FIFTH COSMOS DECLARATION: Negative Curvature of the universe

Still an open question on the morphology of the universe, is which mode fits best. Usually three modes are brought forth. One is the (closed) positive curvature which would result eventually with the universe recompressing as if returning to the initial hot inflation or big bang. Second is the apparently supported flat universe which in certain ways do not change, with the properties of the cosmos on knife edge and may end in an eternal expansion. Third mode is negative curvature where likely there is no constraint on expansion and the universe is open and forever expanding. The latter two would tend to ensue an also very cold cosmos as the heat sources die out. Einstein and De Sitter favored a positive curvature which would be a cylindrical cosmos. Space would have a constant curvature which does not change with time, "wherein the radius of the curvature is associated with the total mass of matter located in space" is a wont of Einstein but not of de Sitter.

 As is seen presently, the universe expanded at least once to range from big bang by a repulsive energy to its present size, which is not static. Perhaps it should be recalled that Einstein did not favor infinities. He banished infinities from his calculations to make them more manageable.

One take may be that indeed the universe had the capacity to expand, as from the big bang, and then another energy takes over some 9-6 billion years ago to accelerate the expansion. Neither expansion event are well understood presently. "The most basic component of modern cosmology is the expansion of the universe.", wrote Lawrence M. Krauss and Robert J. Scherrer. A key feature, which is fairly recently explored, and believed found in the universe is an important nature to how we see the universe today and likely tomorrow. Expansion is not fully explained but the uncovered nature of 'empty space' provides an amazing insight how expansion may proceed.

'Empty space' is depicted, and associated with Dark Energy, as the source that provides the energy to expand the universe. It seems to be depicted as self propagating. It is like there are packets of energy alone and they simply are the sole source to process and propagate by itself more energy whereupon there is further expansion. This relies on no other nature process to proceed. "As space expands, each new volume of space contains more of the energy that the constant provides.", "every cubic centimeter of space abounds with invisible energy! This energy, utterly invisible and unperceivable, has no 'useful' form. Its sole function and result is to push space apart, and as is it does so, new space comes into existence, itself just as rich in energy as the space that gave it birth.", wrote Donald Goldsmith in "the runaway universe."

So, the universe is found expanded. How is that explained? The space is filled with 'empty space' and so that is providing capacity to expand. It is likely the universe will keep expanding, as recent studies, including COBE satellite in 2006 and more, which indicate there probably is not enough matter density to halt and overcome and reverse the expansion tendency.* The tendency is toward dissipation, as from negative pressure, not contraction. Therefore, expansion is untethered.

Questions to be asked are:

1) if the universe was not to expand or have a limited expansion, i.e. fluctuating , then why is there 'empty space' to augment expansion?

2) May there be any constraint to this expansion or again, is it untethered as there indeed is almost nothing to hinder its process as it has done for billions of years.

3) For what other reason would there be ongoing acceleration of the cosmos but for an open mode?

4) If it is a flat universe, then the total energy should already be measurable. But with a curved universe it would be open and more difficult to measure.

'The sea that I am entering has never been crossed', a line that Alexander Friedmann II liked to use.

Alexander Friedmann started review of the question in Dec. 6, 1922 letter to Einstein, "it will follow from the world equations derived by you that the radius of curvature of the world is a quantity independent of time." That was to be clarified to Einstein in May 1923, "where Friedmann showed by direct calculations that the necessary condition for the disappearance of the divergence of the matter tensor, which was pointed out by Einstein in his note, by no means implies that the radius of the curvature of the world is constant in time." 

Alexander A. Friedmann II, mathematician and astronomer working on general relativity solutions, reasoned that the X and Y axis can extend forever. So why should not the universe have the same capacity, especially in an unchecked mode since its assumed micro-inception. Yakov D. Tamarkin suggested that negative constant curvature should be considered as a viable model. Friedmann then made a case for a nonstationary universe. He published a paper on that in 1924, "it will be shown that  it is really possible to obtain a world with constant negative curvature of space from Einstein's cosmological equations". In which case the universe would be depicted as open or negative curvature, especially if 'empty space' can keep producing. "We saw that the Einstein cosmological equations have solutions describing a world with constant negative curvature of space. This fact shows that the cosmological equations alone are not sufficient to answer the question of the finiteness of our world. ...It is clear from here that before discussing the finiteness of the world, it should be clarified what points should be regarded as coincident and what as different." This may be the first time that negative curvature constant was computed. What would obviate that? His further computations came up with, "from this that non-stationary worlds with constant negative curvature of space and with positive density of matter could exist." A dust universe would best fit with this model.

Friedmann, finishing his work on obviating the static universe, next turns to more deeper meanings of the cosmos structure, the proper understanding of finite and infinite. He says, "I consider it particularly necessary to treat the question about the Universe in due detail, because...distorted ideas have spread about finiteness, closedness, curvature and other properties of our space, which are supposedly established by the relativity priniciple." He draws up, "one misunderstanding repeated not only in popular science articles and books, but also in more serious and specialist works on the relativity principle. I mean the notorious question of the finiteness of the Universe, i.e. of the finiteness of our physical space filled with shining stars." (In those days, the distant objects regarded viewed in the heavens were depicted as mere nebulae and not as distinct stars and galaxies.) Here, he addresses the positive curvature and may be attributed to it, like a finite, straight line and the volume of the Universe is also finite. "These by no means follow from the metric of the world, but the metric can only be derived from the world equations."

Friedmann co-authored a book "Fundamentals of the Theory of Relativity", part 1, published by Academia Publishers with physicist Vsevelod K. Frederiks, who had returned around 1921 from being detained during WWI at Gottingen, Germany and worked then with David Hilbert.

On the then existing ideas of the general structure of the universe, Friedmann assessed that those assumptions do not follow from the general theory of relativity and that they cannot follow from it because the general theory is incapable of studying the global, only local properties. He transferred the focus away from local toward global, topological study of the world structure. He did not want the world described only in Euclidean methods, he sought to expand parameters to match the universe. He also used Riemannian geometry and Weyl to that end.

Using the Ricci tensor and scalar for radius of curvature (R), Friedmann chose to diverge from others' reliance on energy density via rho (p) for determining  the way the universe would proceed or expand. He made it an independent parameter.

Friedmann utilized the Ricci scalar, "R", to determine the curvature of the universe. The Ricci scalar: "represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of Euclidean space...As such, it provides one way of measuring the degree to which a given Riemannian metric might differ from that of ordinary Euclidean n-space." In relativity, the Ricci tensor is part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time. Friedmann used non-zero values in his equations to depict the speed of the universe and its evolutionary pattern.

It does not make a lot of sense to hedge for a knife edge existence (and unchanging ?) when there is abundance of 'empty space' noted and to not think that all of that expansion would limit itself. It seems the outcome of an abundance and self increasing 'empty space' alludes to the likelihood already if not the destination that the universe is moving into or will be negative constant curvature or open (-k). Negative curvature (and energy) is associated with an open universe. It can be found then and now that several science studies imply many elements support an unbounded expansion. (Actually, scientists have narrowed it down to just two basic options, contraction to a big crunch or open expansion.) "Today we know that the Universe is infinite, so it must always have been infinite, even at the beginning. What relevance, then, can the Schwarzchild radius have to the cosmic spacetime? " wrote Fulvio Melia around 2014. The influence of gravity is also seen as dwindling. The ninety degree angles that depict the flat universe are not fixed and can morph.

"If one constructs a triangle on a sphere from arcs of great circles, the sum of its angles will exceed two right angles."

"Curvature is a proper, invariant property of space, therefore its quantitative measure does not depend upon the system of coordinates chosen." " Curvature can be measured if one knows the space metric tensor. Although the type of a metric tensor depends very strongly on what system of coordinates is used, the curvature calculated from it is free from such dependence."

Friedmann had noted on world metric, "the interval should be purely imaginary at all points of the world and for all events, having the same space coordinates (the square of the interval in this case should be negative)." It is noted from his computations, "So, it does not follow from the constancy and positiveness of the curvature of the Universe at all that our Universe is finite." 

Further, "Friedmann stresses that the density in his solutions is positive, as it should be: 'From this follows the possibility of nonstationary worlds with constant negative space curvature and positive matter density'.

"On the assumption that there is not such vacuum, [antigravity] it was asserted during the 1960's  that the data on the low density of matter implied not only irreversible expansion, but also negative space curvature. The negative curvature was commonly associated with an infinite volume of the Universe," "the new geometrical picture disproves finiteness of volume for spaces with negative constant curvature."

"In his new  work he introduces a new geometry, its two-dimensional analogs no longer being similar to a sphere in terms of their general structure: paraboloid as well as pseudosphere are surfaces which are infinitely extendable, so their area should be considered as infinite. If one transfers this property of two-dimensional spaces of negative curvature directly to a three-dimensional space of the same type, one should ascribe an infinite value to its volume." 

It is noted that General Relativity could totally resolve features of the cosmos. But the formulae does not congruently fit a flat, planar cosmos. A broadened parameter set will better suit the observations with the formulae 

Albert Einstein, clinging to Ernst Mach, held that space and time could not exist without having matter present to measure it. Unless the supply of mass was infinite, the extent of spacetime could not be infinite, unless there is no mater.

de Sitter in 1916 described a universe infinite in extent. Einstein had to drop Mach's idea from general relativity. as incongruent. He had to accept that space is separate from matter. This latter is reinforced by Alexander Friedmann's paper in 1922.

It is said the data favors a flat universe for as far as we can see. Yet, how to fully explain that the universe is only now flat. Did it start out as flat and how did it evolve? It could be that is all we read currently. The current universe may be perceived as flat but a better collection of data may show that universe is limited or isolated. There are currently at least two science papers depicting that while the universe is perceived as flat, they imply there are curves at the outer range. It is noted, from Friedmann's works, that there are 18 different varieties of topology in Euclidean space, and for curved spaces the number of allowable topological realizations could be infinite. 

An earlier paper involving work on curvature and different possibilities of interpretation of them including open universe was produced in 1988 by J. D. Cohn and D.I. Kaiser in  "Phyiscal Review D, 58, "Where do all the Supercurvatures modes go?" "Rather, it is important to understand the dynamics of these modes within region R, corresponding to our observable, open universe." "These non-Cauchy surfaces (fixed time in the flat coordinates) extend into the open universe."

It is so far determined that there is nothing outside the frontier (or "boundaries") of the universe that would impede expansion, so negative constant curvature could be applicable to this situation. There likewise is no stated surety, yet, that hot inflation necessarily follows the alleged vacuum virtual particles inception, nor is there an argument that determines hot inflation must follow such developments in any other scenarios of inception of a cosmos. (The cosmos as such would be in deep freeze and conflagration or hot inflation chances arising would likely be negligible; one would have to explain how a conflagration would materialize and provide multiple instances for more like firecrackers sequentially igniting in the dimension of unlit, dense cold, to which the universe is returning.)

There is a factor of processes of the universe, e.g. particle physics. It is noted that some processes are likely to be ongoing for trillions of years. If the universe is "closed" or would end completely before the processes, that are identified, then the processes would not have opportunity to complete. With an open universe, then there will be ample opportunity for these processes to complete.

  A greatly improved collection of data might show that the flat universe is encompassed by a larger field which may not register readily with calculations. It is possible the flat universe is surrounded by this large field which may appear to be devoid of anything that will register. Friedmann had early in 20th century introduced the term "hypersurface" to describe an embedded universe. A very recent paper published in Feb. 2018 lead by Anders Andreassen at Harvard implies negative energy will sweep through the cosmos as a bubble as result of an adjustment. It may be noted here that that there is not a verified direct link with negative energy and negative curvature. A dynamic treatment is given at arxiv.org: 1402.4522. A paper of Feb.2, 2008 by Lawrence M. Krauss and James Dent touched briefly on negative energy.

'The deepest breakthroughs in physics took place when a contradiction forced researchers to re-evaluate their assumptions.', Nathan Selberg recently said.

There are, of course, other scientists and people who have dwelt in the issue of open universe and negative curvature.  There are many other different ways of approaching this subject. For one example, one modification from Stephen Hawking and Thomas Hertog paper is recently published  in 2017 and 2012. Sean Carroll has a work up at the turn of the century, as did also Lawrence M. Krauss and Glenn D. Starkman. The team of K.Dutta, Ruchika, A. Roy worked up an analysis with possible observational data included in August 2018 (plus some of their references include Yuting Wang), Arxiv.org 1808.06623.

Some say recently that there is even room in data from the Planck satellite for a marginally open universe.

Liddle and Cortes considered possibilities of negative curvature in their 2013 paper.  It is said, "The researchers showed that the curvaton field would generate the lopsided density fluctuations that have been observed if space had a slightly negative curvature on large scales." This would develop, perhaps during bubble nucleation, while the universe is quite young and the scale would grow from there probably from asymmetry that allows a dipolar modulation.

* Even in mid twentieth century it was evident there would be constant expansion as was written by George Gamow's book, "One Two Three...Infinity" p. 333, " Is there any possibility that the expanding masses of the universe will turn back on us and squeeze our stellar system, the Milky Way, the sun, the Earth, and the humanity on Earth,  into a pulp with nuclear density?

According to conclusions based upon the best available information, this will never happen.  Long ago, in the early stages of evolution, the expanding universe broke all of the ties that might have held it together and is now expanding into infinity"