Markus Hanke

The source term in the gravitational field equations isn't mass, energy or even relativistic mass - it is the full stress-energy-momentum tensor. This is a frame-independent quantity, so all observers agree on it.

That's not correct - in fact, if the body is spherically symmetric and the field is irrotational, then, in three dimensions, you are automatically dealing with an inverse-square law. Since this implies that the field must have vanishing divergence outside the source, a perfect sphere is exactly equivalent to a point source, so far as the form of the force law is concerned. As swansont has correctly pointed out, this is precisely Gauss's theorem, which is in turn a special case of Stoke's theorem, which follows from elementary topological considerations.

I think that, within the context of physics ( as opposed to other domains of enquiry ), the numerical outcome of measurements taken is precisely what constitutes objective reality, and that premise is what the scientific method is based on. One can philosophise about some type of "hidden" reality somewhere "behind" the outcome of measurements taken, but that is at best metaphysics, at worst full-blown philosophy. This may be a tenable point of view if one constricts himself to the classical domain only, but it demonstrably fails if you look at the bigger picture and consider the quantum domain also. Here, if one simultaneously assumes both Einstein locality and counterfactual definiteness ( i.e. the notion that there is an objective reality independent from any measurements performed ), there is an upper bound as to how strongly measurement outcomes can possibly be correlated - that's precisely the Bell inequalities. It is experimental fact that in the real world those inequalities are violated, which means we need to give up either Einstein locality or counterfactual definiteness ( = realism ) in order to correctly model the universe. It is difficult to imagine how giving up Einstein locality can be meaningfully reconciled with the observed causal structure of spacetime - especially given the fact that Lorentz invariance, and hence CPT invariance, appear to be fundamental symmetries -, so my argument stands that we are pretty much forced to let go of the idea that there is some type of reality separate from what we can measure, unless some future model ( quantum gravity ? ) completely overturns our understanding of spacetime and causality, which of course I can't rule out. None of this is really relevant for GR as such, but it does become relevant when one looks beyond GR at the bigger picture.

While I understand what you are trying to say, I nonetheless disagree with you. "Reality" in the context of physics is what we can measure with our instruments; in the specific case of GR, reality is what clocks and rulers measure. We can very easily compare our map ( = curved spacetime ) against real, physical measurements taken of space ( rulers ) and time ( clocks ). That is why I mentioned the Pound-Rebka experiment, since it is a nice case-in-hand to demonstrate this.

@StringJunky : Thank you To put this even more succinctly - if you don't believe that the contour lines on a map are "real", then strap a 50-pound backpack onto yourself and meet me on the summit. We'll talk then

All my life I have been an avid outdoorsman - I hike, I kayak, I camp, and all of it in really remote areas. When I set off on a hike I always take detailed topographic maps of the area with me ( not a fan of GPS, except in emergencies ); the map shows me the structure of the landscape, and by the density of altitude lines I can judge just exactly how steep an ascent or descent is going to be. It is obvious that the map is not identical with the terrain itself - I hike in the real world, not on a map. Nonetheless, there is a clear and very definitive relationship between the map and the territory it represents - the map isn't the territory, but it accurately reflects all its relevant features, such as the shape and geometry of rivers, lakes, hills and mountains, and the steepness of inclines. These features are quite real. The same is true for GR. It is a mathematical model that allows us to describe the relationship between events in spacetime, by considering these events to be points on a 4-dimensional manifold. The presence of curvature just means that events are related differently to one another, as compared to regions which are not curved. This is an abstract description ( =map ), yet once again there is a definitive relationship between the map and the territory it depicts, because when I go and take clocks and rulers to physically measure those relationships, then I am going to find exactly what my map tells me I would find, just like when I go hiking. So is curvature real or not ? Curved spacetime ( = the map ) is not identical to the territory it describes ( = the real universe ), but it captures and accurately describes all relevant features of the real world, just like any really good map would. In that sense, curvature is quite real, because when I go and measure with clocks and rulers ( e.g. Pound-Rebka ), I find relationships between events that correspond exactly to what my map tells me. The precise relationship between map and territory, and what constitutes "reality" in that context and what does not, is largely a philosophical matter. Physics does not do philosophy, it concern itself only with making increasingly accurate models / maps of the universe around us. As such, GR is pretty much as "real" as it gets.

That's your problem right there in a nutshell. Having ideas, imagination, and enthusiasm is good and important in science just like in all other domains of enquiry; however, if the prerequisite knowledge of the subject matter is missing, then none of it is of any value. My recommendation to you would be to first learn physics in considerable detail - maths and all -, because that will enable you to make use of your imagination from a position of knowledge, rather than ignorance; more importantly, it will enable you to judge which ideas may hold some value, and which don't. Only then do you have a chance, however small, to make a meaningful contribution in the field. As it stands, what you have written can only be described as word salad, with a generous bit of dressing on top - and no, I am not being rude, just brutally honest.

Ok no worries, I was just curious My own background is mostly in General Relativity, so the focus is almost always on the electromagnetic field tensor rather than the A field, but I can see your point.

Not quite, or at least that depends on what you mean by "nature". Basically, what I was attempting to point out is that - given the very fact that electromagnetism exists - the general form of the laws that govern the structure and evolution of electromagnetic fields follows from fundamental principles of topology. Now, topology ( being the study of shapes and transformations thereof ) seems at first glance to have very little to do with anything in physics, so this fact is quite surprising. Maxwell himself would not have been aware of this, as the necessary formalism to reveal those relationships was not developed until after his death. What's more, it turns out that Einstein's General Relativity ( the theory of gravity ) is also based on that very same topological principle. I would say that this principle constraints the form that the laws governing those concepts can take, rather than the nature of the concepts themselves. Since we do not yet have a model that can describe both fundamental particles and the microscopic structure of spacetime under one common umbrella, I don't think that the question as to the fundamental nature of electromagnetism can be properly answered. But of course that does not mean that we don't understand how these fields behave. Suppose you have a ball - the boundary of a ball is its surface, which we call a sphere. What is the boundary of a sphere ? Right - it doesn't have one, it is a continuous manifold wherever you go. The boundary of a boundary of the ball is zero, i.e. it doesn't exist. Suppose you have a pyramid. What is the boundary of the pyramid ? It's the four triangles making up its surface area. What is the boundary of that surface ? Again, it doesn't exist - the surface of the pyramid does not end anywhere since the four triangles are seamlessly joined together into a continuous surface, you can take a pen and start drawing a line, and keep going into all eternity without ever having to lift the pen off that surface. The boundary of the boundary of the pyramid is zero. It doesn't exist. And so on. I think you get the general idea. That is the principle in topology that "the boundary of a boundary is zero". The above are both very simple examples, but the principle can be generalised to more complex scenarios in arbitrarily many dimensions, and can be made mathematically precise so long as certain conditions hold. It then makes an appearance in algebraic form, in that if you have two boundary operators on an integral, or two exterior derivative operators together, the result will always be identically zero. This is what I used in post #133. I am not sure I follow you here. Do I understand you correctly in that you would not identify the term "electromagnetic field" with the Faraday tensor / 2-form F and its dual *F, but rather with the electromagnetic 4-potential / 1-form A ?

I think that is a matter of convention rather than physics, but to me, the "electromagnetic field" is the Faraday tensor field, and I think a lot of texts follow that nomenclature - probably for historic reasons. But I guess one could make a case for the connection form to be regarded as "the field". Ultimately though the maths are clear and unambiguous; I only mentioned these things to point out that Maxwell's theory ( or even QED for that matter ) is not just some ad-hoc invention, but based on fundamental principles of nature.

Actually, there is more going on here than immediately meets the eye. The field itself isn't fundamental, it arises from a more fundamental entity called the electromagnetic 4-potential A via exterior differentiation : F=dA. Stoke's theorem then implies that, for the boundary of a small volume V : [latex]\displaystyle{\int_{\partial V}F=\int_{\partial V}dA=\int_{\partial \partial V}A\equiv 0}[/latex] Or, written more succinctly : dF=ddA=0. Physically this means that magnetic field lines have no boundary, meaning they do not end anywhere and can thus only form closed loops; in other words, there are no magnetic charges. That is precisely the "magnetic" part of Maxwell's equations. In the above expression, the last integral identically vanishes because of a very fundamental principle of topology called "the boundary of a boundary is zero". Therefore, the structure of electromagnetism as a model is not just some arbitrary ad-hoc construct that someone came up with, but it is actually based on powerful and very elementary topological principles which at first seem to have nothing to do at all with physics. The electromagnetic 4-potential itself then arises due to a symmetry breaking process from whatever it is that will enable us to eventually unify all the fundamental forces - in that context it is important to remember that the Standard Model as it stands is only an effective field theory, and by no means the last word. P.S. It may be interesting to mention that the same topological principle ( "the boundary of a boundary is zero" ) also underlies the theory of General Relativity, though it is less obvious and requires more complicated maths to demonstrate.

I should point out here that quantum fields are conceptually quite different from classical fields; they are not "supported" by anything, have no source, and do not actually describe the "state" of anything. They are simply a collection of operators attached to each event in spacetime, which may differ in phase from point to point - if you so will, they are a supporting structure which allows us to perform certain calculations at those events. The Standard Model contains at least 17 of those fields ( depending on how you count them ), all of which coexist at each event, and the excitations of which are precisely our elementary particles. As such, they cannot be considered a "medium" in any reasonable sense of the word.

That depends what you mean by "experience". For example, take three test particles which are spaced radially, and let them freely fall towards a central mass, like so : As time goes by, they all approach the central body, but they also increase the distance between them. Why ? Not because there are any forces acting on them ( remember there is no acceleration, so a=0, and hence F=ma=0 on all these particles ), but because they are in a spacetime that is not flat, so as they age into the future ( they can't do anything else ! ), their spatial position will change. That's curvature right there for you - it is not just some abstract, theoretical concept, but something quite tangible, and it can be easily observed. The "engine" that drives all dynamics here is the fact that everything ages into the future, and, because time and space are intrinsically linked, spatial positions and distances change as that happens. This of course also works if you arrange the particles horizontally instead of radially : While the mathematical description might be quite involved ( I plead no contest in that regard ), the actual meaning of the equations is really quite simple, and anyone can grasp it - they are just descriptions of what happens to test particles.

The meaning of "straight" in this context is different than just being "straight in space". What it means is that no forces act on the satellite anywhere along its trajectory - you can easily see this by placing an accelerometer on board, which will read zero at all times, so there are no forces in its rest frame. In other words, the satellite is in free fall at all times. If you plot its world line in spacetime ( not just its trajectory in space ! ), you find that between any two events you choose, this world line is such that it traces out the longest proper time - this is called the "principle of extremal ageing", and in that sense the world lines of test particles in free fall constitute the most direct and straight connection between events in spacetime, and such world lines are called geodesics. This does not imply that the test particle's spatial trajectory ( = the orbit of the satellite ) is a straight line - it implies only that, given initial conditions, it is the trajectory that allows the satellite to trace out the most proper time as recorded by its own clocks. Always remember that we are in spacetime, not just in space.

I don't really understand the point in this - even the standard field equations admit dynamic ( i.e. time-dependent ) metrics as valid solutions, so you don't even need the ADM formalism for this. Why is that so special ?

That is interesting, I have not come across this statement before. Could you elaborate on this a bit ? Let's say I have a composite system of two fermions, and I create an entanglement relationship between them. I am interested only in their spin state, so either particle can be either spin-up or spin-down. Where does uncertainty come into this ?

You are right in that as of now there is no empirical evidence for the existence of wormholes and similar topological constructs; nonetheless, the fact remains that these are mathematically valid solutions to the gravitational field equations. Since those very same equations are otherwise in extremely good accord with experiment and observation, I think a case can also be made for not readily dismissing these phenomena purely on ideological grounds. There is certainly no scientific reason to a priori rule them out. In fact, as recent work on the ER=EPR conjecture has shown, such concepts may well have a role to play in the fundamental makeup of spacetime.

While I do not advocate any particular alternative of GR over another, in this context I thought it appropriate to mention that there is a class of bimetric theories of gravity which closely mimic weakly interacting dark matter effects, without requiring any new particles ( which would need to fit neatly into the Standard Model, with all associated difficulties ). One example - among others - is Sabine Hossenfelder's proposal : https://www.researchgate.net/publication/235472718_Bimetric_theory_with_exchange_symmetry This is of course all conjecture, but IMHO does warrant further study.

The fundamental principle you are referring to is the fact the laws of physics are the same in all inertial frames; the invariance of c is a consequence of this. This principle is not violated in your example, since no matter which rest frame you choose, no relative velocity will exceed the speed of light.

You seem to be asking whether or not the spin state of the particle, which is definite after the measurement, changes over time. The answer is that this depends on whether or not the particle interacts with other particles, but this information isn't given in your setup. You also need to bear in mind that after the initial measurement at t=0, the particle pair is no longer entangled, since wave function collapse always destroys entanglement. The other subtle issue I see here is the question of simultaneity, since t=0 may not necessarily mean the same thing to Alice and Bob, depending on the spatial and temporal relationships between the rooms.

Perhaps it would help to start at the most basic entity for a system in motion, being its Lagrangian. The action for a relativistically moving particle is of the form [latex]\displaystyle{S=\int \left ( -mc^2\sqrt{1-\frac{v^2}{c^2}} \right )ds}[/latex] From this, you can then derive a quantity called the 4-momentum by taking [latex]\displaystyle{p_{\mu}=-\frac{\partial S}{\partial x^{\mu}}=\left ( \frac{E}{c},-\mathbf{p} \right )}[/latex] The usual energy-momentum relation is then quite simply the norm of this 4-vector : [latex]\displaystyle{p^{\mu}p_{\mu}=\left | \mathbf{p} \right |^2-\frac{E^2}{c^2}=-m^2c^2}[/latex] So, the basic idea is that energy-momentum is a 4-vector, and the norm of that 4-vector is defined to be the proper mass of the particle. Because we are in flat Minkowski spacetime, calculating the norm of a 4-vector is equivalent to applying the Pythagorean theorem to the three elements of the equation ( proper mass, energy = time component of vector, and momentum = magnitude of spatial component of vector ). The first step ( Lagrangian to 4-momentum ) can also be made mathematically precise via the calculus of variations.

Harry Potter ? He sure knew his teleportation techniques

Based on what you have written here, what I think is that you know very little about quantum mechanics; as such you quite simply are not in a position to advance any meaningful "theories". What I suggest you should do is engage in a serious and very detailed study of existing quantum theory - this will put you in a position of knowledge and understanding, and that is the only position from which it is possible to make meaningful contributions to physics. Having imagination is great, and it is important, but in the context of physics it is useless without knowledge.

Xinhang, in post #47 you say : which is evidently in direct contradiction to empirical data, and hence nonsense. In post #51 then you suddenly say : which seems to imply that you think time dilation is after all a real phenomenon. You are contradicting yourself here, you can't have it both ways. So does - according to you - time dilation exist, or not ? If you decide to respond to this, then bear in mind that in physics "time" is defined as being what clocks measure; you cannot decouple this concept from its physical measurement, or else you are no longer doing physics.

@geordief, you might like this interactive Minkowski diagram generator - it's great for playing around with various parameters in order to get a "feel" for what is going on : http://www.trell.org/div/minkowski.html