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 { l a , n a , m a , m ¯ a } {\displaystyle \{l^{a},n^{a},m^{a},{\bar {m}}^{a}\}} , where { l a , n a } {\displaystyle \{l^{a},n^{a}\}} is a pair of real null vectors and { m a , m ¯ a } {\displaystyle \{m^{a},{\bar {m}}^{a}\}} 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 = m ¯ a m ¯ a = 0 ; {\displaystyle l_{a}l^{a}=n_{a}n^{a}=m_{a}m^{a}={\bar {m}}_{a}{\bar {m}}^{a}=0\,;} l a m a = l a m ¯ a = n a m a = n a m ¯ a = 0 ; {\displaystyle l_{a}m^{a}=l_{a}{\bar {m}}^{a}=n_{a}m^{a}=n_{a}{\bar {m}}^{a}=0\,;} l a n a = l a n a = − 1 , m a m ¯ a = m a m ¯ a = 1 ; {\displaystyle l_{a}n^{a}=l^{a}n_{a}=-1\,,\;\;m_{a}{\bar {m}}^{a}=m^{a}{\bar {m}}_{a}=1\,;} g a b = − l a n b − n a l b + m a m ¯ b + m ¯ a m b , g a b = − l a n b − n a l b + m a m ¯ b + m ¯ a m b .

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