Olanic

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Hello!

I am truing to simplify kretchmann variable in the following worksheet:

 

 
M > 

# Obtaining Ricci and Kretchmann;
with(DifferentialGeometry):with(Tensor):

DGsetup([t, r, theta, phi], M);
g := evalDG(-(1-2*M*mu/r)^(1/mu)*dt &t dt+(1-2*M*mu/r)^(-1/mu)*`&t`(dr, dr)+r^2*(1-2*M*mu/r)^(1-1/mu)*(`&t`(dtheta, dtheta)+sin(theta)^2*`&t`(dphi, dphi)));
C := Christoffel(g):

`frame name: M`

 

_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], -(-(2*M*mu-r)/r)^(1/mu)], [[2, 2], (-(2*M*mu-r)/r)^(-1/mu)], [[3, 3], r^2*(-(2*M*mu-r)/r)^((mu-1)/mu)], [[4, 4], r^2*(-(2*M*mu-r)/r)^((mu-1)/mu)*sin(theta)^2]]])

(1.1)

Rie := CurvatureTensor(C):
R := RicciScalar(g,Rie);
h := InverseMetric(g):
kretchmann := ContractIndices(RaiseLowerIndices(g, Rie, [1]), RaiseLowerIndices(h, Rie, [2, 3, 4]), [[1, 1], [2, 2], [3, 3], [4, 4]]);

2*(-(2*M*mu-r)/r)^(1/mu)*M^2*(mu^2-1)/(r^2*(2*M*mu-r)^2)

 

4*(-(2*M*mu-r)/r)^(-2*(mu-1)/mu)*M^2*(M*mu^2+2*M*mu+M-2*r)^2/(r^6*(2*M*mu-r)^2)+4*(-(2*M*mu-r)/r)^(2/mu)*M^2*(M*mu^2+M*mu-r)^2/((2*M*mu-r)^4*r^4)+20*(-(2*M*mu-r)/r)^(2/mu)*(M*mu+M-r)^2*M^2/((2*M*mu-r)^4*r^4)+4*(-(2*M*mu-r)/r)^(-2*(mu-1)/mu)*M^2*(M*mu^2+M*mu-r)^2/(r^6*(2*M*mu-r)^2)+4*(-(2*M*mu-r)/r)^(-2*(mu-1)/mu)*M^2*(M*mu+M-r)^2/(r^6*(2*M*mu-r)^2)

(1.2)
M > 

# simplification

M > 

simplify(normal(R),symbolic)

2*(-1)^(1/mu)*(2*M*mu-r)^((1-2*mu)/mu)*r^((-1-2*mu)/mu)*M^2*(mu^2-1)

(1.3)
M > 

simplify(kretchmann,size,symbolic)

4*(-(2*M*mu-r)/r)^(-2*(mu-1)/mu)*M^2*(M*mu^2+2*M*mu+M-2*r)^2/(r^6*(2*M*mu-r)^2)+4*(-(2*M*mu-r)/r)^(2/mu)*M^2*(M*mu^2+M*mu-r)^2/((2*M*mu-r)^4*r^4)+20*(-(2*M*mu-r)/r)^(2/mu)*(M*mu+M-r)^2*M^2/((2*M*mu-r)^4*r^4)+4*(-(2*M*mu-r)/r)^(-2*(mu-1)/mu)*M^2*(M*mu^2+M*mu-r)^2/(r^6*(2*M*mu-r)^2)+4*(-(2*M*mu-r)/r)^(-2*(mu-1)/mu)*M^2*(M*mu+M-r)^2/(r^6*(2*M*mu-r)^2)

(1.4)
M > 

 


 

Download RicciScalarKretchmann.mw

The problem is that I cannot obtain a good form of it. With Mathematica FullSimplify[] function I got the following form (LaTeX code incoming): $K =& 4 M^2 \Bigl(A-B r+C r^2\Bigr)(r-2 M \mu)^{\frac{2}{\mu}-4}r^{-\frac{2}{\mu}-4},\
    A =&M^2 (\mu (3 \mu+2)+7) (\mu+1)^2,\,B = 8 M (\mu+2) (\mu+1),\, C = 12$, i.e. terms $(r-2 M \mu)$ and $r$ got fully factorized. However, I could never achieve the same form in Maple. Any help?


I am sorry if this is a silly and many-times-answered question, but I tried consulting with Maple help and googling solutions without any success.

Regards,
Nick

Hello!

I am trying to determine the type of Lie algebra originating from Killing vectors. However, Maple says that some variable x is protected (though, I guess, it shouldn't even be in the L). Is there any way to solve this issue?

Best regards, 
Nick

error.mw

restart

with(DifferentialGeometry); with(Tensor); with(LieAlgebras); with(GroupActions); with(Library)

DGsetup([x, y], M)

g := evalDG((`&t`(dx, dx)+`&t`(dy, dy))/y^2)

_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 1/y^2], [[2, 2], 1/y^2]]])

(1)

kv := KillingVectors(g)

[_DG([["vector", M, []], [[[1], (1/2)*x^2-(1/2)*y^2], [[2], y*x]]]), _DG([["vector", M, []], [[[1], x], [[2], y]]]), _DG([["vector", M, []], [[[1], 1]]])]

(2)

L := LieAlgebraData(kv, Poincare)

_DG([["LieAlgebra", Poincare, [3]], [[[1, 2, 1], -1], [[1, 3, 2], -1], [[2, 3, 3], -1]]])

(3)

DGsetup(L); MultiplicationTable("LieTable")

Query(Poincare, "Semisimple")

true

(4)

CSA := CartanSubalgebra(Poincare)

[_DG([["vector", Poincare, []], [[[2], 1]]])]

(5)

RSD := RootSpaceDecomposition(CSA)

Error, (in MultiSet:-Entries) attempting to assign to `x` which is protected.  Try declaring `local x`; see ?protect for details.

 

``


 

Download error.mw

 

Hello! 

I tried to find Killing vectors of a certain metric, but Maple gives an error, which is reproduced with the following piece of code (the metric which I need is more difficult, so I would like to refrain from using "Physics" package). Could anybody tell me what the problem is and how I should solve it, please?

 

exampleError.mw
 

restart

with(DifferentialGeometry); with(Tensor); with(LieAlgebras)

DGsetup([xi, chi, u1, u2, v1, v2], P)

`frame name: P`

(1)
P > 

g := evalDG(1/2*(`&t`(dxi, dxi)+`&tensor`(du1*v1+du2*v2+dchi, du1*v1+du2*v2+dchi)*exp(-2*xi)+`&t`(dv1, dv1)+`&t`(dv2, dv2)))

_DG([["tensor", P, [["cov_bas", "cov_bas"], []]], [[[1, 1], 1/2], [[2, 2], (1/2)*exp(-2*xi)], [[2, 3], (1/2)*exp(-2*xi)*v1], [[2, 4], (1/2)*exp(-2*xi)*v2], [[3, 2], (1/2)*exp(-2*xi)*v1], [[3, 3], (1/2)*exp(-2*xi)*v1^2], [[3, 4], (1/2)*exp(-2*xi)*v1*v2], [[4, 2], (1/2)*exp(-2*xi)*v2], [[4, 3], (1/2)*exp(-2*xi)*v1*v2], [[4, 4], (1/2)*exp(-2*xi)*v2^2], [[5, 5], 1/2], [[6, 6], 1/2]]])

(2)
P > 

KV := KillingVectors(g)

Error, (in DifferentialGeometry:-Tensor:-Christoffel) expected 1st argument to be a metric tensor. Received: _DG([["tensor", P, [["cov_bas", "cov_bas"], []]], [`...`]])

 
P > 

``


 

Download exampleError.mw

 

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