On singular semi-Riemannian manifolds

Stoica, Ovidiu Cristinel

doi:10.1142/s0219887814500418

Cited by 37 publications

(76 citation statements)

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“…In fact, contrary to what is widely believed, we will see that the singularities of the FLRW model are easy to understand and are not fatal to General Relativity. In [24] we presented an approach to extend the semiRiemannian geometry to the case when the metric can become degenerate. In [25] we applied this theory to the warped products, by this providing means to construct examples of singular semiRiemannian manifolds of this type.…”

Section: Contentsmentioning

confidence: 99%

“…In [24] we introduced a way to extend semi-Riemannian geometry to the degenerate case. There is a previous approach [34,35], which works for metric of constant signature, and relies on objects that are not invariant.…”

Section: 2mentioning

confidence: 99%

“…Our need was to have a theory valid for variable signature (because the metric changes from being non-degenerate to being degenerate), and which in addition allows us to define the Riemann, Ricci and scalar curvatures in an invariant way, and something like the covariant derivative for the differential forms and tensor fields which are of use in General Relativity. After developing this theory, introduced in [24], we generalized the notion of warped product to the degenerate case, providing by this a way to construct useful examples of singularities of this well behaved kind [25].…”

Section: 2mentioning

confidence: 99%

“…In [24,25], it is developed the geometry of manifolds endowed with metrics which are not necessarily non-degenerate everywhere. A special type of singularities, named semi-regular, are shown to have nice properties.…”

Section: Perspectivesmentioning

confidence: 99%

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The Friedmann-Lemaître-Robertson-Walker Big Bang Singularities are Well Behaved

Stoica

2015

Int J Theor Phys

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Abstract. We show that the Big Bang singularity of the Friedmann-Lemaître-Robertson-Walker model does not raise major problems to General Relativity. We prove a theorem showing that the Einstein equation can be written in a non-singular form, which allows the extension of the spacetime before the Big Bang. The physical interpretation of the fields used is discussed.These results follow from our research on singular semi-Riemannian geometry and singular General Relativity. The FLRW model shows that the universe should be, at a given moment of time, either in expansion, or in contraction. From Hubble's observations, we know that the universe is currently expanding. The FLRW model shows that, long time ago, there was a very high concentration of matter, which exploded in what we call the Big Bang. Was the density of matter at the beginning of the universe so high that the Einstein's equation was singular at that moment? This question received an affirmative answer, under general hypotheses and considering General Relativity to be true, in Hawking's singularity theorem [9,10,11] (which is an application of the reasoning of Penrose for the black hole singularities [12], backwards in time to the past singularity of the Big Bang). ContentsGiven that the extreme conditions which were present at the Big Bang are very far from what our experience told us, and from what our theories managed to extrapolate up to this moment, we cannot know precisely what happened then. If because of some known or unknown quantum effect the energy condition from the hypothesis of the singularity theorem was not obeyed, the singularity might have been avoided, although the density was very high. One such possibility is explored in the loop quantum cosmology [13,14,15,16,17,18], which leads to a Big Bounce discrete model of the universe.Not only quantum effects, but also classical ones, for example repulsive forces, can avoid the conditions of the singularity theorems, and prevent the occurrence of singularities. An important example comes from non-linear electrodynamics, which allows the construction of a stress-energy tensor which removes the singularities, as it is shown in [19] for black holes, and in [20] for cosmological singularities.We will not explore here the possibility that the Big Bang singularity is prevented to exist by quantum or other kind of effects, because we don't have the complete theory which is supposed to unify General Relativity and Quantum Theory. What we will do in the following is to push the limits of General Relativity to see what happens at the Big Bang singularity, in the context of the FLRW model. We will see that the singularities are not a problem, even if we don't modify General Relativity and we don't assume very repulsive forces to prevent the singularity.One tends in general to regard the singularities arising in General Relativity as an irremediable problem which forces us to abandon this successful theory [21,22,23,15]. In fact, contrary to what is widely believed, we will see that the singularities of...

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Section: Contentsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Perspectivesmentioning

confidence: 99%

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The Friedmann-Lemaître-Robertson-Walker Big Bang Singularities are Well Behaved

Stoica

2015

Int J Theor Phys

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“…Benign singularities turn out to be, in many cases, manageable [39][40][41]. The infinities simply disappear, if we use different geometric objects to write the equations and describe the phenomena.…”

Section: Two Types Of Singularitiesmentioning

confidence: 99%

The Geometry of Black Hole Singularities

Stoica

2014

Advances in High Energy Physics

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Recent results show that important singularities in General Relativity can be naturally described in terms of finite and invariant canonical geometric objects. Consequently, one can write field equations which are equivalent to Einstein's at nonsingular points but, in addition remain well-defined and smooth at singularities. The black hole singularities appear to be less undesirable than it was thought, especially after we remove the part of the singularity due to the coordinate system. Black hole singularities are then compatible with global hyperbolicity and do not make the evolution equations break down, when these are expressed in terms of the appropriate variables. The charged black holes turn out to have smooth potential and electromagnetic fields in the new atlas. Classical charged particles can be modeled, in General Relativity, as charged black hole solutions. Since black hole singularities are accompanied by dimensional reduction, this should affect Feynman's path integrals. Therefore, it is expected that singularities induce dimensional reduction effects in Quantum Gravity. These dimensional reduction effects are very similar to those postulated in some approaches to make Quantum Gravity perturbatively renormalizable. This may provide a way to test indirectly the effects of singularities, otherwise inaccessible.

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And the Math Will Set You Free

Stoica¹

2016

The Frontiers Collection

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h o l o t r o n i x @ g m a i l . c o m IntroductionHow mathematical is the physical world? Is mathematics essential to describe the universe, or it is just a tool which is not even necessary? There is no unanimously agreed answer to this question. The spectrum of opinions ranges from "mathematics is just a tool helping us to classify our observations" to "the world is nothing but a mathematical structure". Like for any debate, part of the tension between different views is the implicit usage of different definitions. So, to avoid possible confusion, we should define both mathematics and physics.Is mathematics discovered, or invented? Are mathematical structures entities which have their own existence, eternal and unchanging? Then, how can we know them? If we can access them with our thoughts, shouldn't they then be connected to our minds somehow? Can we discover and explore them? Or it is us who invent them? Or maybe we invent the axioms, and then we discover the consequences? Let's try to find out. A universe in a dotThink at a number. Did that number exist before you picked it, or it is you who invented it? Don't rush with the answer, because it is not as easy as it might seem.Consider the most brilliant text ever written. Is it the Bible, or the complete works of Shakespeare, or perhaps the complete collection of arXiv articles? Any of these can be written in a computer. The computer uses a code to assign numbers to letters, for example A → 65, B → 66, ..., Z → 90, a → 97, ..., z → 122. Internally, it represents these numbers in binary form, as strings of 0 and 1, so that for example A becomes 01000001, B becomes 01000010 etc. Let's take any text and assign to it a number between 0 and 1, of the form 0.d 1 d 2 . . ., where d 1 d 2 . . . is the decimal representation of the binary representation of the text.So, for example, the King James Bible starts with 1. In the beginning God created the heaven and the earth. Therefore, its number is 0.192110078742304518747878005390999191523668554050387237..., which obviously is between 0 and 1 (see fig. 1).

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On singular semi-Riemannian manifolds

Cited by 37 publications

References 60 publications

The Friedmann-Lemaître-Robertson-Walker Big Bang Singularities are Well Behaved

The Friedmann-Lemaître-Robertson-Walker Big Bang Singularities are Well Behaved

The Geometry of Black Hole Singularities

And the Math Will Set You Free

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