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{{Inline|date=July 2012}}
{{General relativity|cTopic=Phenomena}}
A '''gravitational singularity''' or '''spacetime singularity''' is a location where the quantities that are used to measure the [[gravitational]] field become [[infinity|infinite]] in a way that does not depend on the coordinate system. These quantities are the scalar invariant [[Curvature of Riemannian manifolds|curvature]]s of spacetime, which includes a measure of the density of matter.
 
For the purposes of proving the [[Penrose–Hawking singularity theorems]], a [[spacetime]] with a singularity is defined to be one that contains [[Geodesic (general relativity)|geodesics]] that cannot be extended in a [[Smooth function|smooth]] manner.<ref>{{cite web|last=Moulay|first=Emmanuel|title=The universe and photons|url=http://fqxi.org/data/forum-attachments/Photon.pdf|publisher=FQXi Foundational Questions Institute|accessdate=26 December 2012}}</ref> The end of such a geodesic is considered to be the singularity. This is a different definition, useful for proving theorems.
 
The two most important types of spacetime singularities are '''''curvature singularities''''' and '''''conical singularities'''''.<ref name=uggla>{{cite web|last=Uggla|first=Claes|title=Spacetime singularities|work=Einstein Online|publisher=Max Planck Institute for Gravitational Physics|accessdate=26 December 2012}}</ref>  Singularities can also be divided according to whether they are covered by an [[event horizon]] or not ([[naked singularity|naked singularities]]).<ref>{{cite book|last=Patrick Di Justo, Kevin Grazier|first=Patrick and Kevin Grazier|title=The Science of Battlestar Galactica|year=2010|publisher=John Wiley & Sons|location=New York|isbn=978-0470399095|pages=181|url=http://books.google.com/books?id=iK1YbKrNRcoC&printsec=frontcover#v=onepage&q&f=false}}</ref>   According to [[general relativity]], the initial state of the [[universe]], at the beginning of the [[Big Bang]], was a singularity. Both [[general relativity]] and [[quantum mechanics]] break down in describing the Big Bang,<ref>{{cite web|last=Hawking|first=Stephen|title=The Beginning of Time|url=http://www.hawking.org.uk/the-beginning-of-time.html|work=Stephen Hawking: The Official Website|publisher=Cambridge University|accessdate=26 December 2012}}</ref>  but in general, quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths.<ref>{{cite book|last=Zebrowski|first=Ernest|title=A History of the Circle: Mathematical Reasoning and the Physical Universe|year=2000|publisher=Rutgers University Press|location=Piscataway NJ|isbn=978-0813528984|pages=180|url=http://books.google.com/books?id=2twRfiUwkxYC&printsec=frontcover#v=onepage&q&f=false}}</ref>  Another type of singularity predicted by general relativity is inside a [[black hole]]: any [[star]] collapsing beyond a certain point (the [[Schwarzschild radius]]) would form a black hole, inside which a singularity (covered by an event horizon) would be formed, as all the matter would flow into a certain point (or a circular line, if the black hole is rotating).<ref>{{cite web|last=Curiel|first=Eric and Peter Bokulich|title=Singularities and Black Holes|url=http://plato.stanford.edu/entries/spacetime-singularities/|work=Stanford Encyclopedia of Philosophy|publisher=Center for the Study of Language and Information, Stanford University|accessdate=26 December 2012}}</ref>  This is again according to general relativity without [[quantum mechanics]], which forbids wavelike particles entering a space smaller than their wavelength. These hypothetical singularities are also known as curvature singularities.
 
==Interpretation==
 
Many theories in physics have [[mathematical singularities]] of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the [[ultraviolet catastrophe]], [[renormalization]], and instability of a hydrogen atom predicted by the [[Larmor formula]].
 
In [[supersymmetry]], a singularity in the [[moduli space]] happens usually when there are additional [[mass]]less degrees of freedom in that certain point. Similarly, it is thought that singularities in spacetime often mean that there are additional [[Degrees of freedom (physics and chemistry)|degrees of freedom]] that exist only within the vicinity of the singularity. The same fields related to the whole spacetime also exist; for example, the [[electromagnetic field]]. In known examples of [[string theory]], the latter degrees of freedom are related to [[String (physics)#Types of strings|closed string]]s, while the degrees of freedom are "stuck" to the singularity and related either to [[String (physics)#Types of strings|open string]]s or to the twisted sector of an [[orbifold]].
 
Some theories, such as the theory of [[loop quantum gravity]] suggest that singularities may not exist. The idea is that due to [[quantum gravity]] effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter.
 
The [[Einstein–Cartan theory|Einstein-Cartan]]-Sciama-Kibble theory of gravity naturally averts the gravitational singularity at the Big Bang.<ref>{{cite journal |author=Poplawski, N. J. |authorlink=Nikodem Popławski |year=2012 |title=Nonsingular, big-bounce cosmology from spinor-torsion coupling |journal=[[Physical Review D]] |volume=85 |pages=107502 |doi=10.1103/PhysRevD.85.107502|arxiv = 1111.4595 |bibcode = 2012PhRvD..85j7502P }}</ref> This theory extends general relativity to matter with intrinsic angular momentum ([[spin (physics)|spin]]) by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the [[torsion tensor]], as a variable in varying the action. The minimal coupling between torsion and [[Dirac spinor]]s generates a spin–spin interaction in [[fermion]]ic matter, which becomes dominant at extremely high densities and prevents the scale factor of the Universe from reaching zero. The Big Bang is replaced by a cusp-like [[Big Bounce]] at which the matter has an enormous but finite density and before which the Universe was contracting.
 
==Types==
===Curvature===
 
Solutions to the equations of [[general relativity]] or another theory of [[gravity]] (such as [[supergravity]]) often result in encountering points where the [[Metric (mathematics)|metric]] blows up to infinity. However, many of these points are completely [[Smooth function|regular]], and the infinities are merely a result of using an inappropriate [[coordinate system]] at this point. In order to test whether there is a singularity at a certain point, one must check whether at this point [[Diffeomorphism invariance|diffeomorphism invariant]] quantities (i.e. [[scalar (physics)|scalar]]s) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.
 
An example is the [[Schwarzschild metric|Schwarzschild]] solution that describes a non-rotating, [[Electric charge|uncharged]] black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the [[event horizon]]. However, [[spacetime]] at the event horizon is [[Smooth function|regular]].  The regularity becomes evident when changing to another coordinate system (such as the [[Kruskal coordinates]]), where the metric is perfectly [[Smooth function|smooth]]. On the other hand, in the center of the [[black hole]], where the metric becomes infinite as well, the solutions suggest singularity exists. The existence of the singularity can be verified by noting that the [[Kretschmann scalar]], being the square of the [[Riemann tensor]] i.e. <math>R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}</math>, which is diffeomorphism invariant, is infinite.
While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity". In a rotating black hole, also known as a [[Kerr black hole]], the singularity occurs on a ring (a circular line), known as a "[[ring singularity]]". Such a singularity may also theoretically become a [[wormhole]].<ref>If a rotating singularity is given a uniform electrical charge, a repellent force results, causing a [[ring singularity]] to form. The effect may be a stable [[wormhole]], a non-point-like puncture in spacetime that may be connected to a second ring singularity on the other end. Although such wormholes are often suggested as routes for faster-than-light travel, such suggestions ignore the problem of escaping the black hole at the other end, or even of surviving the immense [[tidal force]]s in the tightly curved interior of the wormhole.</ref>
 
More generally, a spacetime is considered singular if it is [[Geodesic (general relativity)#Geodesic incompleteness and singularities|geodesically incomplete]], meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the [[event horizon]] of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the [[Big Bang]] [[physical cosmology|cosmological]] model of the [[universe]] contains a causal singularity at the start of [[time]] (''t''=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite space-time curvature.
 
===Conical===
 
A conical singularity occurs when there is a point where the limit of every [[Diffeomorphism invariance|diffeomorphism invariant]] quantity is finite, in which case [[spacetime]] is not smooth at the point of the limit itself. Thus, spacetime looks like a [[Cone (geometry)|cone]] around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere if a suitable [[coordinate system]] is used.
 
An example of such a conical singularity is a [[cosmic string]].
 
===Naked===
{{main|Naked singularity}}
 
Until the early 1990s, it was widely believed that [[general relativity]] hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the [[cosmic censorship hypothesis]]. However, in 1991, physicists Stuart Shapiro and [[Saul Teukolsky]] performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed.
 
==Entropy==
{{see|Black hole|Hawking radiation|Entropy}}
Before [[Stephen Hawking]] came up with the concept of Hawking radiation, the question of black holes having entropy was avoided. However, this concept demonstrates that black holes can radiate energy, which conserves entropy and solves the incompatibility problems with the second law of thermodynamics. Entropy, however, implies heat and therefore temperature. The loss of energy also suggests that black holes do not last forever, but rather "evaporate" slowly. Small black holes tend to be hotter whereas larger ones tend to be colder. All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, so they are all gaining energy. They will not begin to lose energy until a cosmological redshift of more than one million is reached, rather than the thousand or so since the background radiation formed.
 
==See also==
* [[Penrose-Hawking singularity theorems]]
* 0-dimensional singularity: [[magnetic monopole]]
* 1-dimensional singularity: [[cosmic string]]
* 2-dimensional singularity: [[Domain wall (string theory)|domain wall]]
* [[Fuzzball (string theory)]]
 
==Notes==
{{reflist|2}}
 
==References==
{{refbegin}}
* {{cite journal | last = Shapiro | first = Stuart L. | authorlink = | coauthors = [[Saul Teukolsky|Teukolsky, Saul A.]] | title = Formation of naked singularities: The violation of cosmic censorship | date = 1991 | journal = [[Physical Review Letters]] | volume = 66  | issue = 8 | pages = 994–997 | doi = 10.1103/PhysRevLett.66.994 | pmid = 10043968 | bibcode=1991PhRvL..66..994S}}
* {{cite book | author = [[Robert Wald|Robert M. Wald]] | title = [[General Relativity (book)|General Relativity]] | publisher =  [[University of Chicago Press]] | year = 1984 | isbn = 0-226-87033-2 }}
* {{cite book | author = [[Charles W. Misner]] | coauthors = [[Kip Thorne]] & [[John Archibald Wheeler]] | title = [[Gravitation (book)|Gravitation]]  | publisher = [[W. H. Freeman]] | year = 1973 | isbn = 0-7167-0344-0 }} §31.2 The nonsingularity of the gravitational radius, and following sections; §34 Global Techniques, Horizons, and Singularity Theorems
{{refend}}
* Roger Penrose(1996)"[http://www.ias.ac.in/jarch/jaa/17/213-231.pdf Chandrasekhar, Black Holes, and Singularities]"
* Roger Penrose(1999)"[http://www.ias.ac.in/jarch/jaa/20/233-248.pdf The Question of Cosmic Censorship]"
* Τ. P. Singh"[http://www.ias.ac.in/jarch/jaa/20/221-232.pdf Gravitational Collapse, Black Holes and Naked Singularities]"
 
==Further reading==
 
* ''[[The Elegant Universe]]'' by [[Brian Greene]]. This book provides a layman's introduction to string theory, although some of the views expressed are already becoming outdated. His use of common terms and his providing of examples throughout the text help the layperson understand the basics of string theory.
 
{{Relativity}}
 
[[Category:Concepts in physics]]
[[Category:General relativity]]
[[Category:Lorentzian manifolds]]
[[Category:Physical paradoxes]]

Revision as of 13:51, 3 February 2014

Template:Inline Diving Coach (Open water ) Dominic from Kindersley, loves to spend some time classic cars, property developers in singapore house for rent (Source Webpage) and greeting card collecting. Finds the world an interesting place having spent 8 days at Cidade Velha. A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter.

For the purposes of proving the Penrose–Hawking singularity theorems, a spacetime with a singularity is defined to be one that contains geodesics that cannot be extended in a smooth manner.[1] The end of such a geodesic is considered to be the singularity. This is a different definition, useful for proving theorems.

The two most important types of spacetime singularities are curvature singularities and conical singularities.[2] Singularities can also be divided according to whether they are covered by an event horizon or not (naked singularities).[3] According to general relativity, the initial state of the universe, at the beginning of the Big Bang, was a singularity. Both general relativity and quantum mechanics break down in describing the Big Bang,[4] but in general, quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths.[5] Another type of singularity predicted by general relativity is inside a black hole: any star collapsing beyond a certain point (the Schwarzschild radius) would form a black hole, inside which a singularity (covered by an event horizon) would be formed, as all the matter would flow into a certain point (or a circular line, if the black hole is rotating).[6] This is again according to general relativity without quantum mechanics, which forbids wavelike particles entering a space smaller than their wavelength. These hypothetical singularities are also known as curvature singularities.

Interpretation

Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the ultraviolet catastrophe, renormalization, and instability of a hydrogen atom predicted by the Larmor formula.

In supersymmetry, a singularity in the moduli space happens usually when there are additional massless degrees of freedom in that certain point. Similarly, it is thought that singularities in spacetime often mean that there are additional degrees of freedom that exist only within the vicinity of the singularity. The same fields related to the whole spacetime also exist; for example, the electromagnetic field. In known examples of string theory, the latter degrees of freedom are related to closed strings, while the degrees of freedom are "stuck" to the singularity and related either to open strings or to the twisted sector of an orbifold.

Some theories, such as the theory of loop quantum gravity suggest that singularities may not exist. The idea is that due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter.

The Einstein-Cartan-Sciama-Kibble theory of gravity naturally averts the gravitational singularity at the Big Bang.[7] This theory extends general relativity to matter with intrinsic angular momentum (spin) by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a variable in varying the action. The minimal coupling between torsion and Dirac spinors generates a spin–spin interaction in fermionic matter, which becomes dominant at extremely high densities and prevents the scale factor of the Universe from reaching zero. The Big Bang is replaced by a cusp-like Big Bounce at which the matter has an enormous but finite density and before which the Universe was contracting.

Types

Curvature

Solutions to the equations of general relativity or another theory of gravity (such as supergravity) often result in encountering points where the metric blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.

An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, spacetime at the event horizon is regular. The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly smooth. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest singularity exists. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i.e. RμνρσRμνρσ, which is diffeomorphism invariant, is infinite. While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity". In a rotating black hole, also known as a Kerr black hole, the singularity occurs on a ring (a circular line), known as a "ring singularity". Such a singularity may also theoretically become a wormhole.[8]

More generally, a spacetime is considered singular if it is geodesically incomplete, meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time (t=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite space-time curvature.

Conical

A conical singularity occurs when there is a point where the limit of every diffeomorphism invariant quantity is finite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere if a suitable coordinate system is used.

An example of such a conical singularity is a cosmic string.

Naked

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Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed.

Entropy

Template:See Before Stephen Hawking came up with the concept of Hawking radiation, the question of black holes having entropy was avoided. However, this concept demonstrates that black holes can radiate energy, which conserves entropy and solves the incompatibility problems with the second law of thermodynamics. Entropy, however, implies heat and therefore temperature. The loss of energy also suggests that black holes do not last forever, but rather "evaporate" slowly. Small black holes tend to be hotter whereas larger ones tend to be colder. All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, so they are all gaining energy. They will not begin to lose energy until a cosmological redshift of more than one million is reached, rather than the thousand or so since the background radiation formed.

See also

Notes

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References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 §31.2 The nonsingularity of the gravitational radius, and following sections; §34 Global Techniques, Horizons, and Singularity Theorems

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Further reading

  • The Elegant Universe by Brian Greene. This book provides a layman's introduction to string theory, although some of the views expressed are already becoming outdated. His use of common terms and his providing of examples throughout the text help the layperson understand the basics of string theory.

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  3. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. Template:Cite web
  5. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  7. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  8. If a rotating singularity is given a uniform electrical charge, a repellent force results, causing a ring singularity to form. The effect may be a stable wormhole, a non-point-like puncture in spacetime that may be connected to a second ring singularity on the other end. Although such wormholes are often suggested as routes for faster-than-light travel, such suggestions ignore the problem of escaping the black hole at the other end, or even of surviving the immense tidal forces in the tightly curved interior of the wormhole.