This question is on the construction of the Einstein Field Equation.(adsbygoogle = window.adsbygoogle || []).push({});

In my notes, it is said that

>The most general form of the Ricci tensor [itex]R_{ab}[/itex] is [tex]R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}[/tex]

where [itex]R[/itex] is the Ricci scalar.

Why is this the most general form (involving up to the second derivative of the metric --- by definition of [itex]R_{ab}[/itex])? I suppose there are symmetry and degrees of freedom arguments. But I can only see why the LHS is a *possible* form. I can't se why it is the most general form...

>Taking the covariant derivative $\nabla_a[/itex] of the expression above gives [tex]C=\frac{1}{2}[/tex].

This I understand.

>Compare the resulting expression with the Poisson equation gives [tex]A=\frac{8\pi G}{c^4}[/tex].

This I *don't* understand --- perhaps I am being silly again... but still.

I assume the "Poisson equation" referred to here is [tex]E^i{}_i=4\pi\rho G[/tex]

where [itex]E^i{}_i[/itex] is the tidal tensor and may be expressed as [tex]E^i{}_i=R^i{}_{aib}T^aT^b[/tex] where [itex]T^a[/itex] is the tangent vector of the geodesic.

So contracting [tex]R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}[/tex] with [itex]T^aT^b[/itex] gives [tex]4\pi\rho G=AT_{ab}T^aT^b[/tex].

But then what? Or perhaps I have already made a mistake? I only know the energy-momentum tensor [itex]T_{ab}[/itex] to be of the form

[tex]\begin{pmatrix}H&\pi_i\\\frac{s_i}{c}&T_{ij}\end{pmatrix}[/tex] where [itex]H[/itex] is the energy density, [itex]\pi_i[/itex] is the momentum density, [itex]s_i[/itex] is the energy flux.

But I don't understand how it leads to [tex]A=\frac{8\pi G}{c^4}[/tex].

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# Constructing the Einstein Field Equation

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