d'Alembertian differential operator with respect to the spacetime coordinates - Maple Programming Help

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Physics[dAlembertian] - d'Alembertian differential operator with respect to the spacetime coordinates

Calling Sequence

dAlembertian(A)

dAlembertian(A, [X])

Parameters

A

-

any mathematical expression or relation between expressions

[X]

-

(optional) list of differentiation variables with respect to which the d'Alembertian is to be computed

Description

• 

The dAlembertian command is a computational representation for the d'Alembert differential operator, displayed as a square box . The definition used is

A=μμA=gμ,ννμA

  

where the differential operator ∂__mu, represented in Maple by d_, is defined by

μ=xμ

  

and where xμ represents the contravariant spacetime vector of a coordinate system defined using Coordinates or Setup.

• 

In a galilean system (Euclidean or Minkowski), both xμ and the differential of the coordinates xμ are vectors (tensors with 1 index), and so μ is a vector representing

μ=xμ

  

where xμ=gμ,νxν is the covariant spacetime vector, and so μμA=μμA.

• 

Unlike the galilean case, in a curvilinear system of coordinates, neither x__mu nor μ are vectors and the formula above for μ loses its meaning; instead, the convention used in the Physics package (it becomes the one above only in the galilean case) is

μ=gμ,νν

  

Hence in a non-galilean spacetime μμAμμA, because the metric gμ,ν depends on the coordinates and so the two factors in the right-hand-side above do not commute, and the formulas relating the second application of d_[mu] to the dAlembertian are

μμA=A

μμA=μgμ,ννA+A

  

reflecting the nontensorial character in curvilinear coordinates of μ  and of the dAlembertian.

• 

The %dAlembertian command is the inert form of dAlembertian; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command. To obtain the form of dAlembertian or its inert representation as a sum of diff constructions, use convert/diff.

• 

As in the case of d_[mu], when only one argument is given to dAlembertian, the differentiation variables are the ones set by the Setup command, typically x1,x2,x3,x4 (x0 is automatically assigned to x4), represented by X (see Coordinates). To change these default differentiation variables, see Setup.

• 

Regardless of the existence of default differentiation variables, you can always call dAlembertian with two arguments, where the second argument is a list with the differentiation variables you want; in this case, the list should have as many symbols as the spacetime dimension, which by default is 4 but can be set to any value by the Setup command.

• 

Some automatic simplifications are carried out each time dAlembertian(A) is called, as follows:

  

- If A does not depend on the differentiation variables, then 0 is returned.

  

- If A is an unknown function (the rule for its derivative is unknown), a Dirac delta function, or a derivative, then the result is returned unevaluated, as dAlembertian(A).

  

- If A is of the form d_μB, then d_[mu](dAlembertian(B)) is returned.

  

- Otherwise, dAlembertian(A) is computed calling d_ as in μμA.

• 

In general, to accomplish differentiation, dAlembertian calls d_, which in turn makes calls to the Physics/diff command, which in turn uses the standard Maple diff command with appropriate arguments. In this way, any user-defined differentiation rule in the library or that you created, such as for a function foo of the form `diff/foo`, is automatically taken into account by dAlembertian.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Set the default differentiation variables for dAlembertian and d_, define a spacetime tensor function A, and use the enhanced display scheme of the differential equation packages.

Setupdifferentiationvariables=X

Default differentiation variables for d_, D_ and dAlembertian are:X=x1,x2,x3,x4

Systems of spacetime coordinates are:X=x1,x2,x3,x4

Default differentiation variables for d_, D_ and dAlembertian are:X=x1,x2,x3,x4

differentiationvariables=X

(2)

DefineA,B

Defined objects with tensor properties

A,B,γμ,σμ,Xμ,μ,gμ,ν,εα,β,μ,ν

(3)

PDEtoolsdeclaref,A,B,ΦX

fXwill now be displayed asf

AXwill now be displayed asA

BXwill now be displayed asB

PhiXwill now be displayed asΦ

(4)

The dAlembertian operator is the double application of d_[mu], with the index contracted.

d_μd_μfX

dAlembertianfX,X

(5)

In the default 4 = 3+1 dimensional Minkowski spacetime, the form of dAlembertian as a sum of diff constructions is

convert,diff

diffdifffX,x1,x1diffdifffX,x2,x2diffdifffX,x3,x3+diffdifffX,x4,x4

(6)

The dAlembertian deals normally with derivatives expressed in terms of any of the Maple differential operators, including D, diff, and d_. It also distributes over sums and products.

d_μfXAμX+BμX

d_μfX,XA~muX+fXd_μA~muX,X+d_μB~muX,X

(7)

dAlembertian

dAlembertianAμX,Xd_~mufX,X+2d_μd_νfX,X,Xd_~nuA~muX,X+d_μdAlembertianfX,X,XA~muX+dAlembertianfX,Xd_μA~muX,X+2d_~nufX,Xd_νd_~muAμX,X,X+fXd_~mudAlembertianAμX,X,X+d_μdAlembertianB~muX,X,X

(8)

d_μfXAνX

d_μfX,XAνX

(9)

dAlembertian

d_μfX,XdAlembertianAνX,X+d_μdAlembertianfX,X,XAνX+2d_~alphaAνX,Xd_αd_μfX,X,X

(10)

Because dAlembertian is a second order differential operator (dAlembertian(f) = d_[mu](d_[mu](f))), the differentiation rule, when applied to a product as in above, requires decomposing the dAlembertian operation into the sequence of d_[mu] operations. Note also the introduction of a dummy index alpha, which could be any spacetime index; these indices are chosen after checking that they are not assigned and not already present in the given expression at the time of introducing them.

Regardless of having set the default differentiation variables to X, you can call dAlembertian or d_ with other coordinates as differentiation variables. For example, set one more coordinate system:

CoordinatesY

Systems of spacetime coordinates are:X=x1,x2,x3,x4,Y=y1,y2,y3,y4

X,Y

(11)

dAlembertianfX

dAlembertianfX,X

(12)

dAlembertianfY,Y

fY,Y

(13)

Note that the output above displays the differentiation variables y1,y2,y3,y4. This is so because they are not the "default differentiation variables;" if you set them to be y1,y2,y3,y4, then the omitted differentiation variables are y1,y2,y3,y4, and the other differentiation variables (namely, x1,x2,x3,x4) are displayed.

Setupdifferentiationvariables=Y

Default differentiation variables for d_, D_ and dAlembertian are:Y=y1,y2,y3,y4

differentiationvariables=Y

(14)

dAlembertianfX

0

(15)

dAlembertianfX,X

dAlembertianfX,X

(16)

dAlembertianfY,Y

fY

(17)

The dAlembertian enters the classical field equations in various models; this is the electromagnetic field tensor.

Setupdifferentiationvariables=X

Default differentiation variables for d_, D_ and dAlembertian are:X=x1,x2,x3,x4

differentiationvariables=X

(18)

Fμ,νd_μAνXd_νAμX

d_μAνX,Xd_νAμX,X

(19)

Maxwell equations result from taking the functional derivative of the Action. Use delay evaluation quotes to display the operation unevaluated, then perform the operation on the next line.

'Fundiff'IntcFμ,ν2,X,AρY

FundiffIntIntIntIntd_μAνX,Xd_νAμX,X2,x1=∞..∞,x2=∞..∞,x3=∞..∞,x4=∞..∞,AρY

(20)

subsY=X,

2d_μd_νA~nuX,X,X2dAlembertianAμX,Xg_~mu,~rho+2dAlembertianAνX,X+2d_μd_νA~muX,X,Xg_~nu,~rho

(21)

To simplify the contracted spacetime indices, use the Simplify command.

Simplify

4dAlembertianA~rhoX,X+4d_νd_~rhoA~nuX,X,X

(22)

The Lagrangean of the lambdaPhi^4 model, the corresponding Action, and the field equations:

L1d_μΦXd_μΦX2m2ΦX22+λΦX44

12d_μΦX,Xd_~muΦX,X12m2ΦX2+14λΦX4

(23)

'Fundiff'IntcL,X,ΦY

FundiffIntIntIntInt12d_μΦX,Xd_~muΦX,X12m2ΦX2+14λΦX4,x1=∞..∞,x2=∞..∞,x3=∞..∞,x4=∞..∞,ΦY

(24)

subsY=X,

dAlembertianΦX,XΦXΦX2λ+m2

(25)

See Also

Coordinates, d_, Define, diff, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup