An exact solution of the Einstein field equation is a Lorentz metric that corresponds to a physically realizable energy-momentum tensor. In the study of exact solutions of the field equations, it is sometimes convenient to decompose the Riemann tensor into its trace and trace-free parts. This is accomplished by taking the definition of the Weyl tensor in terms of the Riemann and Ricci tensors and making the Riemann tensor the subject of the formula. In four dimensions, this gives:
where the Weyl tensor is the trace-free part (as it satisfies Cabad = 0) and the tensors G and E have the following components:
- Gabcd = ga[cgd]b
![E_{abcd}=\tilde{R}_{a[c}g_{d]b}+\tilde{R}_{b[d}g_{c]a}](a/../upload/math/b1446f39b59c42403d674e54d58c83f1.png)
where 
are the components of the trace-free Ricci tensor.
Exact solutions for which the energy-momentum tensor is identically zero in the region under consideration are termed vacuum solutions and represent the gravitational field in a region of spacetime where there are no material gravitational sources; strictly speaking, as the gravitational field can do work (in moving planets around the Sun, for example), the field possesses energy (although determining the precise location of this energy in the field is still a problem) and therefore by E = mc2 has an effective mass which thereby creates another gravitational field. This is one of the major difficulties in finding exact solutions of the field equations, and quite often simplifying assumptions such as linearising the field equations are made.
References
- Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. and Herlt, E. Exact Solutions of Einstein's Field Equations (2nd edn.) (2003) CUP ISBN 0521461367
- Adler, R., Bazin, M. and Schiffer, M. Introduction to General Relativity (2nd edn.) (1975) McGraw-Hill New York ISBN 0-07-000423-4