The energy momentum tensor is a symmetric, rank-2 contravariant tensor $T$ which determines the right-hand side of the Einstein field equations.

If FieldType = "DiracWeyl", then the additional arguments for EnergyMomentumTensor are: a solder form (compatible with the metric $g$), a rank 1 covariant spinor $\mathrm{\ψ}$, and the complex conjugate $\overline{\mathrm{\ψ}}$.

If FieldType = "Dust", then the additional arguments for EnergyMomentumTensor are: a vector field $U$, a scalar $\mathrm{\μ}$ (energy density).

If FieldType = "Electromagnetic", then the additional arguments are either: a 1-form $A$ (the electromagnetic 4-potential), or a skew-symmetric rank 2 tensor $F$ (the field strength tensor).

If FieldType = "PerfectFluid", then the additional arguments for EnergyMomentumTensor are: a vector field $U$, and scalars $\mathrm{\μ}$ (energy density) and $p$ (pressure).

If FieldType = "Scalar", then the additional argument for EnergyMomentumTensor is a scalar $\mathrm{\φ}$.

If FieldType = "NMCScalar", then the additional argument for EnergyMomentumTensor is a non-minimally coupled scalar $\mathrm{\φ}$.

See the Details help page for the explicit formulas used to calculate the various energy-momentum tensors, the matter field equations and the divergence identities.

These commands are part of the DifferentialGeometry:-Tensor: package, and so can be used in the form EnergyMomentumTensor(...), MatterFieldEquations(...), DivergenceIdentities(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. They can always be used in the long form DifferentialGeometry:-Tensor:-EnergyMomentumTensor, DifferentialGeometry:-Tensor-MatterFieldEquations, DifferentialGeometry:-Tensor:-DivergenceIdentities.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tools}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,N\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{g1}≔\mathrm{evalDG}\left(x^4\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g1}≔x^4\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{OTetrad}≔\mathrm{evalDG}\left(\left[\frac{1}{x^2}\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\right)$

$\mathrm{OTetrad}≔\left[\frac{1}{x^2}\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{\σ1}≔\mathrm{SolderForm}\left(\mathrm{OTetrad}\right)$

$\mathrm{\σ1}≔\frac{x^2\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}+\frac{x^2\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}+\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{\ψ1}≔\mathrm{evalDG}\left(h\left(x\right)\mathrm{dz1}-f\left(x\right)\mathrm{dz2}\right)$

$\mathrm{\ψ1}≔h\left(x\right)\mathrm{dz1}-f\left(x\right)\mathrm{dz2}$

$\mathrm{barpsi1}≔\mathrm{evalDG}\left(h\left(x\right)\mathrm{dw1}-f\left(x\right)\mathrm{dw2}\right)$

$\mathrm{barpsi1}≔h\left(x\right)\mathrm{dw1}-f\left(x\right)\mathrm{dw2}$

$\mathrm{T1}≔\mathrm{EnergyMomentumTensor}\left(''DiracWeyl''\,\mathrm{g1}\,\mathrm{\σ1}\,\mathrm{\ψ1}\,\mathrm{barpsi1}\right)$

$\mathrm{T1}≔-\frac{\sqrt{2}\left({f\left(x\right)}^2-{h\left(x\right)}^2\right)}{x^3}\mathrm{D\_t}\mathrm{D\_y}+\sqrt{2}\left(-h\left(x\right)f\prime \left(x\right)+f\left(x\right)h\prime \left(x\right)\right)\mathrm{D\_x}\mathrm{D\_y}-\frac{\sqrt{2}\left({f\left(x\right)}^2-{h\left(x\right)}^2\right)}{x^3}\mathrm{D\_y}\mathrm{D\_t}+\sqrt{2}\left(-h\left(x\right)f\prime \left(x\right)+f\left(x\right)h\prime \left(x\right)\right)\mathrm{D\_y}\mathrm{D\_x}$

$\mathrm{E1}≔\mathrm{MatterFieldEquations}\left(''DiracWeyl''\,\mathrm{g1}\,\mathrm{\σ1}\,\mathrm{\ψ1}\,\mathrm{barpsi1}\right)$

$\mathrm{E1}≔\frac{\frac{\mathrm{I}}{2}\sqrt{2}\left(f\prime \left(x\right)x+f\left(x\right)\right)}{x}\mathrm{D\_w1}-\frac{\frac{\mathrm{I}}{2}\sqrt{2}\left(h\prime \left(x\right)x+h\left(x\right)\right)}{x}\mathrm{D\_w2},-\frac{\frac{\mathrm{I}}{2}\sqrt{2}\left(f\prime \left(x\right)x+f\left(x\right)\right)}{x}\mathrm{D\_z1}+\frac{\frac{\mathrm{I}}{2}\sqrt{2}\left(h\prime \left(x\right)x+h\left(x\right)\right)}{x}\mathrm{D\_z2}$

$\mathrm{Div1},\mathrm{RHS1}≔\mathrm{DivergenceIdentities}\left(''DiracWeyl''\,\mathrm{g1}\,\mathrm{\σ1}\,\mathrm{\ψ1}\,\mathrm{barpsi1}\,\mathrm{T1}\,\mathrm{E1}\right)$

$\mathrm{Div1},\mathrm{RHS1}≔\frac{\sqrt{2}\left(-xh\left(x\right)f″\left(x\right)+xf\left(x\right)h″\left(x\right)+2f\left(x\right)h\prime \left(x\right)-2h\left(x\right)f\prime \left(x\right)\right)}{x}\mathrm{D\_y},\frac{\sqrt{2}\left(-xh\left(x\right)f″\left(x\right)+xf\left(x\right)h″\left(x\right)+2f\left(x\right)h\prime \left(x\right)-2h\left(x\right)f\prime \left(x\right)\right)}{x}\mathrm{D\_y}$

$\mathrm{Div1}-\mathrm{RHS1}$

$0$

$\mathrm{map}\left(\mathrm{DGsimplify}\,\mathrm{eval}\left(\left[\mathrm{E1}\right]\,\left[f\left(x\right)=\frac{1}{x}\,h\left(x\right)=\frac{1}{x}\right]\right)\right)$

$\left[0\mathrm{D\_z1}\,0\mathrm{D\_z1}\right]$

$\mathrm{DGsimplify}\left(\mathrm{eval}\left(\mathrm{Div1}\,\left[f\left(x\right)=\frac{1}{x}\,h\left(x\right)=\frac{1}{x}\right]\right)\right)$

$0\mathrm{D\_t}$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-t^2\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g2}≔\mathrm{dt}\mathrm{dt}-t^2\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{u2}≔\mathrm{evalDG}\left(\cosh \left(f\left(t\right)\right)\mathrm{D\_t}-\frac{\sinh \left(f\left(t\right)\right)}{t}\mathrm{D\_x}\right)$

$\mathrm{u2}≔\cosh \left(f\left(t\right)\right)\mathrm{D\_t}-\frac{\sinh \left(f\left(t\right)\right)}{t}\mathrm{D\_x}$

$\mathrm{TensorInnerProduct}\left(\mathrm{g2}\,\mathrm{u2}\,\mathrm{u2}\right)$

$1$

$\mathrm{\μ2}≔h\left(t\right)$

$\mathrm{\μ2}≔h\left(t\right)$

$\mathrm{T2}≔\mathrm{EnergyMomentumTensor}\left(''Dust''\,\mathrm{g2}\,\mathrm{u2}\,\mathrm{\μ2}\right)$

$\mathrm{T2}≔h\left(t\right){\cosh \left(f\left(t\right)\right)}^2\mathrm{D\_t}\mathrm{D\_t}-\frac{h\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)}{t}\mathrm{D\_t}\mathrm{D\_x}-\frac{h\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)}{t}\mathrm{D\_x}\mathrm{D\_t}+\frac{h\left(t\right)\left({\cosh \left(f\left(t\right)\right)}^2-1\right)}{t^2}\mathrm{D\_x}\mathrm{D\_x}$

$\mathrm{E2}≔\mathrm{MatterFieldEquations}\left(''Dust''\,\mathrm{g2}\,\mathrm{u2}\,\mathrm{\μ2}\right)$

$\mathrm{E2}≔\frac{h\left(t\right)\cosh \left(f\left(t\right)\right)+\stackrel{\mathbf{.}}{h}\left(t\right)\cosh \left(f\left(t\right)\right)t+h\left(t\right)\sinh \left(f\left(t\right)\right)\stackrel{\mathbf{.}}{f}\left(t\right)t}{t},\frac{{\cosh \left(f\left(t\right)\right)}^2-1+\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)\stackrel{\mathbf{.}}{f}\left(t\right)t}{t}\mathrm{D\_t}-\frac{\cosh \left(f\left(t\right)\right)\left(\sinh \left(f\left(t\right)\right)+\cosh \left(f\left(t\right)\right)\stackrel{\mathbf{.}}{f}\left(t\right)t\right)}{t^2}\mathrm{D\_x}$

$\mathrm{Soln}≔\left[h\left(t\right)=\frac{\mathrm{\_C2}}{{\left(1+t^2{\mathrm{\_C1}}^2\right)}^{\frac{1}{2}}}\,f\left(t\right)=\mathrm{arcsinh}\left(\frac{1}{t\mathrm{\_C1}}\right)\right]$

$\mathrm{Soln}≔\left[h\left(t\right)=\frac{\mathrm{\_C2}}{\sqrt{1+t^2{\mathrm{\_C1}}^2}}\,f\left(t\right)=\mathrm{arcsinh}\left(\frac{1}{t\mathrm{\_C1}}\right)\right]$

$\mathrm{simplify}\left(\mathrm{eval}\left(\left[\mathrm{E2}\right]\,\mathrm{Soln}\right)\,\mathrm{symbolic}\right)$

$\left[0\,0\mathrm{D\_t}+0\mathrm{D\_x}\right]$

$\mathrm{Div2},\mathrm{RHS2}≔\mathrm{DivergenceIdentities}\left(''Dust''\,\mathrm{g2}\,\mathrm{u2}\,\mathrm{\μ2}\,\mathrm{T2}\,\mathrm{E2}\right)$

$\mathrm{Div2},\mathrm{RHS2}≔\frac{2h\left(t\right){\cosh \left(f\left(t\right)\right)}^2-h\left(t\right)+\stackrel{\mathbf{.}}{h}\left(t\right)t{\cosh \left(f\left(t\right)\right)}^2+2h\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)\stackrel{\mathbf{.}}{f}\left(t\right)t}{t}\mathrm{D\_t}-\frac{2h\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)+\stackrel{\mathbf{.}}{h}\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)t+2h\left(t\right){\cosh \left(f\left(t\right)\right)}^2\stackrel{\mathbf{.}}{f}\left(t\right)t-h\left(t\right)\stackrel{\mathbf{.}}{f}\left(t\right)t}{t^2}\mathrm{D\_x},\frac{2h\left(t\right){\cosh \left(f\left(t\right)\right)}^2-h\left(t\right)+\stackrel{\mathbf{.}}{h}\left(t\right)t{\cosh \left(f\left(t\right)\right)}^2+2h\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)\stackrel{\mathbf{.}}{f}\left(t\right)t}{t}\mathrm{D\_t}-\frac{2h\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)+\stackrel{\mathbf{.}}{h}\left(t\right)\cosh \left(f\left(t\right)\right)\sinh \left(f\left(t\right)\right)t+2h\left(t\right){\cosh \left(f\left(t\right)\right)}^2\stackrel{\mathbf{.}}{f}\left(t\right)t-h\left(t\right)\stackrel{\mathbf{.}}{f}\left(t\right)t}{t^2}\mathrm{D\_x}$

$\mathrm{Div2}\&minus\mathrm{RHS2}$

$0\mathrm{D\_t}$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g3}≔\mathrm{evalDG}\left(x^2\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g3}≔x^2\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{A3}≔\mathrm{evalDG}\left(\mathrm{f1}\left(x\right)\mathrm{dt}+\mathrm{f2}\left(x\right)\mathrm{dy}\right)$

$\mathrm{A3}≔\mathrm{f1}\left(x\right)\mathrm{dt}+\mathrm{f2}\left(x\right)\mathrm{dy}$

$\mathrm{T3}≔\mathrm{EnergyMomentumTensor}\left(''Electromagnetic''\,\mathrm{g3}\,\mathrm{A3}\right)$

$\mathrm{T3}≔-\frac{{\mathrm{f2}\prime \left(x\right)}^2{x}^2+{\mathrm{f1}\prime \left(x\right)}^2}{2x^4}\mathrm{D\_t}\mathrm{D\_t}+\frac{\mathrm{f1}\prime \left(x\right)\mathrm{f2}\prime \left(x\right)}{x^2}\mathrm{D\_t}\mathrm{D\_y}+\frac{-{\mathrm{f2}\prime \left(x\right)}^2{x}^2+{\mathrm{f1}\prime \left(x\right)}^2}{2x^2}\mathrm{D\_x}\mathrm{D\_x}+\frac{\mathrm{f1}\prime \left(x\right)\mathrm{f2}\prime \left(x\right)}{x^2}\mathrm{D\_y}\mathrm{D\_t}-\frac{{\mathrm{f2}\prime \left(x\right)}^2{x}^2+{\mathrm{f1}\prime \left(x\right)}^2}{2x^2}\mathrm{D\_y}\mathrm{D\_y}-\frac{-{\mathrm{f2}\prime \left(x\right)}^2{x}^{}}{}$