 e_ - Maple Help

Physics[Tetrads][e_] - represents and computes a tetrad (vierbein) and the corresponding null vectors of the Newman-Penrose formalism

Physics[Tetrads][eta_] - represents the (tetrad) metric of a local system of references Calling Sequence e_[a, mu] e_[a, mu, keyword] e_[keyword] eta_[a, b] eta_[a, b, keyword] eta_[keyword] Parameters

 _mu - a spacetime index related to a global system of references, these are names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves _a, b_ - the tetrad indices related to a local system of references, as names representing integer numbers the same way as the spacetime indices keyword - optional, it can be definition, matrix, nonzero, and can be given alone or together with covariant or contravariant indices. Description

 • The e_[a, mu] and eta[a, b] commands respectively represent the tetrad (also vierbein; by default, this is an orthonormal tetrad) and the tetrad metric, that is, the metric of the local frame - which by default is inertial, of Minkowski type. These two tensors are defined in terms of each other by ${𝔢}_{a,\mathrm{\mu }}{𝔢}_{b}^{\mathrm{\mu }}={\mathrm{\eta }}_{a,b}$.
 • Both e_ and eta_ accept the keywords accepted by the other tensors of the Physics package, these are definition, matrix and nonzero, that can be given with or without indices. If given with indices, the corresponding output takes their character (covariant or contravariant) into account. In the case of e_, you can also use the keyword nullvectors, to see the null vectors corresponding to a given tetrad. Note anyway that these null vectors are available as commands of the Tetrads package; these are the l_, n_, m_ and mb_ commands. Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$$\mathrm{with}\left(\mathrm{Tetrads}\right)$
 ${}{}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}{}\mathrm{lowercaselatin_ah}{}\mathrm{letters to represent}{}\mathrm{tetrad}{}\mathrm{indices}$
 $\mathrm{Defined as tetrad tensors}{}\left(\mathrm{see ?Physics,tetrads}\right){},{}{𝔢}_{a,\mathrm{\mu }}{},{}{\mathrm{\eta }}_{a,b}{},{}{\mathrm{\gamma }}_{a,b,c}{},{}{\mathrm{\lambda }}_{a,b,c}$
 $\mathrm{Defined as spacetime tensors representing the NP null vectors of the tetrad formalism}{}\left(\mathrm{see ?Physics,tetrads}\right){},{}{l}_{\mathrm{\mu }}{},{}{n}_{\mathrm{\mu }}{},{}{m}_{\mathrm{\mu }}{},{}{\stackrel{&conjugate0;}{m}}_{\mathrm{\mu }}$
 ${}{}\mathrm{_______________________________________________________}$
 $\left[{\mathrm{IsTetrad}}{,}{\mathrm{NullTetrad}}{,}{\mathrm{OrthonormalTetrad}}{,}{\mathrm{PetrovType}}{,}{\mathrm{SegreType}}{,}{\mathrm{TransformTetrad}}{,}{\mathrm{WeylScalars}}{,}{\mathrm{e_}}{,}{\mathrm{eta_}}{,}{\mathrm{gamma_}}{,}{\mathrm{l_}}{,}{\mathrm{lambda_}}{,}{\mathrm{m_}}{,}{\mathrm{mb_}}{,}{\mathrm{n_}}\right]$ (1)
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (2)

In a flat space, the spacetime ${g}_{\mathrm{\mu },\mathrm{\nu }}$ and tetrad ${\mathrm{\eta }}_{a,b}$ metrics are the same, so the orthonormal tetrad $\mathrm{𝔢__a,μ}$ is just the identity

 > $\mathrm{g_}\left[\right]$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{rrrr}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]\right)$ (3)
 > $\mathrm{eta_}\left[\right]$
 ${{\mathrm{eta_}}}_{{a}{,}{b}}{=}\left(\left[\begin{array}{rrrr}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]\right)$ (4)
 > $\mathrm{e_}\left[\right]$
 ${{\mathrm{e_}}}_{{a}{,}{\mathrm{μ}}}{=}\left(\left[\begin{array}{rrrr}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\right)$ (5)

In a curved spacetime, for instance, set a Local Rotational Symmetry metric metric:

 > $\mathrm{g_}\left[\left[13,7,5\right]\right]$
 ${}{}\mathrm{_______________________________________________________}$
 ${}{}\mathrm{Systems of spacetime coordinates are:}{}{}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 ${}{}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}{}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{The}{}{}\mathrm{metric in coordinates}{}\left[x{,}y{,}z{,}t\right]$
 $\mathrm{Parameters:}{}\left[\mathrm{\epsilon }{,}A{}\left(t\right){,}B{}\left(t\right){,}\mathrm{A1}\right]$
 $\mathrm{Comments:}{}ⅇpsilon=1 or ⅇpsilon=-1$
 $\mathrm{Resetting the signature of spacetime from}{}\left(\mathrm{- - - +}\right){}\mathrm{to}{}\left(\mathrm{+ + + -}\right){}\mathrm{in order to match the signature in the database of metrics}$
 ${}{}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}{}\mathrm{lowercaselatin_is}{}\mathrm{letters to represent}{}\mathrm{space}{}\mathrm{indices}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}\mathrm{ε}{}{A{}\left(t\right)}^{2}& 0& 0& 0\\ 0& {B{}\left(t\right)}^{2}{}{ⅇ}^{2{}\mathrm{A1}{}x}{}\left(2{}{\mathrm{cosh}{}\left(x\right)}^{2}-1\right)& -2{}{B{}\left(t\right)}^{2}{}{ⅇ}^{2{}\mathrm{A1}{}x}{}\mathrm{cosh}{}\left(x\right){}\mathrm{sinh}{}\left(x\right)& 0\\ 0& -2{}{B{}\left(t\right)}^{2}{}{ⅇ}^{2{}\mathrm{A1}{}x}{}\mathrm{cosh}{}\left(x\right){}\mathrm{sinh}{}\left(x\right)& {B{}\left(t\right)}^{2}{}{ⅇ}^{2{}\mathrm{A1}{}x}{}\left(2{}{\mathrm{cosh}{}\left(x\right)}^{2}-1\right)& 0\\ 0& 0& 0& -\mathrm{ε}\end{array}\right]\right)$ (6)
 > $\mathrm{PDEtools}:-\mathrm{declare}\left(A\left(t\right),B\left(t\right)\right)$
 ${A}{}\left({t}\right){}{\mathrm{will now be displayed as}}{}{A}$
 ${B}{}\left({t}\right){}{\mathrm{will now be displayed as}}{}{B}$ (7)

The default orthonormal tetrad is now

 > $\mathrm{e_}\left[\right]$
 ${{\mathrm{e_}}}_{{a}{,}{\mathrm{μ}}}{=}\left(\left[\begin{array}{cccc}A{}\left(t\right){}\sqrt{\mathrm{ε}}& 0& 0& 0\\ 0& \sqrt{\mathrm{cosh}{}\left(2{}x\right)}{}B{}\left(t\right){}{ⅇ}^{\mathrm{A1}{}x}& -\frac{B{}\left(t\right){}{ⅇ}^{\mathrm{A1}{}x}{}\mathrm{sinh}{}\left(2{}x\right)}{\sqrt{\mathrm{cosh}{}\left(2{}x\right)}}& 0\\ 0& 0& \frac{B{}\left(t\right){}{ⅇ}^{\mathrm{A1}{}x}}{\sqrt{\mathrm{cosh}{}\left(2{}x\right)}}& 0\\ 0& 0& 0& -\sqrt{\mathrm{ε}}\end{array}\right]\right)$ (8)

The following null vectors correspond to this tetrad:

 > $\mathrm{e_}\left[\mathrm{nullvectors}\right]$
 ${{l}}_{{\mathrm{\mu }}}{=}\left[\begin{array}{cccc}\frac{{1}}{{2}}{}\sqrt{{2}}{}{A}{}\left({t}\right){}\sqrt{{\mathrm{ε}}}& {0}& {0}& {-}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\epsilon }}}}{{2}}\end{array}\right]{,}{{n}}_{{\mathrm{\mu }}}{=}\left[\begin{array}{cccc}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{A}{}\left({t}\right){}\sqrt{{\mathrm{ε}}}& {0}& {0}& {-}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\epsilon }}}}{{2}}\end{array}\right]{,}{{m}}_{{\mathrm{\mu }}}{=}\left[\begin{array}{cccc}{0}& \frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{\mathrm{cosh}}{}\left({2}{}{x}\right)}{}{B}{}\left({t}\right){}{\mathrm{exp}}{}\left({\mathrm{A1}}{}{x}\right)& \frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{B}{}\left({t}\right){}{\mathrm{exp}}{}\left({\mathrm{A1}}{}{x}\right){}\left({-}{\mathrm{sinh}}{}\left({2}{}{x}\right){+}{I}\right)}{\sqrt{{\mathrm{cosh}}{}\left({2}{}{x}\right)}}& {0}\end{array}\right]{,}{\stackrel{{&conjugate0;}}{{m}}}_{{\mathrm{\mu }}}{=}\left[\begin{array}{cccc}{0}& \frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{\mathrm{cosh}}{}\left({2}{}{x}\right)}{}{B}{}\left({t}\right){}{\mathrm{exp}}{}\left({\mathrm{A1}}{}{x}\right)& {-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{B}{}\left({t}\right){}{\mathrm{exp}}{}\left({\mathrm{A1}}{}{x}\right){}\left({\mathrm{sinh}}{}\left({2}{}{x}\right){+}{I}\right)}{\sqrt{{\mathrm{cosh}}{}\left({2}{}{x}\right)}}& {0}\end{array}\right]$ (9)

You can compute these null vectors directly since these are also part of the Tetrads package:

 > ${\mathrm{l_}\left[\mathrm{\mu }\right]}^{2},\mathrm{l_}\left[\mathrm{\mu }\right]\mathrm{n_}\left[\mathrm{\mu }\right],\mathrm{l_}\left[\mathrm{\mu }\right]\mathrm{m_}\left[\mathrm{\mu }\right],\mathrm{l_}\left[\mathrm{\mu }\right]\mathrm{mb_}\left[\mathrm{\mu }\right]$
 ${{l}}_{{\mathrm{\mu }}}{}{{l}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{,}{{l}}_{{\mathrm{\mu }}}{}{{n}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{,}{{l}}_{{\mathrm{\mu }}}{}{{m}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{,}{\stackrel{{&conjugate0;}}{{m}}}_{{\mathrm{\mu }}}{}{{l}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}$ (10)
 > $\mathrm{map}\left(u↦u=\mathrm{SumOverRepeatedIndices}\left(u\right),\left[\right]\right)$
 $\left[{{l}}_{{\mathrm{\mu }}}{}{{l}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{0}{,}{{l}}_{{\mathrm{\mu }}}{}{{n}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{-1}{,}{{l}}_{{\mathrm{\mu }}}{}{{m}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{0}{,}{\stackrel{{&conjugate0;}}{{m}}}_{{\mathrm{\mu }}}{}{{l}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{0}\right]$ (11)

You can query about the definition of any of these tensors in the same way you can now query any other tensor:

 > $\mathrm{m_}\left[\mathrm{definition}\right]$
 ${{m}}_{{\mathrm{\mu }}}{}{{l}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{0}{,}{{m}}_{{\mathrm{\mu }}}{}{{n}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{0}{,}{{m}}_{{\mathrm{\mu }}}{}{{m}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{0}{,}{{m}}_{{\mathrm{\mu }}}{}{\stackrel{{&conjugate0;}}{{m}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}{1}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}{-}{{l}}_{{\mathrm{\mu }}}{}{{n}}_{{\mathrm{\nu }}}{-}{{l}}_{{\mathrm{\nu }}}{}{{n}}_{{\mathrm{\mu }}}{+}{{m}}_{{\mathrm{\mu }}}{}{\stackrel{{&conjugate0;}}{{m}}}_{{\mathrm{\nu }}}{+}{{m}}_{{\mathrm{\nu }}}{}{\stackrel{{&conjugate0;}}{{m}}}_{{\mathrm{\mu }}}$ (12)
 > $\mathrm{eta_}\left[\mathrm{definition}\right]$
 ${{\mathrm{\eta }}}_{{a}{,}{b}}{=}{{𝔢}}_{{a}{,}{\mathrm{\mu }}}{}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}$ (13)
 > $\mathrm{e_}\left[\mathrm{definition}\right]$
 ${{𝔢}}_{{a}{,}{\mathrm{\mu }}}{}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}{=}{{\mathrm{\eta }}}_{{a}{,}{b}}$ (14)

Verify the definition of the tetrad $\mathrm{𝔢__a,μ}$ given above

 > $\mathrm{TensorArray}\left(,\mathrm{simplifier}=\mathrm{simplify}\right)$
 $\left[\begin{array}{cccc}1=1& 0=0& 0=0& 0=0\\ 0=0& 1=1& 0=0& 0=0\\ 0=0& 0=0& 1=1& 0=0\\ 0=0& 0=0& 0=0& -1=-1\end{array}\right]$ (15)
 > See Also Compatibility