Markus Hanke

The appearance of a singularity in a model of physics generally means that the model has broken down because it has been extend beyond its domain of applicability. It does not mean that the model actually predicts a physical singularity to occur. In that sense, singularities - whether gravitational or at the BB - (almost) certainly are not actual, physical objects; they are more like flags saying “we don’t know yet what happens here”. Mass as a property of elementary particles only appeared at and after electroweak symmetry breaking (~10^-35s) when the Higgs mechanism kicks in; prior to that, all particles would have been massless. So yes, the very early universe contained only various forms of energy - which, however, still has a gravitational effect of course.

This seems like an awfully complicated way to do this. Why not just use Helmholtz’s Theorem? We know that the curl of the potential field gives the magnetic field (by definition!), so this is already fixed. The potential field is also invariant under certain gauge transformations (I think it’s the addition of a scalar field gradient, but I’d have to check that), hence we will always be free to make the divergence vanish, simply be choosing a suitable gauge, without affecting the curl. So in essence, under Helmholtz’s Theorem, the divergence has no physical relevance at all in this.

! Moderator Note Moved to Speculations.

I would just like to add a remark here, perhaps some readers may find it helpful. I am a regular and committed meditator - I practice several hours of formal meditation every day, and have done so for some years. Many of the perceptions described here are common and well known phenomena that naturally arise when the mind settles and becomes concentrated; in the Pāli language they are called nimittā. This can be anything from a slight tickling sensation somewhere, to pins and needles, to a sensation of something moving as a current through the body, to various pains, to full blown auditory and/or visual hallucinations, among other things. A sensation of electrical currents in parts of the body is especially common, from what I have seen. A had a period a few years back when I used to get this regularly, and the sensation of electricity sometimes got strong enough to cause me considerable discomfort, and gave me twitches and involuntary muscle spams during meditation sits. I have heard of people for whom this becomes so strong that they suffer intense pain, muscle cramps, and involuntary movements - they literally “jump” on their meditation cushions. Some people need to temporarily stop sitting because of this. As described here, with a little practice it is easy to induce these sensation at will, and control them to some extent - one can move them around the body, make them stronger or weaker, change their qualities etc etc. I cannot speculate what the underlying mechanisms are, as the human body is not my area of scientific expertise. What is clear though is that body and mind are not separate things, they are intimately connected, so it isn’t surprising that such things may occur. These phenomena are quite natural, and very common among meditators; there are specific ways and methods to address these things, in the context of an ongoing meditation practice. The general advice is to not pay too much attention to them, since directing the focus of attention towards these phenomena will strengthen them and make them occur over and over again. In many specific practice frameworks the occurrence of such phenomena is in fact taken as a sign of progress, since they naturally develop when concentration and single-pointedness become stronger. They can also become a hindrance though, because they can distract from practicing the main technique, and some people become infatuated with these sensations, as they also can feel very pleasant at times. So most of what has been described here is natural and quite well known, and not a cause for concern. This, however, is not: As someone with experience as a volunteer in the emergency services, this would have me concerned; anisocoria isn’t normal (unless you were born with it, which does happen), so I would strongly advise a precautionary trip to your doctor, to rule out other underlying issues.

There are no guarantees here. The best we can do is make the assumption that whatever finds the message has a roughly similar sensory apparatus as we do, and that their mental processes are roughly similar to our own; we can then attempt to construct a pictorial or auditory message in the most general and (to us) universal of forms, and hope for the best. Over and above that, all bets are off. The thing is that all languages are social constructs - words, sounds and pictograms mean to us what we take them to mean because everyone within our social context agrees that they do mean that, and we have been taught those particular conventions in early childhood. Even amongst us humans it can sometimes be very difficult to communicate certain ideas and concepts outside of a given social context, and our attempts at communication with other species in the animal kingdom have met with at best limited success. Communicating to an alien species that may share few or even none of our cultural and social conventions could be exponentially harder still - and potentially disastrous, should we get it wrong. In the worst case, the alien race may be sufficiently different in terms of sensory apparatus and mental processes that there isn’t even a common channel for communication, never even mind a common language. I don’t know how such an encounter would pan out.

Presumably an omnipotent being would have no need to observe the quantum system, he could have knowledge of its entire history without having to collapse it first. Since that knowledge is not accessible to us, this case would be indistinguishable from God not existing.

I do not really wish to get involved in this discussion, as I believe that understanding the human condition should not become a partisan issue. But I do wish to offer two observations: 1. It seems that almost everyone here equates religion with theism, or (even more narrowly) with Christianity. This is misguided - all theistic world views are to some degree religious, but not all religions are theistic, or even supernatural. There are religious systems that are expressly empirical, right here and now in this lifetime. I think it is important to clarify what you all actually mean by 'religion', in the context of this debate. 2. For those of you who know me from here and other forums, you will probably agree that I am all about science - it's a huge part of my life, and I spend a lot of time researching and teaching myself physics, and that's not likely to ever change. Nonetheless, there is also a religious dimension to me - in fact, I live full-time in a monastery, and have plans to ordain as a monk in a contemplative order next year. This dimension is equally as important to me as is science. For me personally, there has never been a conflict between the scientific and the religious/spiritual sides of me. I understand them as complementary domains of enquiry, that ask different questions about the same human condition. My scientific enquiries have helped me gain insights on my spiritual path, and the spiritual practice has helped me gain new angles on scientific issues. So, for both the religionists who reject science, and the scientists who reject anything religious, you need to ask yourselves the question - why does this need to become/continue to be a partisan issue? The "us vs you" mentality isn't helpful, and can - if taken to the extreme - frequently be dangerous. But when approached with wisdom, the two sides have the potential to coexist harmoniously, and inform each other constructively. Just my humble opinion and experience

Are all affluent people honest?

I think everyone is always responsible for their actions. Whether or not they should be answerable for them is another matter - it essentially boils down to the question of how much choice someone actually had in a given situation. Someone’s social environment, upbringing, mental disposition etc may place strong constraints on their behavioural patterns, so they may not have been as free to choose their actions as we’d think. But then again, this is very difficult to measure objectively, because on the flip side you have plenty of people from extremely difficult backgrounds who are not prone to criminal behaviour at all. So I don’t know what the answer is, but it can’t be a simple one.

I did not make any reference to Newtonian gravity or any particular form of potential, I am only using the fact that the energy-momentum tensor has to be locally conserved. The relation I gave follows from Noether’s theorem, and not any particular theory of physics. The point was simply that, if you allow c to vary, this conservation law no longer holds, because the underlying symmetry that gives rise to the conserved quantity is no longer there. If c is not constant, energy-momentum cannot be conserved, irrespective of what else you attempt to change.

Again, gravitational potential - if it can be meaningfully defined at all - is a gauge field with a gauge freedom to choose a zero point, whereas Planck’s constant obviously isn’t. It is not physically meaningful to relate the two in this manner.

Yes it will be. If you look at the above equation, if c is variable, the covariant derivative will contain extra terms including derivatives of c. These terms don’t cancel out, so there is no way to not violate the relation.

So does the idea that c is a variable. Consider the local conservation of the energy-momentum tensor in the presence of gravity: \[T{^{\mu}}{_{\nu ||\mu}}=0\] Since the covariant derivative depends on the metric, which explicitly contains c, and because in your idea c varies in a way that is not covariant, the above relationship ends up being no longer valid. This whole idea puts you in a situation where there is no longer any conservation of energy-momentum, not even locally. This is clearly in direct contradiction to experiment and observation.

Just to add to what has already been said by other contributors here: 1. First and foremost, the notion of "gravitational potential" can only be defined in spacetimes that are stationary (more precisely: those which admit a time-like Killing vector field) and asymptotically flat. It cannot be generalised to more general spacetimes, which makes it useless so far as a general model for gravity is concerned 2. Gravitational potential itself is not an observable, only differences in potential can be observed and measured. This is because the potential has a gauge freedom, in that one can freely choose where the zero point is, without affecting the physics. The same is not true for the speed of light, hence the relation above is trivially and obviously wrong, since it equates two quantities that cannot physically and numerically be equal, on fundamental grounds. 3. A varying speed of light would constitute a violation of Lorentz invariance. This symmetry has been experimentally and extensively tested with modern equipment to extremely high precision, both here on Earth and in the vacuum of space - needless to say, no such violations have ever been found. Given the degree of precision of these tests, any variability in the speed of light can effectively be ruled out far beyond the usual 5 sigma threshold. 4. A variable speed of light would also break CPT symmetry, which underlies the Standard Model of Particle Physics. Since we continue to successfully use and test this model in particle accelerators on pretty much a daily basis, any variability in c can also effectively be ruled out on that ground. 5. Neither classical Maxwellian electrodynamics nor quantum electrodynamics allow for varying values of permittivity and permeability (in the same medium of course). Hence the notion of a varying speed of light is actually in direct contradiction to what we know about electrodynamics. 6. As has been pointed out on another recent thread, a scalar field theory such as this one is fundamentally incapable of capturing all required degrees of freedom of gravity; there is more to gravity than just time dilation! I could probably go on, but these are the points that immediately come to mind without thinking about the issue too much. I'll leave it at this.

No, not including Russian.

I don’t really have one, as I have chosen to live in unconventional ways. Currently I am resident in a monastery, and preparing to ordain as a monk in a contemplative tradition, which should happen sometime next year, all going well. I also freelance as an online translator (I speak several languages) on an as-needed basis, to cover the very few expenses I have. In case you meant academic qualifications - I don’t have any, since I never went to university. The things I say here in these discussions reflect my own understanding of the subject matter; it is always up to the reader to verify any information given by consulting established textbooks, before taking them as fact. Online forums in themselves are never valid sources of scientific information.

As has been mentioned earlier on this thread, the concept of ‘gravitational potential’ can only be meaningfully defined in spacetimes that admit a time-like Killing vector field, which is only a small subset of solutions to the field equations - it does not generalise to arbitrary spacetimes.

Yes, and as it happens I am already familiar with some of these sources from my own studies. All of these papers work with highly symmetric, static and stationary spacetimes, mostly Schwarzschild. None of them makes any claim to the effect that the metric can be replaced with a scalar field, in the general case. If you are asking if you can have scenarios where there are gravitational effects without gravitational time dilation being present between reference clocks, then we have already given you several examples. A lot of interior solutions are of this kind, as are some pp-wave vacuum metrics. You can also set up such scenarios in symmetric spacetimes such as Schwarzschild, by looking at geodesics that are not purely radial. Plus many more. The point is simply this - on a 4-dimensional spacetime manifold, you can have ‘curvature in time’ (gravitational time dilation), and ‘curvature in space’ (tidal gravity). Crucially, both of these can (but don’t necessarily have to) be present simultaneously and be mutually dependent in complicated ways - for example, tidal effects don’t need to be static, they can be time-dependent and propagate, and the time dependence can itself by non-trivial. A real-world example would be spacetime in and around a binary star system. It’s due to this inherent complexity and nonlinearity that the 2-body problem does not have a closed analytical “on paper” solution. Thus, in the general case you will need more than a single number to accurately model the situation. That this is so - i.e. that geodesic deviation on this kind of manifold requires a rank-2 tensor - is not specifically linked to GR, it’s just basic differential geometry. As I have pointed out several times already - yes, you can make this work for certain sets of limited and restricted circumstances. The issue is, though, that it doesn’t generalise, so it’s not a “causal mechanism for gravity”, to quote the title of this thread. The only way for you to know for sure is to write down a mathematical model for your idea, and then investigate what kind of predictions it makes in cases other than purely radial in-fall in Schwarzschild spacetime, and comparing those to available data. I can’t stress this enough, and it is the best advice I can give you. I could keep trying to explain things until the cows come home (as they say here where I am), but until you see things with your own eyes in your own mathematical model, you won’t be able to make progress either way. At this point in time, I do not feel I really have anything further of value to add to this discussion.

Apologies, I need to correct myself, I omitted an index. This should have been \[\xi {^{\alpha }}{_{||\tau \tau}} =-R{^{\alpha }}{_{\beta \gamma \delta }} \thinspace x{^{\beta }}{_{|\tau }} \thinspace \xi ^{\gamma } \thinspace x{^{\delta }}{_{|\tau }}\]

This is irrelevant, as it still cannot model tidal effects, for reasons already explained numerous times. The necessary information content just isn’t there in a scalar field, and it won’t magically appear by taking the gradient. I didn’t mention anything about cosmology, the FLRW metric describes the interior of any matter distribution that is homogenous, isotropic, and only gravitationally interacting. Of course it is most often used as a cosmological model, but doesn’t have to be. The frustrating part about this is that you are simply ignoring most of the things we say to you, which makes me feel like I’m wasting my time with this. Also, claiming that you have “explained” something when in fact you haven’t, is also really frustrating. The other thing is that you still haven’t presented an actual model, you just keep verbally describing an idea in your head - there is nothing wrong with that in itself, it is in fact commendable that you spend time thinking about these issues. Nonetheless, until you write down a mathematical model, you can’t be sure just what the implications are - you obviously think you are right, but you won’t know either way until you actually run some numbers. Then I don’t think you really understand what the term “gravity” actually means, because if you did, you would immediately see yourself that this idea of yours cannot work in the general case, and why. Just this one point is already enough; gravity is geodesic deviation. I’ll write it down formally for you: \[\xi {^{\alpha }}{_{||\tau }} =-R{^{\alpha }}{_{\beta \gamma \delta }} \thinspace x{^{\beta }}{_{|\tau }} \thinspace \xi ^{\gamma } \thinspace x{^{\delta }}{_{|\tau }}\] wherein \(\xi^{\alpha}\) is the separation vector between geodesics, and \(x^{\alpha}\) is the unit tangent vector on your fiducial geodesic. Can you find a way to replace the dependence on the metric tensor in these equations with a dependence on just a scalar field and its derivatives, in such a way that the same physical information is captured? If, and only if, you can do so, then you might be onto something with your idea.

Simplify the expression all the way to the end, given the relations you posted earlier: \[g=\frac{c^{2}}{r}\left( 1-\left(\frac{t_{0}}{t_{f}}\right)^{2}\right) =\frac{c^{2}}{r}\left( 1-\left(\frac{t_{f}\sqrt{1-\frac{2GM}{rc^{2}}}}{t_{f}}\right)^{2}\right) =\frac{2GM}{r^{2}}\] As r->0, the gravitational acceleration increases without bound, and diverges at r=0. This is clearly not what we physically observe, since a test particle at r=0 experiences no net acceleration at all; yet it is still time dilated wrt to some external reference clock at infinity. I’ve been thinking about this some more, and I was actually wrong on something, and need to go back on it - even in Schwarzschild spacetime, you cannot specify all aspects of gravity with time dilation alone; you need at least a vector field of some kind. Consider two test particles (with their own gravitational influence being negligible) which fall freely side by side, but separated by some distance, towards a central mass. They fall at the same rate, so at every point their radial distance to the central mass is the same, hence they experience no gravitational time dilation with respect to each other. However, as they fall, their trajectories will start to converge, i.e. they approach each other as they fall towards the central mass, and eventually collide near r=0. There will be relative acceleration between the test particles perpendicular to their radial in-fall, even though they are not time dilated wrt to one another. This is because even in Schwarzschild spacetime there is tidal gravity - all radial free fall geodesics converge at r=0. You can capture purely radial in-fall via time dilation alone, but not these tidal effects. So even in simple Schwarzschild spacetime this idea ultimately fails; if you use a single scalar field to model gravity, you do not obtain the correct free-fall geodesics which we observe in the real world (unless the free fall is purely radial, which is trivial anyway). In fact, if you write the proper equations of motion for light using only a scalar model, you will find that there is no gravitational bending of light around massive objects, which is of course contrary to observational evidence (see Misner/Thorne/Wheeler, Gravitation, §7.1).

As I have attempted to explain at length, this is true only in Schwarzschild spacetime, since that is a 1-parameter family of metrics. It does not generalise to any other case. I don’t think you have understood much of what I spent considerable time trying to explain. Neither time dilation nor gravitational acceleration are variables in the field equation, and for good reason. Gravity, in GR, is geodesic deviation - the failure of initially parallel world lines to remain parallel in the presence of gravitational sources. It’s a geometric property of spacetime. In 4-dimensional spacetime, you cannot describe geodesic deviation by just a scalar quantity, it requires a higher rank object. This is nothing to do with GR specifically, it’s just basic differential geometry. You can write a scalar field model for the case of Schwarzschild spacetime (simply define a gravitational potential as function of r), but that is only because it is a highly symmetric case - this does not generalise to gravity as an overall concept. So if Schwarzschild spacetime is all you are interested in, then there is not actually an issue; you just can’t claim it is a causal mechanism for gravity in the general case, because it evidently isn’t, for all the many reasons already pointed out in previous posts. As for your last question, I already gave an example earlier - in FLRW spacetime, you have relative acceleration between test particles due to expansion or contraction of the spatial part of the metric, but no gravitational time dilation between those same test particles. Any metric where the temporal part is constant, but the spatial part is not, will be of that nature.

Just to be extra clear - the scenario can of course be treated via GR, it’s just that it’s not possible to do so via pen-and-paper methods. You would need to feed this into specialised software, and let a computer run the numbers. I do not have access to such software, so I can’t give you an outright answer to your original question. You are right, it is a pretty fundamental problem - but many fundamental problems in physics can only be solved numerically. Even in simple Newtonian gravity, if there are more than 2 gravitating bodies, the system can only be treated numerically. It is actually not surprising to me at all that this can’t be done on paper, given that the Einstein equations are a system of 16 highly nonlinear, coupled, partial differential equations. It’s more surprising to me that it can be done if one of the two masses is negligible, giving the Aichlburg-Sexl ultraboost solutions. Spacetime curvature overall is a rank-4 tensorial quantity, the Riemann curvature tensor - it describes how geodesics deviate in any arbitrary 4-dimensional spacetime. Time dilation is only a subset of that geometrical information; essentially, you can think of time dilation as ‘curved time’, and tidal gravity as ‘curved space’. Unless you have very special, highly symmetric circumstances (as e.g. in Schwarzschild spacetime), you cannot separate these two aspects - which is why, after taking account of all the various index symmetries, there are a total of 20 functionally independent components in the Riemann tensor, and you need them all to uniquely determine all aspects of a spacetime’s geometry in the general case. Time dilation alone is not enough, i.e. you can’t replace a rank-4 object that has 20 functionally independent components with just one scalar quantity, and expect to be able to capture the same information. So the answer is no, for the general case you cannot separate time dilation from the rest of your spacetime’s geometry in any meaningful way. This being said, as you introduce symmetries into your spacetime, the amount of information required to uniquely determine its geometry decreases. For Schwarzschild spacetime, you are dealing with a highly special case that is spherically symmetric, static, stationary, a vacuum, and asymptotically flat. Because it admits a time-like Killing vector field, you are able to define the notion of ‘gravitational potential’ here - the Schwarzschild geometry then simply is a family of nested surfaces (spheres) of gravitational equipotential. So for this special case, you can in fact write down a scalar field that is simply a gravitational potential with respect to some reference point (usually the center of the gravitating mass). But this is only possible because Schwarzschild spacetime is so highly symmetric - this does not generalise to more general spacetime, and most certainly not to the set of all possible spacetimes. And even then, the simple-looking form of the Schwarzschild metric is somewhat deceptive, because once you do actual calculations with it (e.g. how long it takes for a test particle to fall along a certain trajectory), things can become fairly complicated fairly quickly, since you need to integrate the relevant parts of the line element.

It’s momentum flux, not energy flux. What you are describing here is a relativistic 2-body problem, for which there is no closed analytical solution to the field equations; you can only treat this case via numerical methods. I don’t know what exactly happens here in terms of GR; I have never done this simulation myself. However, if we slightly change the scenario, then I can give you a definitive answer: let’s say there is only one (spherically symmetric) gravitating body plus an observer whose own gravitational influence is negligible. Spacetime around this mass is simply the Schwarzschild metric. If we now introduce relativistic motion (i.e. mass and observer move at nearly the speed of light with respect to one another), how will that change the gravity exerted by the mass? The appropriate solution to the Einstein equations for this case is called the Aichlburg-Sexl Ultraboost - at first glance this metric looks very different from the Schwarzschild metric, however, closer inspection reveals that these two metrics are actually just diffeomorphisms of each other. In other words, we are dealing with the same physical spacetime, it’s just that events in it are labelled differently. All curvature invariants are the same (this can be explicitly shown, though it is tedious) between these two solutions. Thus, relative motion does not increase gravity; you are still in the same spacetime with the same geometry, it is just “seen” differently (roughly analogous to how different inertial frames in SR are related by a simple rotation of the coordinate system about some hyperbolic angle in spacetime). If this weren’t so, you could construct unresolvable paradoxes just by introducing relative motion, and the model would not be internally self-consistent. I should also remind you that, if we are looking at the vacuum outside the mass, the energy-momentum tensor is always zero there. It is only non-vanishing in the interior of the mass distribution. Therefore, whether there is relative motion or not, you are actually solving the same equation: \(R_{\mu \nu}=0\); the only thing that changes are initial and boundary conditions.

Kinetic energy is an observer-dependent quantity, so it is best understood as a relationship between the two reference frames in spacetime. It is in itself not a source of gravity. Neither one of these are in themselves sources of gravity. What enters into the field equations as part of the energy-momentum tensor are momentum density and momentum flux. These are neither linear nor angular (the distinction is just a convention anyway). If there is any kind of momentum present in a gravitational source, then it will contribute to one or both of the aforementioned quantities, but the way it does so is not always trivial; in fact, finding the energy-momentum tensor for a given distribution of matter-energy can be a very difficult task, particularly if the distribution is not static or stationary. If the kinetic energy is evenly (statistically) spread out over the entire distribution, then you can sometimes simplify things by letting it enter as a contribution to another component of the tensor, the energy density. This is just the last case I mentioned above - refer to equation (16) in that paper. The kinetic energy becomes a contribution to the energy density term of the tensor. Physically this means you are describing a different system (one that has a higher temperature as compared to a reference system), not the same system in motion. That’s because you haven’t produced a model yet, you have thus far only described an idea you have had, and how you yourself understand that idea. The next step from here would be for you to actually write down a model - i.e. a field equation for the time dilation field you are proposing -, and then see what kind of predictions that model yields, and how they compare against experiment and observation. Remember, it is always good to have ideas, but it is for yourself to investigate the scientific value of that idea - you can’t just assume your idea is “right”, and then ask for others to show you wrong. Yes, that is the right approach