 computes different forms of a null tetrad for the current spacetime metric - Maple Programming Help

Physics[Tetrads][IsTetrad] - returns true or false according to whether a given matrix is a null or orthonormal tetrad and indicates what kind of the tetrad is

 Calling Sequence NullTetrad(method = ..., firstvector = ...) OrthonormalTetrad(method = ..., firstvector = ...) IsTetrad(T, orthonormal, null, quiet)

Parameters

 method = .. - optional, indicates the method to be used, that could be GramSchmidt or Eigenvectors firstvector = .. - optional, a Vector or a list with the vector components, as many as the dimension of spacetime; indicates the first departing vector to be used with the GramSchmidt method T - a Matrix representing a tetrad orthonormal - optional, to indicate to IsTetrad to check only for an orthonormal tetrad null - optional, to indicate to IsTetrad to check only for an null tetrad quiet - optional, to indicate to IsTetrad to avoid displaying the type of tetrad on the screen

Description

 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 (transformations, 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 permit computing different tetrads, that may result more convenient in different contexts, for example when computing the Weyl and Ricci scalars or performing a Petrov classification.
 • 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.

Examples

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

Set the spacetime metric to something non-flat, for instance take the metric [13,7,5] of the Exact solutions book referenced at the end of this page:

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

Compact the display of the functions $A\left(t\right),B\left(t\right)$:

 > $\mathrm{PDEtools}:-\mathrm{declare}\left(A\left(t\right),B\left(t\right)\right)$
 ${A}\left({t}\right){\mathrm{will now be displayed as}}{A}$
 ${B}\left({t}\right){\mathrm{will now be displayed as}}{B}$ (3)

 > $\mathrm{e_}\left[\right]$
 ${{\mathrm{e_}}}_{{a}{,}{\mathrm{μ}}}{=}\left(\left[\begin{array}{cccc}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& 0\\ 0& \sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}& -\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\mathrm{sinh}\left(2x\right)}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& 0& \frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& 0& 0& -\sqrt{\mathrm{ε}}\end{array}\right]\right)$ (4)

It is possible to test whether these tetrads satisfy the tetrad definition

 > $\mathrm{e_}\left[\mathrm{definition}\right]$
 ${{𝔢}}_{{a}{,}{\mathrm{\mu }}}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}{=}{{\mathrm{\eta }}}_{{a}{,}{b}}$ (5)

 > $\mathrm{IsTetrad}\left(\right)$
 $\mathrm{Type of tetrad:}{}\mathrm{orthonormal}$
 ${\mathrm{true}}$ (6)

 > $\mathrm{Setup}\left(\mathrm{tetradmetric}=\mathrm{null}\right)$
 $\left[{\mathrm{tetradmetric}}{=}\left\{\left({1}{,}{4}\right){=}{-1}{,}\left({2}{,}{3}\right){=}{1}\right\}\right]$ (7)

Note: this form of the null tetrad metric is consistent with time in position 4 (default) - and not with time in position 1. To see the more frequent form of the tetradmetric

 > $\mathrm{Matrix}\left(4,\left\{\left(1,2\right)=-1,\left(3,4\right)=1\right\},\mathrm{shape}=\mathrm{symmetric}\right)$
 $\left[\begin{array}{rrrr}0& -1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$ (8)

you can either set the signature of the underlying inertial system of references (Minkowski) indicating time (different sign) in the first place, or set the tetradmetric directly using Setup(tetradmetric = the_matrix_above).

So now the metric of the local (tetrad) system of references is:

 > $\mathrm{eta_}\left[\right]$
 ${{\mathrm{eta_}}}_{{a}{,}{b}}{=}\left(\left[\begin{array}{rrrr}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ -1& 0& 0& 0\end{array}\right]\right)$ (9)

 > $\mathrm{e_}\left[\right]$
 ${{\mathrm{e_}}}_{{a}{,}{\mathrm{μ}}}{=}\left(\left[\begin{array}{cccc}-\frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& -\frac{1}{2}\sqrt{2}\sqrt{\mathrm{ε}}\\ 0& \frac{1}{2}\sqrt{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}& -\frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(\mathrm{sinh}\left(2x\right)-I\right)}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& \frac{1}{2}\sqrt{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}& -\frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(\mathrm{sinh}\left(2x\right)+I\right)}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ \frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& -\frac{1}{2}\sqrt{2}\sqrt{\mathrm{ε}}\end{array}\right]\right)$ (10)
 > $\mathrm{IsTetrad}\left(\right)$
 $\mathrm{Type of tetrad:}{}\mathrm{null}$
 ${\mathrm{true}}$ (11)

You can compute different forms of an orthonormal tetrad or a null tetrad using the OrthonormalTetrad and NullTetrad commands, requesting the use of different methods, or passing a starting vector. For instance, this is the default form of an orthonormal tetrad for the current spacetime metric:

 > $\mathrm{OrthonormalTetrad}\left(\right)$
 $\left[\begin{array}{cccc}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& 0\\ 0& \sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}& -\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\mathrm{sinh}\left(2x\right)}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& 0& \frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& 0& 0& -\sqrt{\mathrm{ε}}\end{array}\right]$ (12)

There are two methods available for computing these tetrads: the GramSchmidt (generalization to curved spaces) and Eigenvectors (based on computing the Eigenvectors of a related Matrix). By default, the Physics package routines decide on what method is more convenient to use for the the spacetime metric set. In this case the routines use the GramSchmidt method; you can check the other method via:

 > $\mathrm{OrthonormalTetrad}\left(\mathrm{method}=\mathrm{Eigenvectors}\right)$
 $\left[\begin{array}{cccc}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& 0\\ 0& \frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& \frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& 0\\ 0& -\frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& \frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& 0\\ 0& 0& 0& \sqrt{\mathrm{ε}}\end{array}\right]$ (13)

If this is an orthonormal tetrad you prefer, you can set it via Setup(e_ = ...), where ... is the matrix you prefer (see the previous output). To see what the null tetrads corresponding to these two orthonormal tetrads would be:

 > $\mathrm{NullTetrad}\left(\right)$
 $\left[\begin{array}{cccc}-\frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& -\frac{1}{2}\sqrt{2}\sqrt{\mathrm{ε}}\\ 0& \frac{1}{2}\sqrt{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}& -\frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(\mathrm{sinh}\left(2x\right)-I\right)}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& \frac{1}{2}\sqrt{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}& -\frac{1}{2}\frac{\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(\mathrm{sinh}\left(2x\right)+I\right)}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ \frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& -\frac{1}{2}\sqrt{2}\sqrt{\mathrm{ε}}\end{array}\right]$ (14)
 > $\mathrm{NullTetrad}\left(\mathrm{method}=\mathrm{Eigenvectors}\right)$
 $\left[\begin{array}{cccc}-\frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& \frac{1}{2}\sqrt{2}\sqrt{\mathrm{ε}}\\ 0& -\frac{1}{2}\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(I\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}-\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\right)}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& \frac{1}{2}\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(I\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}+\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\right)}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& 0\\ 0& \frac{1}{2}\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(I\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}+\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\right)}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& -\frac{1}{2}\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\left(I\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}-\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\right)}{\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}+2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}\sqrt{2{\mathrm{cosh}\left(x\right)}^{2}-2\mathrm{cosh}\left(x\right)\mathrm{sinh}\left(x\right)-1}}& 0\\ \frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& \frac{1}{2}\sqrt{2}\sqrt{\mathrm{ε}}\end{array}\right]$ (15)

When using the GramSchmidt method you can also specify the first vector. Recalling, the method works iterating the computation of the next vector (line of the tetrad) starting from a first vector. For example, instead of starting from $\left[\sqrt{\mathrm{\epsilon }}A\left(t\right),0,0,0\right]$, start from $\left[0,0,0,1\right]$ compare with default result of OrthonormalTetrad lines above.

 > $\mathrm{OrthonormalTetrad}\left(\mathrm{firstvector}=\left[0,0,0,1\right]\right)$
 $\left[\begin{array}{cccc}A\left(t\right)\sqrt{\mathrm{ε}}& 0& 0& 0\\ 0& 0& 0& I\sqrt{\mathrm{ε}}\\ 0& -\frac{B\left(t\right){ⅇ}^{\mathrm{A1}x}\mathrm{sinh}\left(x\right)\sqrt{2}\sqrt{\mathrm{cosh}\left(2x\right)}}{\sqrt{-1+\mathrm{cosh}\left(2x\right)}}& \frac{2\mathrm{cosh}\left(x\right){\mathrm{sinh}\left(x\right)}^{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}\sqrt{2}}{\sqrt{-1+\mathrm{cosh}\left(2x\right)}\left(2{\mathrm{cosh}\left(x\right)}^{2}-1\right)}& 0\\ 0& 0& -\frac{IB\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\end{array}\right]$ (16)
 > $\mathrm{IsTetrad}\left(\right)$
 $\mathrm{Type of tetrad:}{}\mathrm{orthonormal}$
 ${\mathrm{true}}$ (17)
 > $\mathrm{NullTetrad}\left(\mathrm{firstvector}=\left[0,0,0,1\right]\right)$
 $\left[\begin{array}{cccc}-\frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& -\frac{\frac{1}{2}I\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\\ 0& -\frac{IB\left(t\right){ⅇ}^{\mathrm{A1}x}\mathrm{sinh}\left(x\right)\sqrt{\mathrm{cosh}\left(2x\right)}}{\sqrt{-1+\mathrm{cosh}\left(2x\right)}}& \frac{2I\mathrm{cosh}\left(x\right){\mathrm{sinh}\left(x\right)}^{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{-1+\mathrm{cosh}\left(2x\right)}\left(2{\mathrm{cosh}\left(x\right)}^{2}-1\right)}& \frac{1}{2}I\sqrt{2}\sqrt{\mathrm{ε}}\\ 0& \frac{IB\left(t\right){ⅇ}^{\mathrm{A1}x}\mathrm{sinh}\left(x\right)\sqrt{\mathrm{cosh}\left(2x\right)}}{\sqrt{-1+\mathrm{cosh}\left(2x\right)}}& -\frac{2I\mathrm{cosh}\left(x\right){\mathrm{sinh}\left(x\right)}^{2}\sqrt{\mathrm{cosh}\left(2x\right)}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{-1+\mathrm{cosh}\left(2x\right)}\left(2{\mathrm{cosh}\left(x\right)}^{2}-1\right)}& \frac{1}{2}I\sqrt{2}\sqrt{\mathrm{ε}}\\ \frac{1}{2}\sqrt{2}A\left(t\right)\sqrt{\mathrm{ε}}& 0& -\frac{\frac{1}{2}I\sqrt{2}B\left(t\right){ⅇ}^{\mathrm{A1}x}}{\sqrt{\mathrm{cosh}\left(2x\right)}}& 0\end{array}\right]$ (18)
 > $\mathrm{IsTetrad}\left(\right)$
 $\mathrm{Type of tetrad:}{}\mathrm{null}$
 ${\mathrm{true}}$ (19)
 >