaystyle \{t,r,\theta ,\phi \}}$, ${\displaystyle \{u,r,\theta ,\phi \}}$ or ${\displaystyle \{v,r,\theta ,\phi \}}$ coordinates where the radial behaviors can be well described, with ${\displaystyle u}$ and ${\displaystyle v}$ denote the outgoing (retarded) and ingoing (advanced) null coordinate, respectively.

Example: Null tetrad for Schwarzschild metric in Eddington-Finkelstein coordinates reads

${\displaystyle ds^{2}=-Fdv^{2}+2dvdr+r^{2}(d\theta ^{2}+\sin ^{2}\!\theta \,d\phi ^{2})\,,\;\;{\text{with }}F\,:=\,{\Big (}1-{\frac {2M}{r}}{\Big )}\,,}$

so the Lagrangian for null radial geodesics of the Schwarzschild spacetime is

${\displaystyle L=-F{\dot {v}}^{2}+2{\dot {v}}{\dot {r}}\,,}$

which has an ingoing solution ${\displaystyle {\dot {v}}=0}$ and an outgoing solution ${\displaystyle {\dot {r}}={\frac {F}{2}}{\dot {v}}}$. Now, one can construct a complex null tetrad which is adapted to the ingoing null radial geodesics:

${\displaystyle l^{a}=(1,{\frac {F}{2}},0,0)\,,\quad n^{a}=(0,-1,0,0)\,,\quad m^{a}={\frac {1}{{\sqrt {2}}\,r}}(0,0,1,i\,\csc \theta )\,,}$

and the dual basis covectors are therefore

${\displaystyle l_{a}=(-{\frac {F}{2}},1,0,0)\,,\quad n_{a}=(-1,0,0,0)\,,\quad m_{a}={\frac {r}{\sqrt {2}}}(0,0,1,i\sin \theta )\,.}$

Here we utilized the cross-normalization condition ${\displaystyle l^{a}n_{a}=n^{a}l_{a}=-1}$ as well as the requirement that ${\displaystyle g_{ab}+l_{a}n_{b}+n_{a}l_{b}}$ should span the induced metric ${\displaystyle h_{AB}}$ for cross-sections of {v=constant, r=constant}, where ${\displaystyle dv}$ and ${\displaystyle dr}$ 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

${\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \gamma =0}$
${\displaystyle \rho ={\frac {-r+2M}{2r^{2}}}\,,\quad \mu =-{\frac {1}{r}}\,,\quad \alpha =-\beta ={\frac {-{\sqrt {2}}\cot \theta }{4r}}\,,\quad \varepsilon ={\frac {M}{2r^{2}}}\,;}$

${\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0\,,\quad \Psi _{2}=-{\frac {M}{r^{3}}}\,,}$

${\displaystyle \Phi _{00}=\Phi _{10}=\Phi _{20}=\Phi _{11}=\Phi _{12}=\Phi _{22}=\Lambda =0\,.}$

Example: Null tetrad for extremal Reissner–Nordström metric in Eddington-Finkelstein coordinates reads

${\displaystyle ds^{2}=-Gdv^{2}+2dvdr+r^{2}d\theta ^{2}+r^{2}\sin ^{2}\!\theta \,d\phi ^{2}\,,\;\;{\text{with }}G\,:=\,{\Big (}1-{\frac {M}{r}}{\Big )}^{2}\,,}$

so the Lagrangian is

${\displaystyle 2L=-G{\dot {v}}^{2}+2{\dot {v}}{\dot {r}}+r^{2}({\dot {\theta }}^{2}+\sin ^{2}\!\theta \,{\dot {\phi }}^{2})\,.}$

For null radial geodesics with ${\displaystyle \{L=0\,,{\dot {\theta }}=0\,,{\dot {\phi }}=0\}}$, there are two solutions

${\displaystyle {\dot {v}}=0}$ (ingoing) and ${\displaystyle {\dot {r}}=2F{\dot {v}}}$ (outgoing),

and therefore the tetrad for an ingoing observer can be set up as

${\displaystyle l^{a}\partial _{a}\,=\,{\Big (}1\,,{\frac {F}{2}}\,,0\,,0{\Big )}\,,\quad n^{a}\partial _{a}\,=\,{\Big (}0\,,-1\,,0\,,0{\Big )}\,,}$

${\displaystyle l_{a}dx^{a}\,=\,{\Big (}-{\frac {F}{2}}\,,1\,,0,0{\Big )}\,,\quad n_{a}dx^{a}\,=\,{\Big (}-1\,,0\,,0\,,0{\Big )}\,,}$

${\displaystyle m^{a}\partial _{a}\,=\,{\frac {1}{\sqrt {2}}}\,{\Big (}0\,,0\,,{\frac {1}{r}}\,,{\frac {i}{r\sin \theta }}{\Big )}\,,\quad m_{a}dx^{a}\,=\,{\frac {r}{\sqrt {2}}}\,{\Big (}0\,,0\,,1\,,i\sin \theta {\Big )}\,.}$

With the tetrad defined, we are now able to work out the spin coefficients, Weyl-NP scalars and Ricci-NP scalars that

${\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \gamma =0}$
${\displaystyle \rho ={\frac {(r-M)^{2}}{2r^{3}}}\,,\quad \mu =-{\frac {1}{r}}\,,\quad \alpha =-\beta ={\frac {-{\sqrt {2}}\cot \theta }{4r}}\,,\quad \varepsilon ={\frac {M(r-M)}{2r^{3}}}\,;}$

${\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0\,,\quad \Psi _{2}=-{\frac {(Mr-M)}{r^{4}}}\,,}$

${\displaystyle \Phi _{00}=\Phi _{10}=\Phi _{20}=\Phi _{12}=\Phi _{22}=\Lambda =0\,,\quad \Phi _{11}=-{\frac {M^{2}}{2r^{4}}}\,.}$

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–Penrose formalism