A nice article on the issue:

What are singularities? Do they exist in nature?

http://www.askamathematician.com/2012/09/q-what-are-singularities-do-they-exist-in-nature/

In a 1936 paper, Einstein himself (along with co-author Nathan Rosen) wrote that “a singularity brings so much arbitrariness into the theory that it actually nullifies its laws. … Every field theory, in our opinion, must adhere to the fundamental principle that singularities of the field are to be excluded.”

As far as contemporary physics is concerned, until we have an understanding of how gravity works in the quantum world (quantum gravity), we will never know whether singularities actually exist in physical reality.

When using Newton law for two stellar bodies and wants answers outside the radius of these two bodies, we simplify the solution of the problem by assuming that all the matters is concentrated into a single point. We frame the problem in a way that is only valid outside the radius of the stellar bodies. If we want to predict gravity within the radius then we use mass density distribution and then we do not get any infinite value of gravity.

Bing Bang theory is framed with a singularity although we know that the equations that are solved are not valid in the near region of the singularity. The results are thus valid up to 13.7 billion years ago but not further back. So Bing Bang theory is a good theory for the last 13.7 billion years but not really a theory of origin. If one day we get a quantum gravity theory we may go further back. I cannot conceived a theory of origin because such a theory would have to explain where it comes from. We may succeed to get another theory, but again we will have to answer where it comes from, and so on to infinity. Assuming any point of departure raise the question of where this departure come from. We can only walk this endless road to the beginning.

Dear Louis,

You said:" The results are thus valid up to 13.7 billion years ago but not further back". That makes no sense since the age of the universe 13.7 billion years is an approximation and physics today can do far better than this! What you should have said is: Within the framework of the laws of physics as we understand them today, we can say only that the universe came into existence when it already had an age of 10-43 seconds-the Planck time. Beyond this the laws of physics as we know them today break down.

... perhaps this is just a stupid comment.. but when we start approaching a singularity, usually we start looking for the Holy Grail.. multiscale physics.. we want to go from something macroscopic (where we all use continuum approximations) to something smaller and smaller and smaller and smaller... and we sooner or later find out (as Louis also pointed out) that we need to carefully consider the initial hypotheses in order to study that particular point (for the crack, as well as e.g. for dislocations, the singularities are usually due to the continuum vs discrete approach). Guess what, gauge theory can be used to treat the dislocation case...

The classical Big Bang model indicate a beginning of the universe at a Hawking-

Penrose singularity, but this was mitigated by 'contemporary' Big Bang model, which opened up the possibility of an early quantum cosmological era and an inflationary dynamic allowing our universe to be but one amidst a multiplicity of possible universes within a theoretical multiverse.

However, this mitigating view was itself subsequently mitigated by the discovery of Brode, Guth and Vilinkin that every inflationary model universe(and/or multiverse) must have a beginning. Since this indicates an edge of time (prior to which there is no time), the conclusion of Borde, Guth, and Vilenkin point strongly to a creation of the universe (from no previously existing physical matter-energy). The cause of such a creation would have to transcend our universe(and any multiverse in which it may be situated).

If all dimensional matter is disintegrated and ejected from black holes while their unbound mass-energy is retained, imparted to the local curvature of spacetime, couldn't a true singularity exist as merely the focal point of spacetime curvature?

Issam,

I disagree with this estimation of 10-43 seconds-the Planck time because it is based on the assumption that the model is valid in the initial instant and we know it is not.

Issam,

Beyond the issue of precise measurements it is interesting that there is not necessarily an edge to time.

Time works in odd ways - for instance, so-called 'delayed choice experiments' tell us something essential about how time can play out. A single universe could conceivably itself be a large-scale "delayed choice experiment" - in effect, the universe would bootstrap itself into existence by , for instance, creating 'backwards in time' a quantum fluctuation that would give rise to itself.

The only way this can work, though, is if there is *something* outside of time..... (this is a book length argument though, well beyond the space available here)

I think singularity comes from our lack in knowlege in describing certain physical phenomenon that push us to assume models or laws that contain singular points. In real world I think there is no singular points even the Big Bang theory ought to be modefiedto agree with Inflation because its not logical that Inflation happened from singular point. even if inflation happened after the big bang why the big bang didnt consume the whole energy? why there id need to Inflation this indicates that the big bang happened not from a singular point but from a difused body of particles

Chris

This discussion seem to be going outside the scope of the question of this thread. And yes you are right arguments for and against the concept of "before the big bang" and the origin of the universe are highly complex and lengthy. It would not be fair to the author of the question in this thread to diverge from the main issue.

If I have understood correctly the original question and the various answers given by the contributors of this thread, then the main implication is that singularities seem to be pointing out the weaknesses and the limitations of available models Consequently, studying these singularities would probably yield better improved models. However, can we ever manage to avoid completely these singularities just by improving our models while using the same mathematical tools or do we need to go a step beyond that and bring in new mathematics or else the adequate tools are already available but are not being used ?

Great question HELehtihet, there are different kinds of singularities which tend to be all lumped into the same bag because they're poorly understood and hard to apprehend directly - these (fundamentally) different singularities (e.g. quantum fluctuations, black hole formations, Penrose bouncing universes) can give rise to the same ostensible phenomenon (such as a Big Bang)

I would tend to believe that for most types of singularities we already have the adequate math, although we may not have the proper data points (there are examples of lacking data points even though ascertaining such data points is wholly feasible indirectly, from induced phenomena) . For exotic singularities, then indeed perhaps our math is not up to the task

Singularities are features that exists in the mathematical descriptions. They are artefact of the language as presently developed. Without real numbers the concept of singularity cannot exist. But a real number cannot exist in reality given the infinite amount of information each of them requires. If like me you consider information as physical then a real number is not an acceptable way to model reality.

Real numbers are real.

Nev,

The number pi would requires more information to be represented than all the information that exist in the visible universe. If it exists, it has to exist in this world and for that the information of the number has to be a physical reality that exists in this world.

It is generally true and maybe always so that the presence of a singularity in a finished physical calculation indicates some approximation is being made. In the case of the sun's gravitational field, the approximation is assuming that the sun's mass is concentrated at a point. This approximation doesn't work if the test object is close. That's why satellite orbits must take into account the shape of the earth. Also in the case of the sun if you do take its mass to be distributed in a sphere, then you find that there is no field at the center of mass.

An exception is if you're dealing with distributions, one definition of which is the integrand of an integral. If executing the integral does away with the singularity, fine. If it doesn't you have a problem.

True, Louis, and pi does exist in the world as a physical constant, or the ratio (whole to part) between the circumference of a circle and its diameter. Furthermore, all numbers may be expressed as a ratio, that is, a whole physical object to its part, that ratio turning out to be a number constantly, which is again, a ratio between two physical entities.

Nev,

Why do you think pi exists in the world? Where? Physical constants exist in the physical world, the world of mathematical models empirically tested. The world of physics is a subset of the mathematical world which is part of our culture that help us describe/understand the world/REALITY/NOUMENAL. the physical world STRICKLY SPEAKING IS NOt THE WORLD.

The question of this thread is like asking: Do singularities have a "real" as opposed to mathematical or idealized existence? I recommend this reference (technical but very relevant and highly motivating):

http://physics.stackexchange.com/questions/3892/do-singularities-have-a-real-as-opposed-to-mathematical-or-idealized-existence

Issam,

Interestingly enough this boils down to the same question that always rears its head again in leading edge physics / nature of reality issues: is math actual reality, or is reality only described by and independent of math? (aka the Aristotelian vs. the Platonic view of reality)

As you are well aware, there are proponents on both sides, with very strong arguments on both sides (say, Dieter Zeh for Aristotelian and Max Tegmark for Platonic) - However the wording of your question above "Do singularities have a real as opposed to mathematical existence" presupposes that you have solved the issue, because the Aristotelian view is implicit in it.

I have not and certainly do not even attempt to claim so. It is just that while reading about this very interesting topic I came across the reference above and wanted to share it with you. I entirely agree with you that each side has his strong argument . I am not siding with anyone, I believe we should keep an open mind.

Further, the wording of the question is not mine! It is by somebody called Gordon and the question was actually asked in January 2011.

Sorry Issam, my misapprehension :-(

Louis, I think this because universal physical laws or phenomena and intrinsically their constantly corresponding numerical constants exist independently of or a priori any culture. In other words, they existed before there ever was any culture, e.g., at the 'Big Bang' or in prehistoric times or in text books, and will continue to exist whether or not highly fickle cultures exist. These laws with their constants didn't suddenly come onto the scene when a human decided to have table manners, nor do they fail to exist on some galaxy one of our gadgets has yet to reach. They were always there and at any second, an asteroid following those laws can wipe out dinosaurs or humanity and, God forbid, my little dog, too. The laws, however, I am sure will find a way to continue on without us, and perhaps as part of those laws, we may continue on as well in another new and hopefully improved form.

While both math and experiment are necessary, they are a bit less co-equal than indicated. If the math disagrees with experiment and you haven't misinterpreted the experiment then the math must yield. That's physics.

Dear Langworthy.

Let me joke a little.

There is a universal value called "Smith's constant". It is defined as follows.

Smith's constant is the quantity that must be added to an experiment result in order to agree with the corresponding theoretic value.

That is a typical mathematician sense of humor. I have said this before and I shall say it again I have always liked your sense of humor professor:)

Dear Issam,

Thank you. Perhaps sense of humor is the main sign of intelligence.

A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.

John E Littlewood (1885 - 1977)

This is not a joke! Bayesian probability makes the point that observed experimental values always differ from the intended measurement value.

Perhaps it is approperiate here to present a quote by Albert Eistein:

"No amount of experimentation can ever prove me right; a single experiment can prove me wrong."

---Albert Einstein

Now that's relativity, that is physics.

Maybe I can attempt to answer the actual question. The problem of the Sun's gravitational field is that of the field generated by an approximate sphere. It is, of course, not singular, either in theory or practice. The fact that outside the sphere the field is exactly M/d^2 is a theorem due to Newton. Inside the sphere it is, of course, different. In the real world, the Sun is not perfectly spherical and such issues must be taken into account. A singularity in the interaction only arises between elementary particles. Some evidence of at least near singularities arise in the Coulomb interaction between elementary particles, where, however, it does not cause serious problems due to quantum mechanics: even the model of the hydrogen atom with an exact singularity has a well-defined solution.

As to the Big Bang, I do not know enough to say for sure. It is my understanding that it has been shown that, within the classical context, using General relativity, there must be a real singularity. But of course classical mechanics is not the whole truth, though in the case of gravity we do not have a well-understood quantum theory.

In physics, problems are always simplified. For instance, the gravitational field M/d^2 is obtained assuming spherical symmetry. In addition, the amount of energy consumed in tides is disregarded. I know that this energy can be disregarded at the present, but its effect for a long time amount cannot be ignored.

A simple interpretation of things suggests that there is an infinite density occurring at the centre of every black hole, which is just like the infinity at the end of the Universe. But a black hole creates a horizon around this phenomenon: not even light can escape from its vicinity. So we are insulated, we cannot see what goes on at those places where the density looks as though it's going to be infinite. And neither can the infinity influence us. These horizons protect us from the consequences of places where the density might be infinite and they stop us seeing what goes on there,

I think that even black holes have no real singularity if quantum theory of gravity may arise this can put an end of singularities in gravity because the singularity in gravity is associated with singlularity in time which is too hard to imagine because logically time can not stopped by a celecial body.

General relativity has been successful in describing the macroscopic properties of black holes. However, at the microscopic level, it predicts that black holes have a singularity at their cores: a region where the gravitational field is infinitely strong. Within the picture of gravity presented by general relativity, such a singularity would destroy all information about the quantum states of matter falling into a black hole. Yet one of the basic tenets of quantum mechanics is that information is preserved. The loss of information in a singularity is thus paradoxical and points to a fundamental incompatibility between general relativity and quantum mechanics. A long-standing hope has been that the application of a quantum theory of gravity to the description of black holes would resolve these contradictions.See one of my my earlier post.

Now, writing in Physical Review Letters, Rodolfo Gambini at the University of the Republic in Uruguay and Jorge Pullin at Louisiana State University, Baton Rouge, show that the quantization of a special class of black holes—known as spherically symmetric black holes—is indeed possible within a framework for quantum gravity known as loop quantum gravity. Their analysis shows that a region of highly curved spacetime (where quantum effects of gravity can be manifest), rather than a singularity, is what makes up the core of a black hole. While this promising theory removes the singularity implied by classical general relativity, further research will be needed to establish whether these results solve the information loss paradox and if the approach may be generalized to other classes of black holes.

-----------------------------------------------------------------------------------

Loop Quantization of the Schwarzschild Black Hole

Rodolfo Gambini and Jorge Pullin

Phys. Rev. Lett. 110, 211301, 23 May 2013

Published May 23, 2013

Jose, true, and from the natural numbers, geometry (point, line, square, etc.) and other branches of mathematics were constructed deductively (i.e., deductive reasoning -- reasoning at least considered a priori humans, so mirroring of reality self-similarly through note our spherically-shaped eyes evolved among other senses together with and not contrary, although easily to be slightly mistaken (or made/invented complicated) perhaps, to reality), e.g., from a center point, assigning radius of 1 to the unit circle, and as well other self-similarly and formulaically-deduced constructions of solids. So not only are points or discrete objects rooted and abstracted from those roots or actual 'points' into actual physical entities in reality, the names given to those root or foundational entities are numbers, real and natural (part of the reals) and in curved space (as note the unit circle is, etc.), I dare say, complex or 'imaginary.'

Joseph, Actually?

Joseph,

If you roll your cursor over the top right corner of your message & click on the little menu tab, you can select 'Delete' to delete the duplicate postings...

We must realize that the universe is more complicated than our mathematics. We need to develop our describing to physical phenomena so that no singular points may appear.

Mathematical models of reality are statements on reality. If by the actuality of a model we mean that relations in the model are accurately close to the relations between the measurements then it is impossible to verify such assertion since we cannot make measurements at the singularity. If by the actuallity of a model we mean that the model IS reality then this is a metaphysical statement that cannot be tested.

Singularities are the problems of theoretical modelling but not of mathematical techniques.

Dear Issam,

It depends on what you mean as mathematical techniques. Recall those integration methods for complex-valued functions based on singular points and Riemann surfaces.

Dear Sadeem,

>> "We must realize that the universe is more complicated than our mathematics."

Possibly but not necessarily. Sometimes, quite simple underlying physical laws may lead a system to exhibit a very complex behavior.

Since the projection of a simple curve could produce the singularities one could get rid of (resolve) the singularities by going to the upper "bigger" space.

Dear Lehtihet but dont you agree that must singularities come from the physical laws themselves . because many of them had been put in a similar form of the inverse square law.

Dear Gro that 's nice can I add the boundary value problems may be another option to get rid of singularities.

Dear Professor Juan-Esteban

In applied mathematics the term singularity is used to indicate that a conventional way of modelling a certain physical process mathematically leads to consequences which for variuos reasons cannot be accepted. This definition implies that the assessment is made on the basis of some physical criteria associated with what features of the process should be modelled, i.e. the criteria which lie outside the mathematics of the problem, though in some cases non-existence of a solution makes the formulation of the problem unacceptable also from the purely mathematical point of view.

If a solution exists but some physical arguments indicate a `singularity', one has to find out whether it appeared due to simplifications and/or additional assumptions made in the process of obtaining the solution or whether it is inherent in the very formulation of the problem. In the former case, the singularity can be removed mathematically by relaxing `extra' constraints, i.e. without altering the underlying model. In the latter case, one has to look for those physical processes which are not accounted for in the conventional formulation and generalize the model itself.

In comptational fluid dynamics, singularities pose immense challenge. It is here that singularities are problems of mathematical modelling and not of mathematical techniques. Any attempt to bypass the difficulty using some ad hoc criteria would reduce the modelling to a semiempirical level thus making the whole numerical exercise entirely pointless.

For more on this and review of cases please refer to:

1-Singularities in Mathematical Models of Physical Phenomena, by Yulii D. Shikhmurzaev:

http://web.mat.bham.ac.uk/Y.D.Shikhmurzaev/singul.htm

2-Goldshtik, M. A. 1990 Viscous-flow paradoxes. Annu. Rev. Fluid Mech.22, 441-472.

Dear Issam,

we are talking about different topics. What I have mentioned before is the integration method based upon residues. See http://en.wikipedia.org/wiki/Residue_%28complex_analysis%29

Notice that, by no means, I have mentioned them as instance of the only possible singular point model.

Yes professor you are right, we are talking about different topics and this is why I had to explain in more detail my earlier short statement.

Jose, your remarks resonate with my own thoughts. Indeed, new maths will reveal as real (or actual), precise order of deemed chaos, anomalies, singularities, paradoxes, or the 'imperfect,' along with our old tools of self-fulfilling prophecies/probabilities.

Juan-Esteban,

Going back to your original question posting, you state:

"At d=0 the field becomes infinite."

Actually, in accordance with http://en.wikipedia.org/wiki/Shell_theorem, at d=0 the force field field amplitude becomes "zero by symmetry at the centre of mass." This doesn't likely affect the discussion, since shell theorem is an idealized mathematical analysis of a perfectly spherically symmetrical distribution of mass. However, it may be possible that, at the extreme energy of a black hole that perfectly spherical symmetry (at least as an abstracted origin point for the curvature of spacetime) might be achieved. If such were the case, would it be valid to ask how large the origin point of a non-linearly curved spacetime might be? If so there might even be physical implications... Sorry - I can't evaluate.

Dear James,

You have said,

Going back to your original question posting, you state:

"At d=0 the field becomes infinite."

Have you read the whole original question?

See a copy-paste of the original question I had written:

>> Notice that the well-known gravitational field of Sun M/d^2 works fine when

>> calculating planet orbits. At d=0 the field becomes infinite. It is a singular point.

>> However, we know that at d = 0 by no means the field is infinite.

Have you read the last text line?

However, when we calculate planet orbits in the Newton's equation the considered force is M/d^2. ¿Can you prove that d=0 is not a singular point of the function M/d^2?

Do you mean that the limit M/d^2 when d tends toward 0 is not infinite?

It is just the core of my question that we calculate orbits using the field M/d^2, which possesses a singular point; however, near of d=0 the field becomes linear, that is to say, with no singular points. In other words, the singular point d=0 is virtual, and this why the original question is stated in order to extend this virtuality to every singular point.

I suppose that you have not read the complete original question; otherwise I cannot understand what you mean.

"By contrast, it is constant inside the Sun." Inside a sphere of uniform density, the gravitational field strength is a linear function of distance from the center. Outside the sphere, the inverse square law holds.

It looks like the discussion went the way of theology. To me the problem with the Sun gravitation is trivial : one laws applies in the outside area (Coulomb-like force), and the other -- inside the Sun (linear with the distance; essentially the same as for a harmonic oscillator). Nothing here to be falling into a philosophical stupor. Perhaps a distinctly relevant as far as physics, is a question of how far we can go in reducing the spacial size of observable effects (the same about the time). Big Bang size is a question from the same basket. Well, we do know that the smallest spatial "grain" or "pixel" (which is a "Big Bang size" too), is a so called Planck length sqrt ( hbar G / c^3), or about 1.6 10^(-23) Angstrem (for the time "pixel", or Palnck time, it is about 10^(-43) second). Beyond those bounds, according to superstring theories, we cannot distinguish between time and space -- actually instead of 4 dimensions we are supposed to see many more (from 10 to 20+, depending on specific models...)

Take a simple convex lens, by changing its focal length decreasing its curvature from K to zero and across to -K, we have a singularity in focal length approaching infinity for K=0...

So we have a singularity, and it is an actual one...

Is it a problem, well not because we can think in curvature (unit 1/m) or optical power (unit diopters=1/m) and the singularity disappears...

but thinking in radius or focal length the infinity term (ie the singularity) is real

Perhaps I'm confused, but as I understand, d (or r) represents not the distance between the surfaces of two objects but the separation distance between two point masses representing the presumedly spherically symmetrical objects being evaluated. Please see http://en.wikipedia.org/wiki/Universal_gravitation.

It is the problem of renormalization theory also present at Physics of Elementary Particles. All theories have discontinuities at the origin and that is their crucial drawback: they define infinite quantities without them to exist in reality, just because of the mathematical properties of the used formulas.

I liked the "Smith's constant", it reminded me the models used in Econometrics: choose that with the least sum of squares departure from real data.

Whole disciplines of Science have been built with such naive assumptions, not only Physics!

To go the discussion a little bit away imagine that for the Standard Model all particles are point singularities and if such singularities existed in reality our world would be a mass production of universes every time, every moment due to that enormous presence of energy density!

So, it is time to get rid of the worship to our 'best' theories: if they were ok then we should never had been created due to high energy explotions...

I think there are different kinds of singularities some of them is mathematical due to origin assumtions like the problem of gravity of the sun can be solved. using Dirack delta function , neverthless there are complicated problems like black holes singularity which are not easy like that. I think such singularities are related to physical assumtions, usually the physical singularities are more complicated than mathematical ones. Big bang singularities can. be one of them. these singularities either contains a physical meaning or they indicates a fault in the theory at such ranges near the singularity. in all cases I think singularity in the universe is only theoreical and represent no real fact.

Dear Martin,

Excellent contribution.

Thank you.

An indicator for the non existence of singular points is recent experiments at LHC: recall the discussion about the possibility of creating a local singularity that could absorb whole Earth: nothing at all happened just because the concept of local singularity maybe has not the meaning we believe it has.