# A Singular Question

1. ### TalonD

178
What are the differences or similarities of the singularity of the big bang vs the singularity of a black hole? in each case it is matter condensed or collapsed to a very dense state. When a star collapses and forms a black hole, does the matter still have some size or is it infinately small, is there some limit to how much matter can be compressed?

2. ### mathman

6,517
The concept of singularity in both cases comes from pushing general relativity to its extreme. However, this concept is incompatable with quantum theory. Any description of what the state of the universe at the start of the big bang or conditions inside black holes is not known.

Big bang theory describes what happens after the big bang, but not at time t=0.

3. ### Chronos

9,986
I favor the planck density as the maximum possible density of matter. It is nearly, but not quite infinitely dense.

4. ### mysearch

522
Not sure this is a singular question, but it does raise number of very interesting issues. First, I am no expert, so my comments just reflect some of the things I have read. However, as touched on my Mathman and Chronos, a singularity also appears to be a point where general relativity and quantum theory collide, therefore we are probably discussing issues of speculative hypothesis rather than scientific fact.

Whether a black hole is a singularity depends on whether we fully understand the nature of matter and its degeneracy under gravitational collapse. As I understand it, there are 2 accepted states of degeneracy after the radiation pressure of star fusion stops, i.e. electron and neutron. After this a neutron star is assumed to collapse into a black hole, although there are several speculative hypotheses relating to quark degeneracy.

As such, physics may have a working hypothesis of how a black hole singularity might be formed, although possibly little idea as to what happens beyond its event horizon. So what inferences can be made about the BB singularity?

This seems an altogether more expansive question, which may not the purpose of this thread, so have simply provided the following link as an example of but one of many ideas about the wider issues surrounding the nature of the BB singularity.
http://www.space.com/scienceastronomy/white_hole_030917.html

To counter this speculative idea, here is a possibly more mainstream view:
http://www.mathpages.com/HOME/kmath339.htm

Finally, here is a link to an archived discussion in this forum
https://www.physicsforums.com/archive/index.php/t-202260.html

5. ### Chronos

9,986
A singularity is not necessary to generate an event horizon.

6. ### TalonD

178
what would have an event horizon besides a colapsed star / black hole? Can something be massive enough and yet not be gravitationally colapsed?

7. ### marcus

24,510
Singularities occur in specific theories. They're places where a particular theory breaks down and they normally get fixed by improving the theory. So a singularity in some other theory wouldn't have anything to do with GR. Here we are talking about cosmo/black hole singularities. These are places where GR breaks down.

It is not just a collision with QM, we know GR must be wrong simply because it fails to compute.

I urge you to read "A Tale of Two Singularities" at the einstein-online link I have in my sig. It's an excellent discussion and clears up some potential confusion. Astronomers use the blow-up point of GR as a convenient time-marker without excluding the possibility that time-evolution extends back further. In nature there may be no actual singularity (it would be kind of surprising if there were) but in cosmo the t=0 failure of classic GR is still an excellent time-marker from which to date things. So one refers to it as if it were an actual event. "One second after the singularity." "Two minutes after the singularity."

That's true in spades! One shouldn't assume that at the center of a BH there is an actual singularity. Singularities are unphysical.

The question is what is there instead. The BH singularity occurs only in the vintage-1915 theory. Improved BH models eliminate the singularity---but still of course have the event horizon.

So what you say is absolutely true. Logically/mathematically a singularity isn't required because indeed we have models which have the event horizon and outwardly resemble classic black hole, but don't blow up.

Now the job is to find ways to rule out some models and narrow down. Witnessing collapse events---the gammaray and gravitational radiation (if it can ever be detected!) may help. It's very hard, but not theoretically impossible.

Nonsingular models (both cosmo and BH) have become a major research interest just in the years we have been discussing at PF.

I think Chronos was talking about what you say: a collapsed star/black hole. That's the kind of thing that has an event horizon.

but in NATURE such a thing would not be likely to have a singularity.
I'm not sure what an actual singularity in nature would look like, probably impossible.
The singularity is a blow-up in a particular model (classic 1915 GR) where the model says nonsense, like infinite density and infinite curvature.

I think the point is that with new modeling techniques we can model the collapsed star successfully without the theory blowing up.

The problem now is to choose between several different nonsingular models. It's challenging. Some observational tests must be found. We'll be in suspense until some way is found to throw out some and narrow the range of possibilities.

Just to mention one---just Tuesday a couple of days ago there was an international seminar and Param Singh gave a talk on singularity resolution, based on a some analytical models. There the typical thing that happened in collapse was (so to speak) a bounce out the back door. Collapse leading to reexpansion of a fresh tract of space. In his analysis the bounce typically occurred at about 40 percent of planck density.

8. ### TalonD

178
Marcus
I read the EOL two singularities. I can understand the singularity of the early universe is just where the math breaks down but the matter,energy,space,time or what ever it was back then, continues on before that point in time. right? Whatever is inside a black hole seems to me similiar to whatever was the early universe in that both are very dense. So I was wondering if that is as far as the similarity goes? or are they both the same? Also if there are ways to avoid the singularity of a BH can that also be used to avoid the singularity of the universe? I would guess not or you would have said so but I cant help asking, considering the obvious similarity of the two at least from an averge Joe's point of view.
I have a related sort of question but it probably belongs in the GR forum.

9. ### marcus

24,510
Talon, just to get a little hands-on experience with Planck density, try putting this into the google window

c^6 /G^2/(hbar*c) in g/cc

then press return. It should give you planck density in grams per cc.
That is as compared with the normal density of water.

If you leave off the "in g/cc" and just type in this:
c^6 /G^2/(hbar*c)
then google will calculate it for you in kilograms per cubic meter.

If you want the Planck energy density, or equivalently, the Planck unit of pressure
(both of which are relevant to big bang or bounce conditions) then type in the same thing but with an extra factor of c^2 (that's to turn the kg of mass into joules of energy)

c^8/G^2/(hbar*c)

This should give you a figure expressed in force per area (newtons per m^2 = pascal)

The google calculator knows the values of things like hbar, and G, and c. So you never have to look up stuff in a handbook, or type in numerical values of the constants.

Something simple underlies the bewildering looking formula: two facts
c^4/G is the Planck unit of force

hbar*c is the Planck unit of force x area

Those two things are all you need to know to figure out all the Planck quantities.

For instance: force^2/(force x area) = force/area = pressure

So the unit of pressure
has to be (c^4/G)^2 divided by hbar*c. Which is what I said before, for pressure.

If you don't like to calculate with physical constants, don't worry about it. But if you do, you might enjoy calculating some Planck constants like this on your own, instead of looking them up.

Planck units like this are the units that take over when people study what could actually be going on in places where the 1915 theory blows up and stops computing. They are the relevant units, so it's good to begin getting acquainted.

Last edited: Nov 20, 2008
10. ### marcus

24,510

My personal hunch agrees with yours, and lot of working cosmologists certainly would be surprised if it turned out that all that stuff actually stops at t=0. So right, but before we begin to sound confident, though, we have to see one of the nonsingular models be a winner, and some of the others get shot down. There has to be an empirical reason to favor one over the others.
Another challenging job is to relax the assumptions of isotropy and homogeneity.

Param Singh, in his Tuesday seminar talk, emphasized that as a major caveat. Nonsingular cosmo models have to be extended to cover the case where there is less symmetry and uniformity. They have to check that the bounce still occurs when things are lopsided and lumpy.

Oddly enough, there has been more progress with getting rid of the cosmological singularity than with the black hole singularity. Many cases have been studied and a bounce replacing the cosmo singularity seems to be comparatively robust conclusion.

Intuitively, just as you say, the black hole situation ought to be similar. But things have not worked out as well there, so far. Not as many papers. Not as much consistency in the conclusions.

Part of the reason is that far more effort and talent have gone into developing nonsingular quantum cosmology---probably for several reasons, but there is one outstanding reason. In the cosmology case we can see the results of the bounce (if a bounce is what happened.)
Early universe structure formation, nucleosynthesis, microwave background, eventually relic neutrinos, and probably more stuff. There is a lot of potential evidence to check ideas against.

Anyway, for whatever reason, more talent and effort has gone into studying the nonsingular BB than the nonsingular BH, and for whatever reason more progress is seen there. Inspite of the obvious similarity, which I agree is there. I see it too.

11. ### TalonD

178
fascinating.

and that google calculator is amazing!

I understand simple algebra so I can play around with those planck values.

Now that makes me wonder something else. I don't know or understand the math that leads to the singularity, does that mean that calculations go to infinity? like the GR formula for increasing mass with increasing speed where it is infinite at C? Even though the planck density is very small it is still finite right?

12. ### marcus

24,510
What you say about it still being finite is right. Intutively, the Planck length unit is very small, so the volume is very very very small (it's the cube of the length).
So the density has to be stupendously large, because it represents a unit of energy (or inertia if you mean mass density) packed into a really tiny volume.

What do you get when you put this into google calculator?

(c^4/G)^2/(hbar*c)

If I divide out c^2 so the answer will be a mass density, I get a huge number like 10^96 kg per cubic meter

Or if I specify I want the answer in grams per cc, then I get something on the order of 10^93

That is when I put this in

(c^6/G^2)/(hbar*c) in g/cc

Just as a check, please give it a try and let me know what you get.

13. ### TalonD

178
c^6 /G^2/(hbar*c) in g/cc
I get 5*10 to the 93

((c^4/G)^2/(hbar*c))/c^2
5*10 to the 96 kg/m3

c^6 /G^2/(hbar*c)
5*10 to the 96 kg/m3

c^8 /G^2/(hbar*c)
4.6*10 to the 113 pascals

twenty three years since college, I need to brush up if I'm going to do any real math or physics lol

Last edited: Nov 20, 2008
14. ### marcus

24,510
Excellent! Thanks. So Google calculator is working for you. I get the same answers as you do.

In case you or anyone is curious here is Param Singh's talk from 18 November
titled "Loop quantum cosmos are never singular"
http://relativity.phys.lsu.edu/ilqgs/

I'm not recommending it because it is technical and he talks very fast. But you can download the slides PDF separately and get a quick impression of what it is about. And see his caveats, where he qualifies and gives reservations. Not something to spend a lot of time on, but a sample of current work.

He does give the figure that the bounce seems always to happen when the density reaches 40 percent Planck.
Or more precisely 0.41 Planck.

So how big the universe is at the time it bounces depends on how much matter was in it in the first place.

I mean, if the universe consisted of 5x10^96 kilograms of mass, then it would only manage to collapse down to a couple of cubic meters volume!

Let's use what Param says and suppose that the density at bounce is 40 percent planck. So that is 2x10^96 kilograms per m^3.

Google calculator knows "mass of sun". You can check me on this, but I think that what this means is that 10^66 solar masses collapse down to a cubic meter volume. Lord goshamighty great thundering bullfrogs! That is a lot of stars. It makes me think I've done some awful arithmentical error. Well, see whether it checks out for you. I am bothered by the absurdly large numbers. Maybe Param is using a different definition of planck density. or I have made some blunder. Let's treat this as highly tentative and let me think about it for a day or so.

Last edited: Nov 20, 2008
15. ### Chronos

9,986
The Planck density is key to understanding event horizons. An event horizon forms at the surface of any such entity. The mass itself need not, and probably should not continue to collapse beyond that point if current thinking is on the right track. The implications are very interesting. At the Planck density, the estimated mass of the known [observable] universe could reside in an unbelievably small volume, as marcus noted.

16. ### marcus

24,510
Chronos, I decided to pursue this a bit more on calulator level.
Taking radius of observable to be 45 billion lightyears, and critical density to be 0.85 joule per km^3, of which 0.27 is dark and ordinary matter

(4/3) pi (45 billion lightyears)^3 (0.27*0.85 joule per km^3)

Google tells me that the energy equivlent of all the stuff in the observable universe is about 10^71 joules
(more precisely it says 7.4 x 10^70 joules)

Now the reduced* planck energy density is (c^4/(8 pi G))^2/(hbar*c)

So let's see what size volume would be occupied by 10^71 joules at that 0.41 planck density.

10^71 joules/( 0.41(c^4/(8 pi G))^2/(hbar*c)) in cc

It gives me 3 x 10^-34 cc

YUK. what a disgustingly small volume
Still, it is a large volume by Planck standards. It is about 10^63 times the planck volume.

*I think they tend to use the so-called reduced planck units in this context, which makes a difference of 8 pi
in some things. You simply use 8 pi G, instead of plain G. I personally have no preference, but I'm trying to
match what I think is done in the source articles.

Last edited: Nov 24, 2008
17. ### mysearch

522
Could I try to clear up a few questions related to the Planck scale and your calculations? Wikipedia give all the planck scale equations, so I will simply cite the link for reference: http://en.wikipedia.org/wiki/Planck_units. This link doesn’t define the planck volume directly, but I assume that it is the cube of the planck length, which in google calculator form is:

((h * G) / (2 * pi * (c^3)))^0.5 = 1.61609735 × 10^-35 meters

Therefore, if I cube the previous result, I arrive at the planck volume?

(((h * G) / (2 * pi * (c^3)))^0.5)^3 = 4.22087562 × 10^-105 m^3

You then state the critical density as "0.85 joule per km^3". However, while I agree with this figure could you clarify a few of questions:

o Is Hubble’s constant [H] primarily established based on observational measurements and then the critical density calculated from Friedmann’s equation?

o Is it true that for the universe to be spatially flat [k=0], the component energy densities would have to be very close to the critical density?

o Is the figure of 4% baryonic matter based on estimates of the amount of observable matter?

o Is the figure of 23% cold dark matter based on estimates necessary to correct gravitational anomalies, e.g. galactic rotations?

o While the figure of 73% is clearly the remaining portion of the critical density; is there any other observational evidence for dark energy other than the acceleration issue?
Based on the present critical density above, the energy contained in the observable universe of radius of 45 billion lightyears is:

(4 / 3) * pi * ((45 billion lightyears)^3) * (0.85 (joule per (km^3))) = 2.7472062 × 10^71 joules

Presumably, you only scale 27% of this energy because of the assumption that dark energy does not change its energy density with expansion. However:

o Does this imply that the energy of the observable universe is now larger than at earlier times, when the relative density of dark energy falls to ~0%?

Last edited: Nov 24, 2008
18. ### marcus

24,510
Yes.

Yes, in total.
I don't know the whole story of how it is estimated. There is a lot of gas and dust. Different kinds of observations (radio telescopes see cold gas clouds, Xray telescopes see hot clouds etc etc) and different kinds of expert inference. A lot of work on this, but ultimately can only involve guesswork. Yes the 4% figure is based on observation.

Indeed I believe it is largely based on things like galaxy rotation curves and the apparent stability of clusters of galaxies. But it is confirmed on other grounds as well. For instance there is structure formation in the early universe. The models of structure formation work because there is enough dark matter to start the process of coagulation.

In the computer models it is the dark matter that starts the process of clumping and forming cobwebby strands separated by voids. The ordinary matter was too dispersed to get started curdling.

So dark matter actually forms the skeleton of structure in the universe and without it we would not be as well clumped.

This was a bit of serendipity, because they first estimated how much dark matter there would have to be to make presentday galaxies and clusters stable, and then that turned out to be the right amount of dark matter to explain the observed rate of structure formation in the early universe----starting with the amount of irregularity seen in the CMB.

As a rule, no one observation ever clinches an argument, it is always a process of fitting together several pieces.

Same thing here. They got the original estimate (equivalent to 0.62 joule per cubic km) by how much would be needed to explain the acceleration measured with supernovae up to and including 1998.

But then by good luck or serendipity that turned out to be the amount needed to explain spatial flatness, given estimates of the amount of dark and ordinary matter made on other grounds!

And then some other observations also supported that estimate (approx 0.62 joule per km^3) like the integrated Sachs-Wolfe effect, and further, more distant Supernovae.

I think near flatness is an important part. One can check near flatness using galaxy counts and also independently using the CMB. Then one can measure H(t=present) and use that to estimate rho_crit, (about 0.85 joule per km^3) the density needed for spatial flatness. Then one can estimate how much dark and ordinary matter (say 0.23 joule per km^3).
And the rest, needed to get approximate flatness, turns out to agree with what was estimated for dark energy.

The arguments are all a bit guessy and iffy, but they mesh, coming from different directions. So one can provisionally work with the consensus model (as many do) without complete certainty, always keeping an alert lookout for inconsistent evidence that might refute it, or alternatives that might do as good a job at accounting for all the myriad different kinds of data.

Again a web of supporting or consistent observation, no one single clincher. IMHO.

I believe is an unresolved puzzle related to this.

However the observable universe is not an isolated subsystem, What it includes changes with time, so perhaps it does not provide the best example of an apparent violation of conservation of energy.

One could try taking any comoving volume. Any volume which expands along with the regular increase of distances (expanding in concert with the Hubble flow) and the energy of all the forms we know about or at least usually measure will be increasing due to the constant dark energy density---always proportional to volume.

Also a comoving volume experiences a net loss of CMB energy due to redshifting, and expansion gradually drains kinetic energy like that of the neutrino background. To me this has always seemed a puzzle. Where does it go?

I hear some people say that the lost energy of the CMB (it's photons have lost 99.9% of their energy) somehow goes into gravitational energy.

Anyway, as far as I am concerned there are unanswered questions about this. One doesn't know the extent to which the universe as a whole conserves energy. It seems one may only have conservation laws in local coordinates and for isolated subsystems. Maybe the total global energy of the universe is not even mathematically well-defined, and a comoving volume isn't adequately isolated and energy can always flow in and out of the box.

One way to think about it is this: Laws should have an operational definition. Energy conservation law basically prohiibits someone from building a perpetual motion machine? Can you think of a way to harness the dark energy (the amount of which grows as the volume expands)? If no observer can harness it, then maybe it is not violating any law.

Last edited: Nov 24, 2008
19. ### Chronos

9,986
When space expands, a photon redshifts. This nicely fits the conservation of energy principle. The total energy of the photon is diluted across a larger volume of space. Incidently, this phenomenon refutes the notion of space as a propogation media for photons [i.e., aether]. If space were such an entity, photons would blue shift when it was tensioned - like the note emitted by a guitar string being tightened. If space behaves more like a fluid, you should get shear when it is stretched. In that sense, dark energy could be equated with turbulence.

Last edited: Nov 24, 2008
20. ### mysearch

522
Marcus,
I hope you don’t mind, but I thought your previous response was so useful I made a reference to it in the sticky thread at the beginning of this forum. See following link: https://www.physicsforums.com/showpost.php?p=1974939&postcount=51. As implied, I found this to be a very useful summation of the state-of-play concerning what appears to be some of the more critical assumptions of the LCDM model.

I might still be a little sceptical of the serendipity theory regarding the energy density of dark energy at this time, but need to research some of the background issues a little more before commenting. Therefore, started to do some figures on the energy issues related to the observable universe now and at CMB decoupling, i.e. z-1090, as the following figures can be checked against the cosmological calculators. However, I was wondered if anybody had any comments on the results so far?

Energy Density at z=0;
Observable radius = 4.558e10 LY = 4.312e26 m
Observable volume = 3.965e32 LY^3 = 3.358e80 m^3

Baryons = …4%; ……….3.41e-11 joules/m^3; …1.145e70 joules
CDM = …...23%; ………1.96e-10 joules/m^3; …6.581e70 joules
Radiation = 0.00824%; …7.03e-14 joules/m^3; …2.360e67 joules
Lambda = …73%; …...…6.23e-10 joules/m^3; …2.092e71 joules
Total = …..100%; ……...8.53e-10 joules/m^3; ….2.864e71 joules

Energy Density at z=1090
Observable radius = 4.180e7 LY = 3.954e23
Observable volume = 3.059e23 LY^3 = 2.589e71 m^3

Baryons = ..11%; …4.53e-2 joules/m^3; ….1.172e70 joules
CDM = …..64%; …2.60e-1 joules/m^3; …. 6.731e70 joules
Radiation = 25%; …1.02e-1 joules/m^3; …. 2.640e70 joules
Lambda = …0%; ….6.23e-10 joules/m^3; …1.612e62 joules
Total = ….100%; ….4.08e-1 joules/m^3; ….1.056e71 joules
So we are considering a comoving volume expanding from 41.80 million lightyears to 45.58 billion lightyears. Therefore, we can estimate the unity change in energy of this volume as follows:

Baryon:… 1.145e70 / 1.172e70 = 0.977
CDM: .…. 6.581e70 / 6.731e70 = 0.977
Radiation: 2.360e67 / 2.640e70 = 0.00089
Lambda:…2.092e71 / 1.612e62 = 1.297e9
Total: ……2.864e71 / 1.056e71 = 2.712
So under expansion and within the limits of accuracy of the calculation, the baryon and cold dark matter energy within this comoving volume remains unchanged, as we would expect. The radiation energy falls due to the additional (1/a) wavelength expansion factor, while there is an exponential increase in dark energy because it scales with volume. However, the bottom line appears to be that our comoving volume now has 2.7212 times the energy at decoupling, i.e. +370,000 years!

I guess one immediate question that comes to mind is whether this energy analysis should consider any change the gravitational potential energy due to expansion?

I haven’t really had time to consider this issue, but as a generalisation, any increase in potential energy would be negative, therefore I was wondering if this might, in any way, offset the apparent positive increase in energy?
After reading Chronos` post could I clarify a few points:

In the context under consideration, the expansion of space, wherever that means, leads to an increase in the wavelength of a photon in transit. As frequency is the inverse of wavelength, it falls as a function of expansion and so does energy by virtue of E=hf?

Irrespective of the guitar string analogy, the expansion of space does seem to affect the photon. If we assume that the speed of light [c] doesn’t change, then I assume we must assume the product of permittivity and permeability of space doesn’t change. So what caused the photon to change frequency?

Last edited: Nov 25, 2008