muruep00 Posted August 16, 2020 Posted August 16, 2020 I might go straight to the proposal. Consider that the interior of black holes (BH)s only contain negative energy densities, also known as negative masses. These negative masses have been predicted to be repulsive by other famous physicists such as H. Bondi and W. B. Bonnor, but have not been taken seriously due to a paradox called "runaway motion" which takes place between masses of different signs. These negative masses would be created from infalling particles to the BH by a time transformation taking place at the event horizon, which only switches to antigravitational interactions, without violating Einstein's equivalence principle. The original Schwarzschild solution for 1912 is singular at the event horizon, because Schwarzschild did not want to change the signature of the metric when inside the solution, so he chose a radial variable in his metric which is a non-linear auxiliary quantity dependent of the black hole radius and the radial coordinate. This singularity might just be a time transformation, which is not allowed in relativity since the antichronous transformations are arbitrarely prohibited already in special relativity. This is supported by the time transformation of the Feynman Stueckelberg interpretation in relativistic particle physics, and if considered unitary, it implies negative masses (this has been proved by other researchers). The runaway motion would be solved since no interaction is allowed from inside to outside, and all particles crossing the event horizon would suffer this transformation. This violation of the energy conditions clearly solves the central singularity, since no gravitational singularity can arise only from repulsive matter. This solution not only solves the gravitational singularity, the strange theoretical interior of Kerr BHs and the runaway motion, but also it's trivial that the interior of the negative mass BH undergoes an inflation, which is slowed down by time dilation for external observers. This inflation would be noticed by exterior observers, which would perceive BHs as having a greater apparent mass than estimated due to accretion and merging, being this solution another mechanism of growth to be taken into account in studies of supermassive BH formation, which remain a mystery. I can develop it further, but the basic idea is that. What do you think? This proposal has never been published, only negative mass black holes forming from matter which already violates energy conditions outside the black hole (which contradicts observations).
Markus Hanke Posted August 16, 2020 Posted August 16, 2020 22 minutes ago, muruep00 said: The original Schwarzschild solution for 1912 is singular at the event horizon The singularity at the horizon is purely a coordinate singularity, but not a physical one. Spacetime is smooth and regular at the horizon, which you can easily see either by calculating the curvature invariants of the Riemann tensor (they all stay finite and well defined there), or by performing a simple coordinate transformation. This stands in contrast to r=0, where all such invariants diverge, so that one is a physical curvature singularity. Also, Schwarzschild spacetime is a vacuum spacetime, so there is no mass anywhere. The 'M' value in the metric is just a parameter for a 1-parameter family of metrics.
muruep00 Posted August 16, 2020 Author Posted August 16, 2020 (edited) 1 hour ago, Markus Hanke said: The singularity at the horizon is purely a coordinate singularity, but not a physical one. Spacetime is smooth and regular at the horizon, which you can easily see either by calculating the curvature invariants of the Riemann tensor (they all stay finite and well defined there), or by performing a simple coordinate transformation. This stands in contrast to r=0, where all such invariants diverge, so that one is a physical curvature singularity. Also, Schwarzschild spacetime is a vacuum spacetime, so there is no mass anywhere. The 'M' value in the metric is just a parameter for a 1-parameter family of metrics. Thank you for your quick answer. This a loong discussion. But first, I was referring to the original Schwarzschild metric, here you have it: http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Schwarzschild.pdf As you see, this is not the metric we use nowadays (we use the Hilberts metric). In the Schwarzschild metric, the interior is not described by the metric, because coordinate r=0 is the event horizon, and there the metric ends in a singularity. As I said, Schwarzschild did this trying to avoid the change of the signature of the metric, which leads to time and space switching roles and so on. So he invented an auxiliary quantity, called R (page 195), which he uses in the metric. He defined the auxiliary quantity R to be the cube root of the r coordinate cubed plus the schwarzschild radius cubed. As you see, this coordinate transformation he did, was not linear. Now, Im not an expert in this, but I've read that a metric and a non-linear coordinate transformation metric are not the same, and have different geodesics. I am not sure if both the original's 1912 Schwarzschild metric and the one we all study which was done by Hilbert are the same. Hilbert's metric can be checked here: https://www.jp-petit.org/Hilbert-1916-de.pdf in equation (45) and to my understanding, he defines time as an imaginary number (page 70), (isnt that a wick rotation?). Schwarzschild treated time as real coordinate, not an imaginary one. But even if Im wrong and you are right, we don´t know whats inside, so you can actually propose than x thing happens at the event horizon and it does no contradict observations. Proposals of "something happens at the event horizon" are usually not considered serious, just because they violate Einsteins equivalence principle. But if the "change" that you consider only implies gravitational effects, the principle holds. Of course, the Schwarzschild spacetime is vacuum, but we are talking about real black holes. Edited August 16, 2020 by muruep00
Markus Hanke Posted August 17, 2020 Posted August 17, 2020 On 8/16/2020 at 1:33 PM, muruep00 said: As you see, this coordinate transformation he did, was not linear. He didn't do any coordinate transformations, he simply chose a coordinate system in which the determinant of the metric tensor was unity, as an ansatz to solve the field equations. It's essentially just a modified version of polar coordinates, with r=0 being the event horizon. It's a legitimate - but unnecessarily awkward - coordinate choice. On 8/16/2020 at 1:33 PM, muruep00 said: Now, Im not an expert in this, but I've read that a metric and a non-linear coordinate transformation metric are not the same, and have different geodesics. I don't know what you mean by this. The geodesic structure of the spacetime is an invariant, and quite independent of the choice of coordinate systems. On 8/16/2020 at 1:33 PM, muruep00 said: I am not sure if both the original's 1912 Schwarzschild metric and the one we all study which was done by Hilbert are the same. They are the same metrics in the sense that they describe the same physical spacetime. The original is from 1916 though, not 1912. On 8/16/2020 at 1:33 PM, muruep00 said: But even if Im wrong and you are right You haven't actually made any assertions, so I am not saying you are wrong on anything. I only wish to point out that a coordinate singularity does not necessarily imply a physical curvature singularity, simply because the choice of coordinate chart is arbitrary. Coordinate values do not carry any physical meaning, only the events they refer to do. I assume this is what you are getting at, as it is a very common question. The easiest (but not the only) way to tell whether there's a curvature singularity or not is to calculate the curvature invariants of the Riemann tensor. The thing is that Schwarzschild did not know this when he found his solution - GR was a brand new theory back then, and not yet well understood. Hence, there was a lot of debate about the status of the event horizon then. Nowadays though, after studying the model for well over a century, we understand the situation much better, both in terms of physics, and in terms of the maths. On 8/16/2020 at 1:33 PM, muruep00 said: Proposals of "something happens at the event horizon" are usually not considered serious, just because they violate Einsteins equivalence principle. I don't know why you would think that? Spacetime at the horizon is smooth and regular, so the equivalence principle holds there just as it holds everywhere else. On 8/16/2020 at 1:33 PM, muruep00 said: Of course, the Schwarzschild spacetime is vacuum, but we are talking about real black holes. Well, the Schwarzschild solution works well as a rough approximation, and it is a useful teaching tool. However, the boundary conditions used to find this solution aren't physical. In order to have a more accurate model of real-world black holes, you need to look at other (more complicated) solutions to the field equations - such as Vaidya spacetime for example -, or use numerical GR.
muruep00 Posted August 17, 2020 Author Posted August 17, 2020 (edited) 22 minutes ago, Markus Hanke said: He didn't do any coordinate transformations, he simply chose a coordinate system in which the determinant of the metric tensor was unity, as an ansatz to solve the field equations. It's essentially just a modified version of polar coordinates, with r=0 being the event horizon. It's a legitimate - but unnecessarily awkward - coordinate choice. I don't know what you mean by this. The geodesic structure of the spacetime is an invariant, and quite independent of the choice of coordinate systems. They are the same metrics in the sense that they describe the same physical spacetime. The original is from 1916 though, not 1912. You haven't actually made any assertions, so I am not saying you are wrong on anything. I only wish to point out that a coordinate singularity does not necessarily imply a physical curvature singularity, simply because the choice of coordinate chart is arbitrary. Coordinate values do not carry any physical meaning, only the events they refer to do. I assume this is what you are getting at, as it is a very common question. The easiest (but not the only) way to tell whether there's a curvature singularity or not is to calculate the curvature invariants of the Riemann tensor. The thing is that Schwarzschild did not know this when he found his solution - GR was a brand new theory back then, and not yet well understood. Hence, there was a lot of debate about the status of the event horizon then. Nowadays though, after studying the model for well over a century, we understand the situation much better, both in terms of physics, and in terms of the maths. I don't know why you would think that? Spacetime at the horizon is smooth and regular, so the equivalence principle holds there just as it holds everywhere else. Well, the Schwarzschild solution works well as a rough approximation, and it is a useful teaching tool. However, the boundary conditions used to find this solution aren't physical. In order to have a more accurate model of real-world black holes, you need to look at other (more complicated) solutions to the field equations - such as Vaidya spacetime for example -, or use numerical GR. Hello, Markus, thanks again for your response. Yes, you are right he did not make any coordinate transformations. My bad. The coordinate transformation done in the Schwarzschild metric is to show that the event horizon is a coordinate singularity. But this transformation is always done to the Hilbert's metric, not the one Schwarzschild published (correctly as you said, in 1912). I dont know if the Schwarzschild metric and Hilberts metric are the same (might want to check this: https://www.researchgate.net/publication/331936281_Schwarzschild's_family ), as I have not applied a curvature invariants of the Riemann tensor to the original Schwarzschild metric. Im also very skeptical about the interior solution of Schwarzschild metric, since you are building it starting from the assumption that there is actually a central point-like mass. But what if that singularity does not exist in real black holes? The exterior solution might be correct anyways for black holes, but the interior, including what happens at the event horizon, can be something different. My proposal is very clear: there is a gravitational change happening at the event horizon, which does not violate Einstein's equivalence principle. This is the only way to violate energy conditions only at the interior of black holes ir order to prevent the formation of the central gravitational singularity and at the same time be consistent with observations (since we have never observe macroscopic violation of energy conditions in the spacetime we observe, that is, outside black holes). This can only be justified by a time transformation taking place at the event horizon, but GR (neither does SR) does not allow this by arbitrarely decisions of "time transformations are unphysical", done of course long before black holes were studied. What are you thoughts about this idea? Its the only way I found to 1. Be consistent with observations, 2. Include negative masses but solve the runaway paradox, 3. Solve the gravitational singularity and the crazy interior of Kerr BH (wormholes...etc), withour violating Einstein's equivalence principle, 5. Propose a mechanism of BH growth that may solve SMHB formation Edited August 17, 2020 by muruep00
Markus Hanke Posted August 17, 2020 Posted August 17, 2020 5 minutes ago, muruep00 said: (correctly as you said, in 1912) It's 1916 - Einstein didn't find his field equations until 1915. 9 minutes ago, muruep00 said: I dont know if the Schwarzschild metric and Hilberts metric are the same They describe the same physical spacetime, but Schwarzschild's original coordinate chart covers only the exterior region of that spacetime, so of course it does not contain an event horizon. That's not because such a horizon doesn't exist, but because the coordinates used simply don't cover it. 16 minutes ago, muruep00 said: Im also very skeptical about the interior solution of Schwarzschild metric, since you are building it starting from the assumption that there is actually a central point-like mass No, it's the exact opposite - the boundary condition used to derive a maximally extended Schwarzschild solution (which covers the interior region as well) is asymptotic flatness, i.e. it is assumed that far enough away from the black hole gravity is described by Newton's laws. When you solve the field equations, you are then left with a single free parameter in the metric - which, via the asymptotic flatness condition, can be interpreted as mass. It is very important to understand here that unlike in Newtonian physics, the mass in Schwarzschild spacetime is not localisable, meaning it is a property of the entire spacetime. 21 minutes ago, muruep00 said: But what if that singularity does not exist in real black holes? Einstein's GR is a purely classical theory, it does not account for any quantum effects, which during a real-world gravitational collapse cannot be neglected. Hence, in real world black holes there almost certainly are no singularities, we know this already. The question is then, what happens in the central region, if not a singularity? To answer this we need a model of quantum gravity, which we currently don't have yet. To make a long story short, GR is best understood as a classical approximation to a more complete theory of quantum gravity. 24 minutes ago, muruep00 said: This is the only way to violate energy conditions only at the interior of black holes ir order to prevent the formation of the central gravitational singularity Actually, there are many ways to avoid the formation of the central singularity, even in classical physics. For example, one can allow spacetime to have intrinsic torsion - that gives a model called Einstein-Cartan gravity, and no singularities occur there. But such models have other issues that are potentially problematic. 27 minutes ago, muruep00 said: What are you thoughts about this idea? Mathematically speaking it is quite clear that nothing special actually happens at the event horizon - you can see this easily by simply choosing a different coordinate chart for your Schwarzschild spacetime, such as Kruskal-Szekeres coordinates for example. The event horizon isn't really the problem (things would be far more problematic if it wasn't there). The problem is that GR as a theory is purely classical, so its description of the actual collapse process that gives rise to the black hole in the first place is evidently wrong or incomplete, since it can't account for quantum effects. It's simply outside its domain of applicability.
muruep00 Posted August 17, 2020 Author Posted August 17, 2020 4 hours ago, Markus Hanke said: It's 1916 - Einstein didn't find his field equations until 1915. They describe the same physical spacetime, but Schwarzschild's original coordinate chart covers only the exterior region of that spacetime, so of course it does not contain an event horizon. That's not because such a horizon doesn't exist, but because the coordinates used simply don't cover it. No, it's the exact opposite - the boundary condition used to derive a maximally extended Schwarzschild solution (which covers the interior region as well) is asymptotic flatness, i.e. it is assumed that far enough away from the black hole gravity is described by Newton's laws. When you solve the field equations, you are then left with a single free parameter in the metric - which, via the asymptotic flatness condition, can be interpreted as mass. It is very important to understand here that unlike in Newtonian physics, the mass in Schwarzschild spacetime is not localisable, meaning it is a property of the entire spacetime. Einstein's GR is a purely classical theory, it does not account for any quantum effects, which during a real-world gravitational collapse cannot be neglected. Hence, in real world black holes there almost certainly are no singularities, we know this already. The question is then, what happens in the central region, if not a singularity? To answer this we need a model of quantum gravity, which we currently don't have yet. To make a long story short, GR is best understood as a classical approximation to a more complete theory of quantum gravity. Actually, there are many ways to avoid the formation of the central singularity, even in classical physics. For example, one can allow spacetime to have intrinsic torsion - that gives a model called Einstein-Cartan gravity, and no singularities occur there. But such models have other issues that are potentially problematic. Mathematically speaking it is quite clear that nothing special actually happens at the event horizon - you can see this easily by simply choosing a different coordinate chart for your Schwarzschild spacetime, such as Kruskal-Szekeres coordinates for example. The event horizon isn't really the problem (things would be far more problematic if it wasn't there). The problem is that GR as a theory is purely classical, so its description of the actual collapse process that gives rise to the black hole in the first place is evidently wrong or incomplete, since it can't account for quantum effects. It's simply outside its domain of applicability. Thank you for response Markus. Yes, 1916, my bad again... Wouldnt it be better if we use Schwarzschild metric? I mean, we dont know what lies inside black holes. Perhaps there's a singularity solved by quantum gravity, or perhaps its something different... Let me repeat again, the Schwarzschild solution might be valid at the exterior, but in the interior, there could be something different than what you are forcing the Schwarzschild metric to have, that is, a point-like center of mass. What if you dont need quantum gravity to explain the inside of black holes because in reality (independent of what math suggests), something happens at the event horizon and the whole interior solution is different than the Schwarzschild interior solution, not only the singularity. I know about Einstein-Cartan theory, but what Im proposing does not have any issues since the runaway motion paradox is solved and Einsteins equivalence principle holds. Why no body has thought about this idea before? Because you cant be 100% sure that the interior solution only differs from GR to quantum gravity at the singularity, it could be very different entirely! As I said, you are forcing the interior solution to be one way in the math. You can glue the exterior solution to other different interior solution (for instance, an Einstein-Rosen bridge), and now the mathematically speaking, math suggests other different thing (still smooth at the horizon, but the interior is different)... Why not develop a non-singular interior solution and glue it to the exterior solution? The easiest way to do that is by violating energy conditions at all the interior with a perfect fluid of negative energy density, which is my proposal. Having clarified that I know what math suggests (nothing happens at the event horizon, and GR holds until being close the "singularity"), but it can perfectly be other way because it has not been proved by observation, why no body has tried these non-singular interior solutions in a reasonable way (that is, not allowing negative energy densities to exist outside black holes before collapse, which is what most "negative mass black holes" research has been about, check R. Mann)? What if one of these proposals could be verifiable by inconsistent observations such as huge supermassive black holes that no one knows how they formed? What are your thoughts about my proposal?
Markus Hanke Posted August 18, 2020 Posted August 18, 2020 6 hours ago, muruep00 said: Wouldnt it be better if we use Schwarzschild metric? I mean, we dont know what lies inside black holes. Perhaps there's a singularity solved by quantum gravity, or perhaps its something different... You mean the original version? You can of course do that, but personally I don't really see the advantage. But in any case, for the exterior region it makes the same physical predictions. 6 hours ago, muruep00 said: Let me repeat again, the Schwarzschild solution might be valid at the exterior, but in the interior, there could be something different than what you are forcing the Schwarzschild metric to have, that is, a point-like center of mass. As I said, we already know that a point singularity is in all likelihood not what happens in the real world. And even for the exterior region, the Schwarzschild solution is only an approximation. We are not 'forcing' the singularity, it is simply what happens during the collapse process in a purely classical model. Even if we take into account quantum effects as far as we can model them, once you go beyond a certain total mass, there is nothing there that can stop a complete collapse. 6 hours ago, muruep00 said: What if you dont need quantum gravity to explain the inside of black holes because in reality (independent of what math suggests), something happens at the event horizon and the whole interior solution is different than the Schwarzschild interior solution, not only the singularity. 'Independent of what math' suggests is not a very scientific approach. We already know that the event horizon cannot be a curvature singularity (both in terms of theory, and in terms of observational evidence), so it is quite reasonable to work off the assumption that GR gives correct predictions for at least part of the interior region. 6 hours ago, muruep00 said: Why no body has thought about this idea before? Because it's meaningless. 6 hours ago, muruep00 said: As I said, you are forcing the interior solution to be one way in the math No we aren't. The boundary conditions used to derive the solution to the field equations make no reference to singularities, horizons, or interior regions at all. 6 hours ago, muruep00 said: You can glue the exterior solution to other different interior solution Yes, you can 'glue' different regions together, but with two caveats: 1. The overall metric must remain continuous and differentiable everywhere at the boundary 2. The new metric must itself be a valid solution to the field equations In practice, because the field equations are non-linear, this places very stringent constraints on what geometries the regions can have. You have very little freedom in choosing the metrics. In particular, if the exterior region is Schwarzschild, and the interior region beyond the horizon is a vacuum, then the interior geometry must be Schwarzschild as well. There is no other mathematical possibility. 6 hours ago, muruep00 said: (for instance, an Einstein-Rosen bridge) An Einstein-Rosen bridge is already a feature of the maximally extended Schwarzschild metric, it isn't a different interior metric. You just need to pick a suitable coordinate chart that covers the entire spacetime before you can see it in an embedding diagram. 6 hours ago, muruep00 said: Why not develop a non-singular interior solution and glue it to the exterior solution? Can you write down a suggested metric? 6 hours ago, muruep00 said: The easiest way to do that is by violating energy conditions at all the interior with a perfect fluid of negative energy density, which is my proposal. This essentially would mean that the interior region has a non-zero cosmological constant. You can do this of course, but then the exterior region will not have Schwarzschild geometry. 7 hours ago, muruep00 said: but it can perfectly be other way because it has not been proved by observation We have direct observational evidence for how light and massive test particles behave close to the horizon; we also have direct observational evidence of the gravitational wave signature of black hole mergers. Both of these place very stringent constraints on the structure of spacetime near the horizon, and both of these are in perfect accord with what GR predicts in those circumstances. 7 hours ago, muruep00 said: What are your thoughts about my proposal? My thoughts are that you cannot glue an exterior Schwarzschild vacuum to an interior vacuum with negative cosmological constant (which would have the effect of a negative energy density) in any self-consistent way.
muruep00 Posted August 18, 2020 Author Posted August 18, 2020 12 hours ago, Markus Hanke said: We are not 'forcing' the singularity, it is simply what happens during the collapse process in a purely classical model. Even if we take into account quantum effects as far as we can model them, once you go beyond a certain total mass, there is nothing there that can stop a complete collapse. You are assuming that the collapse happens in one particular way. Maybe energy conditions are violated (equivalent to gravity turning to be antigravity) and collapse at the interior solution does not take place, and that is not a quantum effect since it can be modeled in GR. 12 hours ago, Markus Hanke said: 'Independent of what math' suggests is not a very scientific approach. We already know that the event horizon cannot be a curvature singularity (both in terms of theory, and in terms of observational evidence), so it is quite reasonable to work off the assumption that GR gives correct predictions for at least part of the interior region. What is not a scientific approach is to asume that something happens at the interior of black holes because math suggests it, when we have zero experimental proofs of it. Specially when the math you have (GR) gives you a singularity... No measurement has been done inside, period. 12 hours ago, Markus Hanke said: Can you write down a suggested metric? No I cant, but its easy to explain. Its just a spherical space-time with a negative energy density fluid. That glues with the exterior Schwarzschild solution, violates energy conditions, does not violate Einsteins equivalence principle, and its non-singular. Yes it would be similar to a non-zero cosmological constant, although this fuild its a simplification of the mass of the star within the interior solution and not an energy density of the vacuum. 12 hours ago, Markus Hanke said: This essentially would mean that the interior region has a non-zero cosmological constant. You can do this of course, but then the exterior region will not have Schwarzschild geometry. Why not? If this negative-energy density fluid appears right at the same time as the event horizon forms, then the exterior solution is unaltered, since no information (not even gravitational waves) can go inside->outside. 12 hours ago, Markus Hanke said: We have direct observational evidence for how light and massive test particles behave close to the horizon; we also have direct observational evidence of the gravitational wave signature of black hole mergers. Both of these place very stringent constraints on the structure of spacetime near the horizon, and both of these are in perfect accord with what GR predicts in those circumstances. Yes, but that is not the horizon. That only "suggests" the horizon also behaves as our math states, but does not prove the horizon is one way or another. It could be otherwise. 12 hours ago, Markus Hanke said: My thoughts are that you cannot glue an exterior Schwarzschild vacuum to an interior vacuum with negative cosmological constant (which would have the effect of a negative energy density) in any self-consistent way. Why?
Markus Hanke Posted August 19, 2020 Posted August 19, 2020 22 hours ago, muruep00 said: You are assuming that the collapse happens in one particular way. No, we don’t make any assumptions for the collapse process beforehand. What you do is start by writing down the energy-momentum tensor for the interior of the collapsing body (a star, usually), and then impose the necessary condition that the metric is to remain continuous and differentiable at the boundary (the star’s surface). You then insert all of this into the field equations, and work out the solution. There are no assumptions about the collapse process itself. 22 hours ago, muruep00 said: What is not a scientific approach is to asume that something happens at the interior of black holes because math suggests it We already know that GR - which is a mathematical model - works really well in the classical domain, so there is no scientific reason to assume that it won’t work for a collapsing body, up to the point when quantum effects are no longer negligible. The appearance of the singularity means just that - that GR no longer applies in that region, since that stage of the collapse isn’t classical. But it is perfectly reasonable to take GR at its word in that at least for some distance beyond the horizon spacetime remains smooth and regular. The question is really only how far that region extends. On the other hand, suggesting that GR’s modelling of the beginning stages of the collapse is somehow wrong (bearing in mind the aforementioned caveat), and that regular matter is replaced with exotic matter in the process, is, to me, not especially reasonable. 22 hours ago, muruep00 said: No I cant Then how do you know what the geometry of spacetime in that region will be? 22 hours ago, muruep00 said: Its just a spherical space-time with a negative energy density fluid. What is the nature of that exotic matter fluid, how does it arise from the (quite regular) matter of the original collapsing star, and what is its energy-momentum tensor? What holds it in place at the precise radius of the event horizon, since it would be gravitationally self-repulsive? 22 hours ago, muruep00 said: That glues with the exterior Schwarzschild solution, violates energy conditions, does not violate Einsteins equivalence principle, and its non-singular How do you know any of these things, if you don’t have a metric or at least an energy-momentum tensor? We have never observed exotic matter, after all. Gravity is a highly non-linear thing, so it is dangerous to make any kind of assumption about how something will behave, without working through the maths first - remember that energy density is not the only component of the energy-momentum tensor, so it isn’t the only source of gravity here. For exotic matter, the influence of the stress components in the tensor would actually be larger than that of energy-density. 22 hours ago, muruep00 said: Why not? Because you cannot join a region with vanishing cosmological constant to a region with non-vanishing constant, without violating the continuity boundary conditions. There also isn’t any known mechanism for that constant to change from one spatial region to another. 22 hours ago, muruep00 said: If this negative-energy density fluid appears right at the same time as the event horizon forms How does it just “appear”? Where does it come from, and by which physical mechanism does it form? What type of particle is it made of, and how does that fit in with the Standard Model? 22 hours ago, muruep00 said: Yes, but that is not the horizon. That only "suggests" the horizon also behaves as our math states, but does not prove the horizon is one way or another. It could be otherwise. If the interior region wasn’t empty space (as you appear to suggest), then the gravitational wave signature of these mergers would be quite different. In fact, the signature from merging event horizons as modelled by GR is pretty unique, so no, it couldn’t really be otherwise in any significant way. And if the interior was an exotic fluid with negative energy density, then the signature would be very different - in fact, there would be no merger, since two such black holes would be gravitationally repulsive towards one another. 22 hours ago, muruep00 said: Why? I’m somewhat confused now what exactly you really suggest - an empty interior with negative cosmological constant? Or an interior filled with a gravitationally repulsive exotic substance? For the former, you can’t join these regions without violating basic boundary conditions. For the latter, the exterior vacuum cannot be Schwarzschild, since it would have to have a very different geodesic structure - positive mass test particles would actually fall away from the event horizon.
muruep00 Posted August 19, 2020 Author Posted August 19, 2020 (edited) 50 minutes ago, Markus Hanke said: No, we don’t make any assumptions for the collapse process beforehand. What you do is start by writing down the energy-momentum tensor for the interior of the collapsing body (a star, usually), and then impose the necessary condition that the metric is to remain continuous and differentiable at the boundary (the star’s surface). You then insert all of this into the field equations, and work out the solution. There are no assumptions about the collapse process itself. Yes, you assume energy conditions hold! Quote We already know that GR - which is a mathematical model - works really well in the classical domain, so there is no scientific reason to assume that it won’t work for a collapsing body, up to the point when quantum effects are no longer negligible. We know that GR works really well in the classical domain because we compare observations to its predictions. Wherever you dont have observations (inside black holes), you begin assuming that it works. Since particulary that inside is a non-observable region in which GR fails, the scientific approach is assuming nothing! Quote The appearance of the singularity means just that - that GR no longer applies in that region, since that stage of the collapse isn’t classical. But it is perfectly reasonable to take GR at its word in that at least for some distance beyond the horizon spacetime remains smooth and regular. The question is really only how far that region extends. On the other hand, suggesting that GR’s modelling of the beginning stages of the collapse is somehow wrong (bearing in mind the aforementioned caveat), and that regular matter is replaced with exotic matter in the process, is, to me, not especially reasonable. You may assume that, and might be reasonable. But since we have no theory of quantum gravity yet, perhaps we should study other possibilities by not assuming that much, with the tools we have! (GR). Exotic matter is reasonable because it is trivial that it solves the gravitational singularity. In that way, GR would be enough to describe black holes. As you have said, this is a suggestion (hypothesis), not an assumption. Quote What is the nature of that exotic matter fluid, how does it arise from the (quite regular) matter of the original collapsing star, and what is its energy-momentum tensor? What holds it in place at the precise radius of the event horizon, since it would be gravitationally self-repulsive? This is explained in my first post. Regular matter would be transformed into exotic matter by a time transformation at the event horizon. You can prove that if time transformations are linear and unitary, then negative masses appear in relativistic particle physics (you may want to check this: https://arxiv.org/abs/1809.05046 ). For energy-momentum tensor of a perfect negative mass fluid you may want to check https://januscosmologicalmodel.com/pdf/bonnor1989.pdf page 1149, although this paper assumes a whole universe made up of only negative mass, but the interior solution is the same as my proposal. If the event horizon is the division between both positive time spacetime (outside, ours), and negative mass (inside), and nothing inside comes out, both "worlds" are perfectly in place. Quote We have never observed exotic matter, after all. Gravity is a highly non-linear thing, so it is dangerous to make any kind of assumption about how something will behave, without working through the maths first - remember that energy density is not the only component of the energy-momentum tensor, so it isn’t the only source of gravity here. For exotic matter, the influence of the stress components in the tensor would actually be larger than that of energy-density. My model is in agreement with the fact that we have never observed exotic matter, it basically states that it only arises with a time transformation, that is, inside black holes in my model. The way to prove my model does not rely on direct observations of this exotic matter (that could only be done by entering the black hole), but by another mechanism of black hole mass observations as I explained in my first post. Negative mass (exotic matter, negative energy density), pressure associated to it and energy momentum tensors have not only been studied by Bonnor, but also by Bondi: http://ayuba.fr/pdf/bondi1957.pdf Quote Because you cannot join a region with vanishing cosmological constant to a region with non-vanishing constant, without violating the continuity boundary conditions. There also isn’t any known mechanism for that constant to change from one spatial region to another. Well, its not really a cosmological constant, as we have discussed before. In reality, its just the mass of the original star that lies inside the event horizon (and whatever falls in accretion). You can model it as a perfect fluid of negative energy density inside, or as you proposed, a cosmological constant. But its just matter, why cannot join the interior solution with matter to a vaccum exterior solution? Quote How does it just “appear”? Where does it come from, and by which physical mechanism does it form? What type of particle is it made of, and how does that fit in with the Standard Model? Explained in my first post, but also above in this one. Quote If the interior region wasn’t empty space (as you appear to suggest), then the gravitational wave signature of these mergers would be quite different. In fact, the signature from merging event horizons as modelled by GR is pretty unique, so no, it couldn’t really be otherwise in any significant way. What?? That cannot be possible, since no information (not even gravitational waves) can go inside->outside!! The gravitational wave signature must only depend on the interior solution. Quote in fact, there would be no merger, since two such black holes would be gravitationally repulsive towards one another. No, they would not. As I have been explaining, it does not matter if you place repulsive matter at the interior, the exterior solution of black holes in unaltered. Thus, attractive gravity (curvature of spacetime) at the exterior is also unaltered, and black holes would gravitate attractively one another the same way. Quote I’m somewhat confused now what exactly you really suggest - an empty interior with negative cosmological constant? Or an interior filled with a gravitationally repulsive exotic substance? I have always talked about repulsive exotic matter or negative mass (which cannot be other than the one of the original star, which suffered the time transformation I explained in my first post). The easiest way to model this matter is by simplifying it as a perfect fluid of negative energy density, which is of course repulsive. You may want to model it as a cosmological constant, which was an idea of yours, but they are different models, what is important is to understand where does it come from. Quote For the latter, the exterior vacuum cannot be Schwarzschild, since it would have to have a very different geodesic structure - positive mass test particles would actually fall away from the event horizon. What you are describing here are the negative mass black holes of R. B. Mann (https://arxiv.org/abs/gr-qc/9705007) and Bonnor, which are formed by already existing exotic matter (which is in contradiction with observations). In these, there is no transformation from regular matter to exotic matter, this one just already exist around, and collapses. In this kind of black holes, yes, test particles fall away, the horizon is repulsive, and it looks like a white hole (some may also have a naked singularity). These are not the type of negative mass black holes I propose, because I propose a change from regular to exotic matter within the horizon (well, really at the event horizon), so that the exterior solution is unaltered and test particles still fall towards the black hole, because the external spacetime curvature of the original star is remembered after the formation of the event horizon. I very much thank you for your interest in my proposal, although many points have to be clarified. Perhaps its my fault because english is not my first language. I think it is mandatory to be open minded about the fact that energy conditions may not hold once passed the event horizon, because we cannot be 100% sure of what lies inside it without direct observations. Edited August 19, 2020 by muruep00
Markus Hanke Posted August 20, 2020 Posted August 20, 2020 12 hours ago, muruep00 said: Yes, you assume energy conditions hold! Energy conditions are not amongst the boundary conditions used to solve the field equations for the Schwarzschild metric. 12 hours ago, muruep00 said: Wherever you dont have observations (inside black holes), you begin assuming that it works. Yes, we are assuming that it works well up to the point when quantum corrections become non-negligible. 12 hours ago, muruep00 said: Exotic matter is reasonable because it is trivial that it solves the gravitational singularity How do you know this? Energy density/mass isn't the only source of gravity in the field equations. 12 hours ago, muruep00 said: This is explained in my first post. Regular matter would be transformed into exotic matter by a time transformation at the event horizon. Since spacetime in and around the horizon is smooth and regular, no PT reversal can happen there. 12 hours ago, muruep00 said: although this paper assumes a whole universe made up of only negative mass, but the interior solution is the same as my proposal It is not an assumption of the paper, it is a finding - you can not glue a region of negative mass to one of positive mass, without violating basic laws of gravity, which is why the entire universe in the scenario is negative mass. And even if you could, black holes with singularities would still occur, as the author correctly points out, since mass isn't the only source of gravity. 13 hours ago, muruep00 said: My model is in agreement with the fact that we have never observed exotic matter, it basically states that it only arises with a time transformation, that is, inside black holes in my model. There is no such thing as a time transformation (I presume you mean time reversal) at the horizon. And if there were, then you actually had a big problem, because the Standard Model is not invariant under T or PT reversals. 13 hours ago, muruep00 said: But its just matter, why cannot join the interior solution with matter to a vaccum exterior solution? Yes, you can do this, but what I am trying to point out is that the exterior vacuum cannot have Schwarzschild geometry - it will be a different type of spacetime. 13 hours ago, muruep00 said: The gravitational wave signature must only depend on the interior solution. The wave signature depends on the geometry of the entire spacetime, i.e. the overall metric. 13 hours ago, muruep00 said: As I have been explaining, it does not matter if you place repulsive matter at the interior, the exterior solution of black holes in unaltered. Thus, attractive gravity (curvature of spacetime) at the exterior is also unaltered, and black holes would gravitate attractively one another the same way. No. Two bodies composed of exotic matter must always fall away from each other, and their exterior vacuum isn't Schwarzschild. How could it be? The maximally extended Schwarzschild solution (including both the interior of the collapsing star as well as the external vacuum) arises from an energy-momentum tensor for regular matter; if you change this source term, you are going to get a different solution. The other thing is that the 'M' constant in the Schwarzschild metric is a parameter for a 1-parameter family of metrics - so it is a property of the entire spacetime. 13 hours ago, muruep00 said: These are not the type of negative mass black holes I propose, because I propose a change from regular to exotic matter within the horizon (well, really at the event horizon), so that the exterior solution is unaltered and test particles still fall towards the black hole, because the external spacetime curvature of the original star is remembered after the formation of the event horizon. Ok. 13 hours ago, muruep00 said: I think it is mandatory to be open minded about the fact that energy conditions may not hold once passed the event horizon, because we cannot be 100% sure of what lies inside it without direct observations. Energy conditions aren't really the problem here, in my opinion. The main issue is that you are postulating a whole range of things that are very difficult or impossible to reconcile with already known physics, and that you make a number of assumptions that you believe to be true, but haven't actually explicitly checked: 1. That there is a PT or T reversal at the horizon 2. That such a reversal somehow transforms ordinary matter into exotic matter 3. That exotic matter is compatible with the Standard Model 4. That a body made of exotic matter has exterior Schwarzschild geometry 5. That this process avoids the formation of a singularity In the conventional GR picture, we assume only that the classical model remains valid up to the point where quantum effects can no longer be neglected. To be honest, postulating exotic matter creates many more problems than it could possibly solve (and I don't believe it actually solves anything), so it's of not much value. Can I suggest you make an effort to put some maths around your idea, because without a mathematical framework you don't really have a model, you only have a collection of assumptions and conjectures. Once the maths are in place, things can be checked directly. 13 hours ago, muruep00 said: I very much thank you for your interest in my proposal There is a really fine line between asking for honest feedback on an idea, and (perhaps unconsciously) already being convinced that it must be right, and thus finding ways to 'prove' it. I've given you honest feedback (this is from someone who has been working with gravitational models - not just GR - for a long time), and it is up to yourself now what you do with that. My honest opinion is that what you suggest doesn't work, and even if it did, it wouldn't actually solve anything. I'll leave you to it.
muruep00 Posted August 20, 2020 Author Posted August 20, 2020 6 hours ago, Markus Hanke said: Energy conditions are not amongst the boundary conditions used to solve the field equations for the Schwarzschild metric. For the Schwarzschild metric yes, you are right. But for black hole collapse at the interior? Yes, they are assumed, otherwise, simply collapse may not take place! 6 hours ago, Markus Hanke said: Yes, we are assuming that it works well up to the point when quantum corrections become non-negligible. Alright, so, why not assume something else while we look for quantum gravity theory? 6 hours ago, Markus Hanke said: How do you know this? Energy density/mass isn't the only source of gravity in the field equations. I think is trivial, since we are dealing with huge mass density. Because negative energy is needed for negative mass, my idea is that all energies at the interior are negative. 6 hours ago, Markus Hanke said: Since spacetime in and around the horizon is smooth and regular, no PT reversal can happen there. PT reversal cannot happen never in GR, since they are not allowed. First, they should be allowed as transformations in SR. Once we allow those transformations, we could begin talking about whether they can take place at the event horizon or not. The further research in these that I have found is this: https://link.springer.com/article/10.1007/BF00729807 but is very little. 6 hours ago, Markus Hanke said: It is not an assumption of the paper, it is a finding - you can not glue a region of negative mass to one of positive mass, without violating basic laws of gravity, which is why the entire universe in the scenario is negative mass. And even if you could, black holes with singularities would still occur, as the author correctly points out, since mass isn't the only source of gravity. Why? Where does Bonnor state that? If you refer to "if mass were negative the Schwarzschild solution would contain no horizon and Schwarzschild black holes would not exist", those are the black holes of R. B. Mann, not my proposal, because my proposal does contain an horizon. He states that black holes singularities may still occur as charged black holes, but we have never observed those, so no worries with my proposal. 6 hours ago, Markus Hanke said: There is no such thing as a time transformation (I presume you mean time reversal) at the horizon. And if there were, then you actually had a big problem, because the Standard Model is not invariant under T or PT reversals. How are you so sure about that? It could be. The standard model is an incomplete model, which does not include gravity. And we are dealing with gravity and a macroscopic phenomena. 6 hours ago, Markus Hanke said: The wave signature depends on the geometry of the entire spacetime, i.e. the overall metric. That cant be, because in that way, we could tell whats inside. And we cant, no information escapes. 6 hours ago, Markus Hanke said: No. Two bodies composed of exotic matter must always fall away from each other, and their exterior vacuum isn't Schwarzschild. How could it be? The maximally extended Schwarzschild solution (including both the interior of the collapsing star as well as the external vacuum) arises from an energy-momentum tensor for regular matter; if you change this source term, you are going to get a different solution. The other thing is that the 'M' constant in the Schwarzschild metric is a parameter for a 1-parameter family of metrics - so it is a property of the entire spacetime. If you take two black holes, and suddenly you change all their interior-collapsing-star into exotic matter, the exterior solution is unaltered and both black holes are still attractive. I you could tell from outside if this change has taken place, you would be getting information from inside, and that just cannot be done, because no signal (gravitational, luminous...) can go from inside to outside. I think this is very obvious. 6 hours ago, Markus Hanke said: Ok. Energy conditions aren't really the problem here, in my opinion. The main issue is that you are postulating a whole range of things that are very difficult or impossible to reconcile with already known physics, and that you make a number of assumptions that you believe to be true, but haven't actually explicitly checked: 1. That there is a PT or T reversal at the horizon 2. That such a reversal somehow transforms ordinary matter into exotic matter 3. That exotic matter is compatible with the Standard Model 4. That a body made of exotic matter has exterior Schwarzschild geometry 5. That this process avoids the formation of a singularity You may only take number 1. as an assumption. Number 2. is proven here: https://arxiv.org/abs/1809.05046 , number 3. you dont needed to assume it, number 4. is trivial as I explained before and 5. also, but I agree with you that the math should be worked out to prove number 5. Its all really about my assumption number 1. vs. the assumption that GR holds as we know it at the interior until very close to the singularity. The point is, my assumption can be proven right or wrong. The other one you may want to wait for a quantum theory of gravity. 6 hours ago, Markus Hanke said: In the conventional GR picture, we assume only that the classical model remains valid up to the point where quantum effects can no longer be neglected. To be honest, postulating exotic matter creates many more problems than it could possibly solve (and I don't believe it actually solves anything), so it's of not much value. The only problem with exotic matter in GR, is the runaway paradox: https://en.wikipedia.org/wiki/Negative_mass#Runaway_motion . This paradox does not even take place in my proposal due to the characteristics of the event horizon. Let me number the problems my exotic matter proposal solves: 1. Gravitational singularity 2. Interior of Kerr black holes (which are the real ones) 3. Runaway motion 4. Supermassive black hole formation, if my model is solved as I propose. 6 hours ago, Markus Hanke said: Can I suggest you make an effort to put some maths around your idea, because without a mathematical framework you don't really have a model, you only have a collection of assumptions and conjectures. Once the maths are in place, things can be checked directly. I have tried to justify it mathematically, although my work is far from being done. I just wanted to explore this proposal conceptually. 6 hours ago, Markus Hanke said: There is a really fine line between asking for honest feedback on an idea, and (perhaps unconsciously) already being convinced that it must be right, and thus finding ways to 'prove' it. I've given you honest feedback (this is from someone who has been working with gravitational models - not just GR - for a long time), and it is up to yourself now what you do with that. My honest opinion is that what you suggest doesn't work, and even if it did, it wouldn't actually solve anything. I'll leave you to it. I agree with you. But I dont trust my idea, I want to know if there is a way to prove it wrong. That is science. And I have found none. I very much thank you for your interest and the ideas you have posted (for instance, I though it was trivil that a non-charged star cannot collapse if is made up of exotic matter, but as you said, this has to be proven mathematically taking into account other factors such as pressure). What also happens quite often is that researchers only find interesting their ideas, just because they come from them, and not others ideas, even though if they are interesting. We all have to fight our bias
MigL Posted August 20, 2020 Posted August 20, 2020 Interesting discussion. On the one hand, you have Markus' predictive math ( except for the singular region ) arguments. On the other, muruep00's arguments, with an awful lot of 'ifs" . Hard to decide who to agree with. ( no, not at all )
joigus Posted August 20, 2020 Posted August 20, 2020 54 minutes ago, muruep00 said: How are you so sure about that? It could be. The SM's Lagrangian is full of so-called gamma-5 matrices all over the place, just to make it explicitly and unambiguously break CP symmetry maximally. It's not just broken, it's broken by design. All leptons and quarks are lefties in the SM. So... I, for one, am sure about that. It better be, cause maximal parity violation has been shown to hold in EW decays in the laboratory ad nauseam.
muruep00 Posted August 20, 2020 Author Posted August 20, 2020 (edited) 1 hour ago, MigL said: Interesting discussion. On the one hand, you have Markus' predictive math ( except for the singular region ) arguments. On the other, muruep00's arguments, with an awful lot of 'ifs" . Hard to decide who to agree with. ( no, not at all ) Markus' math might give you some predictions inside black holes, but these predictions have not been proven, and they result in a singularity (which you can avoid by stating that math no longer works and you need new math). That math that Markus is using inside black holes also has a lot of "ifs". My only "if" is that a time transformation takes place at the event horizon. As explained in my last post, the fact that a time reversal (linear and unitary) transforms ordinary matter into exotic matter and that the exterior solution is unaltered are not assumptions. But anyway, I suggest a way to prove my model. Since Markus model cannot be proved right or wrong until we get a quantum theory of gravity, the fact that mine is based on more "ifs" is not significant. I would change your word "agree" to "assume" or "belief". Thats better. 11 minutes ago, joigus said: The SM's Lagrangian is full of so-called gamma-5 matrices all over the place, just to make it explicitly and unambiguously break CP symmetry maximally. It's not just broken, it's broken by design. All leptons and quarks are lefties in the SM. So... I, for one, am sure about that. It better be, cause maximal parity violation has been shown to hold in EW decays in the laboratory ad nauseam. Yes, with my sentence "How are you sure about that?" I was referring to Markus sentence: There is no such thing as a time transformation (I presume you mean time reversal) at the horizon. Edited August 20, 2020 by muruep00
joigus Posted August 20, 2020 Posted August 20, 2020 1 minute ago, muruep00 said: Yes, with my sentence "How are you sure about that?" I was referring to Markus sentence: Sorry, but you seemed to be talking about the SM: 1 hour ago, muruep00 said: How are you so sure about that? It could be. The standard model is an incomplete model, which does not include gravity. And we are dealing with gravity and a macroscopic phenomena. I wouldn't disagree too strongly with Markus about that. All I can add is there is no application that I know of in which you invert time at the horizon. He went on to say that this idea doesn't sit too well with the SM. He did say that, and related it to the point you were making. You seem to be suggesting to do away with the SM at the horizon, while retaining GR there. You must have a pretty strong argument if you want to do that. 8 hours ago, Markus Hanke said: There is no such thing as a time transformation (I presume you mean time reversal) at the horizon. And if there were, then you actually had a big problem, because the Standard Model is not invariant under T or PT reversals. There you are, he did mention the SM in connection with the argument. Keep in mind that T, C, and P, are always considered as passive transformations (re-labellings), not as something that you can actually do to a system. I hope to catch up with the main arguments eventually, though. I'm a late comer here.
muruep00 Posted August 20, 2020 Author Posted August 20, 2020 5 minutes ago, joigus said: All I can add is there is no application that I know of in which you invert time at the horizon. Perhaps that is because time inversion is already prohibited in SR, arbitrarily (well, I guess there was no observable evidence for this transformation, so they sentenced it to be unphysical, but we have no observable evidence for the whole universe, for instance, we do not observe the inside of black holes). Maybe if these are allowed, you may be able to build an exterior Schwarzschild solution with a time transformation at the event horizon, and so on with my proposal. And maybe, this is what is happening at black holes right now, but the physics comunity decided to belief that their GR and predictions of black hole interior and collapse process are right, instead of looking for more consistent ideas to solve the gravitational singularity apart from hoping for a theory of quantum gravity that solves it. 5 minutes ago, joigus said: I hope to catch up with the main arguments eventually, though. I'm a late comer here. You are welcomed.
Markus Hanke Posted August 21, 2020 Posted August 21, 2020 15 hours ago, muruep00 said: But I dont trust my idea, I want to know if there is a way to prove it wrong I suggest you should first of all check whether the concept you propose is mathematically consistent with basic laws of physics or not, before you even start to worry about observational evidence. I'll help you out a bit more - consider the energy-momentum tensor for an exotic matter perfect fluid distribution: \[T_{\mu \nu } =-|\rho |u_{\mu } v_{\nu }\] Assuming that this distribution is static, stationary, and spherically symmetric (same conditions as Schwarzschild), it can be shown (Mann 1997) that the only possible exterior vacuum metric that is consistent with the Einstein equations for this scenario has to have the form \[ds^{2} =-A( r,\Lambda ,M) dt^{2} +\frac{1}{A( r,\Lambda ,M)} dr^{2} +r^{2} d\Omega ^{2}\] with the coefficient function \[A( r,\Lambda ,M) =\left(\frac{1}{3} |\Lambda |r^{2} -1+\frac{2M}{r}\right)\] and \(\Lambda<0\). I reiterate again that this is the only possible solution to the Einstein equations which can be matched to a spherical ball of perfect exotic matter fluid - this can be formally proven using a generalised form of Birkhoff's theorem (Bronnikov/Kovalchuk 1980), or alternatively by directly solving the field equations (Mann 1997). It is immediately obvious that this is not a Schwarzschild spacetime - which is why I kept pointing out to you that a Schwarzschild vacuum cannot be glued to this type of energy-momentum distribution. This is actually a rather interesting metric, because it has a fairly complex geometry akin to a Reissner-Nordstroem anti-DeSitter spacetime; there are two event horizons (an inner and an outer one), it is not asymptotically flat, and the global topology is not trivial either. Note that neither one of the horizons is located at what would be the Schwarzschild radius in Schwarzschild spacetime. Because the geodesic structure of such a spacetime is very different from that of Schwarzschild, it would be easy to observationally distinguish the two. If you have the mathematical skills, you could use the above to start your own investigations. Essentially what I am saying to you is that the very idea of having a Schwarzschild vacuum exterior to a distribution of an exotic perfect fluid ball is not consistent with the basic laws of gravity - so the concept is not internally self-consistent, and can't work, irrespective of any finer details. 1
muruep00 Posted August 21, 2020 Author Posted August 21, 2020 (edited) 4 hours ago, Markus Hanke said: I suggest you should first of all check whether the concept you propose is mathematically consistent with basic laws of physics or not, before you even start to worry about observational evidence. I'll help you out a bit more - consider the energy-momentum tensor for an exotic matter perfect fluid distribution: Tμν=−|ρ|uμvν Assuming that this distribution is static, stationary, and spherically symmetric (same conditions as Schwarzschild), it can be shown (Mann 1997) that the only possible exterior vacuum metric that is consistent with the Einstein equations for this scenario has to have the form ds2=−A(r,Λ,M)dt2+1A(r,Λ,M)dr2+r2dΩ2 with the coefficient function A(r,Λ,M)=(13|Λ|r2−1+2Mr) and Λ<0 Hi Markus I dont know why you assume a static/stationary solution. But I just cannot understand why a switch to exotic matter inside black holes is geometrically inconsistent with its exterior solution. No information (no gravitational waves) from inside can affect the spacetime of the exterior solution. You could even eliminate all mass at the interior of black holes, and the exterior solution would be unaltered! As I said, R. B. Mann wanted to solve what an already existing exotic matter collapse looks like. I quote "The purpose of this essay is to demonstrate that a region of negative energy density can also undergo gravitational collapse to a black hole." He begins modeling the region of negative energy density for cloud of freely-falling dust. In that case, yes, the exterior solution would be repulsive. But that has nothing to with what I propose. Can you explain further why is that the only possible solution to the Einsteins equations which can be matched to a spherical ball of perfect exotic matter fluid? I can recall the Wheelers Bag of gold solution that seem to contradict that, for instance Edited August 21, 2020 by muruep00
joigus Posted August 21, 2020 Posted August 21, 2020 (edited) 2 hours ago, muruep00 said: I dont know why you assume a static/stationary solution. Birkhoff's theorem: https://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity) Sorry. I hadn't noticed that Markus had already said it. +1 Edited August 21, 2020 by joigus Addition
muruep00 Posted August 21, 2020 Author Posted August 21, 2020 58 minutes ago, joigus said: Birkhoff's theorem: https://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity) Sorry. I hadn't noticed that Markus had already said it. +1 Why would it be a vacuum solution since we are dealing with exotic matter?
Markus Hanke Posted August 21, 2020 Posted August 21, 2020 2 hours ago, muruep00 said: I dont know why you assume a static/stationary solution. Your original post was about Schwarzschild spacetime, which is static and stationary. If you abandon those boundary conditions, then things are going to become very, very complicated very quickly. 2 hours ago, muruep00 said: But I just cannot understand why a switch to exotic matter inside black holes is geometrically inconsistent with its exterior solution. No information (no gravitational waves) from inside can affect the spacetime of the exterior solution. Gravitational waves have nothing to do with this (the spacetime is stationary!), and no information is being propagated - this isn’t the issue. The issue is rather that you can’t just glue arbitrary regions of spacetime together any which way you want - once you have a specific geometry in one region, then this places very stringent constraints on the kind of geometry adjacent regions can have, because spacetime needs to remain smooth and continuous at the boundary between the two, and the global metric (which covers both regions) has to itself be a valid solution to the field equations. In most cases, this uniquely fixes the entire geometry. In this case, if you have a ball of perfect exotic matter fluid, the exterior can only be an AdS-Reissner-Nordström metric, and nothing else. 2 hours ago, muruep00 said: But that has nothing to with what I propose. The exterior metric is the same whether the exotic matter fluid in the interior collapses, or not - same as in the Schwarzschild case. But it is not the same if you replace ordinary matter with exotic matter, because the two give rise to entirely different geometries (and topologies!). 2 hours ago, muruep00 said: Can you explain further why is that the only possible solution to the Einsteins equations which can be matched to a spherical ball of perfect exotic matter fluid? If you start with the energy-momentum tensor of a ball of perfect exotic matter fluid in a stationary spacetime, and insert it into the field equations, you obtain a single unique interior solution - because of smoothness and continuity boundary conditions, and because all metrics need to be valid solutions to the field equations, this automatically gives rise to a single unique solution for the exterior vacuum - being the metric I quoted. The same is true for Schwarzschild as well - the exterior (vacuum) Schwarzschild metric can only be coupled to an interior (ordinary matter) Schwarzschild metric, and nothing else. You don’t need to take my (or anyone else’s) word for this, you can work through the maths yourself, if you need further proof (but be warned, the maths aren’t trivial!). Start with the energy-momentum tensor given above, and see if you can derive the Schwarzschild metric from it. 2 hours ago, muruep00 said: I can recall the Wheelers Bag of gold solution that seem to contradict that, for instance I’m afraid I fail to see the connection. WBG spacetimes exclusively arise from ordinary matter. I think there is a fundamental misunderstanding here - it seems you think that, just because exotic mass differs only by a sign in the energy-momentum tensor, the resulting exterior metric has to be of the same form as Schwarzschild. But that is not so, because the GR equations are highly non-linear; just flipping a sign in the input does not translate to just flipping a sign at the output. Instead, it leads to a completely different spacetime with a completely different geometry and topology - both in the interior and the exterior. That is the nature of coupled systems of non-linear partial differential equations. 4 minutes ago, muruep00 said: Why would it be a vacuum solution since we are dealing with exotic matter? He’s referring to the exterior vacuum solution outside the fluid ball.
muruep00 Posted August 21, 2020 Author Posted August 21, 2020 3 minutes ago, Markus Hanke said: Your original post was about Schwarzschild spacetime, which is static and stationary. If you abandon those boundary conditions, then things are going to become very, very complicated very quickly. Well, my original post was about a non-static non-stationary solution, I quote myself "the interior of the negative mass BH undergoes an inflation". And the whole process that I propose of the collapse of ordinary matter which is transformed into exotic matter when crossing the event horizon is clearly not stationary. 3 minutes ago, Markus Hanke said: Gravitational waves have nothing to do with this (the spacetime is stationary!), and no information is being propagated - this isn’t the issue. The issue is rather that you can’t just glue arbitrary regions of spacetime together any which way you want - once you have a specific geometry in one region, then this places very stringent constraints on the kind of geometry adjacent regions can have, because spacetime needs to remain smooth and continuous at the boundary between the two, and the global metric (which covers both regions) has to itself be a valid solution to the field equations. In most cases, this uniquely fixes the entire geometry. In this case, if you have a ball of perfect exotic matter fluid, the exterior can only be an AdS-Reissner-Nordström metric, and nothing else. I understand that this only holds for a static solution. 3 minutes ago, Markus Hanke said: The exterior metric is the same whether the exotic matter fluid in the interior collapses, or not - same as in the Schwarzschild case. But it is not the same if you replace ordinary matter with exotic matter, because the two give rise to entirely different geometries (and topologies!). I guess you mean different geometries at the interior. Yes, right. 3 minutes ago, Markus Hanke said: If you start with the energy-momentum tensor of a ball of perfect exotic matter fluid in a stationary spacetime, and insert it into the field equations, you obtain a single unique interior solution - because of smoothness and continuity boundary conditions, and because all metrics need to be valid solutions to the field equations, this automatically gives rise to a single unique solution for the exterior vacuum - being the metric I quoted. The same is true for Schwarzschild as well - the exterior (vacuum) Schwarzschild metric can only be coupled to an interior (ordinary matter) Schwarzschild metric, and nothing else. You don’t need to take my (or anyone else’s) word for this, you can work through the maths yourself, if you need further proof (but be warned, the maths aren’t trivial!). Start with the energy-momentum tensor given above, and see if you can derive the Schwarzschild metric from it. I’m afraid I fail to see the connection. WBG spacetimes exclusively arise from ordinary matter. Okey, I understand static solutions might not be possible. What about time dependent solutions? 3 minutes ago, Markus Hanke said: I think there is a fundamental misunderstanding here - it seems you think that, just because exotic mass differs only by a sign in the energy-momentum tensor, the resulting exterior metric has to be of the same form as Schwarzschild. But that is not so, because the GR equations are highly non-linear; just flipping a sign in the input does not translate to just flipping a sign at the output. Instead, it leads to a completely different spacetime with a completely different geometry and topology - both in the interior and the exterior. That is the nature of coupled systems of non-linear partial differential equations. Well, if you change the parameter M of mass in the Schwarzschild metric, you do get a similar spacetime but outgoing, with no event horizon and with a naked singularity. But apart from that, lets think about the real case of my proposal. You are telling me that if by some mechanism I change the collapsing matter of a black hole at the interior once the horizon has formed into exotic matter, geometry breaks down? Of course no! I might be that no static solution could result from that, but I think there has to be a way to translate my idea into a time dependent metric solution. 3 minutes ago, Markus Hanke said: He’s referring to the exterior vacuum solution outside the fluid ball. Does Birkhoff's theorem apply to solutions that include a fluid ball if the exterior is vacuum?
Markus Hanke Posted August 21, 2020 Posted August 21, 2020 6 minutes ago, muruep00 said: Well, my original post was about a non-static non-stationary solution, I quote myself "the interior of the negative mass BH undergoes an inflation". And the whole process that I propose of the collapse of ordinary matter which is transformed into exotic matter when crossing the event horizon is clearly not stationary. Well, then you need to start with an energy-momentum tensor that isn’t stationary (i.e. some or all of its components will be time-dependent); unfortunately then the resulting metric won’t be stationary either, so it won’t be Schwarzschild. So basically you want to start with ordinary matter (described by some non-stationary non-vacuum metric without cosmological constant), that somehow transforms into exotic matter (necessarily described by a metric with non-vanishing cosmological constant, as quoted above). Even without any further consideration, it is already clear that a process like this is incompatible with GR, because the cosmological constant is a conserved quantity - you can’t have it being zero in one region, and non-zero in another. 16 minutes ago, muruep00 said: Well, if you change the parameter M of mass in the Schwarzschild metric, you do get a similar spacetime but outgoing No you don’t. The ‘M’ is a parameter in a 1-parameter family of metrics, it can’t be negative. It also can’t change from M to (-M), because it is a conserved quantity that is a property of the entire spacetime. If you propose any process where M varies in any way, then you are no longer in Schwarzschild spacetime (such spacetimes are of type Vaidya-Bonner). 19 minutes ago, muruep00 said: You are telling me that if by some mechanism I change the collapsing matter of a black hole at the interior once the horizon has formed into exotic matter, geometry breaks down? No, I am telling you two things: that such a process isn’t physically possible because it is in violation of both GR and the Standard Model; and that, if it were somehow possible, the exterior geometry couldn’t be Schwarzschild. 21 minutes ago, muruep00 said: Does Birkhoff's theorem apply to solutions that include a fluid ball if the exterior is vacuum? It applies only to spacetimes that are vacuum in the exterior. If your fluid ball is embedded in a region of spacetime that isn’t a vacuum (e.g. Vaidya spacetime), then we are again dealing with a completely different situation, that requires detailed analysis.
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