Markus Hanke

The problem here is that those people do not respond to reason, so debating them is pointless. There is literally nothing you could say to them that would change their world view. As the old saying goes, you can't reason someone out of a position that hasn't been arrived at by reason in the first place.

You don't need to reference any particular coordinate system for this, you just look at the curvature invariants of the Riemann tensor. They are all finite at the horizon, unlike is the case for the curvature singularity at r=0. You can also look directly at whether the region is geodesically complete or not, which, again, is independent of any particular coordinate choice. GR is purely classical, so it does not say anything different. I'm at a loss on a different point - the paper talks about a geometry that resembles a 'decaying white hole', but to my understanding there is no white hole counterpart in Vaidya spacetime, unlike in the Schwarzschild case. Or am I getting this wrong?

Your initial claim in this thread was that evaporating black holes cannot exist (see very first sentence of OP). This paper does not support such a claim - in fact it is actually about the information loss paradox. Furthermore, it makes it explicitly clear that evaporating black holes are not Schwarzschild, which is what we have been attempting to explain all along; it does not attempt to dispute that they exist, as you seem to do.

Well, I'll be damned. One wave of the hand, and 5+ billion of us, who just happened to be born into non-Christian cultures, sentenced to burn in hell for all eternity. Makes me wonder just which of those two guys to blame, really.

And to add to this, after all my years on amateur science forums, there are two other crucial problems I see: 1. People often develop an intense focus on one particular or narrow point/source/information, and may even possess a good grasp on it; but then they fail to understand how it fits into a larger context. For example, I have met lots of people who have a good handle on Minkowski spacetime (SR), but then they naively try to add in gravity, and fail to understand why this does not work. Or people who become almost obsessed with one paper by one author, without grasping the context in which it was written, and thus draw the wrong conclusions from it. Nothing in the sciences stands in isolation, knowing and understanding the larger context is as important as any individual piece of knowledge. 2. Too many people seem to be entirely unable to distinguish valid sources of scientific information from pop-sci, personal opinions, or outright woo. Access to information is useless - even dangerous - if one is not equipped to judge the scientific value (or lack thereof!) of it.

Interesting, thanks! I just wonder if there is a mathematic model that describes the evolution and dynamics of the system? Again, just as a matter of curiosity. Just as a quick note - my own personal sense of achievement comes from pursuits that have no monetary value, and my most valuable asset (basic necessities of survival aside for now) is free time that enables me to engage in those pursuits. It is a mistake to assume that a sense of achievement can only result from having paid work; many of the greatest achievements of humankind cannot be measured in monetary terms.

I don’t know anything about global macroeconomics from an academic point of view, but this discussion got me wondering what the underlying symmetries of this closed system actually are, mathematically speaking. My immediate impulse here would be to describe the global flow of funds/money (not the same things, right?) similar to how we do fluid dynamics, using an appropriate differentiable manifold. Has anything like this ever been done? I’d be curious to see the resulting maths - what field equations govern the evolution of the system, what the local and global symmetries are, etc. I would also expect the dynamics of this system to be fully determinate, but having a rapidly increasing uncertainty factor when it comes to predictability, i.e. a chaotic system. Not trying to make any particular point at all here, this is just a matter of curiosity on my part.

If you disregard fatalities due to direct acts of violence, some 1.2 million people died in the four years of the Khmer Rouge regime from hunger, disease, and other ‘natural’ causes linked to malnutrition and inadequate health care. Of course it is difficult to disentangle how much of this is a direct result of economic policies (or rather their absence), but you get the idea - it didn’t end well for the people of Cambodia. I think Google is your friend here, I am not in a position to do that kind of research for you. Keywords such as “communities without money” will get you a lot of search results, both for historical communities, and current ones. They definitely do exist, but once you read between the lines (remember that much of the material will be biased one way or the other), it becomes apparent quickly that the absence of money - just like an abundance of it - is not necessarily correlated to increased happiness. There are always trade-offs of one kind or another, usually related to personal freedom, or opportunities to pursue things other than basic survival. I actually personally know an individual who chooses to live without money for ideological reasons (he isn’t part of any community) - he makes it work for himself, but his days consist of hard toil from morning to late, just in order to secure his basic survival. I know that he has little to no time nor resources for any other type of pursuit. To be honest, that would not be my idea of a fulfilling life, but each to their own.

So did the Khmer Rouge during their regime in 1970s Cambodia - they abolished all currency and pretty much any trace of a money-based economy, and reverted to bartering between communes instead. We all know how this turned out. In my humble and wholly unqualified opinion money is a dangerous thing if it is not seen for what it really is (a social convention), but I would be enough of a realist to recognise that currently there are few if any alternatives that would actually work in practice. Many attempts have been made throughout history to set up communities/societies that don’t use money, and to the best of my knowledge none of them have worked out in the end.

Very well put +1 True, if you have large local variations in curvature (as would be the case in the actual universe), such a global foliation will not in general be possible.

I did not make any reference to Newtonian gravity or any particular form of potential, I am only using the fact that the energy-momentum tensor has to be locally conserved. The relation I gave follows from Noether’s theorem, and not any particular theory of physics. The point was simply that, if you allow c to vary, this conservation law no longer holds, because the underlying symmetry that gives rise to the conserved quantity is no longer there. If c is not constant, energy-momentum cannot be conserved, irrespective of what else you attempt to change.

Yes, well put. Ok, I see. In Vaidya spacetime this issue never arises, since (unlike with Schwarzschild) this coordinate time for a far-away observer remains finite. No, you are correct. What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer. There are infinity many possible observers, and each one of them will use a different foliation scheme; hence the foliation is never objective and shared by everyone, it is always observer-dependent, even if it spans the entire spacetime. There is no such thing as universal time, of course. BTW, slicing up 4D spacetime into an ordered sequence of 3D hypersurfaces is called the ADM formalism of GR.

Again, gravitational potential - if it can be meaningfully defined at all - is a gauge field with a gauge freedom to choose a zero point, whereas Planck’s constant obviously isn’t. It is not physically meaningful to relate the two in this manner.

Well, I am not an expert in philosophy, so I can’t really comment on that. So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold. However, there will always be infinitely many possible foliation schemes, so there is never any preferred notion of time. This is in keeping with the principle of relativity, of course. The relationship between these clocks is simply the coordinate transformation that relates the metrics. For example, the stationary far-away observer can use the Schwarzschild metric, whereas the observer in free fall uses the Gullstrand-Painleve metric. These are simply related by a coordinate transformation, since both frames are of course in the same physical spacetime. But they use different notions of what ‘time’ means, so defining a notion of simultaneity is not in general possible if the observers are separated in time and/or space, unless there are certain very specific symmetries present. At best, it might be possible to foliate spacetime in a manner that both observers can agree upon, using a suitable coordinate system and foliation parameter; but this works only in certain highly symmetric cases, and the foliation parameter is not something that any physical clock would actually show in either of the two frames, so I don’t see how it is helpful here. You may be able to do this in certain special cases that are highly symmetric, which is why I put the qualifier “in general” as part of my original comment. Flat Minkowski spacetime is a trivial example. I don’t think it is possible in Vaidya spacetime though, which is what we would be talking about when it comes to evaporating black holes. Crucially, I don’t think it is helpful to even consider the concept of simultaneity in curved spacetimes, since it is not a generally applicable concept - in my experience, it is bound to lead to more confusion than clarity. I agree, it has little to do with topic of the thread, so I’m not sure why it was brought up at all.

Yes it will be. If you look at the above equation, if c is variable, the covariant derivative will contain extra terms including derivatives of c. These terms don’t cancel out, so there is no way to not violate the relation.

No, we have to take this locally. It’s Schwarzschild coordinate time, so this is what a far-away stationary clock measures locally in its own frame of reference. It is not what physically happens anywhere else. In GR, time is always a purely local concept. Again, this is not possible. Time is a purely local concept in GR; there is, in general, no notion of simultaneity across extended regions of curved spacetime, and you can’t map notions of space and time local to some far-away observer into anything that happens anywhere else. In particular not to test particles in free fall, which aren’t stationary. You can define static hypersurfaces of simultaneity based on the coordinate system you have chosen (in Schwarzschild, these will be nested spheres), but that is not the same thing. It talks about the conventional model for this - that means you use Schwarzschild spacetime, allow the mass terms to vary, and see what happens. This is meant as an approximation and a teaching tool, because the maths are easy to do on paper, unlike is the case for a full description. But as I have attempted to point out several times now, in actual fact any kind of black hole that isn’t stationary can’t physically be Schwarzschild, so it is little surprise that the conventional model does not actually work too well. That was kind of the point of Hawking’s original paper (I recommend you read it) - he started with a conventional Schwarzschild black hole, and examined if and how it is compatible with quantum field theory; and unsurprisingly he found that it isn’t, so Schwarzschild black holes cannot actually occur in nature. At the very least, they have to be generalised to their radiating counterparts, the aforementioned Vaidya black holes. Yes we do. We can use quantum field theory in conjunction with the Vaidya solution to model Hawking radiation in a non-stationary spacetime. This has been done by several authors, e.g. here. However, the result of this is a geometry that is vastly more complex than Schwarzschild, and contains several different types of surfaces - event horizons, apparent horizons, Killing horizons, and trapped surfaces. In particular, in can be explicitly shown that a far-away observer will receive information about the black hole’s state in finite coordinate time, unlike would be the case in Schwarzschild.

So does the idea that c is a variable. Consider the local conservation of the energy-momentum tensor in the presence of gravity: \[T{^{\mu}}{_{\nu ||\mu}}=0\] Since the covariant derivative depends on the metric, which explicitly contains c, and because in your idea c varies in a way that is not covariant, the above relationship ends up being no longer valid. This whole idea puts you in a situation where there is no longer any conservation of energy-momentum, not even locally. This is clearly in direct contradiction to experiment and observation.

Just to add to what has already been said by other contributors here: 1. First and foremost, the notion of "gravitational potential" can only be defined in spacetimes that are stationary (more precisely: those which admit a time-like Killing vector field) and asymptotically flat. It cannot be generalised to more general spacetimes, which makes it useless so far as a general model for gravity is concerned 2. Gravitational potential itself is not an observable, only differences in potential can be observed and measured. This is because the potential has a gauge freedom, in that one can freely choose where the zero point is, without affecting the physics. The same is not true for the speed of light, hence the relation above is trivially and obviously wrong, since it equates two quantities that cannot physically and numerically be equal, on fundamental grounds. 3. A varying speed of light would constitute a violation of Lorentz invariance. This symmetry has been experimentally and extensively tested with modern equipment to extremely high precision, both here on Earth and in the vacuum of space - needless to say, no such violations have ever been found. Given the degree of precision of these tests, any variability in the speed of light can effectively be ruled out far beyond the usual 5 sigma threshold. 4. A variable speed of light would also break CPT symmetry, which underlies the Standard Model of Particle Physics. Since we continue to successfully use and test this model in particle accelerators on pretty much a daily basis, any variability in c can also effectively be ruled out on that ground. 5. Neither classical Maxwellian electrodynamics nor quantum electrodynamics allow for varying values of permittivity and permeability (in the same medium of course). Hence the notion of a varying speed of light is actually in direct contradiction to what we know about electrodynamics. 6. As has been pointed out on another recent thread, a scalar field theory such as this one is fundamentally incapable of capturing all required degrees of freedom of gravity; there is more to gravity than just time dilation! I could probably go on, but these are the points that immediately come to mind without thinking about the issue too much. I'll leave it at this.

That is a really good question, because I still don't know what your position actually is. You seem to be saying that the concept of an evaporating Schwarzschild black hole is self-contradictory - which is trivially true, because Schwarzschild black holes are of course stationary by definition. But evaporating black holes aren't of the Schwarzschild kind, so where exactly is the issue? Then so you also seem to be saying that even ordinary Schwarzschild black holes are self-contradictory, because somehow event horizons can't exist? But I don't understand why you think this, because the reasons you give don't make any sense.

You cannot separate time and space, they always exist together as spacetime, regardless of whether the region in question is a vacuum or not.

To be honest, this isn't how it comes across. You seem to pretty much ignore most of what is said to you, so I don't know what the purpose of this thread is actually supposed to be.

Yes, in classical and conventional GR the singularity is indeed unavoidable. But since the spacetime region containing the final stages of the collapse is outside the domain of applicability of GR, I do not consider this to be an issue. The appearance of a singularity simply means that the domain of applicability of GR is limited - just like is the case with any other model in physics. Prior to the event horizon’s creation, there is no black hole, and you can use the maximally extended Schwarzschild solution to cover both the interior and exterior of the spherically symmetric mass. Once the collapse process passes a critical stage, the horizon appears at some finite distance, and the Schwarzschild metric no longer applies since the spacetime in question is now no longer Schwarzschild. There is no contradiction, because once you have evaporation, the Spacetime is by definition not Schwarzschild. You can’t “look at the Schwarzschild solution” and talk about evaporation at the same time. This is what I have attempted to explain in my previous comments. Yes indeed. At the same time though there is no evidence to definitively rule it out, so it is considered a viable candidate for further research and consideration. Because of the presence of torsion in the interior of the collapsing mass-energy distribution, the geometry of spacetime is different than it would be for the equivalent scenario under GR. Einstein-Cartan just simply does not lead to a run-away collapse resulting in a geodesically incomplete region; it leads to a situation where beyond some critical point the geometry of spacetime is such that - put simply and I’m sure not very correctly - in terms of time evolution, the radial coordinate r trades places with its inverse 1/r. This is analogous to how space and time in some sense trade places below the horizon of a Schwarzschild black hole. The collapsing matter thus rebounds and thus expands back out as it ages into the future, rather than ending up in a singularity. Due to extreme gravitational time dilation in that region, this process would take a very long time, exceeding the total lifetime of the universe by some orders of magnitude, which is why we don’t see this happening all around us. Also, there would be extreme tidal forces in that collapse region, tearing apart everything, so whatever comes back out would be little more than uniform radiation. Note that there are no stationary black holes in Einstein-Cartan gravity, it is always a time-dependent process. At least this is my understanding of the situation, I am not actually much of an expert on Einstein-Cartan gravity.

This is all true. In addition, and perhaps more critically, all these solutions rely on the assumption of asymptotic flatness, i.e. that the respective black hole exists in an otherwise completely empty universe. This is obviously not very physical. Nonetheless, these are all useful approximations to simplify the math for special cases. The issue of singularities is easily resolved by a simple modification of GR, without the need to involve any quantum physics at all. One must simply remove the requirement that the connection be torsion-free, which immediately yields Einstein-Cartan gravity. This model makes the same predictions as conventional GR for vacuum spacetimes, but it now does include a coupling between torsion and intrinsic angular momentum for the interior of mass-energy distributions. It can be shown that gravitational collapse under this model no longer leads to regions of geodesic incompleteness (i.e. there are no singularities); also, the cosmological singularity at the BB no longer occurs. Note that just like GR, this is a purely classical model.

Unfortunately I don’t, sorry. This is not a metric that is often discussed in textbooks - actually, come to think of it, I have never seen it mentioned in any textbook used for teaching. I can’t remember where I came across it first, I think it might have been in Exact Solutions to Einstein’s Field Equations by Stephanie/Kramer. And I just noticed that I misspelled it, this should be Aichelburg-Sexl Ultraboost. No, it’s the latest 17-in-1 vaccine being pushed on the Austrian populace, for the purpose of easier mind control via 5G towers, by their Nazi overlords (who are really Lizard People) from the hollow interior of flat earth. Yes indeed, in Minkowski spacetime they would need to be subject to the SR transformation rules. The comparison was between an observer who is at rest relative to the black hole, and one who is in relative motion at relativistic speeds. The comparison via SR rules is very tricky here, because Vaidya spacetime is not a vacuum - it is uniformly filled with null dust (i.e. radiation), so the spacetime is not asymptotically flat. However, the discrepancy may (!) be small enough to neglect if both observers are very far from the black hole, depending on what degree of accuracy is demanded. I haven’t run the numbers though, hence I may well be wrong on this. Technically speaking though, in Vaidya spacetime there are no regions that can be considered Minkowski. An external observer would not see (as in - visually witness) the infalling object reach the horizon, I think it would fade out and slow down just like in Schwarzschild spacetime, albeit at a different rate. What I am not sure about though is what the external observer would calculate in terms of coordinate infall-time on his own local clock, since Vaidya coordinate time is not the same as Schwarzschild coordinate time. I just tried a quick calculation for this, but it turns out that due to the presence of an off-diagonal term in the metric, this is actually very difficult. I will need to come back to this.

My earlier comments related specifically to Schwarzschild black holes - which are stationary, hence they don’t evaporate. Generalising this to the case of an evaporating black hole is not straightforward or trivial. Such black holes are called Vaidya black holes, and this represents an entirely different solution to the field equations, and thus a spacetime with a different geometry. While Schwarzschild is a vacuum solution, Vaidya is not, because now all of spacetime is filled with radiation, so we are no longer in a vacuum. Adding relative motion between observer and a Vaidya black hole does of course not yield the aforementioned Aichlburg-Sexl ultraboost, but some different metric, which I have not encountered in my studies. Like in the Schwarzschild case, that new metric would be related to the Vaidya metric by some coordinate transformation, but I suspect the transformation will be a lot more complex than in Schwarzschild spacetime, mostly due to the presence of off-diagonal terms in the metric tensor. I’m sure it can be done though, and probably has been, though a quick search does not immediately turn up anything. But the answer to your question will be “no” regardless, because in both cases we are in a curved spacetime, so you can’t naively use the transformation rules of SR. The total time dilation here has a kinetic component from relative motion, and a gravitational component from spacetime curvature. So the lifetime a specific observer calculates for the black hole will be subject not only to his relative speed, but also to the particulars of the geodesic he traces out (i.e. to the initial and boundary conditions of his motion), and where on this geodesic he is when he performs the calculation; so it will be some (probably quite complicated) function of his trajectory and the surface area of the horizon (which will itself be some complicated function, since the horizon is no longer spherical for such an observer). This is a common misunderstanding based on the fundamental error of using the wrong solution to the field equations for the scenario at hand. I know that countless papers have been published showing this calculation, but ultimately these results are not physically meaningful. If you naively go and consider free-fall observers in Schwarzschild spacetime, then yes, the maths will show you that they can’t detect Hawking radiation. However, evaporating black holes are by definition not Schwarzschild, so this point is moot. When done correctly using Vaidya spacetime, the free fall observer will definitely detect radiation (it is a non-vacuum spacetime after all!), just at a different temperature relative to other observers. The type of physical spacetime one is in is characterised by curvature invariants, which is something all observers always agree on, even if they use different coordinate charts to map that spacetime. Like I said, one must use the correct spacetime to model this scenario. How in-fall time is related to evaporation time of a black hole in Vaidya spacetime is a question I can’t answer off hand. It would likely depend on the initial mass of the black hole, its age (as calculated by the in-falling observer), and the in-fall geodesic. For solar-mass (at the time of in-fall) black holes though the lifetime of the black hole will be much longer than the length of most in-fall geodesics, by many orders of magnitude.