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Physics[D_] - indexed covariant differential operator with respect to curvilinear spacetime coordinates

Calling Sequence

D_[mu](A)

D_[mu](A, [X])

D_(A)

D_(A, [X])

Parameters

mu

-

a name representing an integer number between 0 and the spacetime dimension, can also be the number itself or preceded by ~ to represent a contravariant index

A

-

any mathematical expression or relation between expressions, or a set or list of them, or an rtable 

[X]

-

a list of differentiation variables to which the index mu refers

Description

• 

The D_[mu] command is a computational representation for μ, the covariant differential operator, a tensor, the generalization of μ to curvilinear coordinates.

• 

D_ can also be used without an index, as in D_(A) displayed as A, in which case it represents the total differential in curvilinear coordinates, and the output comes automatically expanded as D_[mu](A) * d_(X[~mu]), displayed as AⅆX μ where X is a coordinates system and X[~mu] is the corresponding spacetime vector.

• 

The covariant D_[mu] = μ is defined in terms of the Christoffel symbols as

μAν=μAνΓμ,ναAα

  

when acting on a covariant tensor as Aν, and as

μAν=μAν+Γα,μνAα

  

when acting on a contravariant tensor. For tensors of higher rank, for each covariant index there is one negative Christoffel term as in the formula for Aν and for each contravariant index there is one positive Christoffel term as in the formula for Aν.

• 

To express μAν using this definition in terms of Christoffel symbols use convert to d_ or expand which also know about other identities for μ.

• 

The contravariant gμ,νν=μ is entered as D_[~mu], so using the same D_ just with a contravariant index.

• 

Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices - see `.` and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is Galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.

• 

When only one argument is given to D_ or D_[mu], say as in D_[mu](A), the differentiation variables are the default ones, as indicated by Setup(differentiationvariables). Note you can set various coordinate systems and choose which one is to be considered the differentiation variables for D_ using Setup.

• 

You can also override the default differentiation variables by passing two arguments to D_ or D_[mu], in which case the second argument is expected to be a list with the differentiation variables, as in D_[mu](A, [x1, x2, ...]), and so this list should have as many symbols as the dimension of spacetime, which by default is 4 but can be set to any value with the Setup command.

• 

Some automatic checking and simplifications are carried out each time an operation such as D_[mu](A) is executed. The checking is concerned with possible syntax errors. Regarding the automatic simplifications, a summary is as follows:

– 

If A does not depend on the differentiation variables or is equal to the metric g_, then 0 is returned.

– 

If A is a scalar then d_[mu](A) is returned.

– 

If A belongs to the differentiation variables, then the definition of D_[mu](A) is applied and the resulting value returned.

– 

If A is a sum, product, power, or known function, then the differentiation is distributed accordingly, the same way it is done by d_.

• 

A number of additional identities for covariant derivatives are returned when using expand.

• 

To perform the differentiation, the D_ command makes us of the diff command of the Physics package, which in turn uses the standard Maple diff command, so that any user-defined differentiation rule, such as for a function foo, of the form `diff/foo`, is automatically taken into account by D_.

• 

The %D_ command is the inert form of D_, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Set up a coordinate system to work with - the first one to be set is automatically taken as the differentiation variables for d_, the covariant derivative D_ and the dAlembertian

Setupcoordinatesystems=cartesian

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

coordinatesystems=X

(2)

When Physics is initialized, the default spacetime metric is of Minkowski type. You can see the metric querying Setup, as in Setup(metric);, or directly entering the metric g_ with no indices

g_

gμ,ν=−10000−10000−100001

(3)

For illustration purposes Define an arbitrary spacetime tensor  A

DefineA

Defined objects with tensor properties

A,γμ,σμ,μ,gμ,ν,εα,β,μ,ν,Xμ

(4)

Check the nonzero components of Christoffel: if, as in the Minkowski case, there are none, then μ=μ

Christoffelnonzero

Γα,μ,ν=

(5)

D_X~nu

ⅆXνν

(6)

D_μA~nuX

μAννX

(7)

To set the scenario as an arbitrary non-galilean spacetime set the metric to be arbitrary, that is, depending on 10 arbitrary functions of the spacetime coordinates X

Setupmetric=arbitrary

_______________________________________________________

The arbitrary metric in coordinates x,y,z,t

Signature: - - - +

_______________________________________________________

gμ,ν=f__1Xf__2Xf__3Xf__4Xf__2Xf__5Xf__6Xf__7Xf__3Xf__6Xf__8Xf__9Xf__4Xf__7Xf__9Xf__10X

metric=1,1=f__1X,1,2=f__2X,1,3=f__3X,1,4=f__4X,2,2=f__5X,2,3=f__6X,2,4=f__7X,3,3=f__8X,3,4=f__9X,4,4=f__10X,spaceindices=lowercaselatin_is

(8)

g_

gμ,ν=f__1Xf__2Xf__3Xf__4Xf__2Xf__5Xf__6Xf__7Xf__3Xf__6Xf__8Xf__9Xf__4Xf__7Xf__9Xf__10X

(9)

Related to displaying the components of Christoffel, it is convenient to have a compact display of the 10 arbitrary functions entering the metric as well as of their derivatives; for that purpose use CompactDisplay.

CompactDisplay

f__1x,y,z,twill now be displayed asf__1

f__10x,y,z,twill now be displayed asf__10

f__2x,y,z,twill now be displayed asf__2

f__3x,y,z,twill now be displayed asf__3

f__4x,y,z,twill now be displayed asf__4

f__5x,y,z,twill now be displayed asf__5

f__6x,y,z,twill now be displayed asf__6

f__7x,y,z,twill now be displayed asf__7

f__8x,y,z,twill now be displayed asf__8

f__9x,y,z,twill now be displayed asf__9

(10)

So from herein, in the display of expressions, derivatives are displayed as indexed and the functionality of the arbitrary functions is suppressed.

Check the value of Γ__α,μ,ν for α = μ = ν = 1

Christoffel1,1,1

f__1x2

(11)

So now, for the total differential and the covariant derivative, we have

D_A~nuX

μAννXⅆXμμ

(12)

D_μA~nuX

μAννX

(13)

To express these results using the definition in terms of Christoffel symbols use expand or convert to d_

convert,d_

μAννX+Γνα,μνα,μAααX

(14)

To compute with a representation for  without actually performing the operation, use the inert form %D_. To afterwards perform the operation use value

%D_μg_α,β

μgα,β

(15)

value

0

(16)

The covariant derivative of a scalar is always equal to the  derivative

D_μΦX

μΦX

(17)

D_ΦX

μΦXⅆXμμ

(18)

To reduce the amount of input notation, a scalar constructed with contracted indices in tensors can be entered with all contracted indices covariant, or as a power in the case of a contracted product of a tensor with itself. For example, use AμAμ=Aμ2

Aμ2

AμAμμ

(19)

D_νAμX2

νAμXAμμX+AμXνAμμX

(20)

and as expected in the case of scalars, in the above you see no Christoffel symbols.

Set the spacetime metric g_ by giving the square of the spacetime interval

ds2x2dx2+y2dy2+z2dz2+xydxdydt2

ds2x2dx2+xydxdy+y2dy2+z2dz2dt2

(21)

Setupmetric=ds2

_______________________________________________________

Coordinates: x,y,z,t. Signature: - - - +

_______________________________________________________

gμ,ν=x2xy200xy2y20000z20000−1

_______________________________________________________

metric=1,1=x2,1,2=xy2,2,2=y2,3,3=z2,4,4=−1

(22)

The generalized divergence of a vector in curvilinear coordinates

D_μA~muX

μAμμX

(23)

expand

AμμXμxx+AμμXμyy+AμμXμzz+μAμμX

(24)

Relative scalars, tensors and tensorial expressions

In the context of a curved spacetime, when computing the covariant derivative of a relative scalar, tensor or tensorial expression that has non-zero relative weight w, a relative term is added - see Chapter 4 of [2]. That term can be computed using the Physics:-Library:-GetRelativeWeight command.

g__det%g_:-determinant

g__det g

(25)

Library:-GetRelativeTermg__det,μ

2Γνμ,ννμ,ν g

(26)

Consequently, not just in a flat spacetime where gdet is a constant, but also in a curved spacetime,

%D_μ=D_μg__det

μ g =0

(27)

To understand this zero value on the right-hand side, express the left-hand side in terms of d_

convert,d_

μ g 2Γαα,μαα,μ g =0

(28)

evaluate the inert %d_

factoreval,%d_=d_

g gα,να,νμgα,ν2Γαα,μαα,μ=0

(29)

The factor in parentheses is equal to mugalpha,nu, the covariant derivative of the metric, equal to zero, so

Simplify

0=0

(30)

Set the Schwarzschild metric and consider the covariant derivative of two tensors Tμ and Rμ, defined such that the former has zero relative weight and the latter has relative weight equal to 1. To define them use the Define command

g_sc

_______________________________________________________

Systems of spacetime coordinates are:X=r,θ,φ,t

Default differentiation variables for d_, D_ and dAlembertian are:X=r,θ,φ,t

The Schwarzschild metric in coordinates r,θ,φ,t

Parameters: m

Signature: - - - +

_______________________________________________________

gμ,ν=r2mr0000r20000r2sinθ20000r2mr

(31)

DefineTμ

Defined objects with tensor properties

Aνν,μ,γμ,σμ,Rμ,ν,Rμ,ν,α,β,Tμ,Cμ,ν,α,β,μ,gμ,ν,γi,j,Γμ,ν,α,Gμ,ν,εα,β,μ,ν,Xμ

(32)

DefineRμ,relativeweight=1

Defined objects with tensor properties

Aνν,μ,γμ,σμ,Rμ,Rμ,ν,Rμ,ν,α,β,Tμ,Cμ,ν,α,β,μ,gμ,ν,γi,j,Γμ,ν,α,Gμ,ν,εα,β,μ,ν,Xμ

(33)

Hence

Library:-GetRelativeWeightTμ

0

(34)

Library:-GetRelativeWeightRμ

1

(35)

The corresponding covariant derivatives

%D_ν=D_νTμX

νTμX=νTμX

(36)

expand

νTμX=νTμXΓαμ,ναμ,νTαX

(37)

%D_ν=D_νRμX

νRμX=νRμX

(38)

expand

νRμX=νRμXΓαμ,ναμ,νRαXΓββ,νββ,νRμX

(39)

where in the above we see the additional (relative) term

Library:-GetRelativeTermRμX,ν

Γαα,ναα,νRμX

(40)

See Also

`.`, Christoffel, Coordinates, d_, Define, DifferentialGeometry[Tensor][CovariantDerivative], Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Relative Tensors, Setup, value

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.

  

[2] Lovelock, D., and Rund, H. Tensors, Differential Forms and Variational Principles, Dover, 1989.