Ricci curvature in 2 dimensions: two circular masses make an ugly solution?

Let us have a two-dimensional surface embedded in three spatial dimensions. The surface then has locally the Ricci curvature zero, if we can form the surface by bending a rigid sheet of paper. That is, the curvature can only be zero if the surface is a part of a cone or a "cylinder" locally.

Let us then embed a point "mass" M into the surface. We demand that Ricci curvature is positive at the point M. The obvious solution is a cone whose tip is M.

Note the analogue to Birkhoff's theorem. The theorem states in 1 + 3 dimensions that the only static metric around M is the Schwarzschild metric. Analogously, in two dimensions, the only possible surface around M is a cone.

Let us embed a circularly shaped mass into the 2D surface. The obvious solution is that we cut the head off the cone. The hole is circularly shaped.

    form a half-cone

              ____   circular mass

            /        \   

            \____/

          |            |

          |            |  bend

          |            |

              ____   

            /        \ X singularity?

            \____/   

                       circular mass

    form a half-cone

Let us then have two circular masses on a flat sheet of paper. We begin by bending the paper down along the two vertical lines.

                          /\

                        /    \

                      /        \

                      wedge

  

After that, we can form two half-cones at the top end and bottom end in the diagram by cutting off infinitely many infinitesimally wide wedges whose tip is on either circle.

But the solution is ugly. There is a transition from a cone shape to a flat shape along the egde of a circle. If the transition is sharp, we get a singularity-like point, marked with X in the diagram.

What if the transition is not sharp? Is Ricci curvature still zero outside the circle? Maybe not. At least, the solution is ugly.

Fitting an oversized carpet into a room

             wrinkle

       _______/\________ carpet

Our efforts to make Ricci curvature zero outside matter bring to our mind the carpet analogy from our blog post on November 9, 2023. One can make a part of the carpet flat on the floor, but since the carpet is oversized, there will always be a wrinkle. A wrinkle is a "singularity".

A rubber sheet model of gravity is flexible

The requirement that Ricci curvature is zero outside matter leads us to use a rigid sheet of paper above. It is conceivable that many configurations do not have a solution at all: an attempt to solve the problem inevitably leads to wrinkles in the paper, or to other types of singularities.

A rubber sheet model would be more flexible. Intuitively, we can always find a solution because a rubber sheet can be stretched. Thus, we know, or can guess, that a rubber sheet model always works. This is different from the Einstein field equations, where it is not clear if solutions exist at all for nontrivial cases.

In our blog we have been claiming for two years that the Einstein field equations may be too "rigid", so that they do not have solutions.

Conclusions

Requiring that Ricci curvature is zero outside matter can lead to singularities or ugly solutions. Maybe we have to give up the assumption that Ricci curvature is zero outside matter – also in the case of static configurations? For dynamic changes of pressure we already know (tentatively) that the Einstein field equations break.