Counting the number of Einstein field equations This might be a very naive question but I do not understand how the Einstein's Field Equations give us enough equations to solve for the metric in certain problems. In general, the EFEs have 10 independent equations because both the Einstein tensor and the Stress-Energy are symmetric. This is an adequate number of equations to determine the components of the 10 components of the metric if the stress energy tensor is either a function of the metric or is already known like in the case of the Schwarzschild solution. But as far as I know, we do not know the components of the stress-energy tensor beforehand most of the time. For instance, consider the stress-energy tensor for a collapsing dust: $$T^{\mu \nu} = \rho U^{\mu}U^{\nu}$$
Here, the velocity field of the dust is not known at all points in time and needs to be solved for (with an initial velocity field perhaps). But now we have 4 unknowns from the four velocity and 10 unknowns from the metric. However, we only have 10 Einstein field equations.
Another statement that confuses me is that the condition that the Einstein tensor and the stress energy be divergenceless (which gives us four equations) is inbuilt in the tensors and therefore reduces the number of independent equations from 10 to 6. It seems to me that the number of equations are still 10; if the divergenceless condition gives us no additional information, it should contribute no additional equations but not reduce the number of independent equations in the EFEs.
I understand that the remaining 6 equations are adequate to solve for the metric uniquely (up to a change in coordinates), but I just don't understand how we got here. If there are only 6 independent equations, how do we even solve arbitrary dust problems where there is no symmetry (let's say I were to program this into a computer) and there are 14 unknowns but only 6 equations?
 A: I would like to remind you some points that you already know but you missed.
As you said, the four velocity terms are unknown for all the points in spacetime. Only the initial conditions of the four velocity can be given (and actually, it is enough). More generally, initial conditions for a stress-energy tensor can be given. Given that, you can calculate the initial condition of the metric tensor by solving the Einstein equation. Given the metric tensor, you can solve equation of motion for the matter part at each local coordinates of spacetime. You can choose one from many, depending on your situation of interest. Considering the equation of motion for the matter part, now you are equipped with number of equations more so that the remaining degree of freedom for the matter part is of no problem anymore.
For example, you can solve magneto-hydrodynamic equation of motion for the matter part while you solve the Einstein equation at the same time. In the case when the gravitational interaction is not that serious compared to other types of interactions such as electromagnetic interaction, you can even approximate the effect of gravity as a simple term when you solve the magneto-hydrodynamic equation for simplicity.
As an another example, if your stress-energy tensor contains some scalar field, then you can solve the equation of motion that governs the scalar field, in local coordinates at each point in the space time. This also handles degrees of freedom of the scalar field.
The divergenceless condition provides you a restriction to the equation with too much degrees of freedom so that you can handle and solve the equation, and thus it gives information enough to remove four degrees of freedom.
In the case when you solve the Einstein's equation and the equation of motion of matter analytically, you need many assumptions or approximations because it is extremely hard to handle all the details in the equation. To get numerical solution, on the other hand, you can solve the equation of motion for the matter part and then the Einstein equation for each step while keep updating matter part and metric tensor part. This will give you a numerical solution of the equation.
