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Details for EnergyMomentumTensor, MatterFieldEquations, DivergenceIdentities
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Description
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Here we give the precise formulas for the energy-mometum tensors, matter field equations and divergence identities, as computed by these commands. In the formulas below, the indices are raised and lowered using the metric g, and nabla_i denotes the covariant derivative with respect to the metric g in the direction of the i-th basis vector.
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1. "DiracWeyl". The fields are a solder form sigma, a rank 1 covariant spinor psi and the complex conjungate spinor barpsi. The energy-momentum tensor is the contravariant, symmetry rank 2 tensor
T^{ij} = I/2*[sigma^{iAX'}(psi_A*\nabla^j(\barpsi_X') + nabla^j(\psi_A)*barpsi_X') + (interchange i with j)]
and the matter field equations are the rank 1 contravariant spinors with components
E^{X'} = I*sigma^{iAX'}nabla_i \psi_A and E^{A} = -I*sigma^{iAX'}nabla_i \psi_X'.
The divergence of the energy-momentum tensor is given in terms of the matter field equations by
nabla_j(T^{ij}) = -2\nabla^i(psi_A)*E^A + S^{ijA}_B*psi_A*nabla_j(E^B) + c.c..
Here S is the bivector solder form and c.c. denotes the complex conjugate of the previous terms.
2. "Dust". The fields are a four-vector u, with length +1 or -1 and a scalar mu. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
T^{ij} = mu*u^i*u^j
and the matter field equations consist of the scalar and vector
E = nabla_i(mu*u^i) and V^i = u^j*nabla_j(u^i).
The divergence of the energy momentum tensor is given in terms of the matter field equations by
nabla_j(T^{ij}) = E*u^i + mu*V^i.
3. "Electromagnetic". The field is a 1-form A or its exterior derivative F = F_{ij}dx^i &w dx^j = dA. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
T^{ij} =F^{ih}*F^j_h - 1/4*g^{ij}*F^{hk}*F_{hk}.
and the matter field equations are given by
E^i = nabla_j(F^{ij}).
The divergence of the energy-momentum tensor is given in terms of the matter field equations by
nabla_j(T^{ij}) = F^i_j E^j + A^inabla_j(E^j) .
4. "Fluid". The fields are a four-vector u, with length +1 or -1 and scalars mu and p. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
T^{ij} =( mu +p)*u^i*u^j + p*g^{ij}
The matter field equations are define by the vector
E^i = nabla_j(T^{ij}).
5. "Scalar". The field is a scalar phi. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
T^{ij} = nabla^i(phi)*nabla^j(phi) - (1/2*A + _m^2*\phi^2)*g^{ij} where A = nabla^k(phi)*nabla_k(phi_i),
and _m is a constant. The matter field equations are defined by the scalar
E = g^{ij}*nabla_i(phi)*nabla_j(phi) - _m^2*phi.
The divergence of the energy momentum tensor is given in terms of the matter field equations by
nabla_j(T^{ij}) = nabla^i(phi)*E.
6. "NMCScalar". The field is a scalar phi. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
T^{ij} = (1 - 2*_xi)*nabla^i(phi)*nabla^j(phi) + (2*_xi - 1/2) *A*g^{ij} - 2*_xi*nabla^i(nabla^j(phi)) + 2*_xi*B*g^{ij} + _xi*phi^2*G^{ij} -1/2*_m^2*phi^2*g^{ij},
where: B = nabla^k(nabla^k(phi)); G^{ij} is the Einstein tensor ; and _m and _xi are constants. The matter field equations are defined by the scalar
E = g^{ij}*nabla_i(phi)*nabla_j(phi) - (_xi*S +_m^2)*phi.
where S is the Ricci scalar. The divergence of the energy momentum tensor is given in terms of the matter field equations by
nabla_j(T^{ij}) = nabla^i(phi)*E.
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