Demonstration 2(em):Einstein-Maxwell equations in the Kerr-Newman spacetime.

> restart:

> grtw():

`GRTensorII Version 1.79 (R6)`

`2 February 2001`

`Developed by Peter Musgrave, Denis Pollney and Kay...

`Copyright 1994-2001 by the authors.`

`Latest version available from: http://grtensor.phy...

> grOptionTermSize:=30:

> qload(newkn);

`Default spacetime` = newkn

`For the newkn spacetime:`

Coordinates

x(up)

`x `^a = vector([r, u, phi, t])

`Line element`

` ds`^2 = (r^2+u^2)*` d`*r^`2 `/(r^2-2*m*r+a^2+Q^2)...
` ds`^2 = (r^2+u^2)*` d`*r^`2 `/(r^2-2*m*r+a^2+Q^2)...

Constraints = [u = a*cos(theta)]

`Kerr Newman Solution in Boyer-Lindquist coordinate...

a)Background:

Define the vector potential $A_{a}$ and the electromagnetic field tensor $f_{ab}$:

> grdef(`A{a}:=[0,0,Q*r*(a^2-u^2)/(a*(r^2+u^2)),-Q*r/(r^2+u^2)]`);

Components assigned for metric: newkn

Created definition for A(dn)

> grdisplay(A(dn));

`For the newkn spacetime:`

`A(dn)`

A(dn)

A[a] = vector([0, 0, Q*r*(a^2-u^2)/(a*(r^2+u^2)), -...

> grdef(`f{[a b]}:=2*A{[b ; a]}`);

Created a definition for A(dn,cdn)

Created definition for f(dn,dn)

Two scalar invariants associated with the field tensor are $f_{ab}f^{ab}$ and $f_{ab}f*^{ab}$:

> grdef(`I1:=f{a b}*f{^a^b}`);

Created definition for f(up,up)

Created definition for I1

> grdef(`fstar{a b}:=LevC{a b c d}*f{^c^d}`);

Created definition for fstar(dn,dn)

> grdef(`I2:=f{a b}*fstar{^a^b}`);

Created definition for fstar(up,up)

Created definition for I2

The 4-current is defined by$J^{a}=f^{ab}_{~~;b}/(4\pi)$

> grdef(`J{^a}:=f{^a^b;b}/(4*Pi)`);

Created a definition for f(up,up,cdn)

Created definition for J(up)

and for the field tensor we construct the permutation $f_{ab;c}+f_{bc;a}+f_{ca;b}$.

> grdef(`Id{a b c}:=f{a b ; c}+f{b c ; a}+f{c a ; b}`);

Created a definition for f(dn,dn,cdn)

Created definition for Id(dn,dn,dn)

b)Calculations: First we calculate, simplify and display the field tensor and invariants. The four-current and permutaion of the field tensor are then reduced to zero.

> grcalc(f(dn,dn));

`CPU Time ` = .80e-1

> gralter(_,2,7);

Component simplification of a GRTensorII object:

Applying routine `simplify[trig]` to object f(dn,dn)

Applying routine factor to object f(dn,dn)

`CPU Time ` = .30e-1

> grdisplay(_);

`For the newkn spacetime:`

`f(dn,dn)`

f(dn,dn)

f[r*phi] = -(r-u)*(r+u)*Q*(a-u)*(a+u)/(a*(r^2+u^2)^...

f[r*t] = Q*(r-u)*(r+u)/((r^2+u^2)^2)

f[u*phi] = -2*Q*r*u*(a^2+r^2)/(a*(r^2+u^2)^2)

f[u*t] = 2*Q*r*u/((r^2+u^2)^2)

> grcalc(I1,I2);

`CPU Time ` = .60e-1

> gralter(_,2,7,10);

Component simplification of a GRTensorII object:

Applying routine `simplify[trig]` to object I1

Applying routine `simplify[trig]` to object I2

Applying routine factor to object I1

Applying routine factor to object I2

Applying routine `simplify[sqrt]` to object I1

Applying routine `simplify[sqrt]` to object I2

`CPU Time ` = .60e-1

> grdisplay(_);

`For the newkn spacetime:`

I1

I1 = -2*Q^2*(-u^2+2*r*u+r^2)*(-u^2-2*r*u+r^2)/((r^2...

I2

I2 = 16*(r-u)*(r+u)*Q^2*r*u/((r^2+u^2)^4)

> grcalc(J(up));

`CPU Time ` = .50e-1

> grdisplay(_);

`For the newkn spacetime:`

`J(up)`

J(up)

J^a = `All components are zero`

> grcalc(Id(dn,dn,dn));

`CPU Time ` = .70e-1

> grdisplay(_);

`For the newkn spacetime:`

`Id(dn,dn,dn)`

`Id(dn,dn,dn)` = `All components are zero`

c)Einstein-Maxwell equations: We construct the energy-momentum tensor out of the field tensor ($T_{ab}=(f_{ac}f_{b}^{~c}-g_{ab}f_{cd}f^{cd}/4)/(4\pi)$) and then show that the Kerr-Newman spacetime satifies the Einstein-Maxwell equations.

> grdef(`T{a b}:=(+f{a c}*f{b^c}-g{a b}*I1/4)/(4*Pi)`);

Created definition for f(dn,up)

Created definition for T(dn,dn)

> grdef(`Dif{a b}:=G{a b}-8*Pi*T{a b}`);

Created definition for Dif(dn,dn)

> grcalc(Dif(dn,dn));

`CPU Time ` = .90e-1

> gralter(_,6,7);

Component simplification of a GRTensorII object:

Applying routine expand to object Dif(dn,dn)

Applying routine factor to object Dif(dn,dn)

`CPU Time ` = .10e-1

> grdisplay(_);

`For the newkn spacetime:`

`Dif(dn,dn)`

Dif(dn,dn)

Dif[a]*``[b] = `All components are zero`

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