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Details for NPRicciIdentities and NPBianchiIdentities
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Description
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[a] d(rho) - bardelta(kappa) = rho^2 + sigaa*barsigma + (epsilon + barepsilon)*rho - barkappa*tau - kappa*(3*alpha + barbeta - pi) + Phi00
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[b] d(sigma) - delta(kappa) = (rho + barrho)*sigma + (3*epsilon - barepsilon)*sigma - (tau - barpi + baralpha + 3*beta)*kappa + Psi0
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[c] d(tau) - Delta(kappa) = (tau + barpi)*rho + (bartau + pi)*sigma + (epsilon - barepsilon)*tau - (3*gamma + bargamma)*kappa + Psi1 + Phi01
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[d] d(alpha) - bardelta(epsilon) = (rho + barepsilon - 2*epsilon)*alpha + beta*barsigma - barbeta*epsilon - kappa*lambda - barkappa*gamma + (epsilon + rho)*pi + Phi10
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[e] d(beta) - delta(epsilon) = (alpha + pi)*sigma + (barrho - barepsilon)*beta - (mu + gamma)*kappa - (baralpha - barpi)*epsilon + Psi1
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[f] d(gamma) - Delta(epsilon) = (tau + barpi)*alpha + (bartau + pi)*beta - (epsilon + barepsilon)*gamma - (gamma + bargamma)*epsilon + tau*pi - nu*kappa + Psi2 - Lambda + Phi11
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[g] d(lambda) - bardelta(pi) = (rho*lambda + barsigma*mu) + pi^2 + (alpha - barbeta)*pi - nu*barkappa - (3*epsilon - barepsilon)*lambda + Phi20
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[h] d(mu) - delta(pi) = (barrho*mu + sigma*lambda) + pi*barpi - (epsilon + barepsilon)*mu - pi*(baralpha - beta) - nu*kappa + Psi2 + 2*Lambda
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[i] d(nu) - Delta(pi) = (pi + bartau)*mu + (barpi + tau)*lambda + (gamma - bargamma)*pi - (3*epsilon + barepsilon)*nu + Psi3 + Phi21
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[j] Delta(lambda) - bardelta(nu) = - (mu + barmu)*lambda - (3*gamma - bargamma)*lambda + (3*alpha + barbeta + pi - bartau)*nu - Psi4
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[k] delta(rho) - bardelta(sigma) = rho*(baralpha + beta) - sigma*(3*alpha - barbeta) + (rho - barrho)*tau + (mu - barmu)*kappa - Psi1 + Phi01
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[m] delta(lambda) - bardelta(mu) = (rho - barrho)*nu + (mu - barmu)*pi + mu*(alpha + barbeta) + lambda*(baralpha - 3*beta) - Psi3 + Phi21
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[n] delta(nu) - Delta(mu) = (mu^2 + lambda*barlambda) + (gamma + bargamma)*mu - barnu*pi + (tau - 3*beta - baralpha)*nu + Phi22
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[o] delta(gamma) - Delta(beta) = (tau - baralpha - beta)*gamma + mu*tau - sigma*nu - epsilon*barnu - beta*(gamma - bargamma - mu) + alpha*barlambda + Phi12
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[p] delta(tau) - Delta(sigma) = (mu*sigma + barlambda*rho) + (tau + beta - baralpha)*tau - (3*gamma - bargamma)*sigma - kappa*barnu + Phi02
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[q] Delta(rho) - bardelta(tau) = - (rho*barmu + sigma*lambda) + (barbeta - alpha - bartau)*tau + (gamma + bargamma)*rho + nu*kappa - Psi2 - 2*Lambda
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[r] Delta(alpha) - bardelta(gamma) = (rho + epsilon)*nu - (tau + beta)*lambda + (bargamma - barmu)*alpha + (barbeta - bartau)*gamma - Psi3,
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Here the list of NP Bianchi idenitites, taken from the book of Stewart
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[a] d(Psi1) - bardelta(Psi0) - d(Phi01) + delta(Phi00) = (pi - 4*alpha)*Psi0 + 2*(2*rho + epsilon)*Psi1 - 3*kappa*Psi2 - (barpi - 2*baralpha - 2*beta)*Phi00 - 2*(barrho + epsilon)*Phi01 - 2*sigma*Phi10 + 2*kappa*Phi11 + barkappa*Phi02
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[b] Delta(Psi0) - delta(Psi1) + d(Phi02) - delta(Phi01) = (4*gamma - mu)*Psi0 - 2*(2*tau + beta)*Psi1 + 3*sigma*Psi2 - barlambda*Phi00 + 2*(barpi - beta)*Phi01 + 2*sigma*Phi + (barrho + 2*epsilon - 2*barepsilon)*Phi02 - 2*kappa*Phi12
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[c] d(Psi2) - bardelta(Psi1) + Delta(Phi00) - bardelta(Phi01) + 2*d(Lambda) = - lambda*Psi0 + 2*(pi - alpha)*Psi1 + 3*rho*Psi2 - 2*kappa*Psi3 + (2*gamma + 2*bargamma - barmu)*Phi00
- 2*(alpha + bartau)*Phi01 - 2*tau*Phi10 + 2*rho*Phi11 + barsigma*Phi02
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[d] Delta(Psi1) - delta(Psi2) - Delta(Phi01) + bardelta(Phi02) - 2*delta(Lambda) = nu*Psi0 + 2*(gamma - mu)*Psi1 - 3*tau*Psi2 + 2*sigma*Psi3 - barnu*Phi00 + 2*(barmu - gamma)*Phi01 + (2*alpha + bartau - 2*barbeta)*Phi02 + 2*tau*Phi11 - 2*rho*Phi12
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[e] d(Psi3) - bardelta(Psi2) - d(Phi21) + delta(Phi20) - 2*bardelta(Lambda) = - 2*lambda*Psi1 + 3*pi*Psi2 + 2*(rho - epsilon)*Psi3 - kappa*Psi4 + 2*mu*Phi10 - 2*pi*Phi11 - (2*beta + barpi - 2*baralpha)*Phi20 - 2*(barrho - epsilon)*Phi21 + barkappa*Phi22
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[f] Delta(Psi2) - delta(Psi3) + d(Phi22) - delta(Phi21) + 2*Delta(Lambda) = 2*nu*Psi1 - 3*mu*Psi2 + 2*(beta - tau)*Psi3 + sigma*Psi4 - 2*mu*Phi11 - barlambda*Phi20 + 2*pi*Phi12 + 2*(beta + barpi)*Phi21 + (barrho - 2*epsilon - 2*barepsilon)*Phi22
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[g] d(Psi4) - bardelta(Psi3) + Delta(Phi20) - bardelta(Phi21) = - 3*lambda*Psi2 + 2*(alpha + 2*pi)*Psi3 + (rho - 4*epsilon)*Psi4 + 2*nu*Phi10 - 2*lambda*Phi11 - (2*gamma - 2*bargamma + barmu)*Phi20 - 2*(bartau - alpha)*Phi21 + barsigma*Phi22
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[h] Delta(Psi3) - delta(Psi4) - Delta(Phi21) + bardelta(Phi22) = 3*nu*Psi2 - 2*(gamma + 2*mu)*Psi3 + (4*beta - tau)*Psi4 - 2*nu*Phi11 - barnu*Phi20 + 2*lambda*Phi12 + 2*(gamma + barmu)*Phi21 + (bartau - 2*barbeta - 2*alpha)*Phi22
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[i] d(Phi11) - delta(Phi10) + Delta(Phi00) - bardelta(Phi01) + 3*d(Lambda) = (2*gamma + 2*bargamma - mu - barmu)*Phi00 + (pi - 2*alpha - 2*bartau)*Phi01 + (barpi - 2*baralpha - 2*tau)*Phi10 + 2*(rho + barrho)*Phi02 + barsigma*Phi02 + sigma*Phi20 - barkappa*Phi12 - kappa*Phi21
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[j] d(Phi12) - delta(Phi11) + Delta(Phi01) - bardelta(Phi02) + 3*delta(Lambda) = (2*gamma - mu - 2*barmu)*Phi01 + barnu*Phi00 - barlambda*Phi10 + 2*(barpi - tau)*Phi11 + (pi + 2*barbeta - 2*alpha - bartau)*Phi02 + (2*rho + barrho - 2*barepsilon)*Phi12 + sigma*Phi21 - kappa*Phi22
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[k] d(Phi22) - delta(Phi21) + Delta(Phi11) - bardelta(Phi12) + 3*Delta(Lambda) = nu*Phi01 + barnu*Phi10 - 2*(mu + barmu)*Phi11 - lambda*Phi02 - barlambda*Phi20 + (2*pi - bartau + 2*barbeta)*Phi12 + (2*beta - tau + 2*barpi)*Phi21 + (rho + barrho - 2*epsilon - 2*barepsilon)*Phi22
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