Maxwell Equations in the Schwarzchild Space

Renan Cabrera
cabrer7@uwindsor.ca

Initializations

In[1]:=

Needs["TensorCalculus4`Tensorial`"]

We set base indices for relativity, and define tensor shortcuts for the coordinates x, the 4-current J, the electric field ξ, the metric tensor g, the Maxwell tensor F, the Kronecker δ, and the Christoffel symbols Γ.

In[2]:=

DeclareBaseIndices[{0, 1, 2, 3}]

labs = {x, δ, g, Γ} ;

DefineTensorShortcuts[{{x, J, ξ}, 1}, {{g, F, δ}, 2}, {Γ, 3}]

SetTensorValues[δud[i, j], IdentityMatrix[NDim]]

The metric tensor of this space is given by the components of the following matrix:

In[6]:=

Attributes[M] = {Constant} ;

(cmetric = DiagonalMatrix[{-(1 - 2M/r), 1/(1 - 2M/r), r^2, r^2Sin[θ]^2}])//MatrixForm

(metric = cmetric//CoordinatesToTensors[{t, r, θ, φ}])//MatrixForm

Out[7]//MatrixForm=

( {{-1 + (2 M)/r, 0, 0, 0}, {0, 1/(1 - (2 M)/r), 0, 0}, {0, 0, r^2, 0}, {0, 0, 0, r^2 Sin[θ]^2}} )

Out[8]//MatrixForm=

( {{-1 + (2 M)/x_1^1, 0, 0, 0}, {0, 1/(1 - (2 M)/x_1^1), 0, 0}, {0, 0, (x_1^1)^2, 0}, {0, 0, 0, Sin[x_2^2]^2 (x_1^1)^2}} )

Setting the values to the metric tensor and Christoffel symbol.

In[9]:=

MapThread[SetTensorValues[#1, #2] &, {{gdd[a, b], guu[a, b]}, {metric, Inverse[metric]//Simplify}}] ;

MapThread[SetTensorValues[#1, #2] &, {{Γddd[a, b, c], Γudd[a, b, c]}, CalculateChristoffels[labs]}] ;

Non Homogeneous Maxwell Equations

The following closed cell contains the Tensorial code to derive the equation for the radial electric field in the Schwarzchild metric. Select the cell and evaluate to obtain the derivation. (Select the cell and use Alt C R C to see the code, or use Menu → Cell → Cell Properties → Cell Open.)

The Maxwell electromagnetic tensor

Out[13]=

F_ (μν)^(μν)

The nonhomogeneous Maxwell equations.

Out[15]=

F_ (μν)^(μν) _ (; ν) == J_μ^μ

Expanding the lhs in terms of Christoffel symbols.

Out[17]=

F_ (aν)^(aν) Γ_ (μνa)^(μνa) + F_ (μa)^(μa) Γ_ (ννa)^(ννa) + ∂F_ (μν)^(μν)/∂x_ν^ν == J_μ^μ

Expanding to the array values.

Out[19]=

Eliminating angular components.

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First two equations

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Setting partial derivatives with respect to time and real current to zero

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Symmetrizing the slots of F

Out[27]=

Substituting coordinate values and threading the equations.

Out[30]=

Second equation involves only diagonal components of F, which are zero.

Out[32]=

{-(M F_ (01)^(01))/((1 - (2 M)/r) r^2) + (2 F_ (01)^(01))/r - (2 M F_ (01)^(01))/((2 M - r) r) - (M F_ (10)^(10))/((2 M - r) r) + ∂F_ (01)^(01)/∂r == J_0^0, True}

Out[33]=

-(M F_ (01)^(01))/((1 - (2 M)/r) r^2) + (2 F_ (01)^(01))/r - (2 M F_ (01)^(01))/((2 M - r) r) - (M F_ (10)^(10))/((2 M - r) r) + ∂F_ (01)^(01)/∂r == J_0^0

We obtain a single equation for the radial electric field.

This result can be obtained by a second method. Taking the Divergence of the electric field ξ, and applying similar conditions.

In[34]:=

ξu[i]

CovariantD[%, i] == Ju[0]

MapAt[ExpandCovariantD[labs, a], %, 1]

(%//EinsteinSum[]//EinsteinArray[])/.{PartialD[_][_, xu[0 | 2 | 3]] →0, ξu[2 | 3] →0}

%/.TensorValueRules[x]

Out[34]=

ξ_i^i

Out[35]=

ξ_i^i_ (; i) == J_0^0

Out[36]=

Γ_ (iia)^(iia) ξ_a^a + ∂ξ_i^i/∂x_i^i == J_0^0

Out[37]=

-(M ξ_1^1)/((1 - (2 M)/x_1^1) (x_1^1)^2) + (2 ξ_1^1)/x_1^1 - (M ξ_1^1)/((2 M - x_1^1) x_1^1) + ∂ξ_1^1/∂x_1^1 == J_0^0

Out[38]=

-(M ξ_1^1)/((1 - (2 M)/r) r^2) + (2 ξ_1^1)/r - (M ξ_1^1)/((2 M - r) r) + ∂ξ_1^1/∂r == J_0^0

Where we obtain the same equation provided that  ξ_1^1= F_ (01)^(01), which is according the definition of the Maxwell tensor.

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