Markus Hanke

I haven’t read any of Schwarzschild’s originals papers, since to me these are only of historical interest, and I’m not much into the history of science. My main focus is on the contemporary foundations of GR - its underlying symmetries, why the field equations have the specific form they do, what kinds of solutions they admit and how to find and classify them, what kind of covariant objects can “live” on spacetime, possible candidate models for quantum gravity, generalisations of GR etc etc. Things of that nature. I acquired all my knowledge from a wide range of contemporary texts and sources on physics and maths, with Milner/Thorne/Wheeler being the oldest of them. Both the internal and external Schwarzschild solutions can be found in depth in nearly all these texts, presented in different ways, and maximally extended metrics that cover the entirety of this particular spacetime are also found in many of these texts. I find it unwise to rely too heavily on a single source or author for one’s understanding of GR, so I always make sure I consult multiple sources.

It’s the Dirac delta function, and you have \[\int_{-\infty}^{\infty}\delta(x)dx=1\]

Yes. I should clarify here though that my comments were specifically about the Schwarzschild solution - this wasn’t meant to imply that BHs don’t exist at all in the real world. It’s just that they would be of a different kind than Schwarzschild. At a minimum you have to consider angular momentum and the absence of asymptotic flatness - which leads to Kerr-Vaidya spacetime as a starting point. That’s a considerably more complex geometry than Schwarzschild.

The relativistic formulation of quantum mechanics respects all the usual laws of SR, so there can be no physical interaction (as in: exchange of information) if the events are not within or on each other’s light cones. There can, however, be a correlation between measurements that are space-like separated, as is the case with quantum entanglement. What swansont did yesterday can very well affect your state of affairs today - but not vice versa. So this isn’t a mutual “interaction” as such, but rather a one-way causal influence. Just be careful with the term “event” - in relativity, this term has a very specific meaning, being a point in space at a single instant in time. It doesn’t mean an occurrence with temporal extension, such as a car accident.

But that’s the thing - you can remove the singularity at the horizon by a simple change in coordinates (while all curvature tensors are regular there), so it isn’t a physical singularity, merely a coordinate one. At the same time you can not remove the central singularity in the same way, because the region is geodesically incomplete (all curvature tensors diverge or become undefined there). ‘Real’ in this sense is that which does not depend on choice of coordinates, ie covariant quantities such as tensors, but not coordinate charts and coordinate singularities. While it may conceivably be possible to construct such a solution (eg as a special case of Ellis-Bronnikov spacetime), this would not be Schwarzschild spacetime, but a different type of geometry. Using the boundary conditions for the Schwarzschild solution, it can be shown that the central singularity is in fact inevitable. Schwarzschild black hole, in its collapsed state, is a vacuum solution, it assumes an entirely empty spacetime. Yes, that was my point - in the real world there are all kinds of distant sources, so asymptotic flatness cannot actually occur, meaning Schwarzschild ST is just an approximation that cannot be found in the real world. It’s still very useful though.

I think you should debate this with a mathematician, such as @studiot. 0^0 is an indeterminate form, so whichever value you assign to it will inevitably lead to contradictions in some contexts, while it works just fine in others. So far as I am concerned, in this particular context - algebra and calculus - there are good reasons to treat it as =1, and that works. That’s the consensus. Nonetheless, if you don’t want to use K-S coordinates, then don’t - you are free to choose any coordinate system you like, since this is an additional structure separate from the manifold itself. And that’s the salient point here - as already pointed out multiple times. A coordinate singularity or discontinuity does not necessarily imply that the manifold is singular or discontinuous. Besides, there are coordinate choices that are perfectly smooth and regular at the horizon, such as Gullstrand-Painlevé, or Eddington-Finkelstein. But why do we even need to be talking about specific coordinate choices? That’s precisely why one should use coordinate-independent methods here. The fact of the matter is that the Riemann tensor and its invariants exist and are well-defined on the horizon, so spacetime is necessarily smooth and regular there. That’s all there needs to be said on this. No. Firstly, GR is purely classical, so it is incapable of accounting for quantum effects. Secondly, the field equations are only a local constraint on the metric, but don’t determine it uniquely in and of themselves - you separately need to supply boundary conditions (most notably information about distant sources) in order to obtain specific local solutions. So you get out exactly what you put it - if you supply unphysical boundary conditions, you get metrics that are at best approximations to reality. That’s how it is for Schwarzschild - it assumes, among other things, asymptotic flatness, which isn’t something we find in the real world. So expecting exact Schwarzschild geometry to occur in the universe is foolish. But it is a useful approximation, so long as you understand the limitations. You are in conflict with well-established results then, as can be easily found in most major GR texts. Can you provide a coordinate-independent proof that the event horizon is geodesically incomplete - which is what you seem to be claiming?

...or you can just look at the invariants of the Riemann tensor in that region, in particular the Kretschmann scalar. Since it exists and is regular and well defined on the horizon, spacetime must necessarily be smooth and continuous there.

They are looking at it - if you search for this on arXiv, you’ll find quite a number of papers on this subject. I would imagine it hasn’t become part of the general consensus, because there are also problems and issues with these models. You are right, yes. I think the problem here is that the error bars for the precise value of the Hubble constant are far bigger than the effects of the acceleration, so at the moment we aren’t yet in a position to draw a meaningful graph for this. This is a work in progress.

This is the generally agreed upon definition in algebra. Of course not. The singularity is inevitable in purely classical Schwarzschild spacetime, as can be formally proven using the singularity theorems. As I mentioned earlier, the appearance of a singularity in any theory of physics (not exclusive to GR) generally means that the model has been extended past its domain of applicability. In this case, GR, as being purely classical, fails to account for quantum effects during the collapse. It does most emphatically not mean that we expect a singularity to be a real-world object. If you want to eliminate the singularity, you can choose a connection other than Levi-Civita on your manifold, which allows for the presence of torsion in addition to curvature. This model is called Einstein-Cartan gravity, and is singularity-free. But this is not the same theory as General Relativity. Schwarzschild spacetime is static and stationary by definition. There are no dynamics whatsoever - you actually use this fact as boundary condition to derive the solution in the first place! You can translate the entire manifold in time without changing anything in its geometry. In technical terms, the manifold admits a time-like Killing vector field. When we say that time and space trade places below the event horizon, what we really mean is that ageing into the future inevitably corresponds to a radial decay - meaning there cannot be any stationary frames, no matter how much force you exert in trying to counter gravity. It’s inherent in the causal structure of spacetime itself. This is of course independent of the choice of coordinates. ‘Patch’ is simply the technical term for a particular region on a manifold, it has nothing to do with any manipulations of same.

I don’t know, to be honest - I’ve never looked into these models. Over the last few years my focus has been elsewhere, so I haven’t been keeping up with latest developments as much as I would have liked to. Another one for the future to-do list 👍

joigus has beaten me to it with his excellent answer (+1). As I said earlier, the manifold and a particular coordinate chart chosen on it are not the same things at all - to put it succinctly, having a ‘hole’ in an embedding diagram does not necessarily imply that there is a corresponding ‘hole’ in the manifold, in a topological sense. These are different things. You can in fact have patches (or entire manifolds) without coordinates defined on them. For Schwarzschild spacetime, you need only transform the metric to a different, more complete coordinate basis to see this. But if you want to be absolutely sure and precise, it is always best to use tools that are coordinate-independent. Yes, indeed. Arriving at a precise value is actually not easy, also because external conditions play a role during the collapse. But I think the salient point is that there is such a limit, for any given level of degeneracy. It’s hypothetical to some degree, yes. But just as in the case of quantum gravity, there are good reasons to believe that the GUT domain is quite real, even if we don’t know for sure which of the numerous GUT candidate models will apply. That being the case, quarks and gluons are by-products of a broken GUT symmetry, so once energy levels are high enough, the strong interaction will cease to exist in its ordinary form. In more general terms, I very much agree that singularities are not real-world objects, but artefacts of our models being pushed beyond their domains of applicability.

That’s a really good question! It is indeed possible to vary lambda with time, location, or both - the resulting models are called “agegraphic dark energy models”. There are both advantages and problems associated with these, but I must admit that this isn’t something I’ve been following, so I don’t know where things stand on this. It hasn’t caught on in the mainstream though.

Einstein, despite having come up with the equations initially, didn’t know about the full set of principles underlying their form (the crucial topological concepts underpinning this were worked out by Ellie Cartan at a later date) - so he wouldn’t initially have been aware that the presence of the constant was the ‘normal’ state of affairs, hence it didn’t appear in his original formulation. So unfortunately any fine-tuning to precisely zero still lacks a physical mechanism or reason. Again, I’m not saying it can’t be zero, just that this would be an example of unexplained fine-tuning.

I’m really confused now - could you try to rephrase for me what your main point is? I can’t really make sense of the progression of the last few posts. I should remind you again that an embedding diagram concerns a coordinate chart, which is a separate thing from the manifold itself. To say that any part of an embedding diagram - whether missing or not - is ‘outside the manifold’ is meaningless, since you can have regions that aren’t covered by that particular chart. Schwarzschild spacetime is the simplest and most straightforward solution to the Einstein equations - both its geometry and topology are well understood and have been studied ad nauseam by generations of physicists and mathematicians. Precisely which aspect of it do you think we are misunderstanding?

The trouble with this is that the Einstein equations aren’t just invented out of thin air. There are some fundamental principles of consistency and topology that greatly constrain the form these equations can take (See Meisner/Thorne/Wheeler for details). As it turns out, the equations including the constant are the most general form that fulfils all these conditions - so there needs to be a reason why the constant should be exactly zero. Im not saying it can’t be zero, just that there would have to be a reason for it.

If the embedding diagram terminates in a throat, then that means the coordinate chart isn’t continuous at that point. This doesn’t imply anything about spacetime itself. Ok, but then he would have realised that that spacetime is everywhere continuous and doesn’t terminate at any horizon surface. I don’t really understand the significance of these historical references, to be honest. We nowadays know of a large number of exact solutions to the field equations, including maximally extended metrics that cover the entirety of this particular spacetime, so we know in depth its complete geometry and topology - which is a lot more than was known back in early 1900s. Why do you keep referring back to the state of affairs a hundred years ago? Our state of knowledge and maths has moved on greatly since then. Remember also that metrics aren’t invented - they are derived solutions to the field equations.

Yes, that’s what I meant. Good question! What they did was to approximate the tightly packed neutrons in a neutron star as a special kind of gas, called a Fermi gas. The dynamics of this were fairly well worked out, which made it possible to derive a rough limit for when neutron degeneracy is able to resist gravitation. It’s called the Oppenheimer-Volkoff limit. If that limit is exceeded, gravity will be stronger than the degeneracy pressure. Oppenheimer/Volkoff did not use any results from QCD, which wasn’t fully developed until later. Nowadays we can speculate that there might also be a degeneracy state involving quarks, which then would also have a corresponding limit. This would lead to an astrophysical object called a quark star (purely hypothetical). However, it is safe to say that in this domain no classical approximations will suffice, this is where we need to use quantum gravity - which we don’t yet have. So we can’t say what happens when the quark degeneracy limit gets exceeded. Yes, but if you keep collapsing the body, the pressures and energies will eventually get so high that the fundamental forces will re-unite into a GUT scenario - at which point the concept of ‘quark’ ceases to make sense. It is also not clear that the Pauli exclusion principle is meaningful in a scenario where quantum gravity plays a role. This is inconsistent with GR, because there can be no stationary frames beyond the horizon, due to the fundamental geometry of the spacetime there. So you can’t have stable objects of any kind. In other words, it doesn’t matter what specific mechanism you propose - so long as there is classical spacetime, a full collapse is inevitable (ref also singularity theorems). The only ways to avoid this is to either modify GR, or abandon classical (smooth, continuous) spacetime once certain limits are exceeded. You can calculate them using the laws of QM and QFT: Chandrasekhar limit - electron degeneracy - white dwarfs Tolman-Oppenheimer-Volkoff limit - neutron degeneracy - neutron stars Quark degeneracy (don’t know if there’s a name for it) - quark stars (hypothetical)

I may have misunderstood what you did, then. Apologies.

I don’t know exactly where the confidence level for this stands at the moment, but I think it’s pretty likely that this is real. Not really. It straightforwardly corresponds to having a positive cosmological constant in the Einstein equations. To me, it would actually be much weirder should it turn out that this constant is somehow exactly zero, because there is no a priori reason (that we know of) for that to be the case.

That’s because the rotation is a hyperbolic one, and thus rapidity uses the tanh function - so the difference would only grow large once you have relative speeds close to the speed of light itself.

The current accelerated expansion of the universe is not related to inflation - these are physically different circumstances.

You can also look at this purely geometrically. If O and O’ are in uniform relative motion, then these two coordinate systems are related by a hyperbolic rotation in spacetime (ignore boosts for simplicity) - in other words, a Lorentz transformation is essentially just a simple rotation of the associated coordinates. The speed v is then directly related to the rotation angle by a simple equation; so you can express relative speed as an angle. This is called rapidity.

It’s outside the particular coordinate chart that happened to be used; that doesn’t mean the manifold itself is not smooth and continuous there. I don’t know what this means? There’s no such thing as negative proper mass. The first solutions found in and around 1917 were exterior metrics, meaning they described spacetime in a vacuum, ie outside the central body. Oppenheimer and others later found solutions that describe the interior spacetime of bodies, ie spacetime inside the central body, and by extension a full metric that consistently encompasses both regions. Since, once certain limits are exceeded, it is physically impossible to prevent gravitational collapse, these must contain singularities. There is no such condition once you use a metric that covers both interior and exterior spacetime, which is what they did. When the sign on the squared line element inverts, then that means that time and space trade places - which is to say that ageing into the future becomes equivalent to a diminishing radial coordinate. In other words, there are no longer any stationary frames, and one cannot avoid falling towards the center. The issue here is that Einstein’s GR is a purely classical model of gravity, so it does not and cannot account for quantum effects. Within the framework of the classical model, the appearance of physical singularities is inevitable (this can be mathematically proven). However, the real world isn’t classical below a certain scale, so the current assumption is that quantum gravity will remove such singularities. We don’t have such a model yet, but it’s an era of active research. To remove singularities from classical GR, you can make a small modification to it that leads to a model called Einstein-Cartan gravity. This is free of singularities. However, this modification has other consequences, which to date we haven’t seen in the real world. But you are right of course in that one does not expect singularities to be actual objects that occur in the real world. They are artefacts of the model, and generally mean that it breaks down under the given set of circumstances; they are “physical” only within the context of that model.

What does this mean? Nothing has been ‘corrupted’, it’s just that now, a century later, we have a much better understanding of the foundations of GR than Schwarzschild (or any of his contemporaries) would have had. There was confusion about this only because the model was brand new back then, and it took time to figure things out. Nowadays we are in a much better position. In his original paper, Schwarzschild used a coordinate system that had its origin at the event horizon, so r=0 meant the horizon surface. However, this does not at all mean that there is nothing beyond the horizon, because in GR the choice of coordinates is arbitrary and has no physical significance. Schwarzschild used this convention simply because it made his particular way of deriving the solution mathematically easier. A consequence of this choice is that large parts of the spacetime aren’t covered by any coordinate patch, so, in his notation, there are physical events that cannot be labled by any coordinate. But again, that’s just a convention without physical significance. You can rectify this simply by choosing a different coordinate system - which does not change anything about the actual geometry of the spacetime. This is why there are so many seemingly different metrics (Novikov, Kruskal-Szekeres, Aichlburg-Sexl, etc etc) which all describe the same physical spacetime. To see whether the event horizon is a physical singularity (as opposed to just a coordinate one), and what the nature of spacetime beyond the horizon is, you can use tools that do not depend on the choice of coordinate system at all - such as invariants of the curvature tensors. That way, it’s trivially easy to show that, in classical GR, the horizon as well as all of spacetime in the interior right down to the singularity is in fact perfectly smooth and regular, just like anywhere outside the horizon. This is a standard exercise in pretty much any graduate GR course.

Interesting new paper on anomalies in physical cosmology: https://arxiv.org/pdf/2208.05018.pdf