Markus Hanke

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 I don’t really understand the significance of these historical references, to be honest.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

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

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

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

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

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

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

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

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

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

Well, without a metric we can’t really have a discussion about this. The thing here is that you don’t start with a metric - you begin with an energy-momentum tensor plus boundary conditions, then you use these to solve the Einstein equations. That gives you the metric. All solutions to the Einstein equations are metrics, but not all metrics are valid solutions to the Einstein equations.

Ok, so what is the metric? You haven’t written it down yet.

I believe you are thinking of specific, symmetric solutions such as the Kerr spacetime. In those special cases the situation is indeed unambiguous - but that’s because these cases assume certain symmetries that remove the extra degrees of freedom. The problem referred to in the paper pertains to general regions of curved spacetime, where no symmetries or boundary conditions are assumed. Defining the total energy (not just mass) contained in such a region has been an intractable problem - which this paper now solves. I must look at this in detail, but at the moment I’m engaged in other pursuits.

That’s certainly a factor, but I suspect it’s mostly because a lot of people simply don’t have the ability to step outside the paradigm of what their sensory apparatus tells them about the world - which is essentially Newtonian. Thus, relativity and QM get rejected wholesale, because they “don’t make sense”. Also, believing that you are smarter than a larger than life figure such as Einstein props up people’s egos.

Here’s an article about this paper - it’s actually more about angular momentum (which is just as ambiguous as mass), but the problems are closely related: https://www.quantamagazine.org/mass-and-angular-momentum-left-ambiguous-by-einstein-get-defined-20220713/ I can’t offer any real details yet, since I haven’t studied the paper itself.

You’ve got this backwards - it was you who made the claim that spacetime is a mechanical medium, and that energy-momentum is always conserved. Mainstream physics says no such thing. So the onus is on you to show how your claim is right. Indeed. Yes, that’s right. The problem is that the gravitational field itself carries energy, but this energy isn’t localisable; if you try to account for it, you generally end up with expressions that are observer-dependent. A further problem is that there is more than one way around this, which is why you get different ways to define the energy content of a region of spacetime, like ADM energy, Komar energy etc etc. It’s not immediately clear how to define it in a general, unambiguous way. I believe the problem has recently been solved, though I haven’t had time to look into this new development, so I can’t comment yet.

Spacetime isn’t a mechanical medium, so this is irrelevant. Also, it might surprise you to hear that the law of conservation of energy exists only in flat spacetime - in the presence of gravity, things become rather more complicated.

First of all, spacetime isn’t some kind of physical substance that “expands” in the literal sense, like bread in the oven. Rather, it is a network of causality relations - the set of all points in space at all moments in time, and how they are related to one another. When we speak of “expansion”, then what we mean is that measurements of distance depend on time, ie they change in a certain well defined way according to when they are performed. This is an extremely useful and accurate model, but shouldn’t be reified into something like a physical “substance”. LQG deals with something called “Wilson loops”. These are mathematical objects that are solutions, simply speaking, to an equation that treats spacetime as a quantum state. These loops are not themselves “chunks” of space and time - rather, they form networks called spin foams, which, in the semi classical limit, may become curved spacetimes with a positive cosmological constant. So yes, they do actually describe an expanding spacetime. The point here is that there would be a lower limit to how small intervals of space and time can be - you can’t infinitely subdivide it. Again, this is a mathematical model - like a map of the world. But it’s something that one could, at least in principle, test experimentally, given enough energy. No. They are 1-dimensional objects that live in a background spacetime with 3 macroscopic spatial dimensions, 1 dimension of time, and 7 compactified dimensions. Of course not. It’s the spacetime they live in that expands, not the strings themselves. Physics has nothing to say on any of these issues.

Ah I see. The Klein-Gordon equation is the quantum version of the energy-momentum relation (not the tensor though), and is generally used to describe relativistic fields/particles without spin - scalar particles.

How do you know that they don’t?

Apologies, you are of course correct, “fusion” was what I meant to say (I typed that in a hurry). I’m not sure what you mean here - could you elaborate?

You can’t isolate it (the concept of a single quark doesn’t even make sense), but you can probe its properties by letting it interact with other particles, even while it is bound up in a composite particle. This technique is called deep inelastic scattering.

It’s a bit more complicated than that - a fission bomb fuses hydrogen atoms into heavier elements. Initially you need to supply energy to do this, in order to overcome the electromagnetic repulsion between protons (among other things). However, once the protons are close enough for the residual strong force to kick in, they “fall” into its respective potential well, forming a stable nucleus - which is structured such that you end up with a net surplus energy at the end of the interaction. There’s actually a lot more going on, but that’s the gist of it. I reiterate again that E=mc^2 works only for massive particles at rest - it’s a special case of the energy-momentum relation, which is in turn just the magnitude of the energy-momentum 4-vector.

The source of gravity isn’t just mass (whatever its form), but all sources of energy-momentum. Even an electromagnetic field (which isn’t composed of quarks), and things like pressure, stress and strain have a gravitational effect. Same goes for particles not composed of quarks, such as electrons and neutrinos. These all gravitate.

Elementary particles don’t have an internal structure, because they are local excitations of quantum fields. Such fields don’t have structures. In tech speak, elementary particles are irreducible representations of symmetry groups. There is no physical principle requiring all particles to have internal structure. E=mc^2 has nothing to do with potential energy, which is what you must be referring to in this statement, or else it doesn’t make sense. It’s the energy equivalent of the particle’s rest mass, and this relationship is true only in a massive particle’s rest frame. This has nothing to do with any potentials or internal structures. E=mc^2 has nothing to do with potential energy, nor internal structure. Only with rest mass. No, see above. No, I am saying that there is no internal structure, according to current understanding of the Standard Model. Elementary particles are irreducible - and that’s true for all of them, irrespective of whether they carry electric charge, colour charge, flavour, isospin, or mass. To experimentally verify the elementary-ness of such particles, you use a technique called deep inelastic scattering. This is, however, limited by the available energy of the accelerator. Proposing internal structure for these particles means you need to introduce new physics.