A singularity existed as the initial state of the universe and at the center of a black hole, but quantum mechanics predicts that the smallest possible volume, the Planck volume is 10^-99 cm. Is this a contradiction, or am I not understanding something?
I’m not remotely qualified to answer this, but my understanding is that the Planck volume is derived from physical constants within our universe. If, as many suggest, the laws of physics are variable, perhaps a different universe might only be a Planck length across and still contain galaxies and civilisations.
The standard model of cosmology (i.e. GR) is classical.
Approaching the singularity, QM effects probably become important, but we don’t yet have a satisfactory theory of quantum gravity.
You’ve got right gist of it; but technically, quantum mechanics doesn’t make the prediction in that way. It is known that quantum mechanics and general relativity (Einstein’s “curved spacetime” version of gravity) are inconsistent with each other as we currently understand them. It is also known that if you combine the most important constants in those theories with each other, then you get a quantity with units of length called the “Planck length”, about 10[sup]-33[/sup] cm. It therefore seems like a good guess that this length scale is where quantum gravity effects have to take over: larger than that and the classical theory of general relativity is fine, but smaller than that and you’d have to include quantum mechanics in your predictions. We’re also pretty certain that a singularity is not a physically meaningful object; you can’t get down to an object of “zero size” without taking quantum mechanical effects into account.
The issue is that we don’t know how to do that yet: we don’t have a well-accepted model of “quantum gravity” that allows us to predict what happens as you go from “normal” length scales to scales below the Planck length. All we have is an educated guess that things might get weird around that length scale (or maybe 10 times that length scale, or 0.01 times that length scale).
To expand on what MikeS said: There is no known physical significance to the Planck length (nor the Planck time, nor the Planck mass or energy). We know that there must be some situations where both general relativity and quantum mechanics are relevant. We know that neither theory by itself could be completely applicable to such a situation, and that there must be some new theory combining both which describes such situations. We do not know what such a theory would look like: Even our best guesses at such are woefully incomplete. We suspect that such a theory would involve quantization of space-time, since space-time is what’s relevant for GR. If space-time is quantized, it might be quantized in such a way that there’s a smallest possible distance: Quantization of angular momentum and of electric charge both work that way… but then again, quantization of energy does not work that way, so we can’t be sure. And if there is a smallest possible distance, then we don’t know what it is: Our best guess is that it’s somewhere in the vicinity of the Planck length (mostly, because we don’t even have any means available for making any other guess), but even if it is in that vicinity, it might not be exactly the Planck length: Nobody would be in the least surprised if it turned out to be half the Planck length or pi times it or something like that (well, beyond our pleasant surprise at finding it at all, of course).
All that said: We also wouldn’t be particularly surprised if, once we developed a theory of quantum gravity, it predicted that singularities don’t actually exist. But we don’t have such a theory yet, and the best theories we do currently have still predict that they do exist, so there’s not much more we can do there right now.
I believe one of the big hurdles in formulating Grand Unified Theory is the fundamental differences between Quantum mechanics and Relativity: QM assumes matter and energy
are quantized…reducable to fundamental, irreduceable chunks. Relativity is a continuum theory…it is assumed that space, time, matter and energy is continuous from the infinite to the infinitesimal.
Earlier posts of mine regarding the OP:
http://boards.straightdope.com/sdmb/showthread.php?t=807193&highlight=planck
http://boards.straightdope.com/sdmb/showthread.php?t=668009&highlight=planck
Yes, I imagine that there would be a problem with angular momentum when r=0
Macroscopic angular momentum depends on r, but at the quantum mechanical level, there are two different kinds of angular momentum: There’s orbital angular momentum, which works more-or-less like the classical kind and also depends on r, but there’s also what’s called “spin” (which is only very loosely analogous to the classical phenomenon by the same name), which doesn’t depend on r at all, and which applies even to particles which have, so far as we can determine, zero size. In any event, both kinds of angular momentum (and their sum, the total angular momentum) have a minimum possible nonzero value.
It is important to remember that all ideas on what happens near and past the event horizon are purely speculative theories, and every proposed speculative theory is currently untested.
There are also several types of black holes and the OP uses a few terms in a way that I cannot eliminate a significant number of options to try and butcher an analogy on what may happen from a GR context.
As the OPs other thread was considering a distant observer, and this question relates to black holes that have existed from the birth of the universe I will try to assume the most generic case.
Note this is just one possibility in a sea of “what ifs”
In the most basic case in GR, and if light wasn’t gravitationally redshifted to invisibility any external observer an object that crossed the event horizon would appear to just stop, for the remainder of history. From the reference frame of the object entering the black hole both sides of the spacetime interval would become negative. This would invert the space like and time like axes in that objects view, meaning that they would have the freedom to move “through time” while the attraction towards the singularity would become a causal movement similar to the arrow of time for us. Under this model, and assuming the black hole was not of a configuration where spaghettification would happen it is quite possible that no matter how long you could wait you would never reach the singularity.
But just as backwards in time is not an option for movement under our reference frames, radially outward would just become a non-option.
The implications of this is that the “singularity” could just be a single location and purely an artifact of our chosen coordinate system but that objects radially entering a black hole will never actually reach it.
This singularity under Penrose/hawking may be spacelike, timelike, orbifold, or a jump discontinuity in the metric. It is not an absolute that this geodesic incompleteness is in the form of a dimensionless singularity with an infinitesimal small single point, also called a space-like singularity. There are a few speculative theories that result in a time-like singularity which wouldn’t pose an issue for the plank length, but may have issues with the plank time etc…
There is a lot of fun reading and learning in this area, but as this is GQ I would caution the OP and other readers to revisit my opening statement.
While there has been some promising work, this is an area of unsettled science.
Currently we only have purely speculative theories.
I love trying to follow this stuff woefully equipped for it though I be. Could you clear one thing up? When you speak of a spacelike singularity and a timelike one I’m confused as I keep reading that space and time are identical with relativity, ie spacetime. So how are they differentiated here?
This is confusing to everyone, we are not prepared to think in 4D, and on the quantum scale you actually have to think in 6 dimensions to consider the implications of degenerate matter and the phase states. In a neutron star the matter actually occupies all of the possible phase states and the pauli exclusion principle is the limiting factor and neutrons exist in the “same place” but in different phases. If we are lucky some individual will at some point in the future will have the ability to intuitively visualize this but it is unlikely.
We humans lack the ability to visualize the full impacts of spacetime, but at least in our universe they are not exactly the same. We cannot move backwards in the time dimension and our interactions are with other portions of spacetime are limited by our light cone.
But an example is the theory that spinning charged black hole (which should be most of them) would actually have an even horizon that is ring shaped. If you were to pass through the event horizon in that case your worldline would end from the perspective of the rest of the universe when you crossed it, as would the world line for an observer which crossed it on the other side. Those terminations may happen at the same time for an external observer but those locations would also appear to be dimensionally separated for that observer.
It will require more advances in physics and some form of test will need to be figured out before we know, but those possibilities are not purely restricted to a dimensionless single location like the classic concept of a singularity suggests. But note that these ideas do not necessary exclude a dimensionless single point singularity, and we may find that both exist. We simply just do not know right now.
History is full of theories where the math worked but were invalidated through experiment. I hope that we find ways to test these concepts within our lifetimes but it is also likely that we will never have a way to test these theories.
It is fun to think about and a fun hobby, to be honest it is a lot nicer once you can start to so some of the math as that is the way to get past our limitations in conceptualizing the claims and ideas on how this may work.
I like this because in context it is both physical/experimental (and may [probably?] be so in principle as well as currently so conceptually/mathematical/theoretical.
A rare situation in modern science, I should think, and well stated by Chronos.