The stress-energy tensor of a perfect fluid in its rest frame is:(adsbygoogle = window.adsbygoogle || []).push({});

(1) Tij= diag [ρc2, P, P, P]

where ρc2 is the energy density and P the pressure of the fluid.

If Tij is as stated in eq.(1), the metric tensor gij of the system composed by an indefinitely extended perfect fluid in its rest frame must be the solution of the corresponding Einstein’s equation:

(2) Rij –1/2 gij R = (8πG/c4) Tij

The metric tensor

(3) gij= diag [1, -1, -1, -1]

cannot be the solution of eq.(1) because from eq.(3) it follows that:

(4) Rij –1/2 gij R = 0

If the foregoing argument is true, man cannot say that in a generic frame, where the fluid’s four-velocity is Ui and the metric tensor is gij, the stress-energy tensor is:

(5) Tij= (ρ +P/c2) Ut Ut + P gij

Where am I wrong?

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# B Metric tensor of a perfect fluid in its rest frame

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