There is a paper here which does something like this, and draws pretty pictures. The metric has been "absolutized" by replacing the negative coeffecients with their absolute values, so it becomes Riemannian instead of Lorentizian, but the diagram is then annotated with a bunch of yellow triangles showing "the direction of time", and together these two things apparently contain all the information of the original spacetime. For the spacetime around an ordinary planet all the triangles point in the same direction, so this Riemannian version seems like a vali... (read more)
There is a paper here which does something like this, and draws pretty pictures. The metric has been "absolutized" by replacing the negative coeffecients with their absolute values, so it becomes Riemannian instead of Lorentizian, but the diagram is then annotated with a bunch of yellow triangles showing "the direction of time", and together these two things apparently contain all the information of the original spacetime. For the spacetime around an ordinary planet all the triangles point in the same direction, so this Riemannian version seems like a vali... (read more)