by multiplying by the metric, which will be useful when working with Einstein's field equations, and by further decomposition, This tensor is called the Ricci tensor which can also be derived by setting and in the Riemann tensor to the same indice and summing over them. Then the curvature scalar can be found by going one step further, So now we have 3 different objects, the Riemann curvature tensor: or the Ricci tensor: the scalar curvature: all of which are useful in calculating solutions to Einstein's field equations. The energy–momentum tensor The sources of any gravitational field (matter and energy) is represented in relativity by a type symmetric tensor called the energy–momentum tensor. It is closely related to the Ricci tensor. Being a second rank tensor in four dimensions, the energy–momentum tensor may be viewed as a 4 by 4 matrix. The various admissible matrix types, called Jordan forms cannot all occur, as the energy conditions that the energy–momentum tensor is forced to satisfy rule out certain forms. Energy conservation In special and general relativity, there is a local law for the conservation of energy–momentum. It can be succinctly expressed by the tensor equation: This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'. The Einstein field equations The Einstein field equations (EFE) are the core of general relativity theory. The EFE describe how mass and energy (as represented in the stress–energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor). In abstract index notation, the EFE reads as follows: where is the Einstein tensor, is the cosmological constant, is the metric tensor, is the speed of light in vacuum and is the gravitational constant, which comes from Newton's law of universal gravitation. The solutions of the EFE are metric tensors. The EFE, being non-linear differential equations for the metric, are often difficult to solve. There are a number of strategies used to solve them. For example, one strategy is to start with an ansatz (or an educated guess) of the final metric, and refine it until it is specific enough to support a coordinate system but still general enough to yield a set of simultaneous differential equations with unknowns that can be solved for. Metric tensors resulting from cases where the resultant differential equations can be solved exactly for a physically reasonable distribution of energy–momentum are called exact solutions. Examples of