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General Relativity using Computer Algebra

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General Relativity using Computer Algebra

 

Problem: for the spacetime metric,

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))

 

a) Compute the trace of

 

"Z[alpha]^(beta)=Phi R[alpha]^(beta)+`𝒟`[alpha]`𝒟`[]^(beta) Phi+T[alpha]^(beta)"

 

where `≡`(Phi, Phi(r)) is some function of the radial coordinate, R[alpha, `~beta`] is the Ricci tensor, `𝒟`[alpha] is the covariant derivative operator and T[alpha, `~beta`] is the stress-energy tensor

 

T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))

b) Compute the components of "W[alpha]^(beta)"" ≡"the traceless part of  "Z[alpha]^(beta)" of item a)

 

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)" obtained in b)

 

Background: The equations of items a) and b) appear in a paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by Plamen Fiziev, a Maple user.  These equations model a problem in the context of a Branse-Dicke theory with vanishing parameter "omega." The Brans–Dicke theory is in many respects similar to Einstein's theory, but the gravitational "constant" is not actually presumed to be constant - it can vary from place to place and with time - and the gravitational interaction is mediated by a scalar field. Both Brans–Dicke's and Einstein's theory of general relativity are generally held to be in agreement with observation.

 

The computations below aim at illustrating how this type of computation can be performed using computer algebra, and so they focus only on the algebraic aspects, not the physical interpretation of the results.

a) The trace of " Z[alpha]^(beta)=Phi R[alpha]^(beta)+`𝒟`[alpha]`𝒟`[]^(beta) Phi+T[alpha]^(beta)"
 

b) The components of "W[alpha]^(beta)"" ≡"the traceless part of " Z[alpha]^(beta)"
 

c) An exact solution for the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)"