set and query the computational environment for the Physics package commands

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Physics[Setup] - set and query the computational environment for the Physics package commands

	Calling Sequence
	Setup( ) Setup(parameter) Setup(parameter = value, other_options)

Parameters

parameter	-	(optional) name (can be the keyword query); query the value of the specified parameter
parameter = value	-	(optional) equation of the form name = value; set the value of one or many of the parameters of the computational environment
other_options	-	(optional) any of the keywords: advanced, additionally, clear, conventions, default, keywords, query, quiet, redo, savecurrent

Description

•

The Setup command is used to set, clear and query the computational environment used by the commands of the Physics package. To avoid having to remember the relatively large number of parameters that can be set, if you type the parameter misspelled, or just a portion of it, a matching against the existing parameters is performed, and when there is only one match, the input is automatically corrected. For example, to see the list of settable parameters, enter Setup(keywords), or simply pass a portion of the word , such as Setup(k).

•

Alternatively, you can launch the graphical interface of the package, a setup applet, and set the values of these parameters directly from it. For this purpose, call Setup with no arguments, as in Setup(), and after closing the applet the entire setup will be returned on the screen as a list of equations with the keywords on the left-hand side and their values on the right-hand side.

•	When called with the argument conventions, Setup displays a brief summary of the computational conventions used by the Physics and Physics[Vectors] packages.

•	A summary of the parameters that can be set, the corresponding values when Physics is loaded, and the expected types of new values when you set them by using Setup are as follows:

Parameter	Value	Expected type of new value

abbreviations	= true	truefalse
algebrarules	= none	set((Anti)Commutator = algebraic)
anticommutativeprefix	= none	{symbol, set(symbol)}
bracketbasis	= none	name
bracketrules	= none	set(Bracket(Bra, Ket) = algebraic)
coordinatesystems	= none	{symbol, set(symbol)} # Capital letters only
Dgammarepresentation	= standard	identical(standard, chiral, Majorana)
differentiationvariables	= none	symbol # Capital letter
dimension	= 4	And(posint, satisfies(d -> d > 1))
gaugeindices	= none	identical(greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, uppercaselatin)
hermitianoperators	= none	{name, set(name)}
levicivita	= galilean	identical(galilean, nongalilean)
mathematicalnotation	= none	truefalse
metric	= Minkowski	<keyword, line_element, Matrix, set(nonzero components)>
noncommutativeprefix	= none	{name={algebraic, range}, set(name={algebraic, range})}
quantumbasisdimension	= none	{name, set(name)}
quantumcontinuousbasis	= none	{name, set(name)}
quantumdiscretebasis	= none	{name, set(name)}
quantumoperators	= none	{name, set(name)}
signature	= `-`	identical(`+`, `-`)
spacetimeindices	= greek	identical(greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, uppercaselatin)
spaceindices	= none	identical(greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, uppercaselatin)
spinorindices	= none	identical(greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, uppercaselatin)
usephysicsevaluator	= true	truefalse
vectorpostfix	= none	symbol

Note: when the Vectors subpackage is loaded, or any of its commands is called by their long forms, the vectorpostfix is automatically set to the underscore .

•

Mathematical notation: in the standard GUI, when Physics is loaded and mathematicalnotation is set to true, anticommutative and noncommutative variables are displayed in different colors, mathematical functions are displayed in textbook notation, if Physics[Vectors] is loaded, non-projected vectors and unit vectors are displayed with an arrow and a hat on top, respectively, the vectorial differential operators (Nabla, Curl, etc.) with an upside down triangle, and more. The same enhancement in the display is obtained by using the Options dialog (go to Tools -> Options, select the Display tab, and set the Typesetting level dropdown to Extended; see also interface). The notational aspects related to the Physics package can be changed to some point; the corresponding starting and expected values are:

Parameter	Value	Expected type (for details see below)

anticommutativecolor	= purple	identical(none, symbol) # one of 16 colors
noncommutativecolor	= olive	identical(none, symbol) # one of 16 colors
unitvectordisplay	= hat	identical(none, hat)
vectordisplay	= arrow	identical(none, arrow, bold, halfarrow)

•	To set the values of these parameters (could be many at the same time), call the Setup command with equations having the parameter keywords on the right-hand side, and their value on the right-hand side. The output of Setup is a list of equations of the same kind, echoing the values set.

•	When setting values, some explanatory messages are frequently displayed on the screen. To avoid that verbosity (will not avoid the list output), use the optional argument quiet.

•	To query the current value of any of these parameters, enter Setup(parameter) or Setup(query, parameter), where parameter is one of the keywords in the first column above. Entering Setup(query) returns the current values for all of the parameters.

•	To clear the value of one or many of these parameters, pass the keyword clear together with the parameter(s) that you want to clear; the value is set to none.

•

For most parameters, when setting their values, their previous settings are not discarded. For example, when you set some symbols to represent quantum operators, the previous quantum operators remain set as such. When you want to set the value of a parameter and discard the previous setting, use the optional argument redo.

•	When setting a parameter automatically discards the previous setting, but you want to preserve that setting, use the optional argument additionally.

•

You can also load an advanced setup (available but not loaded automatically when you load Physics) and save the setup existing at any moment, so that the next time Physics is loaded, it automatically loads with the saved setup. For these purposes, call Setup with the optional argument advanced or savecurrent, respectively. To restore the starting setup, enter Setup(default). These are the values loaded with the advanced setup:

Parameter	Advanced value

abbreviations	= true
algebrarules	= {_ = 0,
	_ = 0,
	_ = }
anticommutativeprefix	= {theta, Q}
bracketrules	=
coordinatesystems	= {X}
Dgammarepresentation	= standard
differentiationvariables	= [X]
dimension	= 4
gaugeindices	= uppercaselatin
hermitianoperators	= {R, P}
levicivita	= galilean
mathematicalnotation	= true
metric	Minkowski
noncommutativeprefix	= {Z}
quantumcontinuousbasis	= {R, P}
quantumdiscretebasis	= {u}
quantumoperators	= none
signature	= `-`
spacetimeindices	= greek
spinorindices	= lowercaselatin
usephysicsevaluator	= true
vectorpostfix	= _

The parameters of the computational environment

A description of the parameters and the setting action associated with each of them is as follows.

abbreviations

true or false.

The default for this option is true, and so the abbreviations for KroneckerDelta and for LeviCivita are set when you load the package. Therefore, and KroneckerDelta can be used to perform the same operation. When abbreviations is set to false, these two abbreviations are unset. Alternatively, for these two or other Maple commands, you can set a macro to achieve the same results.

algebrarules

An equation with a Commutator or AntiCommutator on the left-hand side and an algebraic expression representing the result for that operation on the right-hand side.

The operands in the (Anti)Commutator in the left-hand side of the equations are expected to be names, functions, or indexed functions, in which case the indices and functionality will be taken as parameters of the operation and expected to appear when arguments are passed to perform the operation (see examples of this in Commutator). It is also possible to pass a set of (Anti)Commutator rules at once, as a set (by enclosing them between { }). These are used by Commutator and AntiCommutator to return results accordingly. Tip: when setting algebra rules, use the inert forms %Commutator and %AntiCommutator instead of the active forms Commutator, AntiCommutator; this may help to prevent a premature evaluation of the Commutator or AntiCommutator.

anticommutativecolor

aqua, black, blue, fuchsia, gray, green, lime, maroon, navy, olive, purple, red, silver, teal, or yellow.

When mathematicalnotation is set to true, the anticommutativecolor setting causes the Maple Standard GUI to display anticommutative variables in the selected color.

anticommutativeprefix

Any valid sequence of characters.

Sets the prefix of characters entering the symbols representing anticommutative variables; that is, variables and for which = . Note: in the context of the Physics package, the multiplication operator, * is commutative, anticommutative, or noncommutative, depending on the corresponding commutation properties of its operands (see type/commutative).

bracketbasis

A name .

represents a space of quantum states, and the corresponding vectors (labelled with the same name) will be used when calling the Bracket command with a single argument. For example, will be computed as .

bracketrules

An equation with a Bracket on the left-hand side and an algebraic expression representing the result for that bracket (scalar product) on the right-hand side.

The bracket can be of the form or for some Bra and Ket and , and the quantum operator should have all the quantum numbers of and represented by symbols. The right-hand side is an algebraic expression depending on these symbols (see examples in Bracket). It is also possible to pass a set of bracket rules at once, as a set (by enclosing them between { }), as in . As a shortcut, it is valid to pass the right-hand side directly, without using an equation with the keyword bracketrules. These bracket rules are saved internally as procedures having the quantum numbers as parameters, and are used by Bracket when computing brackets. Tip: when setting bracket rules, use the inert form %Bracket instead of the active form Bracket; this may help to prevent a premature evaluation of the bracket.

coordinatesystems

Any of the keywords cartesian, cylindrical or spherical, or a capital letter, or a set of them.

Aliases are set by using the alias command so that the capital letter represents the sequence of associated coordinates, e.g., the lowercase letters postfixed with a positive integer ranging from 1 to the dimension. For example, coordinatesystems = X sets the alias , and sets . See also Coordinates

Dgammarepresentation

standard, chiral, or Majorana.

Sets the representation to be used for the Dirac matrices. Note: the representation depends on the dimension of spacetime, and for a dimension lower than 4, only the standard representation is implemented.

differentiationvariables

A capital letter, or a set of them.

Sets the default differentiation variables for the d_, D_ and d'Alembertian differentiation commands. These variables are automatically set when you define the first system of coordinates. If the setting for differentiationvariables is cleared (options clear, differentiationvariables), you can still use these three differentiation commands, provided you explicitly indicate the differentiation variables as a second argument.

dimension

A positive integer greater than 1.

Sets the dimension of the spacetime manifold. When this parameter is set, all of the definitions sensitive to the dimension are reset, such as aliases set by Coordinates, the representation for the Dirac matrices, and all tensor definitions introduced by Define. The behavior of the spacetime manifold depends also on the signature, which is another of the parameters that can be set (see below). As a shortcut, you can specify both the dimension and signature in one equation, with a list on the right-hand side, as in , where is the dimension and is one of `+`, `-`, representing the signature, associated with a Euclidean or Minkowski spacetime, respectively. For example, the default value when you load the Physics package is , representing a 3 + 1 Minkowski spacetime with signature (-,-,-,+). Remark: the conventions for the dimension only affect the results of computations with objects defined by you using Define or known because they were defined automatically when Physics was loaded.

gaugeindices

greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, or uppercaselatin.

Sets the identifier for gauge indices in that every index found in a tensor object defined by the Define command that has the gaugeindices prefix as a root and is followed by a positive integer will be interpreted as a gauge index.

hermitianoperators

A name, or a set of them.

Indicates that these names represent linear hermitian quantum operators. This information is taken into account when computing, for example, the Dagger of the operator. When an operator is set to be Hermitian, it is also automatically set to be a noncommutative quantum operator (see quantumoperators below).

levicivita

galilean or nongalilean

Indicates that the LeviCivita (pseudo)tensor represents either the galilean totally antisymmetric symbol , whose components are all equal to 0, 1 or -1, or its generalziation to curvilinear coordinates (see definition in LeviCivita).

mathematicalnotation

true or false.

Provides a noticeable improvement in the display of expressions by using traditional mathematical notation for Physics objects. For example, vectors are displayed with an arrow on top, the vectorial differential operators (such as Nabla and Laplacian) are displayed with an upside down triangle, anti and noncommutative variables, and so quantum operators, are displayed in different colors, mathematical functions are displayed using textbook notation, and more. The same enhancement in the display can be obtained by using the Options dialog (go to Tools -> Options, select the Display tab, and set the Typesetting level dropdown to Extended; see also interface). To change this mathematical notation to your preferred one, see anticommutativecolor, noncommutativecolor, and vectordisplay.

metric

<keyword, line_element, Matrix, set(nonzero components)>

Indicates the spacetime metric; instead of the word metric the left-hand-side can also be the metric tensor itself, as in g_[mu, nu]. The right-hand-side can be a keyword, for instance any of arbitrary, Euclidean, Minkowski, Schwarzschild, Tolman or related to the metrics of Chapter 12 of Exact Solutions of Einstein's Field Equations - see references at the end. Note that you can pass the keyword incomplete or misspelled, in which case a searchtext is performed among the keywords understood by g_ and if a match is found the metric is set accordingly, or if many matches are found then corresponding information is displayed on the screen. See g_ for examples.

The right-hand-side in metric = ... or g[mu, nu] = ... can also be a line element, quadratic in the differentials of the coordinates (you need to set the Coordinates first), passed using either of d_ or %d_", e.g. as in d_(x), d_(y), or by concatenating d' with the coordinate, e.g. as in dx, dy, ...

Finally, the right-hand-side in metric = ... can also be a 2 x 2 symmetric Matrix, in which case Setup extracts from it the nonzero elements using ArrayElems and sets the metric with that information; or it can also be the set returned by ArrayElems itself.

noncommutativecolor

aqua, black, blue, fuchsia, gray, green, lime, maroon, navy, olive, purple, red, silver, teal, or yellow.

When mathematicalnotation is set to true, the noncommutativecolor setting causes the Maple Standard GUI to display noncommutative variables in the selected color.

noncommutativeprefix

Any valid sequence of characters; the same as anticommutativeprefix.

Sets the identifier for noncommutative variables, that is, variables that are neither commutative nor anticommutative.

quantumbasisdimension

An equation with a Ket-label on the left-hand side and a positive integer, an algebraic expression representing it, or a range (of the form where and can also be half-integers) on the right-hand side; or a set of such equations.

It is valid to state different dimensions for different quantum numbers of Kets of a single basis. For that purpose, if the label of the basis is A, you can set the dimension of each quantum number entering , where and are positive integers, algebraic expressions representing them, or ranges of the form . This dimension of a quantum basis is used to construct projectors when you call the Projector command.

quantumcontinuousbasis

A name, or a set of them.

Bras and Kets labeled with this name will be considered state vectors of a continuous space of quantum states. This information is taken into account when computing Brackets and Projectors.

quantumdiscretebasis

A name, or a set of them; equivalent to quantumcontinuousbasis.

Indicates that Bras and Kets labelled with the given names belong to a discrete space of quantum states. However, unlike the quantumcontinuousbasis case, when the label (first argument) in a Bra or a Ket has not been set to represent a continuous or discrete space of states, it is assumed that it represents a discrete space.

quantumoperators

A name, or a set of them.

Indicates that these names represent linear quantum operators. This information is taken into account when computing Brackets and scalar products between these quantum operators and Bras or Kets.

signature

`+` or `-`.

This parameter is associated with a Euclidean or Minkowski spacetime manifold, respectively. See dimension above for details.

spacetimeindices

greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, or uppercaselatin; the same as gaugeindices.

Sets the identifier for spacetime indices in that every index found in a tensor object defined by the Define command that has the spacetimeindices prefix as a root and is followed by a positive integer will be interpreted as a spacetime index.

spaceindices

same as spacetimeindices

spinorindices

greek, lowercaselatin, lowercaselatin_ah, lowercaselatin_is, or uppercaselatin; the same as spacetimeindices and gaugeindices.

Sets the identifier for spinor indices.

usephysicsevaluator

true or false.

The default for this option is true, and so each pair of contracted tensor indices in a tensorial expression is automatically rewritten as one covariant and one contravariant. If you are handling tensorial expressions of thousands o terms and experience a slowdown in performance you can try changing this setting to false.

vectordisplay

none, arrow, bold, or halfarrow.

When mathematicalnotation is set to true, the vectordisplay setting causes the Maple Standard GUI to display non-projected vectors with an arrow or halfarrow on top, or in bold, as specified.

vectorpostfix

Any valid sequence of characters.

Sets the identifier for non-projected 3-D vectors of the Physics[Vectors] subpackage.

Examples

(1)

(2)

Query the computational setup that is loaded with the Physics package.

(3)

Compute the Projector for a basis of state vectors that has not been set: it is assumed that the corresponding space is discrete.

Sum(Physics:-`*`(Physics:-Ket(A, n), Physics:-Bra(A, n)), n = 0 .. 2)

(4)

Set a label for a continuous space of quantum states; note you do not need to remember the exact keyword to be used, the matching mechanism will also attempt to match a part of a keyword (returning a list of options when there is more than one partial match).

(5)

The corresponding projector is now a triple integral.

Int(Int(Int(Physics:-`*`(Physics:-Ket(R, x, y, z), Physics:-Bra(R, x, y, z)), x = -infinity .. infinity), y = -infinity .. infinity), z = -infinity .. infinity)

(6)

Note also the difference in behavior between computing the Bracket in state vectors of discrete and continuous spaces of states. Inert representations for these commands have % as a prefix, and the computation can then be performed by using the value command.

(7)

(8)

(9)

(10)

Let and represent "position" and "momentum" state vectors; their scalar product is unknown to the system.

(11)

Let their scalar product be the corresponding normalized plane wave (see references below).

Physics:-Bracket(Physics:-Bra(R, x, y, z), Physics:-Ket(P, p[x], p[y], p[z])) = exp(I*(x*p[x]+y*p[y]+z*p[z]))/(2*Pi)^(3/2)

Physics:-Bracket(Physics:-Bra(R, x, y, z), Physics:-Ket(P, p[x], p[y], p[z])) = (1/4)*2^(1/2)*exp(I*(x*p[x]+y*p[y]+z*p[z]))/Pi^(3/2)

(12)

To set the above as a rule when computing Brackets between the vectors and , use:

$[bracketrules = {%Bracket(%Bra(R, x, y, z), %Ket(P, p[x], p[y], p[z])) = (1/4)*2^(1/2)*exp(I*(x*p[x]+y*p[y]+z*p[z]))/Pi^(3/2)}]$

(13)

Now this rule is used automatically.

(14)

(1/4)*2^(1/2)*exp(I*(a*d+b*e+c*f))/Pi^(3/2)

(15)

(16)

Int(Int(Int(Physics:-`*`(Physics:-Ket(P, p[1], p[2], p[3]), Physics:-Bra(P, p[1], p[2], p[3])), p[1] = -infinity .. infinity), p[2] = -infinity .. infinity), p[3] = -infinity .. infinity)

(17)

Int(Int(Int((1/4)*2^(1/2)*exp(I*(x[1]*p[1]+x[2]*p[2]+x[3]*p[3]))*Physics:-Bra(P, p[1], p[2], p[3])/Pi^(3/2), p[1] = -infinity .. infinity), p[2] = -infinity .. infinity), p[3] = -infinity .. infinity)

(18)

Int(Int(Int((1/4)*Physics:-Ket(P, p[1], p[2], p[3])*2^(1/2)*exp(-I*(x[1]*p[1]+x[2]*p[2]+x[3]*p[3]))/Pi^(3/2), p[1] = -infinity .. infinity), p[2] = -infinity .. infinity), p[3] = -infinity .. infinity)

(19)

In the result above, to compute , the rule for has been used, but you can also state the rule for in addition to the rule for , by using the keyword additionally or stating the two rules at the same time.

A rule can have the same Bra and Ket in its definition and be given using inert commands (same command name preceded by %).

(20)

$[bracketrules = {%Bracket(%Bra(R, x, y, z), %Ket(P, p[x], p[y], p[z])) = (1/4)*2^(1/2)*exp(I*(x*p[x]+y*p[y]+z*p[z]))/Pi^(3/2), %Bracket(%Bra(R, x0, y0, z0), H, %Ket(R, x1, y1, z1)) = E(x1-x0, y1-y0, z1-z0)}]$

(21)

(22)

(23)

Define two quantum operators and a commutator algebra for them.

(24)

(25)

(26)

(27)

In the output of Setup(query), at the beginning, notice that there is no anticommutative prefix set. To work with anticommutative variables, set the prefix to be used to distinguish them; for example, set it to be theta.

$[anticommutativeprefix = {_lambda, theta}]$

(28)

Now theta alone, followed by a positive integer, or indexed as in where is any valid maple object, is interpreted as an anticommutative variable.

(29)

(30)

After loading the Physics package, you can use delay evaluation quotes to prevent the normalization of powers or products. Then simply execute the output again to evaluate.

(31)

(32)

The second mixed derivatives with respect to different anticommutative variables anticommute (note the use of %diff, which could be Diff, the inert representations for diff, turned active by using value).

(33)

(34)

Set the vectorpostfix identifier for non-projected 3-D vectors and load the Physics[Vectors] package.

(35)

(36)

So, in the framework of the Physics[Vectors] package, any symbol ending with the underscore is now interpreted as a non-projected vector.

(37)

(38)

(39)

Changing the vectorpostfix; the matching mechanism is used to avoid typing the entire keyword.

Now, neither nor are vectors, and and are non-projected vectors.

(40)

Error, (in &x) expected a vector as first argument, received: B_

(41)

There is no representation attached to the Dirac matrices Dgamma, so the following command returns itself.

(42)

(43)

Set the Majorana representation for Dirac matrices.

(44)

In the Majorana representation implemented in Maple, all of the components of these matrices are purely imaginary.

array( 1 .. 4, 1 .. 4, [( 4, 1 ) = (0), ( 4, 4 ) = (-I), ( 3, 1 ) = (0), ( 3, 4 ) = (0), ( 1, 3 ) = (0), ( 4, 2 ) = (0), ( 2, 2 ) = (-I), ( 2, 1 ) = (0), ( 2, 4 ) = (0), ( 3, 3 ) = (I), ( 2, 3 ) = (0), ( 3, 2 ) = (0), ( 4, 3 ) = (0), ( 1, 1 ) = (I), ( 1, 4 ) = (0), ( 1, 2 ) = (0) ] )

(45)

Set a working 3-D Euclidean space. The following command is equivalent to Setup(dimension = 3, signature = `+`).

(46)

Now define as an object having tensor properties; in addition, specify that has only one index.

${A[mu], Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, mu, nu]}$

(47)

Differentiate with respect to .

(48)

Because the dimension is equal to 3, this evaluates to:

(49)

or directly differentiate with respect to , so that:

diff(A[mu], A[mu])

(50)

What is known about ?

(51)

When changing the dimension in the middle of the session, the previous definitions set by Define are erased.

(52)

(53)

array( 1 .. 2, 1 .. 2, [( 2, 2 ) = (0), ( 2, 1 ) = (I), ( 1, 1 ) = (0), ( 1, 2 ) = (-I) ] )

(54)

Setting the spacetime metric passing a keyword: it automatically sets the corresponding Coordinates accordingly, and returns the set of nonzero elements of the metric Matrix

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}$

Warning, a representation for the Dirac matrices is not implemented for a spacetime that is not Euclidean or of Minkowski type. Removing the current '%a' representation

$[metric = {(1, 1) = -(diff(R(t, r), r))^2/(1+2*E(r)), (2, 2) = -R(t, r)^2, (3, 3) = -R(t, r)^2*sin(theta)^2, (4, 4) = 1}]$

(55)

To see the matrix form of the metric you can enter g_ without indices

Physics:-g_[mu, nu] = Matrix(%id = 18446790986301524798)

(56)

When there is no exact match, a searchtext is performed against the keywords known, and if there is a single match, the metric is set accordingly

$[metric = {(1, 1) = r/(-r+2*m), (2, 2) = -r^2, (3, 3) = -r^2*sin(theta)^2, (4, 4) = (r-2*m)/r}]$

(57)

When there are many matches, the choices are displayed and you can choose

[12, 34, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous"], "Comments" = ["All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space"]]

[12, 38, 1] = ["Authors" = ["Siklos (1985)"], "PrimaryDescription" = "PureRadiation", "SecondaryDescription" = ["Homogeneous", "PureRadiation"], "Comments" = ["Generic k value, k <> -1 (see [12, 6, 1]), 1/2 (see [12,38,2]), 3/2 (see [12,38, 5]), 2 (see [12,38, 3] and [12, 38, 4]). Note the k = 1/2 is Petrov type "O" ."]]

[12, 38, 3] = ["Authors" = ["Siklos (1985)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["PureRadiation", "Homogeneous"], "Comments" = ["This is _k = 2 and _epsilon = 1 in [12, 38, 1]. Only _epsilon = 1 gives a Maxwell field"]]

[12, 38, 4] = ["Authors" = ["Siklos (1985)"], "PrimaryDescription" = "PureRadiation", "SecondaryDescription" = ["PureRadiation", "Homogeneous"], "Comments" = ["This is _k = 2 and _epsilon = -1 in [12, 38, 1]. Only _epsilon = 1 gives a Maxwell field"]]

Warning, found more than one match for the keyword 'sik', as seen above. Please refine your 'keyword' or re-enter the metric 'g_[...]' with the list of three numbers identifying the metric, for example as in g_[[12, 34, 1]] or Setup(metric = [12, 34, 1])

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (v, y, z, u)}$

$[metric = {(1, 1) = -3*y^(-2+2*k)*epsilon/abs(Lambda), (1, 4) = -(3/2)/(y^2*abs(Lambda)), (2, 2) = 3/(y^2*abs(Lambda)), (3, 3) = 3/(y^2*abs(Lambda))}]$

(58)

Setting the metric by passing the line element, as an expression quadratic in the differentials expressed using d_ or %d_. In order to use as a coordinate, clear first its anticommutative character

$[anticommutativeprefix = {_lambda}]$

(59)

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}$

(60)

(61)

$[metric = {(1, 1) = r/(-r+2*m), (2, 2) = -r^2, (3, 3) = -r^2*sin(theta)^2, (4, 4) = (r-2*m)/r}]$

(62)

The Minkowski line element, expressing the differentials concatenating d to the coordinates

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Setting the metric passing a Matrix or its nonzero components

>	$G := Matrix(4, 4, {(1, 1) = r/(-r+2m), (2, 2) = -r^2, (3, 3) = -r^2sin(theta)^2, (4, 4) = (r-2*m)/r}, shape = symmetric)$

Matrix(%id = 18446790986361127150)

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$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}$

$[coordinatesystems = {X}, metric = {(1, 1) = r/(-r+2*m), (2, 2) = -r^2, (3, 3) = -r^2*sin(theta)^2, (4, 4) = (r-2*m)/r}]$

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>	$Setup(coordinates = cartesian, metric = {(1, 1) = -3y^(-2+2k)epsilon/abs(Lambda), (1, 4) = -3/(2y^2abs(Lambda)), (2, 2) = 3/(y^2abs(Lambda)), (3, 3) = 3/(y^2*abs(Lambda))})$

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x, y, z, t)}$

$[coordinatesystems = {X}, metric = {(1, 1) = -3*y^(-2+2*k)*epsilon/abs(Lambda), (1, 4) = -(3/2)/(y^2*abs(Lambda)), (2, 2) = 3/(y^2*abs(Lambda)), (3, 3) = 3/(y^2*abs(Lambda))}]$

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By default, usephysicsevaluator is set to true so that contracted indices in tensorial expressions are automatically rewritten as one covariant one contravariant.

${A, B, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

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When the indices are already given as one covariant, one contravariant, they are not changed

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If however you are working with tensorial expressions of thousands of terms and experience a slowdown in performance you can try setting the value to false

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The indices now remain the way you enter them (to enter a contravariant index, prefix it with ~)

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	See Also
	Bra, Bracket, Conventions, Coordinates, d_, D_, dAlembertian, Define, Dgamma, g_, Ket, Physics, Physics conventions, Physics examples, Physics/*, Physics[.], Projector, Vectors

References

Cohen-Tannoudji, C.; Diu, B.; and Laloe, F. Quantum Mechanics, Chapter II. Paris, France: Hermann, 1977.

Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. Cambridge University Press, 2003.

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