Markus Hanke

By “global” I meant some region of spacetime that is large enough so that curvature cannot be neglected, not necessarily the entire universe. It is the presence of curvature that precludes the existence of a conservation law in such a region.

Not if the universe has four dimensions, with those fields in it that we know about. It is possible to unify EM and gravity, by introducing a compactified extra dimension, an extra scalar field, plus a few technical assumptions which I’ll skip here - this is called Kaluza-Klein gravity. However, this only has the status of a hypothesis, since no trace of the extra dimension nor the extra scalar field has ever been observed as of yet. There are really more differences than there are similarities.

The concept of “size” does not really make much sense in the context of quantum field theory, because you are dealing with excitations of a field that extends throughout all space and time. The only thing you can do is set up a scenario where you let quantum fields interact - for example, you can shoot another particle at your original electron. In this case there will be a well defined probability to “hit” your electron in a specified region, and by the manner of the resultant inelastic scattering, you can draw conclusions as to the internal structure of the electron. Such experiments have been done, and the result is that there is no indication of them having any kind of internal structure. At the point of scattering (!) they behave like point particles; however, this is only the point where the quantum fields interact, but the fields themselves extend throughout all space and time, so “size” isn’t really a very meaningful concept in this picture. It’s both a point particle, and an extended field, and the two are equivalent descriptions of the same thing.

Unfortunately, he/she has been doing this for as long as I have been on (various different) science forums - and that’s quite a few years now. It’s just the same things being reposted over and over again, without any indication whatsoever that he/she is receptive to what is being said in response. Your post is excellent Mordred, but I think it will - in this case - fall on deaf ears.

Well spotted And that’s exactly the point - most systems aren’t closed ones, and that is why there can be circumstances in which we find highly ordered systems, instead of just random disorder. Because, unlike in closed and isolated systems, in an open and interacting system entropy can decrease locally. That is because entropy is not an absolute value, but a relationship between systems, and it is also observer-dependent. For example, in our solar system, the sun is a source of low entropy relative the rest of the system, so in net terms heat flows only in one direction - from the sun to the planets. At the same time though, its entropy is high when compared to the environment in the early universe. You get the idea.

It is indeed possible to refract gravitational waves (at least in principle); in fact gravitational waves exhibit many of the qualitative properties that “ordinary” waves would also. The main difference is that gravitational waves also self-interact, because gravity is non-linear. This gives them highly complex dynamics, and while aforementioned behaviours exist for them, their quantitative description is very different as compared to ordinary waves. They also exhibit some dynamics that are unique to gravitational waves, and don’t exist for other, linear, wave fields. No. Only regions of geodesic incompleteness (i.e. singularities) are covered by event horizons. Ordinary matter is not.

F is the Faraday 2-form - it represents the electromagnetic field. A is the potential 1-form - the electromagnetic potential field d is the exterior derivative operator.

Not all of curvature per se, but the particular aspect of curvature that is represented by the Einstein tensor. And yes, that is indeed equivalent to energy-momentum. That’s exactly what the equation is telling us: [math]\displaystyle{G_{\mu \nu}=\kappa T_{\mu \nu}}[/math] The two are physically equivalent, up to a proportionality constant. That is so because both the Einstein tensor and the energy-momentum tensor obey the same conservations laws; what’s more, it can be shown that the Einstein tensor is in fact the only tensorial function of the metric that does this. So as you said, they are just two ways to look at the same thing. You don’t need to worry, because it does not need to be treated separately. The physical meaning of gravitational energy is a self-interaction of the gravitational field with itself; colloquially speaking one could say that gravity itself is also a source of gravity. This is encapsulated not as a separate source term (which would be mathematically impossible), but rather in the structure of the field equations themselves. Mathematically this system of equations is highly non-linear, and it is precisely this mathematical non-linearity that physically corresponds to a self-interaction. So you don’t need to worry about it, or include it separately - it’s already accounted for in the field equations themselves.

Since this is the homework help section, I cannot give you a straight answer, but I can point you in the right direction, so that you can go and find the answer through your own research. While the second law of thermodynamics does indeed dictate that entropy (i.e. disorder) can only remain constant or increase, making things more “chaotic” over time, this law does not just apply to any system, but only to systems that have a particular property. Can you do some research and find out what that property is? When you do, the answer to the original question will become immediately apparent.

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.