Markus Hanke

GR uses the Levi-Civita connection, so there is no torsion. Also, if there were any vector fields involved, then those would be 4-vectors, not 3-vectors; and two separate vector fields still do not capture the necessary degrees of freedom. The field equations - like all physical quantities in GR - need to be covariant, so no, you can’t make any kind of explicit reference to an observer. One could also think of it in terms of gravitational radiation fields. These fields extend to infinity, and wave fronts in free space propagate at the speed of light; furthermore you have two distinct polarisation modes. In terms of field quanta, this automatically implies massless spin-2 bosons - which, mathematically speaking, can only “couple” to rank-2 tensors. So this is the lowest rank object that is needed to fully capture all relevant degrees of freedom of gravity.

Un-reasonable statements don’t require reasonable responses. That notwithstanding, I merely applied the author’s very own method of reasoning to a different scenario, in order to highlight the inherent absurdity.

An interesting way to put it! I personally would not consider contentment to be the goal of religious pursuit - but then, what the goal should be is, I think, in many ways a very personal decision. There is no “right” or “wrong” in this, so long as it does not negatively impact on other people or the wider world.

Ok, so the gradient would give you a vector field - which is still insufficient to capture all the necessary degrees of freedom. As I said, at the very least you need a rank-2 tensor field to adequately describe gravity. Time dilation is a relationship between clocks, it’s not a covariant quantity, and it isn’t local either. So, such a thing as a “time dilation field” does not make much physical or mathematical sense. A solution to the Einstein equations is given by a metric - this is the primary and most fundamental mathematical object in GR. Once you have the metric, you can then calculate the relationship between given clocks in spacetime from this.

So if I publish a paper on Invisible Pink Unicorns in my back garden, and no one responds to it, then that means my unicorns must be real? What a ridiculous argument.

The concept of “gravitational potential” can only be meaningfully defined in some spacetimes with very specific symmetries; it, too, does not generalise.

Then that’s perfectly fine, I was not criticising. Generally speaking, the vast majority of people will only engage with the outer form, because that is all they can do and want to do. Only a few choose to go deeper, and a very few dedicate their entire lives to it. To use Christianity as an example - the vast majority of Christians are lay followers, they live normal lives and make their religion part of it. A few Christians then decide to take it a step further and enter priesthood / become preachers, and a very small number go further still and dedicate their entire lives to it by entering a monastic order as monks / nuns. Science is no different in that regard - most people are happy to just engage with a science topic using YouTube videos and pop-sci publications of various kinds. Only a few choose to go deeper. And some choose to go so deep that they make it their life’s work. But not everyone is capable of that, or finds himself in circumstances that are conducive to such an endeavour. It becomes problematic only if people claim that a specific outer form is all there is to a religion, and attach themselves to it. True - and also, most people would not necessarily be capable of going any deeper, even if they wanted to. If you were to send a bunch of random people into the desert or forests to live like hermits do (or even make them join a monastery), it is unlikely that they would make it for very long, and it is even less likely that they would derive any kind of benefit from it. Yes - though I would say that the decision to go into the wilderness generally requires more than “just” unhappiness; it’s a very momentous step. It usually starts with a deep existential crisis of some sort, which leads to equally existential questions being asked; followed by the recognition that leaving those questions unanswered is just simply no longer an option. In the beginning stages, a good teacher is essential and helpful; but once one chooses to go deeper, there comes a point when only direct experiential insight will do, and no teacher can magically convey this. It’s down to our own efforts then. This is again similar to the sciences - a good teacher can teach you the fundamentals, but eventually the point comes when you can only make further progress by thinking about a topic yourself, looking at it from different perspectives, doing your own research...until one day, the penny just drops and you suddenly understand. Teachers can transmit only knowledge, but never understanding. But of course, knowledge is very important since it forms the foundation for all future progress - no understanding can come about if one does not have knowledge of the basics first, so yes, teachers are important.

It is not possible to capture all of gravity’s degrees of freedom with a scalar field theory (or even a vector field); you do require at the very least a rank-2 tensor field to do so. Even if you take just the next “baby step” up from Schwarzschild spacetime to Kerr spacetime, you will find that this concept no longer works. At least in principle you can still derive closed expressions for the relativistic optics in such a spacetime (though the maths are anything but trivial), but they no longer correspond to anything resembling a scalar time dilation field. I invite you to try it out, but be warned - some heavy maths ahead! And the whole thing most certainly does not generalise to arbitrary spacetimes.

Apologies if it came across like that, that was not my intention. I do of course realise that not everyone has religious or spiritual aspirations, or feels any need for those; also, for any given person these inclinations can change over time as life happens. All of this is perfectly fine; religion is not necessarily required to gain an understanding of the human condition. I think what I was attempting to say was that for many people religion/spirituality can be a very helpful tool. But of course (just like anything else) it can also become a trap and a source of enormous suffering, if it is related to wrongly. I personally think this is too simplistic a way to look at it. Of course, most people will relate to their religion in just this way - they attach themselves to their respective outer forms, teachings, rites and rituals, and leave it at this. That’s indeed a “lazy” way of engagement, and requires little else but blind faith. The problem with this is of course that it is not actually transformative, in the sense that rarely does it provide any answers and insights that go deeper than surface level. This is the archetype of your Sunday church goer who shows up for mass, and then goes back to his normal life unchanged and with their same vices, habits, and sufferings. If one seeks real answers to real existential questions, then a far deeper commitment is required; this is why Christian mystics went into the deserts to seek insight, why Buddhist hermits spend decades meditating alone in the forests etc etc. You get the idea. Religion is like a vehicle that can potentially get you to a place of deeper understanding...but that requires deep commitment and lots of effort. You get out exactly what you put in, and that is really not unlike what happens with science. It’s just that the questions that are being asked are different ones.

Just to clarify my earlier comment...it could be true of course, but my point was that we can’t just assume that it is without running the maths first. If I have learned one thing about GR in all my many years of engaging with it, then that would be that it is always good for a surprise or two, and not to trust my “common sense”! Yes, this is a common (and very valid!) source of confusion. The answer to this one lies in realising that within the field equations, the source term for gravity is not energy, but the full stress-energy-momentum tensor. This entity describes the densities, momenta and fluxes of all forms of energy-momentum in spacetime (except nonlinear self-gravitation, which is encoded in the structure of the field equations themselves), and as such it contains contributions of various different kinds, also including angular momentum. To put it very simply, gravitational collapse is governed not just by energy, but by the full energy-momentum tensor; and since it is a tensor, all observers agree on it, and no contradictions arise in the first place. Actually writing down the energy-momentum tensor for a given system (never even mind inserting it into the field equations and solving those) can be a very non-trivial task, however. In the case of a spinning mass on the brink of collapse, I am not aware of any closed analytical treatment; one would need to approach this with numerical methods. Note also that while all observers agree on whether a collapse takes place (or not), they will disagree on when the collapse happens, where it happens (if there is relative motion), and how long it takes.

I consider science and religion (the general concept, not any particular religion) to be different and distinct domains of enquiry, that ask different questions about the world and the human condition. As such, there is no reason why the two can’t coexist harmoniously, in fact it is natural for them to do so. Problems only arise when people start conflating the two - making religious claims about the empirical world, or conversely making “scientific” claims about what is not readily amenable to the scientific method, quickly leads to unnecessary disharmony and arguments (and potentially worse). You don’t use a torque wrench to hammer in a nail - one must pick the right tool for the job at hand. I think both science and religion are equally of value when it comes to understanding what it means to be born human, but they shouldn’t be conflated and confused, because they relate to different aspects of our experience.

This is valid only for spherically symmetric, non-rotating and uncharged gravitational sources in an otherwise empty universe, i.e. in spacetimes that are approximately Schwarzschild. It cannot be generalised to any other case, which is why it is not suitable as a general model of gravity. General Relativity on the other hand represents a general constraint on the metric, i.e. it constrains what form the geometry of spacetime can take, given appropriate initial and boundary conditions. It thus works as a model for gravity regardless of the precise nature of its sources, in any given purely classical scenario.

These scenarios are not physically equivalent - only “us” would measure proper acceleration in our frame, but not the mass in its own rest frame. There is no symmetry between frames once proper acceleration is involved. Do we really know this? There is no known closed analytic interior counterpart to the exterior Kerr metric (unlike is the case for Schwarzschild), so correctly modelling the gravitational collapse of a rotating body is mathematically a difficult problem. Also, the concept of “mass” is not straightforward here, since it is now one parameter in a 2-parameter family of metrics, and hence a global property of the entire spacetime. It’s a gravitational source the exterior vacuum of which can be approximately described by the Kerr metric. It is unlikely that such a thing is feasible, since the kinetics of that system would simply separate the two bodies, long before the gravitational effects would approach criticality. In terms of mathematical analysis, this is a relativistic 2-body problem, which is much more complex than a single rotating mass; such a spacetime is not described by the Kerr metric. I’m sure the analysis can be done, but only numerically.

I am neither Canadian nor American, so this is an outsider’s perspective. I think the wish of culturally separate groups to be self-governing is understandable and bona-fide, since it is based on the hope of preserving said culture, and living in accordance with it. But with every right comes a responsibility, and the responsibility here would be to ensure that the culture actually is being preserved in its original form, and only the members of these groups themselves can do this. This is were many indigenous nations fall down, because they have allowed mainstream culture with all its vices to infiltrate and dilute their own - not intentionally of course, but through negligence. To take Canadian First Nations as an example - Western-styles houses, guns, gambling, cigarettes, booze, gasoline etc etc were never part of their way of life, until they allowed these things to seep in from the outside and undermine their own traditions. If they still lived without any of those things, according to their own traditional ways, there would be no problem; they could self-govern, yet still be part of Canada. Yet now they are neither truly First Nations, nor are they truly Canadian, culturally speaking. Obviously this leads to the conflicts you have mentioned. The same is true for aboriginals in other parts of the world. The above does not come from a place of blame, I just think it is a very difficult issue without straightforward answers. If you look globally, the aboriginal cultures that fare best are those that limit outside influences, even though they exist within a different mainstream culture. Perhaps the Amish people would be a good example here; there are issues with that as well of course, but like I said, there aren’t any easy answers.

GR is hard but not complicated; QM is complicated, but not hard Well said. While these two concepts sometimes coincide (depending on what available information about a quantum system you are looking at), non-separability is more fundamental and more general.

This entire thread is about FTL travel through “warping” spacetime (see OP), so it is reasonable for people to assume that is what you meant. Of course it is possible. Any form of energy-momentum has a gravitational effect - that can be mass, but also less tangible forms of energy-momentum such as electromagnetic fields. You could, for example, alter the geometry of spacetime by generating very strong electromagnetic field (e.g. lasers) in a small region. To give a real-world example, the magnetic fields around magnetars are so strong that they make a non-negligible contribution to the overall gravity of these objects. You don’t need exotic matter to generate gravity. You would only need it in order to set up some very specific geometric configurations, such as a naked Einstein-Rosen-Bridge that is both stable and traversable. For your example with the space station, any form of energy-momentum would work - but you would need a lot of it to have any kind of appreciable effect, which is where the problem lies. It isn’t really feasible from an engineering and resources point of view.

One cannot simply assume that accelerating such a mass will automatically lead to collapse; this is actually a very complicated problem, and would have to be treated mathematically to see what would happen. Even accelerating a fully-formed black hole leads to some non-trivial results (see here for example), and a system just about to undergo collapse is far less trivial still (remember we would need to work with interior metrics here). My intuition is that nothing would happen actually, but I might well be wrong.

Sadly not

This is a reaaalllllyyyy long shot, but...I am planning to take six months out next year to attempt a thru-hike of the Continental Divide Trail in the US (~3100 miles - no small feat!), subject to me being able to raise the necessary funds for the gear I need, and getting a visa. Just wondering if by odd coincidence there is anyone on here who has done this trail, either the whole thing, or a section of it?

You can’t “ride” a gravitational wave, pp-wave, or any other gravitational disturbance for that matter. The wave would pass right through the ship, and, if its amplitude is large enough, would tear it apart or at least cause structural damage of some kind. The other thing of course is that wave-like disturbances propagate at or below the speed of light, so you couldn’t achieve FTL travel this way.

The Gödel metric does not describe the universe we find ourselves in, so it is not really a relevant cosmological solution to the field equations, though it is of academic value. So far as CTCs are concerned, they do also occur in solutions that are much more relevant to us - such as the Kerr metric for example. What I would say here though is that the Kerr metric is an idealised solution, in that its precise boundary conditions are not actually physically realisable. Whether or not that has implications for the presence of CTCs, I don’t know. On a more general level, though allowed by General Relativity, I have would have some fundamental concerns about CTCs. Most notably, in a region of spacetime containing CTCs, the notion of time ordering becomes undefinable, meaning that the evolution of quantum fields and the dynamics of their interactions are no longer unitary. Violations of unitarity are generally a big red flag, and point to something not being right. If I was to hazard a guess, then I would say that topological artefacts such as CTCs only appear in the theory because it is purely classical. Once we have a better understanding of how gravity behaves in the quantum domain, I bet such anomalies will no longer occur.

The source term in the gravitational field equations isn’t density or kinetic energy, but the full stress-energy-momentum tensor. This being a tensorial quantity, it is covariant under changes in reference frame, and thus the same for all observers, regardless of their state of relative motion with respect to the gravitational source. So there isn’t anything that needs to be reconciled - both observers are part of the same spacetime with the same dynamics, they just labels events differently.

As studiot correctly said, the spacetime element is of course also defined in GR (it is, in a sense, the fundamental quantity of the model), the main differences being that in GR it is a purely local quantity, and the coefficients are no longer constants, but functions of the coordinates. Very simply put, this means that, in the presence of gravity, the relationship between two given events will vary depending on where and/or when the events are located. Note that SR is simply a special case of GR, for scenarios where gravitational effects are negligible. They are not two separate models.

That is because those “rates of transfer” are irrelevant to the propagation velocity of the radiation field, which depends only on the permeability and permittivity of the underlying medium: \[ c=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}\] It cannot depend on any other quantity, since that would violate local Lorentz invariance as well as global diffeomorphism invariance. Both emitter and receiver couple to the same radiation field; they just label events in spacetime slightly differently, because their coordinate systems are rotated by some hyperbolic angle.

This directly violates a number of basic principles in physics, most notably: 1. Photons are massless, so they cannot propagate at anything other than exactly c. 2. Under the laws of relativity, light traces out null geodesics in spacetime. This is possible only if their propagation velocity is exactly c 3. The propagation of light is described by Maxwell’s equations - these equations are generally covariant in nature, they take the same form regardless of what the state of relative motion is, or where (in relation to gravitational sources) the experiment takes place. Therefore the propagation velocity cannot depend on the observer. 4. The numerical value of c is a direct function of vacuum permittivity and permeability; both of these are fundamental constants, and do not vary with the state of motion of the observer. Thus the value of c cannot vary either. In a small enough local region, your proposal furthermore amounts to a violation of Lorentz invariance; all available experimental data indicate that no such violations exist.