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.

I think you have a very wrong idea about what scientists do and why. Let me give you a glimpse into it with this quote: P.S. Just posted a quote about representation theory and here in the next post physicists discuss representation theory:

It may be their personal belief. You can also say that some people who reside on fifth floors "do not even believe that the subjective even exists." This is as true.

The fact that scientists do not claim to have a full grasp of reality does not imply "that the domain of the subjective is not required to have a full grasp of reality."

Many scientists claimed to have a full grasp of reality? Who? When? How many?

Swansont claimed to have a full grasp of reality? When? Where?

Not only I know well the history of human innovation as I have studied it professionally (my PhD thesis), I know also that computers do not work like brain (having MSc in Computer Science) and that SCUBA is not a tank with compressed air (being a PADI SCUBA Instructor.) You philosophize without knowledge.

You are playing with words. Meaningless game.

They do not contradict because we do not use "that design in our own way", but use "physical and chemical effects found in nature in our own way." The distinction is between using design vs using effects. This thread is about design.

We do not take design cues from nature, almost never. We imply physical and chemical effects found in nature, but use them in our own way.

Airplanes do not fly like birds. Submarines do not swim like fish. Automobiles do not move like horses. Computers do not work like brains. ... Most of our designs do not have even functional analogs in nature. Telescopes. Spaceraft. Electronics. Assembly lines. Transmitters. Refrigerators. Steam engines. Wheels. Railroad. SCUBA. The list of examples is endless.