Singularities & Quantum Mechanics

[joe1950]

I am wondering if anyone knows if a singularity would be acceptable/possible in the realm of quantum mechanics?

I know little and understand less about either, however I was under the impression that quantum mechanics operates assuming a quanta of anything as the smallest possible reality and does not embrace the idea of a continuum, where space and time can be divided infinitely.

If this is so, and if it is indeed true, singularities, at the big bang and within a black hole cannot exist as being infinitely small and dense.

I hope I've expressed this properly and understandably. I'm not sure I understand the implications myself, but from the limited knowledge I have see a conflict in the two lines of thought.

Thank you. joe

[Linda]

I am not a phycisist at all, so I just have some loose thoughts.

But isn't it so that our dimentions and laws of nature only exist inside our universe and are caused by the big bang. That means the sigularity was before that and didn't have the same laws of nature. I am also thinking that it might have been something in another dimension, or at least outside of ours and that it exploded energi into our dimentions.

[joe1950]

Excellent point, Linda. We certainly must hold open the possibility, perhaps the probability that prior to our universe coming into being the rules were vastly different.

Thank you for the reply and insight! I have to say, for some reason which I cannot comprehend myself, I tend to have trouble with the concept of the infinite, other than a construct of mathematics.

Good day.

joe

[ronin]

Never been a fan of the idea of a singularity.

In physics and maths both zero and infinity are trouble areas, and a singularity manages to encompass both.

I would like to know if at some stage of the mathematics of things they have simplified the maths by saying that the size is so small that zero is a good approximation and this then causes the maths to simplify. This approximation is used in semiconductor doping at a PN junction as I recall. Makes life a lot easier.

The problem then is that once having said that the size is zero you have to stick with it, you cannot make whatever a non-zero size later down the line when the zero size starts to cause problems.

[andrew s]

The expectation is that quantum mechanics will avoid the issues associated with singularities, however, it is only theory there is no evidence one way or the other. The difficulty is that both Quantum Field Theory and General Relativity ( a classical continuum theory) both have passed the tests they have been put to. While most professional physicists think QFT is the more fundamental theory I feel this is just an opinion with no factual basis.

 

Regards Andrew

[joe1950]

Thank you Ronin and Andrew. This must be the area I've often heard of where general relativity and QFT are at odds and give nonsensical results. However, each is amazingly accurate in its respective element. It's very fascinating but probably rather frustrating to the theorists. It s also fortunate, in a practical sense, we can accomplish a great deal using good approximations. 

It seems recent times have witnessed numerous ideas and postulates being offered as scientific theories, but after reading some recent books very critical of the current state of the science, much of these are apparently without testable evidence of any sort. A kind of tabloid physics, if you will. For me, and others may certainly disagree, the idea of a super-multi megaverse with infinite copies of myself wandering about was a bit more than I could tolerate  

Hopefully, with data sources improving as they are with efforts such as the LHC, LIGO, the Webb Infrared telescope, just to mention a few, some solid direction may be found. 

Thank you again for the clearly stated information!

 

 

[Ruud]

The observable universe is finite in size, but the universe as a whole is bigger, maybe even infinite. We just don't know.

The singularity in the context of the universe refers to the beginning of the universe. It does not necessarily refer to its size. After all, if the present universe is infinite, while at the same time its age is finite, it must always have been infinite in size. It can't grow from finite to infinite in a finite amount of time. 

Even if the present universe is finite, physics has its limits. From the beginning of the universe to the end of he Planck epoch (the first 10^-34  seconds) physics can give no description of the cosmos at all.

The singularity of a black hole is a different thing. It is a theoretical construct of which we cannot confirm whether it exists or not. 

When you send a probe into a black hole there are two different reference frames. Seen from the outside, the probe's clock slows down as it approaches the event horizon. At the event horizon, time for the probe (seen from the outside) stops completely. This means that seen from the outside the probe freezes on the event horizon. It never passes it. It almost reaches it and then becomes part of an infinitesimally thin shell which rests on the black hole's event horizon. Seen from the outside the probe never becomes part of any singularity.

In the reference frame of the probe, the clock of the universe speeds up and becomes infinitely fast when the probe reaches the event horizon, i.e. the universe runs through its complete lifespan and dies its heat death. It does not really come to that though. The black hole itself is part of the universe and seen from the probe the black hole 'evaporates' through the process of Hawking radiation as the probe gets really, really close to its event horizon.

This means that neither for an outside observer, nor for the probe, the probe actually passes the event horizon.

Any particle that tries to enter a black hole will behave like the probe. Weirdly enough this may mean that the black hole itself cannot form.

Take the matter that tries to form a black hole. The last particle that is needed to form it already behaves like the probe. Seen from the outside its clock begins to run so slow that seen from the particle the universe dies.  

This means the only thing that truly forms, is a dense region of matter which is infinitesimally larger than its own Schwarzschild radius. The black hole itself cannot form because the last particle that it needs cannot enter it before the universe comes to its end.

If this is true we no longer have to worry about information lost when matter falls into black holes. Black holes only almost  form. Anything that seems to fall into a black hole ends up in a shell just outside the Swarzschild radius of a dense clump of matter which itself never forms a singularity. And when the total mass grows, the Swarzschild radius also grows.

I know. It's a weird idea, but if Hawking can come up with weird ideas anyone can, right?  Hawking actually did come up with a new idea recently. If I remember it well he suggests that event horizons are not closed spheres, but somehow permeable regions through which information can leave a black hole.

 

 

[andrew s]

35 minutes ago, Ruud said:

The observable universe is finite in size, but the universe as a whole is bigger, maybe even infinite. We just don't know.

The singularity in the context of the universe refers to the beginning of the universe. It does not necessarily refer to its size. After all, if the present universe is infinite, while at the same time its age is finite, it must always have been infinite in size. It can't grow from finite to infinite in a finite amount of time. 

Even if the present universe is finite, physics has its limits. From the beginning of the universe to the end of he Planck epoch (the first 10^-34  seconds) physics can give no description of the cosmos at all.

The singularity of a black hole is a different thing. It is a theoretical construct of which we cannot confirm whether it exists or not. 

When you send a probe into a black hole there are two different reference frames. Seen from the outside, the probe's clock slows down as it approaches the event horizon. At the event horizon, time for the probe (seen from the outside) stops completely. This means that seen from the outside the probe freezes on the event horizon. It never passes it. It almost reaches it and then becomes part of an infinitesimally thin shell which rests on the black hole's event horizon. Seen from the outside the probe never becomes part of any singularity.

In the reference frame of the probe, the clock of the universe speeds up and becomes infinitely fast when the probe reaches the event horizon, i.e. the universe runs through its complete lifespan and dies its heat death. It does not really come to that though. The black hole itself is part of the universe and seen from the probe the black hole 'evaporates' through the process of Hawking radiation as the probe gets really, really close to its event horizon.

This means that neither for an outside observer, nor for the probe, the probe actually passes the event horizon.

Any particle that tries to enter a black hole will behave like the probe. Weirdly enough this may mean that the black hole itself cannot form.

Take the matter that tries to form a black hole. The last particle that is needed to form it already behaves like the probe. Seen from the outside its clock begins to run so slow that seen from the particle the universe dies.  

This means the only thing that truly forms, is a dense region of matter which is infinitesimally larger than its own Schwarzschild radius. The black hole itself cannot form because the last particle that it needs cannot enter it before the universe comes to its end.

If this is true we no longer have to worry about information lost when matter falls into black holes. Black holes only almost  form. Anything that seems to fall into a black hole ends up in a shell just outside the Swarzschild radius of a dense clump of matter which itself never forms a singularity. And when the total mass grows, the Swarzschild radius also grows.

I know. It's a weird idea, but if Hawking can come up with weird ideas anyone can, right?  Hawking actually did come up with a new idea recently. If I remember it well he suggests that event horizons are not closed spheres, but somehow permeable regions through which information can leave a black hole.

 

 

Hi Ruud - A number of points in this don't seem to fit with my understanding of Black hole theory in standard GR. For example, I understood that from the point of view of a test particle passing a black hole event horizon, it does not experience any discontinuity at all! I may well be wrong so can you point me at a reference to these ideas.

Thanks Andrew

[Ruud]

59 minutes ago, andrew s said:

from the point of view of a test particle passing a black hole event horizon, it does not experience any discontinuity at all! I may well be wrong so can you point me at a reference to these ideas.

That's true, I've read the same, but as the clock of the test particle starts to run slower (for an outside observer) clocks in the rest of the universe will seem to run faster and faster for the test particle, up to near-infinitely fast when the test particle is about to pass the event horizon. That's why I think there will be a moment, seen from the test particle, that the universe reaches the end of its life.

Unless the universe lives forever, just getting ever more rarified. It might.

However, even if the universe lives forever, the black hole that the particle tries to enter will not live forever due to Hawking radiation. So, seen from the outside the particle freezes on the event horizon, rather than passing it, while the black hole slowly vanishes. Eventually, I suppose, the black hole will set the particle free again as it ceases to exist.

From the standpoint of the particle, the black hole would disappear suddenly as the particle is about to enter it.

All the time, the particle feels fine and it is happy with its own clock. It just never enters. 

Actually I'm making this up as I go along and I would not be surprised if I overlooked something.

EDIT: I realize that I am arguing that black holes don't really exist, just near-black holes, super dense bodies that are infinitesimally larger than their Swarzschild radius. I didn't expect this would happen.

[joe1950]

Well done, Rudd! That certainly gives one pause (no pun intended). I'm certainly not qualified to critique the idea but I have read several accounts where to the outside observer, the particle would never enter the event horizon. So why wouldn't the same be true from the particles point of view but for the opposite effect of time dilation?

Very compelling. Thank you.

 

*For some reason your final statement puts me in mind of a TV advert here, where a doctor in an operating room is operating on a patient and very professionally and effectively giving orders to his staff at the end of the operation. He finally says, "That should do it, finish up." The lead nurse next to him says something to the effect, "That was absolutely brilliant surgery, Doctor!" He replies, "Oh, I'm not a doctor, but I did stay at a Holiday Inn Express last night."

There are others in the series and they are excellent.

So I have to ask you, Rudd, did you stay at a Holiday Inn Express?

[Ruud]

8 minutes ago, joe1950 said:

did you stay at a Holiday Inn Express?

Haha, no!

[andrew s]

Hi Ruud,

To checked my understanding I have looked this up in "Gravity An Introduction to Einstein's General Relativity" by James B Hartle. 

 For a non rotating spherically symmetric black hole, if you consider a test probe that emits light signals at fixed intervals with respect to its proper time (i.e an internal clock) then an external observer will see the signals become dimmer and dimmer and at increasingly long intervals i.e. red shifted as the probe approaches the Schwasrtzchild radius (the event horizon). Once it crosses the horizon no more signals will be detected.

For the probe itself it will see no discontinuity at the Schwasrtzchild radius but the signals will be bent inwards and like the probes world line will terminate on the singularity at the center of the black hole.

In this view there is no build up of material at the Schwasrtzchild radius indeed the radius is given simply by r = 2M where M is the mass of the black hole. If the probe has mass Mp the the radius will expand to r = 2(M + Mp) as the probe cross the Schwasrtzchild radius.

The book does not discuss the view of the remainder of the universe from the probe once it cross the Schwasrtzchild radius as it carries on down and is destroyed at the singularity having suffer major tidal forces.

Regards Andrew

[vlaiv]

Not all things in QM are quantized. Space and time are not for sure. You can have a particle occupy volume of space with certain probability density, but once energy transfer occurs that leads to decoherence (in terms of Copenhagen interpretation measurement is made), particle has more precise position - center of distribution of this more precise position can have any value on real axis so we cannot speak of space being quantized. Energy levels in systems can be quantized, some other properties as well, but not all and it is configuration dependent. Free particle can have continuum of energy states, only bound particle has its energy levels quantized.

Problem with singularities and QM are in the following: lets look at the inverse function (1 over x) as x approaches 0 value of this function shoots to infinity, only being undefined precisely at X equals 0 (for every other value of x you can calculate function value exactly no matter how big its value gets). Particle never has its position defined to single value. It is always range of values, no matter how small it gets (it cannot be as small as you like due to uncertainty principle as momentum would shoot to enormous value - this needs energy - so you will need more and more energy to squeeze particle to smaller and smaller range). There is something called normalization meaning that when you sum probabilities where particle is over this range it should give you 1 (particle is in this range for sure). When you have particle in vicinity of 0 on x axis for above mentioned function you get into trouble, range that you are summing over includes infinity value - this cannot be summed to 1 no matter which scale factor you use - and your calculation fails, particle behaves as it would never the less - it is theory that fails.

There are some sums that even if infinities being involved can be calculated and agree with experiments (one of the funny ones being: sum of all integer values is equal to -1/12 - take a look at this short video). Problem pose other sums that we have not yet found ways to attach meaningful number to (and later confirm by some experiment that is devised from theory behind it). This is being the reason why there is no successful (or accepted) unification of  GR and QM yet.

[joe1950]

Excellent information gents!

The video is very interesting. I subscribed to the channel as there are many others that look to be fascinating.

It's one of those concepts that the more you learn, the more complicated it appears; at least from my perspective level.

Thanks very much!

[Ruud]

Hi Andrew,

3 hours ago, andrew s said:

For a non rotating spherically symmetric black hole, if you consider a test probe that emits light signals at fixed intervals with respect to its proper time (i.e an internal clock) then an external observer will see the signals become dimmer and dimmer and at increasingly long intervals i.e. red shifted as the probe approaches the Schwasrtzchild radius (the event horizon). Once it crosses the horizon no more signals will be detected.

Yes, as the external observer you would notice redshift and time dilation as the blinking clock approaches the Schwarzschild radius (Srad). Ever more. When an object reaches the Srad the redshift will be infinite and the blips of the clock will come to a halt for the external observer. That's when the object freezes if the external observer could still see the clock.

3 hours ago, andrew s said:

For the probe itself it will see no discontinuity at the Schwasrtzchild radius but the signals will be bent inwards and like the probes world line will terminate on the singularity at the center of the black hole.

That's what I've read as well. For some reason I've now come up with a line of reasoning that seems valid but is not.

3 hours ago, andrew s said:

In this view there is no build up of material at the Schwasrtzchild radius indeed the radius is given simply by r = 2M where M is the mass of the black hole. If the probe has mass Mp the the radius will expand to r = 2(M + Mp) as the probe cross the Schwasrtzchild radius.

Not for the probe, but for the external observer the probe comes to a halt at the Srad. Every object with mass has an Srad, btw. It only becomes relevant when you compress M to close to or smaller than its Srad. I think it is calculated as  Srad = 2MG / c² for an object of mass M. I'm not familiar with the book by Hartle, but I just dusted off The Feynman Lectures on Physics. Part one at least touches on the subject and sections of part two may also may contain relevant information. Part three deals with quantum mechanics. Feynman is actually quite accessible. I must have more textbooks on relativity and quantum mechanics/physics  but they're probably somewhere in boxes. I still have them from my time as a student. All of a sudden I realize I have lots of memories associated with these books! Most of the physics I have forgotten though. I went on with mathematics and also took up linguistics.

3 hours ago, andrew s said:

For the probe itself it will see no discontinuity at the Schwasrtzchild radius but the signals will be bent inwards and like the probes world line will terminate on the singularity at the center of the black hole.

Ah, but the outside observer sees the black hole evaporate really slowly. The probe has meanwhile reached extreme time dilation as it is on the edge of the black hole. That's why I think the probe must see the black hole evaporate very fast.

As fast as my physics is rusty. It's an unexpected outcome. Maybe the error lies in ignoring that the black hole itself also experiences extreme time dilation so that the probe and the black hole have their clocks running in sync.

[vlaiv]

When thinking about time dilatation and similar effects (two observers in different non-inertial frames of reference) we must include the fact that there is no simultaneity as we normally think of it. So concept of probe being stuck at event horizon and at the same time something other happening can lead us to wrong conclusions. In order to figure out what would other observer measure in his/her frame of reference there are well established equations that give answer to such questions (more often than not counter intuitive to our belief).

Take for example famous twin paradox of SR, usual question being: if other twin is moving relative to me and his watch is ticking more slowly then mine, what about his view of things, I'm in relative motion to him so my clock must be running more slowly than his is. Right way of thinking is as follows: I perceive (given finite speed of information propagation) that his watch is running more slowly, and he perceives same of my watch but what matters is what clocks show then we are at same space time coordinates and make comparison. In this case equations on my side for my clock and his clock, and equations on his side for his clock and my clock will all agree on observed conditions. Each of us will have simple equation for clock in our reference frame: just delta T, but calculations for other person will be different. Just to give you few pointers how things would differ: in my reference frame he is traveling to distant star that is 1ly away. In his frame of reference due to his motion (we exclude acceleration phase that is usually used to try to explain what is going on, but is not necessary as everything can be explained with constant speed motion and clocks synchronization at times that we share space time coordinates) I'm traveling the opposite way distance that is less than 1ly (length contraction effect). Also I would see him "turning" around when he reaches distant star. At that point he will perceive me as still going in the opposite direction - there is no simultaneity so we must not assume that moment at which I saw him turn around is the same moment at which he saw me "turn around". When combined these effects give us proper answer what will show on each twin's clock when we meet again (did not do maths to give specific answer but plan to do it sometimes in the future so I can do proper blog post and explain this to people willing to read it).

[Ruud]

Funny that you should mention that, Vlaiv. Here's a small excel worksheet that calculates how much time passes on a spaceship that travels a certain distance (in ly) at a constant acceleration of 1g (1.032 ly/y²). At the halfway point the ship tuns around and breaks with the same acceleration.

Some results:

The worksheet: Einstein_Rocket.xls

Once we can make a 1g drive that can work continuously we can travel just about anywhere in the galaxy in a lifetime or less. 

[andrew s]

Sorry Ruud - I have got used to texts that set c = 1 in normal units you are quite correct. In the end the only way to say what happens to an object is to calculate its world line by solving Einstein's field equations and seeing what happens to it.  For me an objects world line defines its history in spite of what external observers observe it to do.

 The evaporation of black holes according to Hawkins is, as you know,  a mix and match of classical and quantum theory, they don't evaporate in standard GR. 

Indeed in some formulations of Quantum Loop Gravity the black hole rebounds on a time scale dependent on its mass and become a white hole. Solid predictions have been made and it is thought they may be one source of gamma ray bursts.

Regards Andrew

[acey]

On 05/03/2016 at 16:23, joe1950 said:

I am wondering if anyone knows if a singularity would be acceptable/possible in the realm of quantum mechanics?

I know little and understand less about either, however I was under the impression that quantum mechanics operates assuming a quanta of anything as the smallest possible reality and does not embrace the idea of a continuum, where space and time can be divided infinitely.

If this is so, and if it is indeed true, singularities, at the big bang and within a black hole cannot exist as being infinitely small and dense.

I hope I've expressed this properly and understandably. I'm not sure I understand the implications myself, but from the limited knowledge I have see a conflict in the two lines of thought.

Thank you. joe

Singularity is a mathematical term and essentially means 1 divided by 0. A black hole corresponds to a solution of general relativity which has a singularity at the centre (infinite density). In quantum theory there are lots of singularities, in the sense of infinities, but they can be cancelled away ("renormalised") to give meaningful predictions of measureable quantities. There's a general hope that in a full theory of quantum gravity the black hole singularity would be finite. The widespread assumption is that at a very small size scale (the Planck length) physics works differently and the classical field theory that leads to the singularity (general relativity) would no longer apply. If space itself is really quantised (i.e. not continuous) the assumption is that this would occur at the Planck scale or smaller. If all of this is true then instead of 1 divided by 0 at the centre of a black hole you would have 1 divided by something very small, which equals something very big but finite.

[joe1950]

Thank you Acey! That's where the idea I recalled originated - Planck length, Planck time. A quantized or granular spacetime replacing the idea of a continuum would perhaps solve the conflicts between GR and QFT.

I likely said something incorrect there, but I know what I mean... I think.  

[joe1950]

Just another thought came to mind (happens now and then). 

Mathematics aside, would it be scientifically possible to have empirical data or the ability to perform an experimental test, needed with the scientific method, anything that is considered possibly infinite? For example, the possibility of the universe being infinite, or there being an infinite number of parallel universes; or the infinite density at a singularity.  

Thanks.

[SilverAstro]

On 06/03/2016 at 15:20, Ruud said:

Once we can make a 1g drive that can work continuously we can travel just about anywhere in the galaxy in a lifetime or less. 

But not good for returning home to show your holiday pictures to your grand kids, nor your great great ,,  !

[vlaiv]

11 hours ago, joe1950 said:

Just another thought came to mind (happens now and then). 

Mathematics aside, would it be scientifically possible to have empirical data or the ability to perform an experimental test, needed with the scientific method, anything that is considered possibly infinite? For example, the possibility of the universe being infinite, or there being an infinite number of parallel universes; or the infinite density at a singularity.  

Thanks.

Not quite sure what you mean by having empirical data of something that is infinite. Either you would have to experiment for an infinite amount of time, or you need infinite amount of paper to write your infinite amount of data. There are "infinite" things that we measure all the time but to a finite precision. Take for example PI - ratio of circumference to diameter. It has infinite decimal places but we measure and calculate it to certain finite precision all the time. You can do it at home, take a piece of string ... you get the picture.

So lets try and give a following answer to this (don't know if it is the correct answer): There are some measures of infinite quantities that you can make (and as any measurement - to a certain precision), but for everything else (there is mastercard? ) we have to solve our cosmology and see if there will be infinite amount of time to do it ...

On a more philosophical note: If you look at what measurement is - comparing two values (one being measured and reference one) and finding relationship of sorts - you can measure infinities all the time and you will get data. If you compare any reference value lying around to infinite one - you can have some sort of relationship (less than, bigger than ...) as your measurement result.

 

[joe1950]

Good explanation Vlaiv. Points I hadn't considered.

I'm thinking, though, that coming to a scientific conclusion, or an acceptable scientific theory that (ex.) the universe is infinite in size or that the density at a singularity is infinite is not possible since neither would be testable. If you can't measure an infinite universe, you could never falsify a theory that it's infinite, I would think.

It's as string theory is but more so. Some say to have empirical data that string theory is an accepted scientific theory would require a nearly impossible technology - from what I understand. Presently, there is no way to test it as a theory. A theory of something infinite would be beyond that, indeed completely impossible to test. So any idea that the universe is infinite, or a singularity has infinite density would remain an untestable idea, with no supporting experimental data.

I may not be clear in my thinking...or my thinking may not be clear! 

Thanks Vlaiv!! 

[andrew s]

6 minutes ago, joe1950 said:

If you can't measure an infinite universe, you could never falsify a theory that it's infinite, I would think.

 

You could falsify it by showing it was finite.  For example I can show my house is finite by measuring it with a ruler.

Regards Andrew