maps are not going to have the same set of features. For example, some maps will map one grid (set) point to many or map many to one. Such maps do not conserve information.

Singular maps have such non-conservation pathology. The term diffeomorphism refers to maps which have a continuous first derivative for all the dimensions involved. They do

not have to have continuous second and higher derivatives, but the continuity of the first derivative ensures that such maps conserve information.

In GR, we have rampant use of non-diffeomorphic maps (coordinate transforms) concocted to remove what are grossly misnamed as "coordinate singularities". Supposedly the

event horizon is merely a coordinate singularity. This is Orwellian level or language abuse since it has nothing to do with the original term in mathematics. The spherical

coordinates do have a coordinate singularity and that is at the poles. A pole is a single point, but the coordinates still apply all 360 degrees of longitude to it. But this

coordinate singularity has no relevance for the event horizon since it is a solution (metric) singularity in radius. There is no coordinate singularity in radius in the spherical

coordinates. GR cultists fervently believe that the event horizon is just some surface in empty space like any other surface.

But the event horizon is not just any surface. It is manifestly a physically singular surface. Photons coming towards the center of mass on radial trajectories will experience infinite

blue shifting. Photons trying to leave on radial trajectories will experience infinite red shifting. So such photons cannot exit the event horizon. Photons traveling tangentially

at the horizon surface are trapped on it. Another show stopper detail is that any massive particle or thing falling towards the "black hole" will reach the speed of light at the

horizon. This follows from the GR metric solution in static coordinates. In GR you can have flowing coordinates (steady state flowing coordinates are called stationary coordinates,

which is rather confusing). In flowing coordinates objects do not reach the speed of light relative to the coordinates. For example, the Painleve-Gulstrand metric involves

perpetual infall of coordinate shells from infinity towards the point mass singularity. At the horizon it is the coordinate shells that are moving at the speed of light and inside

the horizon they are moving faster than the speed of light. But no object is moving faster than the speed of light relative to any of these coordinate shells. The relevance of

such flowing coordinates for physically viable solutions is not clear. If the domain is not infinite then all of space-time will collapse onto black hole singularities in finite time.

Also, note that there is no speed of light limit to coordinate flow. It is just applied to movement by objects relative to those coordinates.

We are also back to aether flow. Funny how a hard core relativist, Einstein, has set up a theoretical framework that recycles aetherist ideas. Like I have posted numerous times,

the aether debate at the turn of the 20th century was an angels on the head of a pin style debate that set physics back. None of the experiments, including the overhyped

Michelson-Morley experiment, establishes relativity as the truth. The touted Lorentz transform can be consistently interpreted as an absolute transform without any

contradiction with empirical observations. When relativist zealots claim there is no absolute rest frame, they are full of shit. The absolute rest frame is the one where

**all**

of the photons live.

Anyway, getting back to black holes. There is a serious problem with the solution inside the event horizon. I will always refer to static space-time solutions where coordinates

do not flow for this discussion. There is a time-reversal at the event horizon. For null (photon) geodesics (space-time world lines, or existence trajectories) the Schwarzschild

metric in spherical coordinates is

0 = A(r)dt^2 + B(r)dr^2 + C(r)dW^2

where dt is the coordinate time increment, dr is the radial increment and dW is the angle increment (latitude and longitude). Consider radial photon trajectories for which dW=0.

The prefactors are

A(r) = 1 - 2GM/r

B(r) = 1/(1 - 2GM/r)

C(r) = r^2

where units such that the speed of light is c = 1 are used.

There is something called a Killing vector which pertains to invariants in the system defined by the metric. The element corresponding to time gives an energy constant of the

motion:

E = (1 - 2GM/r) * dt/dl where l is an implicit parameter describing any geodesic (all geodesics are parametric curves).

If E is a constant of the same positive sign crossing the horizon, then we need to change sign of dt since the 1 - 2GM/r changes sign but dl does not (it is supposed to apply

to the whole geodesic). Cultists will shrill whine that this proves that spherical coordinates are "singular". Total BS. The prefactors, A, B and C are the proper solutions of

Einstein's equations. If B(r) explodes at the horizon (i.e. r = 2GM, or r = 2GM/c^2 in SI units) then that is a feature of the solution and not the reference coordinate system.

Here the cultists will concoct a non-diffeomorphic coordinate transform mapping t, r, W (i.e. theta and phi) to some u, v, w (which are functions of t, r, W). The premier

hack is known as the Kruskal-Szekeres transform which looks like:

ds^2 = 32 (GM)^3 * e^(-r/(2GM))/r * (-dv^2 + du^2) + r^2 dW^2

where

u = (r/(2GM) - 1)^0.5 * e^(r/(4GM)) * cosh(t/(4GM))

v = (r/(2GM) - 1)^0.5 * e^(r/(4GM)) * sinh(t/(4GM))

and r is now defined through

(u^2 - v^2) = (r/(2GM) - 1) * e^(r/(2GM))

This is nothing but pseudo-mathematical masturbation that only physics nincompoops would engage in. It is clearly an attempt to force regularity at the event horizon and

pretend that there is no actual physical singularity there. But this is only "possible" because this transform is singular itself.

So what we most likely have in physical reality is no empty space-time inside the event horizon. The GR solution is only properly defined for r >= 2GM. Kerr's argument is

basically this but for the rotating case which he solved for decades ago. I have only shown the non-rotating Schwarzschild case. Instead of the fetishized point mass

singularity we have some sort of matter distribution over the volume bounded by the horizon. BTW, this volume is not the same as for the alleged point mass case since

there is missing extreme distortion as r -> 0.

What this matter distribution looks like is very interesting. If neutron stars are composed of neutrons and are like giant atomic nuclei, then "black holes" are even more compact

and must involved a compressed form of quark-gluon material.

An issue I have not addressed is how gravity actually works. If it is a quantizable field theory and not a geometric formalism like GR, then we can have a type of asymptotic

freedom at very high matter-energy densities. In quantum chromodynamics which describes the strong force in the nucleus of atoms, the potential is nonlinear. Quarks are

relatively free when they are in close proximity, but if you try to pull two of them apart you simply can't and end up forming two new quark pairs. For gravity we probably have

a weakening of the attraction at high density as gravitons spend more time interacting with each other and not doing their original job of attracting matter-energy. This would

require a nonlinear gravitational potential in the strong-field limit.

The Devil in the GR details is the formulation of the stress-energy tensor (the right hand side) in Einstein's equations. I will not get into details in this post, but Einstein's

formulation allows a pathological self-digging of the gravitational potential (inferred from the space-time deformation). This is not an empirically justified feature. "Black hole"

observations do not confirm GR. They merely establish the existence of high density objects that do not emit light like stars. They also do not establish that these objects

are even consistent with an event horizon. None of the observations are accurate enough to establish a GR consistent radius. That these objects do not emit light (at

wavelengths we can measure) just implies that nuclear chemistry is no longer active. Stars shine because of such chemistry (mostly fusion).