atyy said:I wonder: how do we know the energy-momentum tensor is conserved until we have a solution, since the metric enters the covariant derivative which defines energy momentum conservation?
Einstein's equation, also known as the Einstein field equations, is a set of ten coupled, nonlinear differential equations that describe the relationship between the geometry of spacetime and the distribution of matter and energy within it. It is a cornerstone of Einstein's theory of general relativity.
For an equation to have a solution, it means that there exists a set of values that satisfy the equation and make it true. In the case of Einstein's equation, it means that there exists a set of values for the distribution of matter and energy that accurately describe the geometry of spacetime.
There is no general answer to this question. In some cases, the equations may have a unique solution, while in others, there may be multiple solutions. This depends on the specific conditions and parameters of the system being described.
Scientists use various mathematical techniques, such as numerical simulations and analytical methods, to solve Einstein's equation and determine if a solution exists. They also compare their results with observations and experiments to validate the accuracy of the solutions.
If solutions to Einstein's equation exist, it means that the theory of general relativity accurately describes the relationship between matter, energy, and spacetime. It also has implications for our understanding of the universe, from the behavior of massive objects like black holes to the expansion of the universe and the evolution of the cosmos.