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Physical Mathematics by Kevin Cahill

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11.43 Einstein’s equations

The source of the gravitational field is the energy–momentum tensor Tij. In many astrophysical and most cosmological models, the energy–momentum tensor is assumed to be that of a perfect fluid, which is isotropic in its rest frame, does not conduct heat, and has zero viscosity. For a perfect fluid of pressure p and density ρ with 4-velocity ui (defined by (11.66)), the energy–momentum or stress–energy tensor Tij is

image

(11.370)

in which gij is the space-time metric.

An important special case is the energy–momentum tensor due to a nonzero value of the energy density of the vacuum. In this case p = –ρ and the energy– ...

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