of the Einstein tensor in a minimally coupled massless scalar field solution must have the form In other words, we have a simple eigenvalue and a triple eigenvalue, each being the negative of the other. Multiply out and using Gröbner basis methods, we find that the following three invariants must vanish identically: Using Newton's identities, we can rewrite these in terms of the traces of the powers. We find that We can rewrite this in terms of index gymnastics as the manifestly invariant criteria: Examples Notable individual scalar field solutions include the Janis–Newman–Winicour scalar field solution, which is the unique static and spherically symmetric massless minimally coupled scalar field solution. See also Exact solutions in general relativity Lorentz group References See section 3.3 for the stress-energy tensor of a minimally coupled scalar field. Exact solutions in general relativity