What Lies Within a Star

May 14, 2022

Infinity is infinite.
Sound obvious?
A star collapses.
Event horizon enshrouds a singularity?
"[The star] collapses in on itself, packing first its core, and later all the matter close to it, into a point of infinitesimal dimensions and infinite density — a singularity."
What is infinite is infinite. In other words, for black holes to exist at all, the singularity must be the basis for spacetime itself. The form of the existence of spacetime is the possibility of the singularity. You can not "create infinity". Either it exists or it doesn't, and we know that it does as a matter of logic.
Mar 4, 2020
Not according to modern theory. Space is not infinite. Space came with mass, and it, along with mass is expanding at a very high rate. So, infinity is being filled, now. What is on the outside of space and matter? What is out there, beyond space? How do we know?
Cesium -

Let me offer a few thoughts on your statements.

Fundamental particles don’t have a set diameter, or even even exist as what we normally think of as particles; instead, they are fields of distributed probability that act about a particular locus. When you push these fields really close together (which requires enormous energy) they sometimes convert to other particles, but they don’t just get packed edge to edge like billiard balls in a rack because particles of any size don't have a solid surface.

It is also wrong to think of a black hole as “infinitely dense”, and in fact, really supermassive black holes, which is hypothesized to be at the galactic nucleus, have an average density of less than that of the atmosphere within the event horizon. Instead, what happens is that the slope of the spacetime plenum becomes more and more curved as you approach the hole, until it is such a sharp curvature that you can’t climb back out of the hole.

This is mathematically represented as a radial curve of increasing curvature which becomes infinite at the singularity point, once you pass the event horizon it is no longer useful to say anything about the spatial properties within. In fact, if you were inside of the event horizon of a black hole that was sufficiently large that the gravitational gradients didn’t tear you apart you may not even notice that you were inside except that space may appear to be stretched out a little more in front of you than behind you.

It is more accurate to conceive of the singularity as being an area where space is stretched out to infinity (albeit within a proscribed boundary) than squeezed into an infinitely tiny space.

This applies to a non-charged, non-rotating black hole, and does not include ring singularities and other more exotic discontinuities.

Rotating black holes have ring singularities and yes, they can be passed through and act as wormholes. This is described by the Kerr metric. There are also trajectories which are closed time-like loops.

With that said,
However you should not physically trust in the inner horizon or the inner ergosurface. Although they are certainly there as mathematical solutions of the exact vacuum Einstein equations, there are good physics reasons to suspect that the region at and inside the inner horizon, which can be shown to be a Cauchy horizon, is grossly unstable — even classically — and unlikely to form in any real astrophysical collapse.
See: Visser, Matt. "The Kerr spacetime: A brief introduction." arXiv preprint:
arXiv:0706.0622 (2007).

This problem of stability is often discussed in terms of "mass inflation" which essentially are feedback loops that take even minuscule amounts of matter and energy and blows them up to rival the mass of the black hole very quickly. Since the Kerr metric is a vacuum metric, that geometry will obviously be completely changed if there are such feedback loops. Additionally, there is no 'clean way' to generate the Kerr metric from a real astrophysical collapse.

Presumably, a real rotating black hole would have the outward features of a Kerr black hole such as the ergosphere (a horizon associated with angular momentum) and event horizon, but a much different interior.

The Kerr spacetime has now been with us for some 45 years. It was discovered in 1963 through an intellectual tour de force, and continues to provide highly nontrivial and challenging mathematical and physical problems to this day.

The final form of Albert Einstein’s general theory of relativity was developed in November 1915, and within two months Karl Schwarzschild (working with one of the slightly earlier versions of the theory) had already solved the field equations that determine the exact spacetime geometry of a non-rotating “point particle”. It was relatively quickly realised, via Birkhoff’s uniqueness theorem, that the spacetime geometry in the vacuum region outside any localized spherically symmetric source is equivalent, up to a possible coordinate transformation, to a portion of the Schwarzschild geometry — and so of direct physical interest to modelling the spacetime geometry surrounding and exterior to idealized non-rotating spherical stars and planets. (In counterpoint, for modelling the interior of a finite-size spherically symmetric source, Schwarzschild’s “constant density star” is a useful first approximation. This is often referred to as Schwarzschild’s “interior” solution, which is potentially confusing as it is an utterly distinct physical spacetime solving the Einstein equations in the presence of a speci- fied distribution of matter.)

Considerably more slowly and only after intense debate was it realized that the “inward” analytic extension of Schwarzschild’s “exterior” solution rep- resents a non-rotating black hole, the endpoint of stellar collapse. In the most common form (Schwarzschild coordinates, also known as curvature coordinates), the Schwarzschild geometry is described by the line element:

Screen Shot 2022-05-16 at 5.15.19 PM.png

where the parameter m is the physical mass of the central object.

Physicists and mathematicians looked for such a solution for many years, and had almost given up hope, until the Kerr solution was discovered in 1963 — some 48 years after the Einstein field equations were first developed. From the weak-field asymptotic result we can already see that angular momentum destroys spherical symmetry, and this lack of spherical symmetry makes the calculations much more difficult. It is not that the basic principles are all that different, but simply that the algebraic complexity of the compu- tations is so high that relatively few physicists or mathematicians have the fortitude to carry them through to completion.

Indeed it is easy to both derive and check the Schwarzschild solution by hand, but for the Kerr spacetime the situation is rather different. For instance in Chandrasekhar’s* magnum opus on black holes, only part of which is devoted to the Kerr spacetime, he is moved to comment:
“The treatment of the perturbations of the Kerr space-time in this chapter has been prolixius in its complexity. Perhaps, at a later time, the complexity will be unravelled by deeper insights. But meantime, the analysis has led us into a realm of the rococo: splendorous, joyful, and immensely ornate.”

More generally, Chandrasekhar also comments:
“The nature of developments simply does not allow a presentation that can be followed in detail with modest effort: the reductions that are required to go from one step to another are often very elaborate and, on occasion, may require as many as ten, twenty, or even fifty pages.”

Of course the Kerr spacetime was not constructed ex nihilo. Some of Roy Kerr’s** early thoughts on this and related matters can be found, and over the years he has periodically revisited this theme.

For practical and efficient computation in the Kerr spacetime many researchers will prefer to use general symbolic manipulation packages such as 'Maple', 'Mathematica', or more specialized packages such as 'GR-tensor'. When used with care and discretion, symbolic manipulation tools can greatly aid calculations when there is relatively little textbook coverage dedicated to this topic.

An early discussion can be found in the textbook by Adler, Bazin, and Schiffer [1975] second edition]. The only dedicated single-topic textbook I know of is that by O’Neill. There are also comparatively brief discussions in the research monograph by Hawking and Ellis, and the standard textbooks by Misner, Thorne, and Wheeler, D’Inverno, Hartle, and Carroll.

One should particularly note the 60-page chapter appearing in the very recent 2006 textbook by Pleban ́ski and Krasin ́ski. An extensive and highly technical discussion of Kerr black holes is given in Chadrasekhar, while an exhaustive discussion of the class of spacetimes described by Kerr– Schild metrics is presented in the book “Exact Solutions to Einstein’s Field Equations”.

See: https://www.britannica.com/biography/Roy-P-Kerr

See: https://arxiv.org/pdf/0706.0622.pdf

See: View: https://www.reddit.com/r/askscience/comments/4w0y59/so_i_read_that_rotating_black_holes_have_ring/

See: https://boards.straightdope.com/t/black-holes-atoms-and-infinite-density/560793/7

* In 1931, Subrahmanyan Chandrasekhar calculated that if a star is big enough when its fuel runs out, there is nothing to stop gravity from making its core collapse to create a black hole.

The Chandrasekhar limit determines if a star dies as a white dwarf, or has the mass to exceed this, launching a supernova to create a black hole or neutron star.

Stars are locked in battles against their own gravity, all of which will eventually be lost, leading to violent and radical changes that mark the end of their main sequence lifetimes.

Some of these stars will end their lives as slowly cooling stellar embers known as white dwarfs, but for other stars, this stage merely marks a transition. They will go on to explode in massive cosmic blasts called supernovas creating a neutron star or even a black hole.

See: https://www.space.com/chandrasekhar-limit

** Roy Kerr, in full Roy Patrick Kerr, (born May 16, 1934, Kurow, New Zealand), New Zealand mathematician who solved (1963) Einstein’s field equations of general relativity to describe rotating black holes, thus providing a major contribution to the field of astrophysics.

Kerr received an M.S. (1954) from New Zealand University (now dissolved) and his Ph.D. (1960) from Cambridge University. He served on the faculty of the University of Texas at Austin (1963–72) and, returning to New Zealand, then became a professor of mathematics at the University of Canterbury, Christchurch, in 1972; he retired as professor emeritus in 1993.

See: https://www.britannica.com/biography/Roy-P-Kerr

To conclude, first of all, it is wrong to think of a black hole as “infinitely dense”, and in fact, really supermassive black holes like that inhabiting the galactic nucleus has an average density of less than that of the atmosphere within the event horizon. Instead, the slope of the spacetime plenum becomes more and more curved as you approach the hole, until it is such a sharp curvature that you can’t climb back out of the hole.

Remember that the Schwarzchild radius merely represents the point of no return for a black holes, whereas internally you find the singularity. While their masses are large, they certainly are not infinite. But GR (General Relativity) says that the mass density of singularities is infinite (and the size is zero). But nobody really believes that. This is felt to be a limitation to GR theory.

A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete. The singularity is indicated by a value of infinity for that point, but infinity is not a number and is not a way of measuring any physical quantity. It's simply an out-of-band value used to simplify calculations and should not be mistaken for an actual value.

Concerning spacetime and singularities, we find that all classical spacetime theories of gravitation, the occurrence of singularities form an inevitable and integral part of the description of the physical reality. In the vicinity of such a singularity, typically the energy densities, spacetime curvatures, and all other physical quantities would blow up, thus indicating the occurrence of super ultra-dense regions in the universe.

Again, as seen above, the behavior of such regions may not be governed by the classical theory itself, which may breakdown having predicted the existence of the singularities, and a quantum theory of gravity would be the most likely description of the phenomena created by such singularities.

Spacetime is not created by singularities, but singularities are an inevitable part of spacetime, based on stellar evolution beyond a mass known called the Chandrasekhar limit.

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