Mitcher

I find it strange that you read multiple authors in the domain of BH but not the original, first one of them. It's not at all about historical interest, it is to make sure Schwarshid was not misunderstood. He was, and the recently translated papers can now easily be found. My concern there was about previous comment concerning a "density of some stuff". Properties of ST are mainly properties of mass. Most simplest models of BH, with their "incompressible sphere of matter" or 'point-mass' do not take matter itself into account, only the effects it creates on SP and fields. BH is about matter and energy content: no matter, no BH, hence the density towards their center should probably have something to do with Planck's density, at least in theory since my point here is that matter-energy in its known states cannot realistically exist past the horizon. And central singularities even less as it can be cancelled according to some theories.. I read and assimilated past comments though.

I knew about Peano curves but had never heard of Gabriel's horn, thanks for pointing them to me. Correct, it's equal to 1, not infinity, my mistake.

However one of the main problem in Physics is the discrepancy of 120 order of magnitude between the vacuum energy density at the cosmological scale expressed by the cosmological constant and the quantum vacuum energy density, at the Planck scale. The first one is given in g/cm3 and comes from the content of the Universe. The fact that they are being compared is not insignificant, I'am not sure we can speak about a density of nothing, at least it's not easy to grab. Just curious, hve you read BOTH papers from Schwarschild, with the external but also the internal solution ?

Yes, I forgot the name of that function which is null everywhere excepted on 0 where it is infinite. Yet the integral of that function is infinite. Brilliant.

Specially that BH, if they would exist, would be formed from rotating neutron stars and the rpm's would just instantly go over the hedge. Everything rotates in the Universe. They will say we have to take into account primordial BH's, but since nobody know how they could have formed...

Is this supposed to be Kruskal's new coordinates ? It looks like Weyl's ones to me, leading to dsigma^2 = hd(sigma)^2 + r^2 d(theta)^2 which is about the geometry described by the rotation of a parabola of equation z = (8a(r - 2a))^1/2 in euclidian space with ortho coordinates x, y, z. But as Weyl puts it there is "nothing" inside the sphere with radius 2a. Cosmic acceleration implies the action of a negative pressure and hence of negative energy (energy density by unit volume) so it would not be completely nonsensical to include negative energy states in QFT.

Penrose once said that the probabilities for the Universe to exist were almost zero. However, there it is.

Not at all, I was talking about the change of coordinates Kruskal did specifically to avoid the appearance of a singularity at 2m. The geodesics don't have to end at the throat, which could be seen, with the appropriate metric, as a space bridge, linking two 4D spaces so the geodesics could be prolonged contiuously in the adjacent field. Time and mass would seen to invert there. As a bonus, the central singularity disappears and negative mass could explain negative energy, which we seem to be badly needing now in cosmology.

Sorry, please read non-contractible hypersurface. Yes, that's what I meant, it is the tool to describes physical gravity. I mean the Kruskal ksi function = r + Ln(r - 1) and r + Ln(1 - r) is not regular for r = 2m since the log of something tending towards 0 tends towards minus infinity. I do not understand this. ST would be flat very far away from the Sun if it was not for the planets. If a mere change of variables allows for central or spherical singularities to disappear then what is real ? The proper mass and proper time remain as they are indeed, they change due to a change of coordinates, I agree.

That's a rather profound admittance since GR is a theory expected to describe reality perfectly, and now it would point at an incomplete result ? I do. Kruskal's transcendant f function is not at all smooth on the Schwarzchild's sphere. When the (r - 1) exponent tends to zero from the upper or the lower limit it takes different forms when passing this limit and both sides don't connect with each others. And that's normal since the sphere is not reducible.

If zero power zero = 1 then zero = 1 power(1/0), with undefined exponent. Hence zero power zero cannot equal 1 and is also undefined, excepted in programming usually.

We are ants lost inside a nuclear power station but give us another 50 millions years of intelligent evolution and we willl see it.

One can imagine the acceleration is isotropic for a given age of the Universe but again, how the mainstream is not looking at this, and how it keeps speaking about the Hubble constant ? If it accelerates it is not constant, right ? What am I missing there ? I mean, instead of looking at the Hubble slope I want to see a curve.

As much as Kruskal has been pushing the equations beyond reason (0 power 0 = 1, for one), he still could not get rid of his central singularity. A singularity cannot be real, it is singularly wrong IMO, so I also disagree with the space-time switch. The BH basic model is supposed to be static but it magically becomes dynamic. As far as I know it is impossible to put patches in the space-time itself, maybe an extremely advanced civilisation could it it but this site is not about SF.

The parameter r Schwarschild used has been wrongly interpretated as a radial parameter from the start. If the surface forms a paraboloid of revolution it createss a hole that is not reachable excepted by jumping in a higher dimension, this is easy to understand, like if there is a hole in a sheet of paper it is possible to go on the other side of the surface but impossible to go inside the hole. I mentionned the work of Shakharov, Souriau, Kostant or Petit on this specific.

ok, but this cosmological constant, as well with the Hubble parameter, are prone to varrying with very large scale of time isn't ?

Thanks for the .pdf's, very handy. I have no conceptual problem with c = 1 but within the frame of a numerical sheet it complicates the calculus making it necessary to fudge the equations as to avoid having dx' larger than dt', etc... I'am still working on it and I'am not sure somebody tried to show Lorentz transforms, ds invariance and velocities composition in a single coherent numerical piece of work.

Yes I'am working on it, thanks. Hi. With the help of the many answers I got from people here and the numerical exemple from Peter Collier’s book I was able to close this simple calculation for the first time, founding v’ = - v and ds = ds’ as it should be. Before that I was mixing v, x and t the wrong way. Only glitch is that he is using c = 1 and hence c squared also equals 1 but I think it will work if one use any number larger than 1 and v.

You are right, but Einstein invented his cosmological constant, suggested by Hilbert by the way, before the observed expansion which was discovered with surprise and even stupor. And even Hubble didn't believe in real expansion, he had another explanation for it.

But that's the point, a throat does not terminate, after passing the vertex of a parabola it just continues. The inside of the throat is outside the manifold. He had realized that, I keep suggesting we completely misunderstood Schwarschild's equation I'am afraid.

The manifold can be smooth and continuous without necessarily passing through an event Horizon, it should be the throat circle of a Flamm's paraboloid according to Schwarschild's original metric. There is a thing such as the theoretical apparent negative mass, at least theorized by Shakharov, Souriau, Kostant and others. It is supposed to appear when inverting time. In 1916 Schwarschild wrote a second paper which is less known, in which he wrote the metric inside the Sun. It was only traducted from german to english in 1999 if I remember right. Spherical symmetry does not necessarily imply central symmetry. If it's impossible to penetrate into a BH then it's impossible to be crunched towards the center, be it by following time or space.

No harm done. Interestingly it works almost exactly the same whether you use the angle itself or the sinus.

Why not, afer all it was not an obligation for this constant to exist in the first place, the Universe could have decided to quietly slow down and turn back to the future, and to a big crunch. Who are we to decide ?

It is made clear here that O (or S), is contemplating the two ends of a rod moving at velocity v. Hence x is a moving interval relative to O. We are progressing but that's more or less what I had understood, howecer unable to match some real numbers into the equations. Hi ! when younger I did astrophysics and my predilection subject is cosmology. Thanks for the link, that's exactly what I have been trying to achieve with plain numbers. I don't conform that it's not possible to draw a simple working space-time diagram where Lorentz relationships, velocities addition and conserved ds would be obvious. I understand of course that you would need many more dimensions than a piece of paper can offer.

I, fact no, the difference gets smaller and smaller for v approaching c. The maxima is found at about the 30 to 40 degrees range, where I get 2 degrees or so at most.