Genady

To be precise, this metric is continuous, differentiable, and its first derivative is continuous. It is not twice differentiable though. Of course, it was introduced by hand, as defined in the question: "could one write down a metric for which ..."

lol +1

Today is October 20, 2020?

Developing on my previous post, I think that the metric \[diag(-1,1,1,1+H(z)z^2+H(-z)z^4)\] is not a valid solution to the EFE. *H() is Heaviside step function.

ok

If the metric is twice differentiable everywhere, then its Einstein tensor is everywhere defined, and you can just take this Einstein tensor as the energy-momentum tensor of your equation. Then, this metric is a solution of this equation. P.S. Of course, the metric has to be locally Lorentz to start with.

Both are unwelcomed.

In this example, a smooth curve appears not smooth as one zooms out. The opposite is also possible, i.e., a rugged curve appears smooth as one zooms out.

Space-time geometry is fine with singularities, too. Take a triangle. It is a perfect geometric shape in spite of having three singularities.

Right. But when this happens what fails is GR together with its framework, differential geometry. Not geometry. Geometry is fine with such singularities. Differential geometry has a problem.

This is incorrect. GR requires geometry with certain smoothness. It fails if the geometry is not sufficiently smooth.

I guess I was not clear enough. My claim is that geometry can exist without GR, but GR cannot exist without geometry. Thus, there cannot be an example of GR without geometry. As I said earlier, GR can fail for a reason other than absence of geometry.

@grayson, right?

"Failure of GR does not necessitate failure of geometry" implies geometry without GR. Why would I try to show an example of GR without geometry?

Evidently your perception is mistaken. I assume that it is based on pop-science rather than actual science sources. In my direct experience, science is fun and exciting.