Fluid Solution - Einstein Tensor

Einstein Tensor

The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer.

In the special case of a perfect fluid, an adapted frame

(the first is a timelike unit vector field, the last three are spacelike unit vector fields) can always be found in which the Einstein tensor takes the simple form

where is the density and is the pressure of the fluid. Here, the timelike unit vector field is everywhere tangent to the world lines of observers who are comoving with the fluid elements, so the density and pressure just mentioned are those measured by comoving observers. These are the same quantities which appear in the general coordinate basis expression given in the preceding section; to see this, just put . From the form of the physical components, it is easy to see that the isotropy group of any perfect fluid is isomorphic to the three dimensional Lie group SO(3), the ordinary rotation group.

The fact that these results are exactly the same for curved spacetimes as for hydrodynamics in flat Minkowski spacetime is an expression of the equivalence principle.