The Redefine command redefines the spacetime metric g_ and list of coordinates, or a given tetrad, according to a change in the signature, from any of the four possible signatures (- - - +), (+ - - -), (+ + + -) and (- + + +) to any of the other ones.

When any of fromsignature or tosignature are not given, the corresponding value is the value of the signature set at the moment. To query about this value, enter Setup(signature), or to change it enter Setup(signature = ...) where the right-hand side is any of `- - - +`, `+ - - -`, `+ + + -` or `- + + +`. You can enter the signature leaving or not spaces between the +/- signs, and enclosing them with `` or " ".

The typical scenario for the use of fromsignature is when you change the signature using Setup and notice that the coordinates and metric g_ have not changed accordingly; or you have a tetrad matrix for which Tetrads:-IsTetrad says it is not a tetrad unless you redefine the signature. In these cases entering Redefine(all, fromsignature = previous_signature) shows how would the list of coordinates and metric look if they were redefined from the previous signature to the one you set (i.e., the current one). In the case of a tetrad, entering Redefine(e_[a, mu] = ..., fromsignature = previous_signature) shows how would the passed tetrad would look. In both cases, if the result is as desired, you can use the output to call Setup and set things accordingly, or simpler: in the case of coordinates and metric, instead of all, pass setall, as in Redefine(setall, fromsignature = previous_signature), and that will in addition set things up by automatically calling Setup.

The typical scenario for the use of tosignature is when you are working with a signature and - say to follow a textbook - you want to change to a different signature, and want that not just the signature but also the ordering of the list of coordinates and/or the spacetime metric change accordingly. Here Redefine(all, tosignature = desired_signature) and Redefine(setall, tosignature = desired_signature) will respectively show or also set the new ordering of the list of coordinates and the metric according to the change to the desired_signature. For an example where redefining a tetrad is necessary see this one in the what's new in Physics in Maple 2021.

To restrict these redefinitions to either the coordinates or the metric, instead of all and setall use coordinates and setcoordinates, or metric and setmetric.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{coordinates}=\mathrm{cartesian}\right)$

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

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

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

$G≔\mathrm{Matrix}\left(4\,\mathrm{symbol}=g\,\mathrm{shape}=\mathrm{symmetric}\right)$

$G≔\left[\right]$

$\mathrm{Setup}\left(\mathrm{signature}\,\mathrm{g\_}=G\right)$

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

$\mathrm{Coordinates:}\left[x\,y\,z\,t\right]\mathrm{.\; Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

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

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

$\left[\mathrm{metric}=\left\{\left(1\,1\right)=g_{1,1}\,\left(1\,2\right)=g_{1,2}\,\left(1\,3\right)=g_{1,3}\,\left(1\,4\right)=g_{1,4}\,\left(2\,2\right)=g_{2,2}\,\left(2\,3\right)=g_{2,3}\,\left(2\,4\right)=g_{2,4}\,\left(3\,3\right)=g_{3,3}\,\left(3\,4\right)=g_{3,4}\,\left(4\,4\right)=g_{4,4}\right\}\,\mathrm{signature}=\mathrm{-\; -\; -\; +}\,\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right]$

$\mathrm{g\_}\left[\mathrm{lineelement}\right]$

$g_{3,3}{\mathbf{\ⅆ}\left(z\right)}^2+\left(2\mathbf{\ⅆ}\left(x\right)g_{1,3}+2g_{2,3}\mathbf{\ⅆ}\left(y\right)\right)\mathbf{\ⅆ}\left(z\right)+g_{1,1}{\mathbf{\ⅆ}\left(x\right)}^2+2g_{1,2}\mathbf{\ⅆ}\left(x\right)\mathbf{\ⅆ}\left(y\right)+g_{2,2}{\mathbf{\ⅆ}\left(y\right)}^2+\mathbf{\ⅆ}\left(t\right)\left(\mathbf{\ⅆ}\left(t\right)g_{4,4}+2\mathbf{\ⅆ}\left(x\right)g_{1,4}+2\mathbf{\ⅆ}\left(y\right)g_{2,4}+2\mathbf{\ⅆ}\left(z\right)g_{3,4}\right)$

$\mathrm{Setup}\left(\mathrm{signature}=''+---''\right)$

$\left[\mathrm{signature}=\mathrm{+\; -\; -\; -}\right]$

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

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

$\left\{X\right\}$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Redefine}\left(\mathrm{coordinates}\,\mathrm{fromsignature}=''---+''\right)$

$\left[t\,x\,y\,z\right]$

$\mathrm{Redefine}\left(\mathrm{metric}\,\mathrm{fromsignature}=''---+''\right)$

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

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

$\left\{X\right\}$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Redefine}\left(\mathrm{setall}\,\mathrm{fromsignature}=''---+''\right)$

$\left[X\right],\left[?\right]$

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

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

$\left\{X\right\}$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{g\_}\left[\mathrm{lineelement}\right]$

$g_{2,2}{\mathbf{\ⅆ}\left(y\right)}^2+\left(2\mathbf{\ⅆ}\left(t\right)g_{2,4}+2\mathbf{\ⅆ}\left(x\right)g_{1,2}\right)\mathbf{\ⅆ}\left(y\right)+g_{4,4}{\mathbf{\ⅆ}\left(t\right)}^2+2g_{1,4}\mathbf{\ⅆ}\left(x\right)\mathbf{\ⅆ}\left(t\right)+g_{1,1}{\mathbf{\ⅆ}\left(x\right)}^2+\mathbf{\ⅆ}\left(z\right)\left(\mathbf{\ⅆ}\left(z\right)g_{3,3}+2\mathbf{\ⅆ}\left(t\right)g_{3,4}+2\mathbf{\ⅆ}\left(x\right)g_{1,3}+2g_{2,3}\mathbf{\ⅆ}\left(y\right)\right)$

$\mathrm{normal}\left(-\right)$

$0$

$\mathrm{g\_}\left[\left[12\,21\,1\right]\right]$

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

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

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

$\mathrm{The}\mathrm{McLenaghan,\; Tariq\; (1975)}\mathrm{metric\; in\; coordinates}\mathrm{Tupper\; (1976)}$

$\mathrm{Parameters:}\left[a\,k\,\mathrm{\κ0}\right]$

$\mathrm{Comments:}\mathrm{k\; param\ⅇtriz\ⅇs\; th\ⅇ\; most\; g\ⅇn\ⅇral\; \ⅇl\ⅇctromagn\ⅇtic\; invariant\; with\; r\ⅇsp\ⅇct\; to\; th\ⅇ\; last\; 3\; Killing\; v\ⅇctors}$

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

$\mathrm{Assumptions:}\left[0<a\&comma;0<\mathrm{\&kappa;0}\right]$

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Setup}\left(\mathrm{signature}\right)$

$\left[\mathrm{signature}=\mathrm{+\; +\; +\; -}\right]$

$\mathrm{Redefine}\left(\mathrm{all}\&comma;\mathrm{tosignature}=''+\; -\; -\; -''\right)$

$\left[t\&comma;x\&comma;y\&comma;\mathrm{\phi }\right],\left[?\right]$

$\mathrm{with}\left(\mathrm{Tetrads}\right)\&colon;$

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

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

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

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

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

$\mathrm{g\_}\left[\left[27\&comma;37\&comma;1\right]\right]$

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(z\&comma;\mathrm{zb}\&comma;r\&comma;u\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(z\&comma;\mathrm{zb}\&comma;r\&comma;u\right)\right\}$

$\mathrm{The}\mathrm{Robinson\; and\; Trautman\; (1962)}\mathrm{metric\; in\; coordinates}\left[z\&comma;\mathrm{zb}\&comma;r\&comma;u\right]$

$\mathrm{Parameters:}\left[P\left(z\&comma;\mathrm{zb}\&comma;u\right)\&comma;H\left(X\right)\right]$

$\mathrm{Comments:}\mathrm{a\&DifferentialD;mits\; g\&ExponentialE;o\&DifferentialD;\&ExponentialE;sic,\; sh\&ExponentialE;arfr\&ExponentialE;\&ExponentialE;,\; twistfr\&ExponentialE;\&ExponentialE;\; null\; congru\&ExponentialE;nc\&ExponentialE;,\; rho=-1/r=rho\_b}$

$\mathrm{Domain:}\left[P\left(z\&comma;\mathrm{zb}\&comma;u\right)::\mathrm{real}\right]$

$\mathrm{Signature:}\left(\mathrm{+\; +\; +\; -}\right)$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

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

$H\left(z\&comma;\mathrm{zb}\&comma;r\&comma;u\right)\mathrm{will\; now\; be\; displayed\; as}H$

$P\left(z\&comma;\mathrm{zb}\&comma;u\right)\mathrm{will\; now\; be\; displayed\; as}P$

$\mathrm{Assume}\left(0<P\left(z\&comma;\mathrm{zb}\&comma;u\right)\&comma;H\left(X\right)::\mathrm{real}\&comma;0\le r\right)$

$\left\{u::\mathrm{real}\&comma;z::\mathrm{real}\&comma;\mathrm{zb}::\mathrm{real}\&comma;H::\mathrm{real}\&comma;P::\left(0\&comma;\mathrm{\infty }\right)\right\},\left\{r::\left[0\&comma;\mathrm{\infty }\right)\&comma;u::\mathrm{real}\&comma;z::\mathrm{real}\&comma;\mathrm{zb}::\mathrm{real}\&comma;H::\mathrm{real}\&comma;P::\left(0\&comma;\mathrm{\infty }\right)\right\},\left\{r::\left[0\&comma;\mathrm{\infty }\right)\&comma;H::\mathrm{real}\right\}$

$\mathrm{Setup}\left(\mathrm{tetrad}=\mathrm{null}\right)\&colon;$

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

${𝔢}_{a,\mathrm{\mu }}=\left[\right]$

$\mathrm{e\_}\left[a,\mathrm{\mu }\right]=\mathrm{Matrix}\left(4\&comma;4\&comma;\left[\left[0\&comma;0\&comma;0\&comma;-\mathrm{I}\right]\&comma;\left[0\&comma;\frac{r\mathrm{I}\cdot 1}{P\left(z\&comma;\mathrm{zb}\&comma;u\right)}\&comma;0\&comma;0\right]\&comma;\left[\frac{r\mathrm{I}\cdot 1}{P\left(z\&comma;\mathrm{zb}\&comma;u\right)}\&comma;0\&comma;0\&comma;0\right]\&comma;\left[0\&comma;0\&comma;-\mathrm{I}\&comma;-H\left(z\&comma;\mathrm{zb}\&comma;r\&comma;u\right)\mathrm{I}\right]\right]\right)$

${𝔢}_{a,\mathrm{\mu }}=\left[\right]$

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

$\mathit{Warning,\; the\; given\; components\; form\; a}\left(\mathit{null}\right)\mathit{tetrad,}\mathit{with\; a\; covariant\; spacetime\; index}\mathit{,\; only\; if\; you\; change\; the\; signature\; from}\left(\mathit{+\; +\; +\; -}\right)\mathit{to}\left(\mathit{-\; -\; -\; +}\right)\mathit{.\; You\; can\; do\; that\; by\; entering\; (copy\; and\; paste):}\mathit{Setup}\left(\mathit{signature}\mathbf{=}''-\; -\; -\; +''\right)$

$\mathrm{false}$

$\mathrm{Redefine}\left(\&comma;\mathrm{fromsignature}=''-\; -\; -\; +''\right)$

${𝔢}_{a,\mathrm{\mu }}=\left[\right]$

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

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

$\mathrm{true}$

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

$\left[\mathrm{tetrad}=\left\{\left(1\&comma;4\right)=1\&comma;\left(2\&comma;2\right)=-\frac{r}{P}\&comma;\left(3\&comma;1\right)=-\frac{r}{P}\&comma;\left(4\&comma;3\right)=1\&comma;\left(4\&comma;4\right)=H\right\}\right]$

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

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

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

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

The Physics[Redefine] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.

The Physics[Redefine] command was updated in Maple 2021.