# Is every solution of Einstein field equations unique?

Einstein's equation is

$$8 \pi T_{ab} = G_{ab},$$

where the left side contains the stress-energy tensor and the right side contains the Einstein tensor.

Is there exactly one unique stress-energy tensor corresponding to a given spacetime curvature? Or is it possible for one curvature (i.e. one spacetime metric) to be produced by multiple different mass configurations?

Einstein's equation is

$$G_{ab} = 8 \pi T_{ab}.$$

The left-hand side of the equation, $G_{ab}$, is called the Einstein tensor. It is an expression involving second derivatives of the metric, $g_{ab}.$

The right-hand side of the equation, $T_{ab}$, is called the Stress-Energy tensor. Each stress-energy tensor is associated with a unique matter configuration.

Given a specific matter configuration, $T_{ab}$, we can completely specify $G_{ab}.$ But $G_{ab}$ involves derivatives of the metric, not the metric itself. So specifying $G_{ab}$ does not uniquely specify the metric. In fact, there are multiple different metrics that can give rise to the same Einstein tensor, and therefore to the same matter configuration. So the answer to your first question is no, it is possible to have multiple different metrics with the same matter configuration.

Given a specific metric, $g_{ab}$, we can completely specify $G_{ab}$ by taking derivatives. Since $T_{ab} = G_{ab} / 8 \pi$, specifying a metric also completely specifies the stress-energy tensor, and therefore the matter configuration. So the answer to your second question is no, it is not possible to have one metric corresponding to multiple matter configurations.

Edit: As user Timaeus points out in the comments, the stress-energy tensor $T_{ab}$ is not associated with a unique matter configuration. For example, it carries no information about electric charge. From the perspective of pure GR, this is irrelevant, as it does not effect the curvature of spacetime. However, when considering non-gravitational effects (such as electrostatic forces, etc), the statement "each stress-energy tensor is associated with a unique matter configuration" does not hold.