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Physics[Redefine] - Redefines the spacetime metric, list of coordinates and tetrads according to a change in the signature, from any of the four possible signatures (- - - +), (+ - - -), (+ + + -) and (- + + +) to any of the other ones

 Calling Sequence Redefine(what, fromsignature = ..., tosignature = ...)

Parameters

 what - any of coordinates, metric, all (meaning both); setcoordinates, setmetric, setall, to respectively see how the redefinition changes what, or in addition automatically set (via Setup) the redefined value. fromsignature = ... - optional, to be taken as the starting signature; if not given, it is the signature set at the moment tosignature = ... - optional, to be taken as the desired signature; if not given, it is the signature set at the moment

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true},\mathrm{coordinates}=\mathrm{cartesian}\right)$
 $\mathrm{Systems of spacetime coordinates are:}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{coordinatesystems}}{=}\left\{{X}\right\}{,}{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

Consider the generic form of a spacetime metric. For the purpose of illustrating its redefinition (reordering of lines and columns) when changing the signature, it doesn't need to depend on the coordinates

 > $G≔\mathrm{Matrix}\left(4,\mathrm{symbol}=g,\mathrm{shape}=\mathrm{symmetric}\right)$
 ${G}{≔}\left[\right]$ (2)

Set the metric with this value, note the signature: it is (- - - +):

 > $\mathrm{Setup}\left(\mathrm{signature},\mathrm{g_}=G\right)$
 $\mathrm{_______________________________________________________}$
 $\mathrm{Coordinates:}\left[x{,}y{,}z{,}t\right]\mathrm{. Signature:}\left(\mathrm{- - - +}\right)$
 $\mathrm{_______________________________________________________}$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\right]$
 $\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{space}\mathrm{indices}$
 $\left[{\mathrm{metric}}{=}\left\{\left({1}{,}{1}\right){=}{{g}}_{{1}{,}{1}}{,}\left({1}{,}{2}\right){=}{{g}}_{{1}{,}{2}}{,}\left({1}{,}{3}\right){=}{{g}}_{{1}{,}{3}}{,}\left({1}{,}{4}\right){=}{{g}}_{{1}{,}{4}}{,}\left({2}{,}{2}\right){=}{{g}}_{{2}{,}{2}}{,}\left({2}{,}{3}\right){=}{{g}}_{{2}{,}{3}}{,}\left({2}{,}{4}\right){=}{{g}}_{{2}{,}{4}}{,}\left({3}{,}{3}\right){=}{{g}}_{{3}{,}{3}}{,}\left({3}{,}{4}\right){=}{{g}}_{{3}{,}{4}}{,}\left({4}{,}{4}\right){=}{{g}}_{{4}{,}{4}}\right\}{,}{\mathrm{signature}}{=}{\mathrm{- - - +}}{,}{\mathrm{spaceindices}}{=}{\mathrm{lowercaselatin_is}}\right]$ (3)

Track the line element

 > $\mathrm{g_}\left[\mathrm{lineelement}\right]$
 ${{g}}_{{3}{,}{3}}{{\mathbf{ⅆ}}{}\left({z}\right)}^{{2}}{+}\left({2}{\mathbf{ⅆ}}{}\left({x}\right){{g}}_{{1}{,}{3}}{+}{2}{{g}}_{{2}{,}{3}}{\mathbf{ⅆ}}{}\left({y}\right)\right){\mathbf{ⅆ}}{}\left({z}\right){+}{{g}}_{{1}{,}{1}}{{\mathbf{ⅆ}}{}\left({x}\right)}^{{2}}{+}{2}{{g}}_{{1}{,}{2}}{\mathbf{ⅆ}}{}\left({x}\right){\mathbf{ⅆ}}{}\left({y}\right){+}{{g}}_{{2}{,}{2}}{{\mathbf{ⅆ}}{}\left({y}\right)}^{{2}}{+}{\mathbf{ⅆ}}{}\left({t}\right)\left({\mathbf{ⅆ}}{}\left({t}\right){{g}}_{{4}{,}{4}}{+}{2}{\mathbf{ⅆ}}{}\left({x}\right){{g}}_{{1}{,}{4}}{+}{2}{\mathbf{ⅆ}}{}\left({y}\right){{g}}_{{2}{,}{4}}{+}{2}{\mathbf{ⅆ}}{}\left({z}\right){{g}}_{{3}{,}{4}}\right)$ (4)

Change now the signature by reversing the position of the time-like component, from 4 to 1

 > $\mathrm{Setup}\left(\mathrm{signature}="+---"\right)$
 $\left[{\mathrm{signature}}{=}{\mathrm{+ - - -}}\right]$ (5)

Neither the metric nor the coordinates change: t is still in position 4 and the component (1, 1) of the metric is still ${g}_{1,1}$:

 > $\mathrm{Coordinates}\left(\right)$
 $\mathrm{Systems of spacetime coordinates are:}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\left\{{X}\right\}$ (6)
 > $\mathrm{g_}\left[\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\right]$ (7)

However, after changing the signature using Setup, we may also want to redefine the coordinates accordingly - place t in position 1 - and possibly also the metric reordering its lines and rows. For that purpose you can use Redefine, to which you need to indicate the previous signature (or in a different situation the new signature, to explore the effect of a change before doing it) and an indication of whether you want to redefine the metric, the coordinates or all

 > $\mathrm{Redefine}\left(\mathrm{coordinates},\mathrm{fromsignature}="---+"\right)$
 $\left[{t}{,}{x}{,}{y}{,}{z}\right]$ (8)
 > $\mathrm{Redefine}\left(\mathrm{metric},\mathrm{fromsignature}="---+"\right)$
 $\left[\right]$ (9)

These redefinitions however were not set, the keywords coordinates, metric, or all only trigger the change, t is still in position 4 and the component (1,1) of the metric is still ${g}_{1,1}$

 > $\mathrm{Coordinates}\left(\right)$
 $\mathrm{Systems of spacetime coordinates are:}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\left\{{X}\right\}$ (10)
 > $\mathrm{g_}\left[\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\right]$ (11)

You can now either use the output of this routine to redefine things using the Setup command, or do all in one go using any of the keywords setcoordinates, setmetric, or setall, for example

 > $\mathrm{Redefine}\left(\mathrm{setall},\mathrm{fromsignature}="---+"\right)$
 $\left[{X}\right]{,}\left[{?}\right]$ (12)

Now t is in position 1 and the component (1,1) of the metric equal to ${g}_{4,4}$

 > $\mathrm{Coordinates}\left(\right)$
 $\mathrm{Systems of spacetime coordinates are:}\left\{X=\left(t{,}x{,}y{,}z\right)\right\}$
 $\left\{{X}\right\}$ (13)
 > $\mathrm{g_}\left[\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\right]$ (14)

Note that, despite the reordering of lines and columns in the metric, because we also reordered the variables, the line element has not changed:

 > $\mathrm{g_}\left[\mathrm{lineelement}\right]$
 ${{g}}_{{2}{,}{2}}{{\mathbf{ⅆ}}{}\left({y}\right)}^{{2}}{+}\left({2}{\mathbf{ⅆ}}{}\left({t}\right){{g}}_{{2}{,}{4}}{+}{2}{\mathbf{ⅆ}}{}\left({x}\right){{g}}_{{1}{,}{2}}\right){\mathbf{ⅆ}}{}\left({y}\right){+}{{g}}_{{4}{,}{4}}{{\mathbf{ⅆ}}{}\left({t}\right)}^{{2}}{+}{2}{{g}}_{{1}{,}{4}}{\mathbf{ⅆ}}{}\left({x}\right){\mathbf{ⅆ}}{}\left({t}\right){+}{{g}}_{{1}{,}{1}}{{\mathbf{ⅆ}}{}\left({x}\right)}^{{2}}{+}{\mathbf{ⅆ}}{}\left({z}\right)\left({\mathbf{ⅆ}}{}\left({z}\right){{g}}_{{3}{,}{3}}{+}{2}{\mathbf{ⅆ}}{}\left({t}\right){{g}}_{{3}{,}{4}}{+}{2}{\mathbf{ⅆ}}{}\left({x}\right){{g}}_{{1}{,}{3}}{+}{2}{{g}}_{{2}{,}{3}}{\mathbf{ⅆ}}{}\left({y}\right)\right)$ (15)
 > $\mathrm{normal}\left(-\right)$
 ${0}$ (16)

Redefine is particularly useful when working with metrics from the database of solutions to Einstein's equations, all of which reset the signature to (+ + + -) when loaded. For example:

 > $\mathrm{g_}\left[\left[12,21,1\right]\right]$
 $\mathrm{_______________________________________________________}$
 $\mathrm{Systems of spacetime coordinates are:}\left\{X=\left(x{,}y{,}\mathrm{\phi }{,}t\right)\right\}$
 $\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}\left\{X=\left(x{,}y{,}\mathrm{\phi }{,}t\right)\right\}$
 $\mathrm{The}\mathrm{McLenaghan, Tariq \left(1975\right)}\mathrm{metric in coordinates}\mathrm{Tupper \left(1976\right)}$
 $\mathrm{Parameters:}\left[a{,}k{,}\mathrm{κ0}\right]$
 $\mathrm{Comments:}k paramⅇtrizⅇs thⅇ most gⅇnⅇral ⅇlⅇctromagnⅇtic invariant with rⅇspⅇct to thⅇ last 3 Killing vⅇctors$
 $\mathrm{Domain:}\left[0
 $\mathrm{Assumptions:}\left[0
 $\mathrm{Resetting the signature of spacetime from}\left(\mathrm{+ - - -}\right)\mathrm{to}\left(\mathrm{+ + + -}\right)\mathrm{in order to match the signature in the database of metrics}$
 $\mathrm{_______________________________________________________}$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\right]$ (17)

Note above the message about resetting the signature; query about:

 > $\mathrm{Setup}\left(\mathrm{signature}\right)$
 $\left[{\mathrm{signature}}{=}{\mathrm{+ + + -}}\right]$ (18)

How would this metric [12, 21, 1] and coordinates $\left[t,x,y,\mathrm{\phi }\right]$ would look with the signature (+ - - -) ?

 > $\mathrm{Redefine}\left(\mathrm{all},\mathrm{tosignature}="+ - - -"\right)$
 $\left[{t}{,}{x}{,}{y}{,}{\mathrm{\phi }}\right]{,}\left[{?}\right]$ (19)

By replacing all by setall in the command above not only the list of coordinates and metric matrix form are returned but they are also set in one go.

As an example of redefining tetrads, load Tetrads and the metric [27, 37, 1] of the database of solutions to Einstein's equations

 > $\mathrm{with}\left(\mathrm{Tetrads}\right):$
 $\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_ah}\mathrm{letters to represent}\mathrm{tetrad}\mathrm{indices}$
 $\mathrm{Defined as tetrad tensors}\left(\mathrm{see ?Physics,tetrads}\right),{𝔢}_{a,\mathrm{\mu }},{\mathrm{\eta }}_{a,b},{\mathrm{\gamma }}_{a,b,c},{\mathrm{\lambda }}_{a,b,c}$
 $\mathrm{Defined as spacetime tensors representing the NP null vectors of the tetrad formalism}\left(\mathrm{see ?Physics,tetrads}\right),{l}_{\mathrm{\mu }},{n}_{\mathrm{\mu }},{m}_{\mathrm{\mu }},{\stackrel{&conjugate0;}{m}}_{\mathrm{\mu }}$
 $\mathrm{_______________________________________________________}$ (20)
 > $\mathrm{g_}\left[\left[27,37,1\right]\right]$
 $\mathrm{_______________________________________________________}$
 $\mathrm{Systems of spacetime coordinates are:}\left\{X=\left(z{,}\mathrm{zb}{,}r{,}u\right)\right\}$
 $\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}\left\{X=\left(z{,}\mathrm{zb}{,}r{,}u\right)\right\}$
 $\mathrm{The}\mathrm{Robinson and Trautman \left(1962\right)}\mathrm{metric in coordinates}\left[z{,}\mathrm{zb}{,}r{,}u\right]$
 $\mathrm{Parameters:}\left[P{}\left(z{,}\mathrm{zb}{,}u\right){,}H{}\left(X\right)\right]$
 $\mathrm{Comments:}aⅆmits gⅇoⅆⅇsic, shⅇarfrⅇⅇ, twistfrⅇⅇ null congruⅇncⅇ, rho=-1/r=rho_b$
 $\mathrm{Domain:}\left[P{}\left(z{,}\mathrm{zb}{,}u\right)::\mathrm{real}\right]$
 $\mathrm{Signature:}\left(\mathrm{+ + + -}\right)$
 $\mathrm{_______________________________________________________}$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\right]$ (21)
 > $\mathrm{CompactDisplay}\left(\right)$
 ${H}{}\left({z}{,}{\mathrm{zb}}{,}{r}{,}{u}\right){\mathrm{will now be displayed as}}{H}$
 ${P}{}\left({z}{,}{\mathrm{zb}}{,}{u}\right){\mathrm{will now be displayed as}}{P}$ (22)

The assumptions on the metric's parameters are

 > $\mathrm{Assume}\left(0
 $\left\{{u}{::}{\mathrm{real}}{,}{z}{::}{\mathrm{real}}{,}{\mathrm{zb}}{::}{\mathrm{real}}{,}{H}{::}{\mathrm{real}}{,}{P}{::}\left({0}{,}{\mathrm{\infty }}\right)\right\}{,}\left\{{r}{::}\left[{0}{,}{\mathrm{\infty }}\right){,}{u}{::}{\mathrm{real}}{,}{z}{::}{\mathrm{real}}{,}{\mathrm{zb}}{::}{\mathrm{real}}{,}{H}{::}{\mathrm{real}}{,}{P}{::}\left({0}{,}{\mathrm{\infty }}\right)\right\}{,}\left\{{r}{::}\left[{0}{,}{\mathrm{\infty }}\right){,}{H}{::}{\mathrm{real}}\right\}$ (23)

The null tetrad computed by the Maple system using a general algorithms is

 > $\mathrm{Setup}\left(\mathrm{tetrad}=\mathrm{null}\right):$
 > $\mathrm{e_}\left[\right]$
 ${{𝔢}}_{{a}{,}{\mathrm{\mu }}}{=}\left[\right]$ (24)

The book on solutions to Einstein's equations [1] suggests a different tetrad, free of radicals, which for our illustration purposes is

 > $\mathrm{e_}\left[a,\mathrm{\mu }\right]=\mathrm{Matrix}\left(4,4,\left[\left[0,0,0,-I\right],\left[0,\frac{rI\cdot 1}{P\left(z,\mathrm{zb},u\right)},0,0\right],\left[\frac{rI\cdot 1}{P\left(z,\mathrm{zb},u\right)},0,0,0\right],\left[0,0,-I,-H\left(z,\mathrm{zb},r,u\right)I\right]\right]\right)$
 ${{𝔢}}_{{a}{,}{\mathrm{\mu }}}{=}\left[\right]$ (25)

For this tetrad, however, IsTetrad returns false with the following message:

 > $\mathrm{IsTetrad}\left(\right)$
 ${\mathbit{Warning, the given components form a}}\left({\mathbit{null}}\right){\mathbit{tetrad,}}{\mathbit{with a covariant spacetime index}}{\mathbit{, only if you change the signature from}}\left({\mathbit{+ + + -}}\right){\mathbit{to}}\left({\mathbit{- - - +}}\right){\mathbit{. You can do that by entering \left(copy and paste\right):}}{\mathbit{Setup}}{}\left({\mathbit{signature}}{\mathbf{=}}{"- - - +"}\right)$
 ${\mathrm{false}}$ (26)

So you can change the signature to (---+) as indicated in the message, or instead of that redefine the tetrad, from the signature indicated, to work with the current signature (+++-)

 > $\mathrm{Redefine}\left(,\mathrm{fromsignature}="- - - +"\right)$
 ${{𝔢}}_{{a}{,}{\mathrm{\mu }}}{=}\left[\right]$ (27)

The returned tetrad matches the definition

 > $\mathrm{IsTetrad}\left(\right)$
 $\mathrm{Type of tetrad:}\mathrm{null}$
 ${\mathrm{true}}$ (28)

You can now set it and verify that

 > $\mathrm{Setup}\left(\right)$
 $\left[{\mathrm{tetrad}}{=}\left\{\left({1}{,}{4}\right){=}{1}{,}\left({2}{,}{2}\right){=}{-}\frac{{r}}{{P}}{,}\left({3}{,}{1}\right){=}{-}\frac{{r}}{{P}}{,}\left({4}{,}{3}\right){=}{1}{,}\left({4}{,}{4}\right){=}{H}\right\}\right]$ (29)
 > $\mathrm{e_}\left[\mathrm{definition}\right]$
 ${{𝔢}}_{{a}{,}{\mathrm{\mu }}}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}{=}{{\mathrm{\eta }}}_{{a}{,}{b}}$ (30)
 > $\mathrm{TensorArray}\left(\right)$
 $\left[\right]$ (31)

References

 [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.

Compatibility

 • 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.