Defining a Singularity

[rigney]

How do we define a Singularity? How big does it have to be? How small? What shape? Did our world actually begin as something less than the size of a pin point of light? While I believe our universe was the instantaneous transitional phase of a singularity into becoming matter, could "it" have possibly been billions of miles in diameter at the time?

 

American Heritage, definitions of: SINGULARITY

 

1. The quality or condition of being singular.

2. A trait marking one as distinct from others; a peculiarity.

3. Something uncommon or unusual.

4. Astrophysics: A point in space-time at which gravitational forces cause matter to have infinite density and infinitesimal volume, and space and time to become infinitely distorted.

5. Mathematics: A point at which the derivative does not exist for a given function but every neighborhood of which contains points for which the derivative exists. Also called singular point.

Edited June 3, 2010 by rigney

[insane_alien]

the astorphysics definition is the one you're looking for.

[mulreay]

While I believe our universe was the

instantaneous transition of a singularity into matter, could "it" have been billions of miles in diameter at the time?

 

Just to add that when thinking in terms of the size of the universe 'billions of miles in diameter' is fairly insignificant.

[rigney]

Thanks insane guy!, all information appreciated. Went there and looked at some ideas of which, are only ideas of someone else. Trying not to be a smart a--, but if gravity, known as the weakest of what is "supposidly" the four main forces, pulled everything into such a small space, how? Don't ask what the definitions below mean, I just read them and try to understand. But what wasn't mentioned below was that magnetism has the same reach, only more powerful.

 

Definition::::

 

Gravitation is the force of attraction between particles or objects of matter. It has the greatest reach or range but is also the weakest of the fundamental forces. The gravitational strength is only 6*10−39 of the strength of the strongest nuclear forces.

 

Note: 10−39 equals 1/1039, where 1039 is 1 followed by 39 zeros. That is a very small number.

 

The strength of the gravitational force decreases as the square of the distance between two objects. This means that if you triple the distance, the gravitation will be reduced by 1/9.

 

The force of gravitation is most apparent in objects of large mass, such as planets and stars. Gravitation is what keeps the Earth and other planets in orbit around the Sun.

 

Note: Gravity is defined as gravitation near the surface of the Earth.

Edited June 3, 2010 by rigney

[Martin]

... While I believe our universe was the

instantaneous transition of a singularity into matter,...

 

4. Astrophysics: A point in space-time at which gravitational forces cause matter to have infinite density and infinitesimal volume, and space and time to become infinitely distorted.

...

 

Nowadays there is a widely shared expectation among cosmologists that conditions referred to in #4 did not occur.

 

There are various models of U around time of start of expansion, which so far fit data about equally well. No scientific reason to believe time began at a big bang singularity. We simply do not have answers: various models must be tested, some ruled out, and so forth.

 

The idea that we "know" the U began with a "singularity" some 13.7 billion years ago is a delusion which has been fostered by commercial popularizations. You can sell books and illustrated magazines by peddling halfbake speculation that excites people.

 

There are a few honest public outreach sources. The German research outfit called Max Planck Institute has something called "Einstein Online". They have an essay called "A Tale of Two Big Bangs" which goes into the different ideas people have and the sources of misunderstanding. It says "most cosmologists would be surprised if it actually turned out that the U began in an infinitely dense, infinitely hot, infinitely curved state."

 

In other words this business is still being investigated but the smart money is betting on there NOT being a singularity in the sense of your definition #4.

 

But keep an open mind. People are still working on models, running computer simulations, trying to figure out ways to test, and so on.

 

If you want a taste of the professional literature (mostly too technical for general reader) here is a search covering research just since 2007.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=dk+quantum+cosmology+and+date+%3E2006&FORMAT=WWW&SEQUENCE=citecount%28d%29

 

If you want the public outreach account, google "einstein online cosmology".

That will get you here:

http://www.aei.mpg.de/einsteinOnline/en/spotlights/cosmology/index.html

Then click where it says "A Tale of Two Big Bangs".

Clear, fairly up-to-date, and not so oversimplified as to be meaningless.

Edited June 3, 2010 by Martin

[rigney]

Mulreay, you and I may share a concurring opinion, and while I'd definitely like to get feed back from everyone, your short sentence on "insignificance" intrigued me. Jump back in with both feet, I'm not smart enough to get flustered, just more inquisitive.

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Consecutive posts merged

The U began with a "singularity" some 13.7 billion years ago is a delusion which has been fostered by commercial popularizations. You can sell books and illustrated magazines by peddling halfbaked speculation that excites people.

 

I'll give you a big "Amen" on that Martin. Having a Phd in HK, I have no compunction to questioning everything I read and literally all that I see. Touche!

[ajb]

You have asked quite a technical question, sorry if my answer is a bit technical.

 

In the context of classical general relativity one would like to have the definition 4). However, in practice this is not a very useful one. You can think of the curvature of space-time at a point as a measure of the strength of gravity at that point. The trouble is that that there are several definitions of what we many mean by curvature and more importantly they are dependant on the local coordinates. To some extent this does not matter as we have tensors and know how to relate things in different frames, but what about something that are infinite?

 

Also, the presence of the curvature blowing up to zero means we no longer have the nice smooth structure on space-time. More technically, we do not have a genuine smooth manifold.

 

One way to define a singularity is to use the Kretschmann scalar. It is a scalar and so does not depend on the coordinates used. It is quite a "rough" notion of a curvature.

 

So one definition is that a singularity is a point, or collection of points on space-time for which the Kretschmann scalar is infinite. (Still I am not very happy that we consider something infinite as a scalar, so a better way to think of this is in terms of a limit as we approach the singular points.)

 

A more general definition, which includes examples that are not necessarily "Kretschmann infinite" is in terms of geodesic incompleteness. That is there exists "freely-falling particles" whose motion cannot be determined at a finite time at the point of reaching the singularity.

 

A better, more technical phrasing would be "a space-time is geodesically incomplete if exists at least one timelike or null geodesic that cannot be extended to arbitrarily large values of its affine parameter".

 

You can think of your particle as reaching the "edge of space-time", it cannot go any further. It would "fall of space-time" if it went any further. This "edge" is the singularity.

 

The modern definition of a singular space-time (i.e. we have at least one singularity) is a space-time that is geodesically incomplete. This includes singular spaces as defined by the Kretschmann scalar. (My understanding is that this took a while to become the accepted classification)

 

To get back to a smooth manifold one then removes the point(s) that are singular. Without these points general relativity works fine and we are all happy.

[rigney]

Thanks AJB. While your reply was a bit more than what I'm about, I question everything I read about the cosmology of our universe. And as I find it, knowledge is not an issue when asking questions, but it's nice to question the answers, if only to mull them over. A lot of what you said, I can vaguely relate to; but not in a sound technical sense. My one multiple question now is; Even knowing Einstein was a mathematical genius, how or why did he come up with a space/time warpage analagy and present it as a bowling ball rolling around in a circle on a rubber sheet? And how is it plausible to describe our universe as the huge outer perimeter of, "what", when we can only guess as to how fast it's traveling?

And the tensor thing? That's why I like radii and vectors, neither bend, according to Euclid.

Personally, I like everything about our universe other than having so many different platforms to view it from??

Edited June 4, 2010 by rigney

[mulreay]

I for one would be bored very quickly if the universe did not offer spectacular options

[ajb]

Even knowing Einstein was a mathematical genius, how or why did he come up with a space/time warpage analagy and present it as a bowling ball rolling around in a circle on a rubber sheet?

 

Matter (really energy-momentum) acts as a source of the gravitational field and we think of this field as the local geometry or curvature. Thus, the ball on a rubber sheet is an ok analogy.

 

The presence of the ball (matter) means a dimple in the sheet (local curvature of space-time).

 

I have no idea who first thought of this.

 

And the tensor thing? That's why I like radii and vectors, neither bend, according to Euclid.

 

Tensors are already useful on Euclidean space. The point is that any equation involving tensors written in one set of coordinates also holds in any other set of coordinates.

 

In general relativity, indeed physics as a whole we have the "gauge principle" that states that the physics should not depend on the details of how we describe it. Thus tensors are needed.

 

Though, we should say that there are important non-tensorial objects that are very important in physics. However, they are not direct observables. (Connections spring to mind here)

[rigney]

AJB, I do appreciate your condolence of my ignorance. And while not being submissive, I yeild to your supeior knowledge of the universe. Don't get me wrong, on a given day I'd probably try arm wrestling you one on one to the ground? In all seriousness though, with our earth spinning @ a thousand miles per hr, zipping around our solar system @ 65,000 mph. and tracking through our galaxy @ 600,000 mph?, what can I say??? And at the same time, our galactic universe is racing into infinity at "what speed"? Me, I believe those four quantum speeds that I just mentioned, say it all. If we are in a perpetual rotational time warp or headlong into oblivion, I'd just like to get some sort of idea of where we're going, before we get there? Thanks again.

Edited June 4, 2010 by rigney

[caharris]

Do all objects have a singularity? Not necessarily like the bb one or a black hole, but something similar?

[ajb]

Do all objects have a singularity? Not necessarily like the bb one or a black hole, but something similar?

 

There are space-times that are non-singular.

 

Singular space-times do seem quite unavoidable in general relativity. We have the theorems of Penrose and Hawking (and earlier works) that show that singularities are quite a general feature of general relativity, under some reasonable conditions.

[caharris]

There are space-times that are non-singular.

 

Singular space-times do seem quite unavoidable in general relativity. We have the theorems of Penrose and Hawking (and earlier works) that show that singularities are quite a general feature of general relativity, under some reasonable conditions.

 

What are the space-times that aren't singular? (sorry, I'm new at this as well )

What are some good articles or papers from Penrose or Hawking for this?

[ajb]

What are the space-times that aren't singular?

 

Don't have singularities, they are geodesically complete.

 

What are some good articles or papers from Penrose or Hawking for this?

 

As you claim to be new to this, there is nothing I can really recommend. You could look up the original articles but I am sure you would struggle. Better is the book by Hawking and Ellis [1]. However it is not an introduction to general relativity. It is very difficult.

 

The lecture notes by Carroll [2] I highly recommend as an introduction to general relativity done "proper".

 

After that try Wald [3]. Not really an introduction, but I found it very good. It covers some more advanced stuff.

 

For a mathematical overview of some of the background needed for general relativity: manifolds, fibre bundles, metrics, connections etc I suggest Nakahara [4] and Nash & Sen [5].

 

 

 

---------

[1] S.W. Hawking & G.F.R. Ellis. The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics. Cambridge University Press, New Ed edition (27 Feb 1975)

 

[2] Sean M. Carroll. Lecture Notes on General Relativity. 1997. arXiv:gr-qc/9712019v1

 

[3] R.M. Wald. General Relativity. Chicago University Press (1 Jun 1984)

 

[4] M. Nakahara. Geometry, Topology and Physics (Graduate Student Series in Physics). Taylor & Francis; 2 edition (4 Jun 2003)

 

[5] Charles Nash & Siddhartha Sen. Topology and Geometry for Physicists. Press Inc; New edition edition (Jun 1987) (Also, Dover books is going to release this soon)

[caharris]

Thank you ajb. I will spend many of days reading through these Might as well try and learn it now instead of being bombarded with it at school, huh?

[ajb]

Thank you ajb. I will spend many of days reading through these Might as well try and learn it now instead of being bombarded with it at school, huh?

 

Most undergraduate courses in physics will not cover much, if any of the above.

[caharris]

Most undergraduate courses in physics will not cover much, if any of the above.

 

Oh... Well, at least I find it interesting then I'm still in strong debate over what type of science I plan on studying, so this may very well come up in my future. Singularities have been intriguing me for quite some time, so these should at least quench my thirst for a while.

[ajb]

Singularities are usually seen as the breakdown of a theory, that is one is trying to apply a theory to a scale or scenario it cannot cope with. This is the sign of "new physics".

 

In the context of general relativity, it is generally believed that quantum gravity effects will regulate these singularities. But without any real quantum theory of gravity at our disposal yet this is all speculative.

 

A better example is the classical electron self-energy. In essence in classical electrodynamics it looks like one needs infinite energy to "assemble" an electron. Yet, the universe seems full of them! The solution to this problem can be found in quantum electrodynamics. Thus, the presence is singularities in a classical theory can point to a quantum theory.

 

However, quantum electrodynamics has it's own infinities. These are cured by the procedure of regularisation and renormalisation. If we apply these tools to "quantum general relativity" we end up in a mess, they simply don't work. Simple attempts to quantise general relativity, which itself has infinities produces a theory with infinitely more infinities! So plenty of work left to do...

[caharris]

Singularities are usually seen as the breakdown of a theory, that is one is trying to apply a theory to a scale or scenario it cannot cope with. This is the sign of "new physics".

 

In the context of general relativity, it is generally believed that quantum gravity effects will regulate these singularities. But without any real quantum theory of gravity at our disposal yet this is all speculative.

 

A better example is the classical electron self-energy. In essence in classical electrodynamics it looks like one needs infinite energy to "assemble" an electron. Yet, the universe seems full of them! The solution to this problem can be found in quantum electrodynamics. Thus, the presence is singularities in a classical theory can point to a quantum theory.

 

However, quantum electrodynamics has it's own infinities. These are cured by the procedure of regularisation and renormalisation. If we apply these tools to "quantum general relativity" we end up in a mess, they simply don't work. Simple attempts to quantise general relativity, which itself has infinities produces a theory with infinitely more infinities! So plenty of work left to do...

 

I was actually watching a TV program with Brian Greene (the string theorist) who was talking about this.

But that aside, for the self-energized electron you mentioned, is the singularity that is proposed for the bb the original source for the energy, or is the electron itself considered to be a sort of singularity? Or was that a really stupid question?

[ajb]

But that aside, for the self-energized electron you mentioned, is the singularity that is proposed for the bb the original source for the energy, or is the electron itself considered to be a sort of singularity? Or was that a really stupid question?

 

One would not consider the electron itself to be a singularity. It is the classical energy required to assemble an electron that is infinite. The quantum corrections to this are also infinite leading to a finite answer!

[caharris]

One would not consider the electron itself to be a singularity. It is the classical energy required to assemble an electron that is infinite. The quantum corrections to this are also infinite leading to a finite answer!

 

Okay that makes sense. Thank you for explaining it to me

[eoinmac]

I was under the impression that singularities have no physical shape or anything like that because there is a finite mass smashed into zero volume thus having no dimensions at all?

[ajb]

I was under the impression that singularities have no physical shape or anything like that because there is a finite mass smashed into zero volume thus having no dimensions at all?

 

By singularity one generally means "something that tends to infinity". In the case of curvature singularities it is the curvature at a point that becomes infinite.