Demonstration 1(kerr):Introduction to the Kerr metric, the vacuum ($\Lambda=0$) solution of Einstein's equations for axial symmetry.

(kerr.mpl,newkerr.mpl,npdnkerr.mpl)

> restart:

> interface(labelling=false):

> 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...

`e:/Grtii(6)/Metrics`

The most familiar form of the Kerr metric is in Boyer-Lindquist coordinates. We start by showing that the metric is indeed a vacuum solution (compare Landau and Lifshitz, p.323).

> qload(kerr);

`Default spacetime` = kerr

`For the kerr spacetime:`

Coordinates

x(up)

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

`Line element`

` ds`^2 = (r^2+a^2*cos(theta)^2)*` d`*r^`2 `/(r^2-2...
` ds`^2 = (r^2+a^2*cos(theta)^2)*` d`*r^`2 `/(r^2-2...

`Kerr metric in Boyer-Lindquist coordinates.`

> grcalc(R(dn,dn));

`CPU Time ` = .300

> gralter(_,trig);

Component simplification of a GRTensorII object:

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

`CPU Time ` = .161

> grdisplay(_);

`For the kerr spacetime:`

`Covariant Ricci`

R(dn,dn)

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

The metric is flat for m=0.

> grcalc(R(dn,dn,dn,dn));

`CPU Time ` = .210

> grmap(_,subs,m=0,`x`);

Applying routine subs to R(dn,dn,dn,dn)

> gralter(_,trig);

Component simplification of a GRTensorII object:

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

`CPU Time ` = .30e-1

> grdisplay(_);

`For the kerr spacetime:`

`Covariant Riemann`

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

The time rquired to show that the metric is a vacuum solution depends very much on the coordinates.

For example, under the elementary transformation $u=a*cos(\theta)$ we have:

> qload(newkerr);

`Default spacetime` = newkerr

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

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

`The Kerr metric in Boyer-Lindquist type coordinate...

> grcalc(R(dn,dn));

`CPU Time ` = .81e-1

> grdisplay(_);

`For the newkerr spacetime:`

`Covariant Ricci`

R(dn,dn)

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

The traditional way to argue that r=0 is singular only for $\theta=\pi/2$ is to calculate the Kretschmann scalar (e.g. Wald, p.315 Hawking and Ellis, p.162).

> grcalc(RiemSq);

Created definition for R(dn,dn,up,up)

`CPU Time ` = .280

> gralter(_,6,7);

Component simplification of a GRTensorII object:

Applying routine expand to object RiemSq

Applying routine factor to object RiemSq

`CPU Time ` = .40e-1

We can put the scalar back into the original coordinates ($u=a*cos(\theta)$).

> grmap(_,subs,u=a*cos(theta),`x`);

Applying routine subs to RiemSq

> grdisplay(_);

`For the newkerr spacetime:`

`Full Contraction of Riemann`

`K ` = 48*m^2*(-a*cos(theta)+r)*(r+a*cos(theta))*(a...

Since the solution is vacuum, the only non-vanishing curvature invariants are the Weyl invariants (e.g. Weinberg, p.146).

> grcalc(Winvars);

Scalar invariant library.

Last modified 25 March 1997.

Created definition for C(up,up,up,up)

`CPU Time ` = .471

> gralter(_,7);

Component simplification of a GRTensorII object:

Applying routine factor to object W1R

Applying routine factor to object W1I

Applying routine factor to object W2R

Applying routine factor to object W2I

`CPU Time ` = .60e-1

> grmap(_,subs,u=a*cos(theta),`x`);

Applying routine subs to W1R

Applying routine subs to W1I

Applying routine subs to W2R

Applying routine subs to W2I

> grmap(_,radsimp,`x`);

Applying routine radsimp to W1R

Applying routine radsimp to W1I

Applying routine radsimp to W2R

Applying routine radsimp to W2I

> gralter(_,7);

Component simplification of a GRTensorII object:

Applying routine factor to object W1R

Applying routine factor to object W1I

Applying routine factor to object W2R

Applying routine factor to object W2I

`CPU Time ` = .51e-1

> grdisplay(_);

`For the newkerr spacetime:`

`CM invariant Re(W1)`

`W1R ` = -6*m^2*(a*cos(theta)-r)*(r+a*cos(theta))*(...

`CM invariant Im(W1)`

`W1I ` = -12*r*m^2*a*cos(theta)*(-r^2+3*a^2*cos(the...

`CM invariant Re(W2)`

`W2R ` = -6*r*m^3*(-r^2+3*a^2*cos(theta)^2)*(3*a^6*...

`CM invariant Im(W2)`

`W2I ` = 6*a*cos(theta)*m^3*(-3*r^2+a^2*cos(theta)^...

The coordinates are obviously adapted to two Killing vectors (see further demonstrations under symmetry).

> KillingCoords():

Testing Killing coordinates for newkerr

Created definition for coord1(dn)

Created a definition for coord1(dn,cdn)

Created a definition for coord1(up,cdn)

Created definition for coord2(dn)

Created a definition for coord2(dn,cdn)

Created a definition for coord2(up,cdn)

Created definition for coord3(dn)

Created a definition for coord3(dn,cdn)

Created a definition for coord3(up,cdn)

Created definition for coord4(dn)

Created a definition for coord4(dn,cdn)

Created a definition for coord4(up,cdn)

`CPU Time ` = 1.412

`Killing Coordinate Test Results`

`Coordinate vector ` = [r, u, phi, t]

coord1(up) = [1, 0, 0, 0], ` not a Killing vector.`...

coord2(up) = [0, 1, 0, 0], ` not a Killing vector.`...

coord3(up) = [0, 0, 1, 0], ` a Killing vector.`

coord4(up) = [0, 0, 0, 1], ` a Killing vector.`

We now show by way of the Frobenius theorem that the metric is not static unless a=0 (e.g. Wald p.436):

> grdef(`xi{^a}:=[0,0,0,1]`);

Components assigned for metric: newkerr

Created definition for xi(up)

> grdef(`Xi{a b c}:=xi{[a ;c}*xi{b]}`);

Created definition for xi(dn)

Created a definition for xi(dn,cdn)

Created definition for Xi(dn,dn,dn)

> grcalc(Xi(dn,dn,dn));

`CPU Time ` = .30e-1

> gralter(_,1);

Component simplification of a GRTensorII object:

Applying routine simplify to object Xi(dn,dn,dn)

`CPU Time ` = .10e-1

> grmap(_,subs,u=a*cos(theta),`x`);

Applying routine subs to Xi(dn,dn,dn)

> gralter(_,trig,factor);

Component simplification of a GRTensorII object:

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

Applying routine factor to object Xi(dn,dn,dn)

`CPU Time ` = .40e-1

> grcomponent(Xi(dn,dn,dn),[r,phi,t]);

-1/3*a*(-1+cos(theta))*(cos(theta)+1)*m*(a*cos(thet...

Consider now a covariant NPtetrad. We go directly to the Ricci and Weyl scalars:

> qload(npdnkerr);

`Default spacetime` = npdnkerr

`For the npdnkerr spacetime:`

Coordinates

x(up)

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

`Basis inner product`

eta(bup,bup)

eta^`(a)`*``^`(b)` = matrix([[0, 1, 0, 0], [1, 0, 0...

`Null tetrad (covariant components)`

NPl(dn)

l[a] = vector([1, -(r^2+u^2)/(r^2-2*M*r+a^2), 0, -(...

NPn(dn)

n[a] = vector([1/2*(r^2-2*M*r+a^2)/(r^2+u^2), 1/2, ...

NPm(dn)

m[a] = vector([1/2*sqrt(a^2-u^2)*(I*r+u)*sqrt(2)/(r...

NPmbar(dn)

mbar[a] = vector([-1/2*sqrt(a^2-u^2)*(I*r-u)*sqrt(2...

`Covariant NPtetrad for Kerr metric (u=a*cos(theta)...

> grcalc(RicciSc,WeylSc):

`Basis/tetrad related object definitions`

`Last modified 23 January 2001`

Created a definition for e(bdn,dn,pdn)

`CPU Time ` = .791

> gralter(_,2,7);

Component simplification of a GRTensorII object:

Applying routine `simplify[trig]` to object Phi00

Applying routine `simplify[trig]` to object Phi01

Applying routine `simplify[trig]` to object Phi02

Applying routine `simplify[trig]` to object Phi11

Applying routine `simplify[trig]` to object Phi12

Applying routine `simplify[trig]` to object Phi22

Applying routine `simplify[trig]` to object NPLambda

Applying routine `simplify[trig]` to object Psi0

Applying routine `simplify[trig]` to object Psi1

Applying routine `simplify[trig]` to object Psi2

Applying routine `simplify[trig]` to object Psi3

Applying routine `simplify[trig]` to object Psi4

Applying routine factor to object Phi00

Applying routine factor to object Phi01

Applying routine factor to object Phi02

Applying routine factor to object Phi11

Applying routine factor to object Phi12

Applying routine factor to object Phi22

Applying routine factor to object NPLambda

Applying routine factor to object Psi0

Applying routine factor to object Psi1

Applying routine factor to object Psi2

Applying routine factor to object Psi3

Applying routine factor to object Psi4

`CPU Time ` = .270

> grmap(_,subs,u=a*cos(theta),`x`);

Applying routine subs to Phi00

Applying routine subs to Phi01

Applying routine subs to Phi02

Applying routine subs to Phi11

Applying routine subs to Phi12

Applying routine subs to Phi22

Applying routine subs to NPLambda

Applying routine subs to Psi0

Applying routine subs to Psi1

Applying routine subs to Psi2

Applying routine subs to Psi3

Applying routine subs to Psi4

> grdisplay(_);

`For the npdnkerr spacetime:`

`Ricci Scalar, Phi00`

Phi00 = 0

`Ricci Scalar, Phi01`

Phi01 = 0

`Ricci Scalar, Phi02`

Phi02 = 0

`Ricci Scalar, Phi11`

Phi11 = 0

`Ricci Scalar, Phi12`

Phi12 = 0

`Ricci Scalar, Phi22`

Phi22 = 0

`NPLambda := Ricci Scalar/24`

NPLambda = 0

`Weyl Scalar, NP Psi0`

Psi0 = 0

`Weyl Scalar, NP Psi1`

Psi1 = 0

`Weyl Scalar, NP Psi2`

Psi2 = M/((-r+I*a*cos(theta))^3)

`Weyl Scalar, NP Psi3`

Psi3 = 0

`Weyl Scalar, NP Psi4`

Psi4 = 0

A remarkable property of the metric is the existence of a Killing tensor (e.g. Wald p.321). See the demonstrations under symmetry.

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