Markus Hanke

The concept that all of these are the result of the same underlying mechanism/field is not strange, and has been studied for the past several decades - the result being the five usual superstring theories, which are themselves the limits of M-Theory for different parameters. The trouble is just that, once you start to draw up the mathematics with a view on making it all internally self-consistent and externally fitting the empirical data, you very quickly realise that the devil is in the details; this unification of fermions, bosons, and gravity is a very complicated beast! In fact this is so complicated that it keeps pushing the very limits of our mathematical understanding and abilities. I salute your enthusiasm and dedication...but at the same time I do have to tell you that potential wells, scalar fields, and simple quantum mechanics will not get you very far on this.

Interesting (though the article is behind a paywall) - I didn’t think of Bach-Weyl spacetime, since this is normally used to describe a massive continuous ring, rather than two separate objects. I had never heard of the Israel-Khan solution, so thank you for pointing this out; a very interesting metric. I do not at present have access to any CAM system (I’m just an amateur), and the metric is too complicated to perform the calculation by hand, mostly because there are off-diagonal terms in it, so I can’t immediately answer your original question. I wonder if one could just treat the influence of the second object (approximately) as background curvature to an otherwise standard Schwarzschild metric, in which case the Schwarzschild-deSitter metric could be used as an approximation around the Lagrange point in question. That would not give the precisely correct numerical value, but hopefully a good approximation, and it would be much easier to do. This would give better results the farther the objects are apart from one another.

Well thank you, I shall do my best It must be said though that I am only an amateur/hobbyist, and that my main area of “expertise” (if you can call it that) is really GR and gravitational physics. I know about gauge theories etc, but not on an in-depth level.

Because angular momentum is a conserved quantity. Stars form (roughly speaking) from dust accretion disks, which themselves rotate - this angular momentum has nowhere else to go, so if the disk becomes hot and dense enough, and the star forms, it must rotate too. The same with the planets when they form from the remnants of the disk. So it’s essentially due to conservation laws.

I will skip the mathematical details - but it can be shown that no vector field model (which push gravity would necessarily have to be) can correctly account for gravity as we observe it in the real world ( see Misner/Thorne/Wheeler “Gravitation” for the formal proof). This is essentially because vector fields do not have enough degrees of freedom to do so - you need at least a rank-2 tensor field to correctly model gravity. This rules out anything based on a push or pull concept, except as an approximation in the weak field regime; this is why Newtonian gravity (which is a vector model, but with a pull instead of a push) works fine in the weak field limit, but fails in the strong field domain. If you go about it really smartly, you may perhaps be able to construct some kind of a push model that gives the right results for the weak field limit, but you will find that you come across all sorts of unexpected and cumbersome problems, and in the end you will be left with many more questions than you have answered. Trust me on this, because it has been tried many, many times by different people. Another important point to remember is that gravity is demonstrably non-linear in very complex ways, which you will find very difficult to replicate via mechanical models of push/pull. As for your question how we know that matter is not shrinking - we know this because neither the weak interaction nor the strong interaction are scalable in this way. If matter was shrinking, composite particles such as nucleons and mesons would simply dissolve over time into a gooey mess (or not form in the first place), and decay rates of certain unstable particles would be very different from what they are in the real world. We can also test for such shrinkage directly by examining whether the fine structure constant has been changing (which it would have to, in order for the strength ratio of electromagnetism to the other interactions to remain the same); this has been done in different ways, like this for example: https://en.m.wikipedia.org/wiki/Natural_nuclear_fission_reactor#Relation_to_the_atomic_fine-structure_constant So all in all, what you propose has already been tried many times by different people, and we know that it simply does not work, for a variety of reasons. You simply cannot construct a self-consistent model of push gravity that is both in accord with what we know about the rest of physics, and with experimental data. It can actually be shown that this is not possible, in general ways.

And this, my friend, is how advancements in our understanding of the world are made - by looking at things in whole new ways

This was just really meant as an analogy.

The quantum field is the more fundamental entity here, so a particle is the excitation of a field - like a wave is an excitation of a body of water. You can have a field without any excitations, but you can’t have an excitation (i.e. a particle) without the field. The entire thing - symmetry breaking, the Higgs mechanism, the Higgs field, and its excitation being the Higgs boson - were theoretical predictions made by Prof Higgs long before the particle itself was experimentally found. So basically, he suggested this mechanism based on the question of how fundamental particles in the Standard Model obtain their different masses, and the detection of the Higgs boson later showed that this model was indeed a good one. A field is a mathematical object that assigns some quantity to each point in spacetime. For example, you could decide to assign a single number - such as temperature - to each point on earth’s surface; that would be an example of a scalar field. Or you could decide to assign a force vector to every point in some region of space (maybe outside a magnet etc); that would be an example of vector field. And so on. Quantum fields assign mathematical objects called “operators” to each point in space and time. It depends on exactly how you count them (not as trivial a task as it may naively seem!), but most commonly we are counting 37 quantum fields in the Standard Model as it stands today. That is after symmetry breaking, i.e. when all particles have obtained their respective masses and interactions. It’s essentially one for every fundamental particle, whereby particles and their antiparticles are counted as separate fields. Like I said, the exact number depends on exactly what you consider a single field. Gravity does not fit into this, because it cannot be meaningfully described as a quantum field (well, actually it can, but the model you get is physically meaningless, so we are disregarding this). Yes. In quantum field theory (QFT), everything is described in terms of the fields - they are taken as the fundamental objects.

I am unsure what you are asking, to be honest. In general terms though, a choice of direction is arbitrary only if the situation you find yourself in has a corresponding rotational or translational symmetry. For example, if you start off at the International Space Station and you go DOWN, then you will gain energy (free fall). But if you go UP, you need to invest energy (thrust). So these cases are distinguishable, and hence not equivalent. On the other hand, if you go around a perfectly round and featureless body that is not itself rotating, then which direction is “clockwise” and which one is “anti-clockwise” is a completely arbitrary label, since there is no physical difference at all (assuming there is not some outside point of reference).

It is more complicated than this. Basically, you need to go back to the quantum field theoretic description - each type of particle is an excitation of a quantum field. Mathematically, such a collection of fields is described by a Lagrangian, which will have several terms in it; only terms that are quadratic in the fields correspond to mass. Initially an overall Lagrangian may be invariant under some symmetry group, and not contain such quadratic mass terms; however, if you break the symmetry, suddenly extra terms can appear in the Lagrangian, and some of them may be quadratic, corresponding to massive fields and their interactions. Each of these terms will have a constant in it that is called the coupling strength; and the mass of particles arises from the constant (through some maths). So basically, mass comes from how strongly quantum fields interact, specifically with the Higgs field. This is very inexact and non-specific, but you might get the general idea. No, because the source of gravity is energy-momentum, not just mass. There may not have been rest mass back then, but there definitely was lots of energy! Even today, light (which is massless) has a gravitational effect, due to the fact that it carries energy in the form of momentum,

Surprisingly, no. As a matter of fact, above a certain energy level, all particles are massless and move at exactly c. In a way, masslessness is the natural state of all particles. Mass comes about when ambient energy drops below a certain level, which triggers a process called “spontaneous symmetry breaking”. You then have something called the Higgs mechanism kicking in, which means that most (but not all) particles begin to interact with the Higgs field, which gives them mass and slows them down. So mass is actually an acquired property, it is not intrinsic to fundamental particles; the only intrinsic thing is the degree by which they interact with the Higgs field. What this means is that in the early universe, before a certain point in time, “mass” did not exist - all particles were naturally massless and moving at c. Only when the universe cooled down enough, did mass “happen”.

On a conceptual level, a clock placed at such a Lagrange point will be time-dilated by some amount, as compared to some reference clock far away. However, calculating the exact amount of the time dilation is quite a non-trivial problem, because in GR the metric of such a two-body system is not just a linear combination of two (e.g. Schwarzschild) metrics. In fact, I am not aware of any closed analytical solution to the two-body problem, so this would probably have to be done numerically. Just by thinking about this scenario, I can tell you that the time dilation will not be numerical constant, but a time-dependent function in whatever coordinate system you choose for this scenario. This is because the only way for this system to be stable is if the bodies revolve around a common center of gravity, which means that there is a non-vanishing quadrupole moment, and hence the system will loose energy via gravitational radiation. Therefore the orbits of these bodies will decay over time, affecting the time dilation factor in some non-trivial manner. This one is easier - there will be time dilation compared to a reference clock at infinity, and the dilation factor will be the same at all points within the hollow cavity. This is because spacetime within the cavity is everywhere flat (Birkhoff’s theorem). The time dilation itself depends only on the total mass of the surrounding shell. You can picture this like a tabletop mountain - the top of the mountain is everywhere flat, but it sits at a different level than the surrounding countryside, and the region in between (i.e. the slopes of the mountain) is curved. Hence a clock will be dilated if compared to a far-away clock, even though spacetime in the cavity has no curvature - but of course, spacetime between the cavity and the far-away clock is curved.

I think, based on several bits and pieces that we already know, it is reasonable to assume that in a region where QG effects are non-negligible, spacetime itself will increasingly cease to be smooth and regular. This means that the notion of “separation between events” will become at best fuzzy, at worst meaningless in such regions. That will also mean that the very notion of “speed” and its invariance will loose its meaning. In short - the very concept of Lorentz invariance will likely not make sense in a QG regime, and neither will things such as speed, duration and distance. Note that in the absence of a fully worked out QGT, this is speculation. This is indeed a very reasonable and important question to ask! The short answer is that, for a region of spacetime to be considered special relativistic, its geometry has to be Minkowskian - in practical terms this means that any two neighbouring events will always be related to one another in the same way, regardless of where and when you are within the region in question. This means that a number of restrictions apply: There is no curvature (i.e. the Riemann tensor vanishes everywhere in that region) The temporal and spatial parts of the metric have opposite sign The spacetime is everywhere smooth, continuous, and differentiable Among some other, more technical ones. Essentially, (1) means that gravitation can be neglected in the scenario in question; (2) means we are dealing with a type of hyperbolic geometry, so inertial frames are related by hyperbolic rotations in spacetime; (3) means that spacetime can be described as a manifold, so it’s a continuum of events, and all world lines are continuous and differentiable everywhere. Note that there is no mention here of acceleration, because SR is more general than just inertial frames; it can also handle accelerated observers just fine, so long as the above three criteria apply. For a frame to be inertial is a more restrictive condition, because we also need to add the requirement that proper acceleration vanishes everywhere, meaning we consider only geodesics in spacetime, as opposed to just any arbitrary world line. GR then generalises this by abandoning (1), allowing non-vanishing curvature, and hence non-trivial relationships between events. The limits of GR then would be given by (3) - if we have a situation where energies are so high that spacetime itself becomes subject to quantum fluctuations, we can no longer describe it as a smooth and differentiable manifold, making even simple notions such as distance etc problematic. This is where QG has to step in, then.

Well, I found myself missing the weird and quirky world of amateur science forums

Thanks Well, for all intents and purposes, we can almost use “c” as a synonym for “laws of physics”, since most laws at least in the classical domain explicitly depend on it in some form or another, though that might not always be immediately obvious. It’s just two ways to look at the same thing, really.

You can think of c as short-hand for “causal structure” - if the causal structure between events is the same within two frames, then c is invariant between these frames. This applies principally to inertial frames, since otherwise the relationship between events will explicitly depend on proper acceleration (which is an absolute quantity that all observers agree on). Do note though that sufficiently small patches of spacetime are always approximately inertial; physically this means that the global speed of light may vary in non-inertial situations, but the local speed of light does not. This is true even in the curved spacetimes of GR, and can be directly - and relatively easily - tested. In the specific example I gave about the radiation, the invariance is encapsulated in the d’Alembert operator - if it is invariant between frames (i.e. the same physical propagation of radiation), then so must be c, since it explicitly appears in its definition. The invariance in question is Lorentz invariance, which is specifically an invariance between inertial frames. Its domain of applicability does not extend beyond this. However, since all small enough patches of spacetime are approximately inertial, Lorentz invariance applies to all local regions, everywhere and always, so long as they are small enough. There are no known exceptions to this. As a matter of fact, the Standard Model of Particle Physics - which describes the dynamics of fundamental particles - critically relies on this, since Lorentz invariance gives rise to CPT invariance (and vice versa!), which a basic symmetry underlying the Standard Model. Without Lorentz invariance, the particles we observe could not exist in the form we find them. For example, one immediate consequence of Lorentz invariance being violated locally would be that particles could not have the property of spin. They apply within most of the region below the EH - only in the immediate vicinity of the central region of geodesic incompleteness do they begin to break down. Precisely how far below the horizon you can go before quantum effects become non-negligible depends on the particular spacetime geometry you find yourself in (among other things).

The basic assumption of SR is actually both simpler and more general than the one you have quoted: namely, that the laws of physics are the same in all inertial frames. Practically speaking, this means that there is no local experiment which you can perform in order to distinguish between two given inertial frames - the necessary and sufficient condition for this is that the causal structure of these two local spacetime patches must be identical. The easiest way to experimentally probe the (classical) causal structure of local spacetime is to set up a source of radiation in vacuum, and examine how this radition propagates through space and time. So let us consider some source-free radiation field in vacuum; in any arbitrary inertial frame, this field will obey the general homogenous wave equation [math]\square f(x,t)=0[/math] with some function f(x,t). SR is now telling us that the same is true in all other inertial frames as well, meaning that no matter which inertial frame you look at, the local radiation field will always obey the above wave equation. This necessarily means that c must be invariant between inertial frames. This is something we can of course test, which has been done innumerable times with many different setups, both directly and indirectly. No. But what we can do is look for clues as to whether the fundamental constants have changed over time, or not. One way to do this is to look at the phenomenon of natural nuclear fission reactors, such as the one in Oklo: https://en.m.wikipedia.org/wiki/Natural_nuclear_fission_reactor#Relation_to_the_atomic_fine-structure_constant Since the fine structure constant directly depends on c (and vice versa), this would seem to indicate that - at least in the past 2 billion years or so - there has been no change in the numerical value of c.

That is because we now live in a world where it is easy to access the required information, and learn about the model. Not too long ago, the necessary resources ( i.e. textbooks ) were something only found in major university libraries, so the vast majority of the general populace had little incentive to go and access it; nowadays, most of the needed information is only a Google search away, and hence available to anybody who is interested in the subject matter. Also, at its heart, GR is just really simple geometry - all those tensors with their indices and fancy symbols look intimidating at first, but the underlying principles of geometry are very straightforward, and can be understood by anybody. The major issue I see is that there is a lot of misleading and plainly wrong information about GR out there, and the average Joe will not have the required knowledge to judge the accuracy of any given source. The pop-sci media has a lot to answer for, in that regard.

Spacetime is completely flat in SR, there is no "stretching". The effects of SR aren't merely optical illusions explainable by changes in light, they are real effects with real physical consequences. The "atmospheric muon" case is a good example in that regard.

Either that, or if your free fall frame is small enough so that tidal forces are negligible.

Which two effects do you mean ?

The case of a satellite orbiting a central body that is approximately spherical and has only a small amount of angular momentum, is very straightforward - even without explicit calculation you can easily guess what the outcome would be in general terms. Only if you need actual numbers ( e.g. the amount of clock adjustment necessary for GPS satellites ) would you have to get down and dirty with the maths. That being said, not all scenarios are this straightforward - in more complicated setups such as multi-body systems involving magnetic fields for example, the situation requires some serious calculations involving a lot of computing power to arrive at a conclusion. Correct, but if you mean gravitational time dilation, then it is the other way around - the clock closer to the central body is dilated with respect to a more distant clock. But yes, the situation is not symmetric - if you swap frames, then the more distant clock is "sped up" relative to the close-by clock. For simple scenarios such as stationary spherically symmetric bodies, gravitational time dilation can be taken as a function of gravitational potential ( but remember that this does not generalise to more complex scenarios ).

Better be careful here - when you put an accelerometer into free fall, it will read zero at all times, so where's the "force"

I think I should point out that we need to distinguish between kinematic effects in SR ( due to relative motion ), and gravitational effects in GR ( due to the metric not being constant ). They are best treated as distinct effects, and do not follow the same laws. In SR, relative motion always leads to time dilation and length contraction, never to the opposite, because inertial frames are always symmetric. In GR, there is generally no symmetry between frames, so while one observer may see clocks dilated and lengths contracted, another observer may disagree and say that clocks are sped up and lengths are stretched out. What's more, if you have both gravity and relative motion simultaneously, then these effects will overlap and combine ( but not linearly ), so it is not always easy or straightforward to figure out who measures what; oftentimes brute force is required, in the sense that one needs to put pen to paper and actually do the maths. Once again, this is best avoided from the outset by using covariant quantities.

Yes, they are manifestations of the same phenomenon. Take the atmospheric muon experiment for example - an observer on Earth sees the muon come at him at speed, and thus sees its average decay time to be dilated. In the rest frame of the muon, it is the Earth that approaches at high speed, and hence the distance to the Earth's surface is length-contracted. These are just two sides of the same coin, they are not separate phenomena. The physical outcome then is something both observers agree on - the muon reaches the surfaces and is detected there, which is an empirical finding in both frames.