Markus Hanke

Yes it is. But is very important to remember that this is a purely local conservation law - while it always holds at every point, it may not hold in an extended global region. Yes, it also holds in quantum mechanics, so long as we are dealing with a closed system of course. After collapse, the wave function describes eigenstates of the Hamiltonian operator (which encapsulates the energy dynamics of the system), and the total energy itself takes on allowable values that are eigenvalues of the Hamiltonian. Generally, due to boundary conditions, the spectrum of the Hamiltonian operator is discrete, so the eigenvalues are discrete as well. Yes, you are. That’s the stress-energy-momentum tensor. I am unsure what you mean by this, but it is quite physical in the sense that it has measurable consequences. Yes, exactly correct. Energy-momentum is equivalent to local spacetime curvature, and vice versa, via the Einstein equations. This can be measured, at least in principle. Again you are correct, energy density is an observer-dependent quantity. However, the mathematical object that describes sources of gravity isn’t just energy density, it’s the full stress-energy-momentum tensor (energy density is one of its components). As being a tensorial quantity, all observers agree on it. The energy-momentum tensor, just like all tensors, is a purely local quantity. That means these energies are located exactly where you perform their respective measurements (I know this sounds trivial, but it really isn’t if you think about it in more detail). However, you need to remember that potentials cannot be physically measured, only their gradients can. Furthermore, the energy inherent in gravity itself is not localisable (which is why it isn’t part of the energy-momentum tensor, but encapsulated in the non-linear structure of the field equations themselves). Electromagnetic fields and gravity are quite different in many respects, and this is one of them. You can tell exactly how energy is distributed in an EM field, because electromagnetism obeys a field equation that is linear. The same is not true for gravity, in that the Einstein equations are highly non-linear; hence gravitational self-energy is not localisable. However, you can still localise sources other than gravitational self-interaction, which is everything that is encapsulated in the energy-momentum tensor. This tensor is itself precisely defined via Noether’s theorem, so this is all very well defined mathematically.

While I understand what you are trying to say, I think the above is a little sloppy. The ISS travels on a geodesic in spacetime (it does not fire thrusters at any time), so it is in inertial motion, because an on-board accelerometer will always read exactly zero. The thing is just that spacetime between the ISS and the earth-bound observer is not Minkowski - hence relative motion is not the only factor that needs to be accounted for when calculating the time dilation between the two clocks, because the relationship between these frames is more complicated than just a simple hyperbolic rotation. However, it turns out - due to the symmetries of Schwarzschild spacetime - that the SR and GR effects simply add up to give the total time dilation in this case (note that this is not true in the general case).

Exactly. That being said, on those rare occasions when it is successful, it‘s a very satisfying experience

Just to add to what Eise said - Lorentz invariance (i.e. Special Relativity, with length contraction and time dilation) is a fundamental local symmetry of the physical world, and in some form or another it is a critical part of all other models in physics. This symmetry has been extensively tested, and no violations have ever been observed: https://en.m.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation

I think it is, because in the scenario you suggest the universe does not expand, so the heat has nowhere to dissipate to. It‘s somewhat like a pressure cooker that is powered from its own interior, until there is no information left to be destroyed. I can‘t comment on the rest of that paragraph since it does not seem to be connected to the topic at hand. Because it would essentially be the same scenario as the interior of a shell (even if the shell itself is not thought of as massive) - and we know that this yields a region with uniform gravity, both in Newton as well as in GR. Hence no net gravitational attraction to the boundary anywhere. This alone renders the whole concept inconsistent. I believe he means infinite in the sense that geodesics can be extended indefinitely without ever terminating anywhere, or connecting back on themselves.

This is what I generally try to do - however, it seems that this strategy is successful only on rare occasions.

One very common issue is that people get stuck in a particular paradigm - the most prevalent of which is the notion that human perception and experience is an adequate representation of how the universe works - which of course it isn‘t. That is why we so often see people coming here and elsewhere to reject models such as relativity; because many of its concepts do not make sense in the context of everyday human experience. Essentially, people get stuck in a Newtonian worldview, based on how they experience the world on a day to day basis, and are simply not receptive to the idea that the Newtonian paradigm is very limited in its domain of applicability, and does not apply to the universe at large. It is hence no surprise that many people fight tooth and nail against ideas such as time dilation, length contraction etc etc - because if human perception and experience is your only point of reference, then these things really do not make much sense, because they invalidate the very fundamental notion of there being an absolute time and space. What‘s more, they invalidate the notion that us human beings, and the way we perceive the world, play any kind of privileged role in the universe at all. This can be a very hard pill to swallow for many. The same goes for all of quantum physics, as well as the more technical and advanced models in physics - we just don‘t get that many discussions about them, because people generally don‘t know much about these. Quantum physics in particular pretty much destroys most of what we believe is true about the world, based on human experience - if more people understood what it actually implies, we‘d see no end of „anti-quantum“ discussions here. Addressing this is very difficult, because getting stuck in a paradigm/worldview is a very powerful psychological attachment. If you are truly convinced that time, space and classicality must be absolute and immutable, then no amount of experimental evidence or mathematics - no matter how logical or irrefutable - is likely to sway your mind. I often feel a bit sorry for people like that, because mostly they don‘t realise that they are stuck in a paradigm, so in a certain sense it isn‘t really their fault that they are non-receptive to criticism of their ideas. And even if they realise their being stuck, getting out of the trap generally takes more than just logical arguments. Intellectual knowledge is only the first level of understanding; to be truly convinced of an idea, one has to grasp its paradigm on an deeper, more intuitive level as well. And that can take time and much effort (it does for me, anyway).

Janus has just answered this comprehensively - it‘s because of the issue of simultaneity. May I just add that it can be mathematically shown in a general manner that SR is fully self-consistent, i.e. it is not possible to construct any kind of real paradox using its axioms. This is independent of the specifics of the scenario. That‘s an interesting contradiction, because if relativity did not apply, then the very wire itself could not exist in real world (and neither could you, btw). This is because the quantum field theories that describe the behaviour of all the particles that make up the wire critically depend on the symmetries of relativity. Without it, elementary particles and their composites would either not exist at all, or have very different properties than the ones we observe. As for magnetism specifically, it actually follows from fundamental principles, so you don‘t even need to start with relativity. Suppose we have a potential 1-form A. The source-free part of the electromagnetic field then is, as usual for all such fields, [math]\displaystyle{F=dA}[/math] which is a 2-form. This naturally implies, via Poincare‘s Lemma, that [math]\displaystyle{dF=d(dA)=0}[/math] which is precisely the magnetic part of the Maxwell equations. Since both of the above relations are manifestly Lorentz covariant, on account of the transformation properties of the exterior derivative, the validity of relativity for magnetism is quite a natural consequence of this. It is hence just as correct to say that relativity falls right out of the fact that magnetism exists.

Sorry, but I do not see the connection to the question of whether or not the universe has a boundary. If this were the case, this region would need to be behind an event horizon. The process of anything falling through such a horizon is not unitary - meaning it‘s not a reversible process. Due to Landauer‘s principle, this would imply that this horizon radiates heat, so actually the universe would have to be really, really hot, and continuously heating up further. This is not what we observe. Also, even if you say that GR is not valid for the boundary itself, it would have to be valid for the event horizon, since that is a region of smooth and regular spacetime at some distance from your boundary. And again, I do not think that an event horizon with the global topology and geometry of a flat sheet is compatible with GR. If that were true, then all force vectors would cancel out, leaving zero net force, so there would be no relative motion of galaxies at all. Again, this is not what we observe.

The question is meaningless, because spacetime is not embedded into any higher dimensional manifold, so there simply is no “beyond”. Asking what is beyond is like asking what is north of the North Pole - it does not make any physical or mathematical sense. The existence of a boundary can also be ruled out via other, somewhat more technical arguments. First of all, it is not consistent with the laws of gravity - the Einstein equations have solutions that describe point singularities and ring singularities, but not sheet singularities, being singularities that are spread out like a 2D surface. Your hypothetical boundary would need to be of this kind, since it is by definition a region of geodesic incompleteness. The other main argument here is that, if there is a boundary in all spatial directions, we would essentially have to exist within the interior cavity of an energy-momentum shell of some kind (whether that is massive or not is irrelevant). Now, I do not have a solution to the Einstein equations to hand that describes a dust-filled cavity, but just by briefly thinking about it, and bearing in mind that spacetime in a vacuum cavity is everywhere Minkowski, I can pretty much guarantee that such a spacetime would be very different from the FLRW spacetime which we actually observe around us. In fact, given that ordinary matter density is actually very small over vast distances, I think such a spacetime could reasonably well be modelled by a linearly perturbed Minkowski metric - which is not what we observe. In other words, the FLRW metric that is the best fit for all the experimental data is simply incompatible with the notion of any kind of boundary. And these are only two arguments that immediately come to mind, I could probably come up with many more if I thought about this hard enough.

Spacetime is not the same as ether; it is not a medium with mechanical properties. Of course not, because this falls outside the domain of a classical theory such as General Relativity. You need quantum field theory - specifically, quantum electrodynamics - for a full understanding of this, but that was not developed until well after Einstein’s theory of relativity.

It’s more than just that. Gravity is defined as being geodesic deviation, i.e. a geometric property of spacetime. There is no meaningful distinction between the two. Gravity and electromagnetism are completely different - both in terms of their dynamics, and in terms of their underlying mechanism. There are links between the two, but they are nonetheless distinct phenomena.

Ok, the old German text is “Die neuen Inseln / so hinder Hispanien gegen Orient bey dem Land Indie ligen” I am actually unsure just what it is that they are trying to say here (German is my mother tongue, so I should know lol). I’d translate it something like “The new islands, which lie beyond Spain, in the region of the Orient towards India”. But it’s genuinely not easy to understand, not even for a native speaker.

It’s a very old form of German, written in a cursive script used at that time. The first three words mean “The new islands”, but I have difficulty deciphering the rest - is there no higher resolution image available anywhere?

The gravitational field is spacetime.

Any genuine physicist would be excited to discover evidence of new physics - that myths of “desperately trying to maintain the status quo” that is often bandied about does not make any sense at all. So yes, there are no such forces, nor will there ever be. There is only a healthy scepticism of extraordinary claims, which is how it should be.

Strictly speaking I think one might be able to make that argument. But then the same could be said for any QG model, since technologically speaking we are very far away from being able to experimentally test such models, so even a fully worked out and understood QG model is likely to remain speculation for some time to come. But it should be pointed out that not all speculations are created equal - some are speculative extensions or generalisations of already established models, while others are not rooted in established physics at all.

Indeed - that’s pretty much the point I am trying to make. Yes, I do not deny the importance of physical reasoning. What I am attempting to say is that this will be very difficult in the case of QG, because there is no direct observational data available (just yet), and we also do not know what such a model is even supposed to look like, so physical reasoning is hard. In the case of M-Theory the situation is worse still, because we don’t even have a complete formalism yet, let alone a physical interpretation of it.

There is much tit-for-tat going on with regards to this. Here’s another paper that makes the exact opposite claim, i.e. that observations of this event actually rule out a large number of GR alternatives, including TeVeS: https://arxiv.org/abs/1710.06168 ArXiv is in fact pretty much awash with papers on both sides of the divide. It is difficult for an amateur such as myself to really arrive at a conclusion, but I tend forwards GR as the model that best fits all our data about how gravity behaves. It is also the simplest possible model, and can be constructed more or less from first principles. TeVeS for example would require an extra vector field, two extra scalar fields, and an arbitrary function; that seems very ad-hoc to me, and does not easily relate back to any of our other physics models.

While I understand what you are saying, in terms of logic it is not a valid argument. Just because we do not yet know the full picture of what went on at the time of the Big Bang (but we know some parts of it), does not imply that there must be an outside agent acting on it. For example, in the old days people would get cholera, and put it down to an act of God punishing them for their sins, because they did not know any better. Nowadays of course we know that they got cholera because the water they drank was contaminated. Note though that this does not allow any truth statements either way - the Big Bang (or any other part of science) does not imply a personal God exists, but neither can it definitively rule out that notion. So there is still room for a concept of God, if you so choose. Essentially it always comes down to a personal choice of how you wish to understand the world you live in.

I would disagree with this. Science is an epistemological endeavour, not an ontological one - it is a system to organise knowledge. As such, I would argue that it seeks only to accumulate knowledge about our existence, not an understand of it. This might seem like nitpicking, but it’s actually very important. Especially on science forums such as this one, I very often see people who seem to think that physics (e.g.) is there to seek fundamental truths of the universe - as such, viewpoints can become deeply entrenched, because they are mistaken for absolute truths or falsehoods. But it’s not like that. What we do in physics is make models of the universe, or aspects of it - it’s more like drawing a map of the territory. But such a map is true only insofar as it is a faithful representation of the territory; one can examine how well the map represents the territory, and what limitations the map has. Strictly speaking, saying a map is “true” or “false” does not make much sense, rather, it is a question of the degree of accuracy. For example, Newton is a less accurate representation of gravity than Einsteinian, but it is meaningless to say that either one is true or false.

The former means that the divergence of the gradient of your function vanishes everywhere, so there are no sources or sinks of any gradient (not field) flow. In physical terms, this means that, if you consider a small region centered around some point, the average value of your function in that region must be equal to the value of your function at that point. If the relationship holds everywhere, then you are dealing with a harmonic function, which is physically often a wave field of some sort. Your latter example is a particular form of the Cauchy-Schwarz inequality - it physically means that the inner product of f and g can never be larger than either of these taken in isolation. So in other words, a projection is never larger than either of the vectors/states/functions that are involved in the projection. Both of the above are just common sense, and really quite simple - but you are absolutely right, actually extracting this information from the formalism is a non-trivial task. And that was precisely my point - we can have models of QG that give us a more or less straightforward mathematical statement, and yet we may be unable to physically interpret it. For example, without the entire theory of differential equations, you could not easily extract any physics out of Laplace’s equation, because you would have no way of solving them.

The problem is akin to having a mathematical formalism, but not being able to extract specific predictions from it, because the mathematical tools are missing to work with that formalism. For example, you can know the Einstein field equations, but if you haven’t got a clue how to go about solving them, then you can’t extract any of the physics. So it’s a matter of developing mathematical tools as you go along, and that takes time - which is why String theory appears to have stagnated of late. Actually there is continuous progress, but it’s mostly very technical stuff, and the progress is slow. This issue partly persists even with well-studied models. For example, a complete classification of all possible solutions of the Einstein equations is (to the best of my limited knowledge) still an outstanding problem. Another example is QCD (the strong force) - the field equations are so complex that no closed analytical treatment is possible; we largely rely on numerical simulations as well as simplified approximations. I don’t think there is an alternative to maths when it comes to QG. Of course, it all starts with ideas and approaches, but then these need to be fleshed out with a proper formalism, or else no one will ever know what these models actually say, in physical terms.

There is a substantial number of what I would consider promising approaches, though it is not yet obvious whether a fully self-consistent model of QG is among them - there is always a chance that there isn’t. The trouble is that we have not got the mathematical abilities to fully work out and understand many of these candidate theories, so it is difficult to evaluate their actual value to us. What’s more, we don’t even know what a fully consistent model of QG should look like, and what features it would have. At present the best candidate models would remain Loop Quantum Gravity, Non-Commutative Geometry, Causal Sets, Causal Dynamical Triangulations, Asymptotically Safe Gravity, and M-Theory. This is not a complete list though. M-Theory actually goes a step beyond QG, in that it is a candidate for a “theory of everything” that could model not just gravity, but the entire particle zoo through a unification of all fundamental interactions. There is a candidate theory of QG called Group Field Theory. I can’t really comment on it though, since I am largely unfamiliar with this particular model, except in the broadest of terms. Yes, because before you can even begin to worry about gravity on small scales, you need to first understand the other fundamental interactions, which are orders of magnitude stronger. Doing this leads to quantum field theory, which is the unification of quantum mechanics and special relativity.

Good point...but then, you wouldn’t be looking directly at the mirror either, you’d be looking through an eye piece, which presumably is an arrangement of lenses. Or not? I never owned one of these things.