What restrictions does a perfectly rigid object place on the metric of time?

We need to do a detailed study about what is the metric inside a perfectly rigid object in various cases.

A simple case is a rigid half-sphere whose round border is far away in the Minkowski space. Modifying any of the 3D spatial distances within the object would require an infinite energy. The Einstein-Hilbert action would become infinite through any such modification.

We conclude that the metric of the three spatial dimension must be perfectly flat inside the object. The 3D spatial geometry is euclidean.

What about the 4D geometry which involves time? Is it possible that the time dimension is distorted?

An analogous problem in three dimensions is the case where we know that the round border is in a normal global 3D euclidean geometry where the dimensions are x, y, and z. The middle of the object may have a deformed geometry.

We have a foliation of the object where the round border of each folio f is in the plane z = f. That, is, they are horizontal at the round border but may have a deformation in the middle.

The folios themselves have a flat euclidean 2D geometry.

Is it possible that the folios are deformed in the middle? Yes it is. We cannot prove that making 2 dimensions euclidean forces the 3rd dimension to be euclidean.

In the Schwarzschild solution, both time and the radial coordinate are deformed. But there may be solutions in other cases where just the time is deformed.

Actually, we may define a metric where the 3 spatial dimensions are euclidean but time flows slower for smaller r. Then we can calculate the tensor on the left side of the Einstein equation. If we can construct a system whose stress-energy tensor is equal to this, then we have an example of a euclidean 3D geometry but a deformation of the time dimension.

Minimizing the rubber deformation energy

If we have a rubber object under external stresses caused by weights in it, springs, or whatever, it tries to minimize the energy it has to spend on deforming itself plus the energy in the external stresses.

For example, a rubber sheet will bend under a weight, so that some potential energy of the weight is released, at the cost of an increased deformation energy in rubber.

If we enforce further restrictions on the rubber object by preventing its deformation in certain areas to certain directions, the energy of the system will in most cases increase.

If we move a perfectly rigid object close to a spherical mass, then we restrict the metric which spacetime can assume close to the mass. The deformation energy of spacetime will probably increase.

Can we reduce some energy by moving the rigid object there? The rigid object is weightless. We cannot reduce its potential energy.

We assume that the rigid object originally was not under any stresses. Its deformation energy is zero, and will stay zero because it is perfectly rigid.

This argument suggests that a mass will indeed repel a perfectly rigid weightless object, but we need to study this in more detail.