Final answer:
A dust solution in general relativity is a model where the energy-momentum tensor represents a pressureless fluid with particles moving along geodesic paths. Simply having one non-zero term in the Einstein tensor does not suffice to define it as a dust solution; the entire tensor must represent the characteristics of a pressureless dust.
Step-by-step explanation:
The precise definition of a dust solution in general relativity refers to a specific type of solution to Einstein's field equations, where the stress-energy tensor describes a perfect fluid with zero pressure. In such a solution, the matter content of the spacetime is modeled as 'dust', which means that each particle in the fluid moves along a geodesic path and there are no forces between the particles other than gravity. It is a simplification used in cosmology to model the large-scale distribution of matter in the universe.
Having only the first diagonal term of the Einstein tensor non-zero is not sufficient to define a dust solution. For a space to be considered a dust solution, its energy-momentum tensor must represent a pressureless fluid; this requires more specific conditions than just one non-zero term in the Einstein tensor. In a dust solution, typically all diagonal components representing the energy density and momentum should follow the specific characteristics of dust, and the off-diagonal components associated with pressure should be zero.