NullTetrad and OrthonormalTetrad compute tetrads for the spacetime metric set. Recalling, given the metric of a local (tetrad) system of references, there are infinitely many tetrads (represented in Physics by the Tetrads:-e_ command) relating the components of a tensor in the global (spacetime) and local (tetrad) systems of references. These two commands NullTetrad and OrthonormalTetrad compute different tetrads, that may result more convenient in different contexts, for example when computing the Weyl and Ricci scalars or performing a Petrov classification.

When the tetrad you get from NullTetrad or OrthonormalTetrad is the one you prefer, say E, you can set it, to be used by the system as the current tetrad, by entering $\mathrm{Setup}\left(\mathrm{tetrad}=E\right)$. You can always see the components of the current tetrad by entering e_[].

When called with no arguments, both NullTetrad and OrthonormalTetrad will depart from the current tetrad and compute a corresponding - null or orthonormal - tetrad. The formula used depends on the signature. For the signatures (---+) and (+++-), given an orthonormal tetrad E, NullTetrad returns a null tetrad constructed as M . E, where

Both NullTetrad and OrthonormalTetrad can also be called with a first argument, T, a matrix representing a tetrad, in which case instead of departing from the current components of Tetrads:-e_, they apply the formulas just described to T.

Additionally, instead of constructing tetrads multiplying by M or its inverse, NullTetrad and OrthonormalTetrad can construct tetrads from scratch using two different methods: GramSchmidt or Eigenvectors - see the Examples section.

Independently of NullTetrad and OrthonormalTetrad, you can always set the tetrad to have any preferred matrix form, say E, by entering $\mathrm{Setup}\left(\mathrm{tetrad}=E\right)$, or transform a tetrad into another one using TransformTetrad. Of particular interest is the the ability of TransformTetrad to compute a tetrad in canonicalform, that is a form that results in a canonical form of the Weyl scalars.

IsTetrad is a complementary command: it returns true or false depending on whether the given matrix T is a tetrad, optionally indicating the type of tetrad.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)\;$$\mathrm{with}\left(\mathrm{Tetrads}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{tetrad}\mathrm{indices}$

$\mathrm{Defined\; as\; tetrad\; tensors}\left(\mathrm{see\; ?Physics,tetrads}\right),{𝔢}_{a,\mathrm{\mu }},{\mathrm{\eta }}_{a,b},{\mathrm{\gamma }}_{a,b,c},{\mathrm{\lambda }}_{a,b,c}$

$\mathrm{Defined\; as\; spacetime\; tensors\; representing\; the\; NP\; null\; vectors\; of\; the\; tetrad\; formalism}\left(\mathrm{see\; ?Physics,tetrads}\right),l_{\mathrm{\mu }},n_{\mathrm{\mu }},m_{\mathrm{\mu }},{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{IsTetrad}\,\mathrm{NullTetrad}\,\mathrm{OrthonormalTetrad}\,\mathrm{PetrovType}\,\mathrm{SegreType}\,\mathrm{TransformTetrad}\,\mathrm{WeylScalars}\,\mathrm{e\_}\,\mathrm{eta\_}\,\mathrm{gamma\_}\,\mathrm{l\_}\,\mathrm{lambda\_}\,\mathrm{m\_}\,\mathrm{mb\_}\,\mathrm{n\_}\right]$

$\mathrm{g\_}\left[\left[13\,7\,5\right]\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{The}\mathrm{metric\; in\; coordinates}\left[x\,y\,z\,t\right]$

$\mathrm{Parameters:}\left[\mathrm{\epsilon }\,A\left(t\right)\,B\left(t\right)\,\mathrm{A1}\right]$

$\mathrm{Comments:}\mathrm{\ⅇpsilon=1\; or\; \ⅇpsilon=-1}$

$\mathrm{Domain:}\left[0<t\right]$

$\mathrm{Assumptions:}\left[A\left(t\right)::\mathrm{real}\&comma;B\left(t\right)::\mathrm{real}\&comma;0<\mathrm{A1}\&comma;0<x\&comma;0<A\left(t\right)\&comma;0<B\left(t\right)\right]$

$\mathrm{Resetting\; the\; signature\; of\; spacetime\; from}\left(\mathrm{-\; -\; -\; +}\right)\mathrm{to}\left(\mathrm{+\; +\; +\; -}\right)\mathrm{in\; order\; to\; match\; the\; signature\; in\; the\; database\; of\; metrics}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{\epsilon }{A\left(t\right)}^2& 0& 0& 0\\ 0& {B\left(t\right)}^2{\&ExponentialE;}^{2\mathrm{A1}x}\left(2{\cosh \left(x\right)}^2-1\right)& -2{B\left(t\right)}^2{\&ExponentialE;}^{2\mathrm{A1}x}\cosh \left(x\right)\sinh \left(x\right)& 0\\ 0& -2{B\left(t\right)}^2{\&ExponentialE;}^{2\mathrm{A1}x}\cosh \left(x\right)\sinh \left(x\right)& {B\left(t\right)}^2{\&ExponentialE;}^{2\mathrm{A1}x}\left(2{\cosh \left(x\right)}^2-1\right)& 0\\ 0& 0& 0& -\mathrm{\epsilon }\end{array}\right]$

$\mathrm{CompactDisplay}\left(A\left(t\right)\&comma;B\left(t\right)\right)$

$A\left(t\right)\mathrm{will\; now\; be\; displayed\; as}A$

$B\left(t\right)\mathrm{will\; now\; be\; displayed\; as}B$

$\mathrm{e\_}\left[\right]$

${𝔢}_{a,\mathrm{\mu }}=\left[\begin{array}{cccc}A\sqrt{\mathrm{\epsilon }}& 0& 0& 0\\ 0& B\sqrt{\cosh \left(2x\right)}{\&ExponentialE;}^{\mathrm{A1}x}& -\frac{B\sinh \left(2x\right){\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\\ 0& 0& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\\ 0& 0& 0& -\sqrt{\mathrm{\epsilon }}\end{array}\right]$

$\mathrm{e\_}\left[\mathrm{definition}\right]$

${𝔢}_{a,\mathrm{\mu }}{𝔢}_{b\phantom{\mathrm{\mu }}}^{\phantom{b}\mathrm{\mu }}={\mathrm{\eta }}_{a,b}$

$\mathrm{IsTetrad}\left(\right)$

$\mathrm{Type\; of\; tetrad:}\mathrm{orthonormal}$

$\mathrm{true}$

$\mathrm{Setup}\left(\mathrm{tetradmetric}=\mathrm{null}\right)$

$\left[\mathrm{tetradmetric}=\left\{\left(1\&comma;4\right)=\mathrm{-1}\&comma;\left(2\&comma;3\right)=1\right\}\right]$

$\mathrm{Matrix}\left(4\&comma;\left\{\left(1\&comma;2\right)=-1\&comma;\left(3\&comma;4\right)=1\right\}\&comma;\mathrm{shape}=\mathrm{symmetric}\right)$

$\left[\begin{array}{cccc}0& \mathrm{-1}& 0& 0\\ \mathrm{-1}& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$

$\mathrm{eta\_}\left[\right]$

${\mathrm{\eta }}_{a,b}=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-1}\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]$

$\mathrm{e\_}\left[\right]$

${𝔢}_{a,\mathrm{\mu }}=\left[\begin{array}{cccc}-\frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& 0& -\frac{\sqrt{2}\sqrt{\mathrm{\epsilon }}}{2}\\ 0& \frac{\sqrt{2}B\sqrt{\cosh \left(2x\right)}{\&ExponentialE;}^{\mathrm{A1}x}}{2}& \frac{\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}\left(-\sinh \left(2x\right)+\mathrm{I}\right)}{2\sqrt{\cosh \left(2x\right)}}& 0\\ 0& \frac{\sqrt{2}B\sqrt{\cosh \left(2x\right)}{\&ExponentialE;}^{\mathrm{A1}x}}{2}& -\frac{\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}\left(\sinh \left(2x\right)+\mathrm{I}\right)}{2\sqrt{\cosh \left(2x\right)}}& 0\\ \frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& 0& -\frac{\sqrt{2}\sqrt{\mathrm{\epsilon }}}{2}\end{array}\right]$

$\mathrm{IsTetrad}\left(\right)$

$\mathrm{Type\; of\; tetrad:}\mathrm{null}$

$\mathrm{true}$

$\mathrm{OrthonormalTetrad}\left(\right)$

$\left[\begin{array}{cccc}A\sqrt{\mathrm{\epsilon }}& 0& 0& 0\\ 0& B\sqrt{\cosh \left(2x\right)}{\&ExponentialE;}^{\mathrm{A1}x}& -\frac{B\sinh \left(2x\right){\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\\ 0& 0& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\\ 0& 0& 0& -\sqrt{\mathrm{\epsilon }}\end{array}\right]$

$\mathrm{OrthonormalTetrad}\left(\mathrm{method}=\mathrm{Eigenvectors}\right)$

$\left[\begin{array}{cccc}A\sqrt{\mathrm{\epsilon }}& 0& 0& 0\\ 0& -\frac{B\sqrt{2}{\&ExponentialE;}^{\mathrm{A1}x}}{2\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}}& \frac{B\sqrt{2}{\&ExponentialE;}^{\mathrm{A1}x}}{2\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}}& 0\\ 0& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}\sqrt{2}}{2\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}}& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}\sqrt{2}}{2\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}}& 0\\ 0& 0& 0& \sqrt{\mathrm{\epsilon }}\end{array}\right]$

$\mathrm{NullTetrad}\left(\right)$

$\left[\begin{array}{cccc}-\frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& 0& -\frac{\sqrt{2}\sqrt{\mathrm{\epsilon }}}{2}\\ 0& \frac{\sqrt{2}B\sqrt{\cosh \left(2x\right)}{\&ExponentialE;}^{\mathrm{A1}x}}{2}& \frac{\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}\left(-\sinh \left(2x\right)+\mathrm{I}\right)}{2\sqrt{\cosh \left(2x\right)}}& 0\\ 0& \frac{\sqrt{2}B\sqrt{\cosh \left(2x\right)}{\&ExponentialE;}^{\mathrm{A1}x}}{2}& -\frac{\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}\left(\sinh \left(2x\right)+\mathrm{I}\right)}{2\sqrt{\cosh \left(2x\right)}}& 0\\ \frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& 0& -\frac{\sqrt{2}\sqrt{\mathrm{\epsilon }}}{2}\end{array}\right]$

$\mathrm{NullTetrad}\left(\mathrm{method}=\mathrm{Eigenvectors}\right)$

$\left[\begin{array}{cccc}-\frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& 0& \frac{\sqrt{2}\sqrt{\mathrm{\epsilon }}}{2}\\ 0& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}\left(\mathrm{I}\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}-\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}\right)}{2\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}}& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}\left(\mathrm{I}\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}+\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}\right)}{2\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}}& 0\\ 0& -\frac{B{\&ExponentialE;}^{\mathrm{A1}x}\left(\mathrm{I}\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}+\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}\right)}{2\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}}& -\frac{B{\&ExponentialE;}^{\mathrm{A1}x}\left(\mathrm{I}\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}-\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}\right)}{2\sqrt{\cosh \left(2x\right)-\sinh \left(2x\right)}\sqrt{\cosh \left(2x\right)+\sinh \left(2x\right)}}& 0\\ \frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& 0& \frac{\sqrt{2}\sqrt{\mathrm{\epsilon }}}{2}\end{array}\right]$

$\mathrm{OrthonormalTetrad}\left(\mathrm{firstvector}=\left[0\&comma;0\&comma;0\&comma;1\right]\right)$

$\left[\begin{array}{cccc}A\sqrt{\mathrm{\epsilon }}& 0& 0& 0\\ 0& 0& 0& \sqrt{-\mathrm{\epsilon }}\\ 0& -\frac{B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\mathrm{sech}\left(2x\right)}}& \frac{B{\&ExponentialE;}^{\mathrm{A1}x}\left(-\mathrm{csch}\left(4x\right)+\coth \left(4x\right)\right)}{\sqrt{\mathrm{sech}\left(2x\right)}}& 0\\ 0& 0& \frac{\mathrm{-I}B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\end{array}\right]$

$\mathrm{IsTetrad}\left(\right)$

$\mathrm{Type\; of\; tetrad:}\mathrm{orthonormal}$

$\mathrm{true}$

$\mathrm{NullTetrad}\left(\mathrm{firstvector}=\left[0\&comma;0\&comma;0\&comma;1\right]\right)$

$\left[\begin{array}{cccc}-\frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& \frac{-\frac{\mathrm{I}}{2}\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\\ 0& \frac{-\frac{\mathrm{I}}{2}\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\mathrm{sech}\left(2x\right)}}& \frac{-\frac{\mathrm{I}}{2}\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}\left(\mathrm{csch}\left(4x\right)-\coth \left(4x\right)\right)}{\sqrt{\mathrm{sech}\left(2x\right)}}& \frac{\mathrm{I}}{2}\sqrt{2}\sqrt{\mathrm{\epsilon }}\\ 0& \frac{\frac{\mathrm{I}}{2}\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\mathrm{sech}\left(2x\right)}}& \frac{\frac{\mathrm{I}}{2}\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}\left(\mathrm{csch}\left(4x\right)-\coth \left(4x\right)\right)}{\sqrt{\mathrm{sech}\left(2x\right)}}& \frac{\mathrm{I}}{2}\sqrt{2}\sqrt{\mathrm{\epsilon }}\\ \frac{\sqrt{2}A\sqrt{\mathrm{\epsilon }}}{2}& 0& \frac{-\frac{\mathrm{I}}{2}\sqrt{2}B{\&ExponentialE;}^{\mathrm{A1}x}}{\sqrt{\cosh \left(2x\right)}}& 0\end{array}\right]$

$\mathrm{IsTetrad}\left(\right)$

$\mathrm{Type\; of\; tetrad:}\mathrm{null}$

$\mathrm{true}$