the star, and the pressure must vanish in the limit as the radius approaches . However, the density can be nonzero in the limit from below, while of course it is zero in the limit from above. In recent years, several surprisingly simple schemes have been given for obtaining all these solutions. 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 energy 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. Eigenvalues The characteristic polynomial of the Einstein tensor in a perfect fluid must have the form where are again the density and pressure of the fluid as measured by observers comoving with the fluid elements. (Notice that these quantities can vary within the fluid.) Writing this out and applying Gröbner basis methods to simplify the resulting algebraic relations, we find that the coefficients of the characteristic must satisfy the following two algebraically independent (and invariant) conditions: But according to Newton's identities, the traces of the powers of the Einstein tensor are related to these coefficients as follows: so we can rewrite the above two quantities entirely in terms of the traces of the powers. These are obviously scalar invariants, and they must vanish identically in the case of a perfect fluid solution: Notice that this assumes nothing about any possible