In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid.
The stress–energy tensor of a relativistic fluid can be written in the form[1] Here The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively three and five linearly independent components.
Several special cases of fluid solutions are noteworthy (here speed of light c = 1): The last two are often used as cosmological models for (respectively) matter-dominated and radiation-dominated epochs.
These can always be matched to a Schwarzschild vacuum across a spherical surface, so they can be used as interior solutions in a stellar model.
where the fluid interior is matched to the vacuum exterior is the surface of the star, and the pressure must vanish in the limit as the radius approaches
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
These are the same quantities which appear in the general coordinate basis expression given in the preceding section; to see this, just put
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 equation of state relating the pressure and density of the fluid; we assume only that we have one simple and one triple eigenvalue.
In the case of a dust solution (vanishing pressure), these conditions simplify considerably: or In tensor gymnastics notation, this can be written using the Ricci scalar as: In the case of a radiation fluid, the criteria become or In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to a timelike eigenvector, since there are Lorentzian manifolds, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to a spacelike eigenvector, and these cannot represent radiation fluids.
However, when no adapted frame is evident, these eigenvalue criteria can be sometimes be useful, especially when employed in conjunction with other considerations.