How many equations will we really have to solve?

With two symmetries and three coordinates we can redefine (because we are doing the problem in three spacetime dimensions), we should discover that FIVE of our equations are not needed. But the curvature tensor we'll be using is a symmetric two-index tensor in three dimensions, meaning it has six independent components. Hence we are only starting with SIX equations.

This means when we go to solve this system, we should only find one differential equation for one unknown function. Let's see if things really turn out this way.

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Compute the equations