Redefines the spacetime metric and list of coordinates according to a change in the signature, from any of the four possible signatures (- - - +), (+ - - -), (+ + + -) and (- + + +) to any of the other ones - Maple Programming Help

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Physics[Redefine] - Redefines the spacetime metric and list of coordinates 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 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. In this case entering Redefine(all, fromsignature = previous_signature) shows how would the list of coordinates a metric look if they were redefined from the previous signature to the one you set (i.e., the current one). If the result is as desired, you can use the output to call Setup and set things accordingly, or simpler: 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.

• 

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

withPhysics:

Setupmathematicalnotation=true,coordinates=cartesian

Default differentiation variables for d_, D_ and dAlembertian are:X=x,y,z,t

Systems of spacetime coordinates are:X=x,y,z,t

_______________________________________________________

coordinatesystems=X,mathematicalnotation=true

(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

GMatrix4,symbol=g,shape=symmetric

g1,1g1,2g1,3g1,4g1,2g2,2g2,3g2,4g1,3g2,3g3,3g3,4g1,4g2,4g3,4g4,4

(2)

Set the metric with this value, and in the same call query the value of the signature: it is (- - - +):

Setupsignature,g_=G

metric=1,1=g1,1,1,2=g1,2,1,3=g1,3,1,4=g1,4,2,2=g2,2,2,3=g2,3,2,4=g2,4,3,3=g3,3,3,4=g3,4,4,4=g4,4,signature=- - - +

(3)

Track the line element

g_lineelement

g3,3ⅆz2+2ⅆxg1,3+2g2,3ⅆyⅆz+g1,1ⅆx2+2g1,2ⅆxⅆy+g2,2ⅆy2+ⅆtⅆtg4,4+2ⅆxg1,4+2ⅆyg2,4+2ⅆzg3,4

(4)

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

Setupsignature=+---

Warning, a representation for the Dirac matrices is only implemented for an Euclidean or Minkowski spacetime; removing the previous 'standard' representaton

signature=+ - - -

(5)

Neither the metric nor the coordinates change: t is still in position 4 and the component (1, 1) of the metric is still g1,1:

Coordinates

Systems of spacetime coordinates are:X=x,y,z,t

X

(6)

g_[]

g_μ,ν=g1,1g1,2g1,3g1,4g1,2g2,2g2,3g2,4g1,3g2,3g3,3g3,4g1,4g2,4g3,4g4,4

(7)

This design gives you freedom to set things as necessary. However, after changing the signature we may also want to redefine the coordinates - place t in position 1 - and possibly also the metric reordering its lines and rows accordingly. For this 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

Redefinecoordinates,fromsignature=---+

t,x,y,z

(8)

Redefinemetric,fromsignature=---+

g4,4g1,4g2,4g3,4g1,4g1,1g1,2g1,3g2,4g1,2g2,2g2,3g3,4g1,3g2,3g3,3

(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 g1,1

Coordinates

Systems of spacetime coordinates are:X=x,y,z,t

X

(10)

g_[]

g_μ,ν=g1,1g1,2g1,3g1,4g1,2g2,2g2,3g2,4g1,3g2,3g3,3g3,4g1,4g2,4g3,4g4,4

(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

Redefinesetall,fromsignature=---+

X,g4,4g1,4g2,4g3,4g1,4g1,1g1,2g1,3g2,4g1,2g2,2g2,3g3,4g1,3g2,3g3,3

(12)

Now t in position 1 and the component (1,1) of the metric equal to g4,4

Coordinates

Systems of spacetime coordinates are:X=t,x,y,z

X

(13)

g_[]

g_μ,ν=g4,4g1,4g2,4g3,4g1,4g1,1g1,2g1,3g2,4g1,2g2,2g2,3g3,4g1,3g2,3g3,3

(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:

g_lineelement

g2,2ⅆy2+2ⅆtg2,4+2g1,2ⅆxⅆy+g4,4ⅆt2+2g1,4ⅆtⅆx+g1,1ⅆx2+ⅆzⅆzg3,3+2ⅆtg3,4+2ⅆxg1,3+2g2,3ⅆy

(15)

normal

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:

g_12,21,1

_______________________________________________________

Systems of spacetime coordinates are:X=t,x,y,φ

Default differentiation variables for d_, D_ and dAlembertian are:X=t,x,y,φ

The McLenaghan, Tariq (1975) metric in coordinates Tupper (1976)

Parameters: a,k,κ0

Comments: k paramⅇtrⅈzⅇs thⅇ most gⅇnⅇral ⅇlⅇctromagnⅇtⅈc ⅈnvarⅈant wⅈth rⅇspⅇct to thⅇ last 3 Kⅈllⅈng vⅇctors

Resetting the signature of spacetime from + - - - to - + + + in order to match the signature in the database of metrics

_______________________________________________________

g_μ,ν=1002y0a2x20000a2x202y00x24y2

(17)

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

Setupsignature

signature=- + + +

(18)

Redefineall,tosignature=---+

x,y,φ,t,a2x20000a2x20000x2+4y22y002y1

(19)

How would this metric [12, 21, 1] and coordinates t,x,y,φ would look with the original signature (- - - +) ?

Redefineall,tosignature=---+

x,y,φ,t,a2x20000a2x20000x2+4y22y002y1

(20)

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.

See Also

Coordinates, g_, normal, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup

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.

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