Null Tetrad

$$g_{ij}=\left[{\matrix{0 & 1 & 0 & 0\cr 1 & 0 & 0 & 0\cr 0 & 0 & 0 & -1\cr 0 & 0 & -1 & 0\cr}}\right].$$

It can be expressed as

$$g_{ab}=l_an_b+l_bn_a-m_a\bar m_b-m_b\bar m_a.$$

See also Tetrad

References

d'Inverno, R. Introducing Einstein's Relativity. Oxford, England: Oxford University Press, pp. 248-249, 1992.