Markus Hanke

Yes, in a sense the whole of GR is an approximation to a full theory of quantum gravity - more accurately, GR is the what is called the classical limit of such a theory. But within that classical domain, there is nothing stopping you from making it as accurate as you need to. The obvious limitation here is always computing power, since the less symmetric the problem becomes, the more elaborate the necessary calculations will be.

Very interesting, thanks for sharing it! I hadn’t heard about this paper before, this seems a very reasonable approach to the issue.

While I understand what you are trying to say, may I just point out here that geodesics are never ‘curved’ in the Euclidean sense, since by definition they are world lines that parallel-transport their own tangent vectors at every point, so they are always locally ‘straight’. They also never involve acceleration, since they must be solutions to the differential equation \[x{^\mu}(\tau){_{||\tau \tau}}=0\] I know I’m just nitpicking, but this bit is crucially important in GR

Sorry, I am not certain what you mean by this. It’s more like a conceptual shift...instead of treating the two bodies as separate entities and trying to ‘combine’ them somehow, you describe a single spacetime instead that has more than one gravitational source in it. There is no information lost in this, it’s just a matter of computational effort. Note that the computational resources required increase exponentially with the number and nature of gravitational sources, so if you have more than two ‘simple’ bodies, even powerful computers will take a long time to calculate this. It’s not an approximation, but, like any computation in GR, it is an idealised scenario - at the very least, it will be assumed that the 2-body system is placed in an otherwise completely empty region of spacetime, without other distant sources, and without background curvature. Often, there will be more simplifying assumptions, since each symmetry we impose on the system substantially reduces the computational requirements. Basically, you want to keep it as simple as possible, or else you might be waiting a long time on your computer to finish crunching the numbers.

GR is a nonlinear theory, so gravitational fields do not simply combine. What this means is that, when you have two valid solutions to the field equations and you combine them together, the result you get is generally not itself a valid solution. The only way to describe spacetimes where you have several gravitational sources of comparable “strength” (a relativistic multi-body problem) is to feed the entire setup into the field equations, and work out the appropriate solution from scratch. Unfortunately it turns out that this cannot be done in closed analytical form (i.e. with pen and paper), because the various unknowns in the equations do not neatly separate, as they would for other, simpler scenarios. It therefore has to be done via numerical methods, using computers with considerable computational power.

They fall “down” due to a principle called the principle of extremal ageing. This principle, in simple terms, implies that geodesics in spacetime tend to be the longest (in geometric terms) they can be, given all initial and boundary conditions. Since in an environment such as planet Earth the length of a free-fall geodesic is generally dominated by the time-term within the metric, this implies that such geodesics will tend to be oriented towards regions with higher time dilation, relative to some reference point far away. In other words - closer to the central body. This is why the apple falls down, rather than up - because this is what maximises the geometric length of its world line, given its initial and boundary conditions. It’s important to bear in mind the bit I have highlighted in the end, because initial and boundary conditions do of course play a role in this - for example, if you were to throw the apple upwards with enough acceleration, it might escape to infinity, rather than fall down, because given those initial conditions, that’s the longest possible world line for it. It’s also important to remember that our intuition of what constitutes “long” is once again considerably different from the maths - the purely spatial distance from tree to ground appears very short to us, whereas in fact its geometric length in spacetime is considerable; that’s because in the metric, the time part carries a factor of \(c^2\), which often makes it greater than the spatial part by some orders of magnitude. Mathematically, geodesics are solutions to a set of partial differential equations, hence initial/boundary conditions determine the form of the solution just as much as the equation itself. The equation is, in fact, just the principle of extremal ageing written in mathematical form.

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 \[\xi {^{\alpha }}{_{||\tau \tau}} =-R{^{\alpha }}{_{\beta \gamma \delta }} \thinspace x{^{\beta }}{_{|\tau }} \thinspace \xi ^{\gamma } \thinspace x{^{\delta }}{_{|\tau }}\]

The original derivation of Hawking radiation was done for an idealised black hole called a Schwarzschild black hole, which is determined solely by its mass, and nothing else. In other words, this is a situation where you have an isolated black hole without rotation or electric charge, in an otherwise empty universe with vanishing cosmological constant (= no dark energy). If you change any of these parameters - e.g. by introducing a cosmological constant -, then the Hawking temperature of the black hole will depend on more than just the surface area of the event horizon, and even the horizon structure itself might change. I have not seen a general treatment of this, I only know of an exact solution where global curvature is positive - this is called the deSitter-Schwarzschild black hole. The problem here is that this spacetime is not asymptotically flat, so there is no way to actually define the horizon temperature. Also, there is more than one horizon in this type of spacetime.

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.

What do you mean by “warp bubble”? Are you referring to some specific spacetime geometry here? Is it the Alcubierre warp drive you are alluding to?

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

Every small enough region of spacetime will be locally flat, just as every small enough area of the Earth‘s surface appears locally flat, even though the planet overall is spherical. What “small enough” means depends on the exact circumstances. Over and above that, it is not possible to list “every possible way”, since there are infinity many solutions to the Einstein equations, each of which may or may not contain regions of locally flat spacetime, depending on boundary conditions. Not necessarily. You can have a single mass in otherwise empty space; if you go far enough away from that mass, spacetime will become approximately flat (“asymptotic flatness”). I don’t know what you mean by “warp drive” - we do not have any such device. Sure, just put the object into free fall. The local frame of a freely falling object is an inertial frame, it won’t feel any gravity (“weightless”). Unlike electromagnetic fields, gravity cannot be blocked by any known means. To the best of current knowledge, such a thing as “negative mass” does not exist. If it did, it would free-fall normally towards the gravitational source, just like ordinary matter. However, if the gravitational source is itself composed of ordinary matter, then it will fall away from negative mass; thus, if both masses are of comparable magnitude, you would end up with a run-away effect.

Indeed! I learned all of my physics and maths that go beyond secondary school level in self-study. I felt like it at the time, so no sweat I could do that too I once, just for fun, fully wrote down all of the Einstein equations explicitly without using summation convention - i.e. all the terms involving Christoffel symbols fully written out, without summation over any indices. I can‘t remember how many handwritten pages it took, but it was a huge mess Now image doing that with LaTeX...

Gravitation in GR is geodesic deviation, and thus a geometric property of spacetime; all free-falling test particles experience gravity (they must follow geodesics in spacetime), regardless of whether they have mass or not, and regardless of their internal composition or size. Remember also that within the Standard Model, all fundamental particles are point-like, i.e. any mass distribution is simply a collection of point particles. Relative motion is not a source of gravity; the source term in the field equations is the stress-energy-momentum tensor, which, as being a tensor, is covariant under Lorentz transformations. If that were not so, the theory would not be internally self-consistent. It is important to reiterate that there are two physically distinct types of time dilation - there is kinematic time dilation due to relative motion (which also happens in flat Minkowski spacetime), and there is gravitational time dilation due to curvature of spacetime (which only happens when gravitational sources are present). These two effects can be present simultaneously, but they are nonetheless physically distinct effects.

Spacetime is flat in the interior region enclosed by the shell, not on its surface. The Earth is not a hollow shell, so this case does not apply here. You will then get what is called a relativistic 2-body problem, which is vastly more complicated. If you are a beginner just starting to learn General Relativity (I assume that is what you are), then I would not worry about this particular case, it is better to stick to the basics first. If you have two massive bodies, such as the Earth and the moon, then there will be what is called Lagrange points. To put it very simply, these are locations where the gravitational attraction from the two bodies cancel each other out. If you place a test particle there, it will remain at rest and not be moved by either gravitating body. Spacetime in a small local region around these Lagrange points will also be approximately flat.

I am unsure why you would think that - you have never upset me in the slightest with anything you have said, so no apology is necessary at all. But just to set the record straight anyway: it is highly unlikely that I know more about GR than you do I am merely an interested amateur, and everything I post here - without exception - is entirely self-taught. Unlike Mordred, I have no academic credentials in any area of science; truth be told, I never even went to university at all. My understanding of GR and physics in general is cobbled together from a variety of textbooks over the years. I am on the autism spectrum, and one of the defining characteristics of people on the spectrum is that we tend to get totally absorbed by narrow areas of interest (this is called monotropism) - for me that just happened to be physics, specifically GR, at least in the beginning; so I did a lot of reading and self-study in that area. In recent years my interests have diversified somewhat, and I also got involved with certain areas of philosophy and spirituality. Also, understanding GR is natural and intuitive to me, in a way that does not seem to be the case for most neurotypical people; perhaps people on the spectrum find it easier to step outside established paradigms (in this case Newtonian physics) and look at things from a different angle. We tend to have difficulties with other aspects of life, though. I can only speak for myself here. I have no interest whatsoever in anything to do with politics; I stay as far away from it as I can. Anytime in the past when I needed to get involved in politics (workplace, family, etc) it ended badly for me in some way or another. The main reason would be that, as being on the autism spectrum, I am unable to read social cues and guess at peoples’ social intentions. Social interactions between neurotypical people are a complete mystery to me, I cannot understand them. I function reasonably well in daily life, but that is only because I have learned to mask a lot; it’s not the same thing. To me, politics is a bunch of people with strong opinions, who do not recognise them as being opinions, and mistake them for some kind of reality. So they get terribly agitated when others don’t share them; there is a lot of suffering it it, really, and no one seems to even see that. I have plenty of views and opinions as well, but I tend to be able to recognise them as such (or so I hope), and see how they are changing with time, so I don’t try to push them on other people. They are just constructs of my mind, so ultimately they say more about my mind than they do about the world at large. As for religion and ethics, they are areas of interest to me - but I personally don’t see them as something to be debated or discussed on social media, which is why I don’t participate in those threads. Religion - or rather: spirituality - in particular is something you do, not just some passive view on the world. I see lots of people who call themselves “Christian” or “Muslim” or “Buddhist”, but these are just labels - those same people may speak and act in ways that reveal complete ignorance of the nature of human suffering. And conversely, some of those people I have met who were most at peace with themselves and the world did not label themselves in any way; they just lived a truth that existed within them on a visceral, intuitive level. So religion and spirituality are never external things, they come from the inside; they are lived, not debated. That’s all I can really share with regards to this. Ethics, to me, is the art of finding the path of least suffering, for myself and everyone else who is involved, in any given situation. There is no such thing as “right” or “wrong”, there is only cause and effect. One can write down general principles for this that may hold true in most cultural backgrounds, but ultimately it is again something intuitive and visceral, something that happens inside. Intention has a lot to do with it - if we act from a place that understands the suffering inherent in all sentient life, and consciously choose to act in ways that minimises it to the best of our limited abilities, then the seeds of our actions will generally be wholesome ones. Again, I think it has a lot to do with one’s reasons for being here. I am on this forum for two reasons only - to expand my own knowledge and understanding, and to help others do the same; and very often, these two things are mutually co-dependent, and happen simultaneously. It is no longer about getting anyone else to adopt my own views on things. Ultimately you cannot force someone to understand something; you can only offer them the tools that might enable them to put the causes and conditions in place for such understanding to arise eventually. But different people come from different backgrounds, and they are at different stages of their own journeys when they arrive here on this forum; it does happen that someone just isn’t ready to listen, and then it won’t matter what you say to them, regardless of how rational and scientific it is. They will be unable to see the merit in it. It’s not even their “fault” really, it’s just that the conditions are not right yet for understanding to arise. Getting upset or offended will never help in these situations - most often it is best to simply disengage and walk away. After all, it is their journey, so I don’t need to loose my own balance over it.

There will be different ways to do this, but the simplest one I can think of is a hollow sphere (i.e. a massive shell) in an otherwise empty region of space. Once you go far enough away from the sphere, spacetime will be approximately flat; spacetime in the hollow interior of the shell is also flat (shell theorem); but the region in between is not flat - it has exterior Schwarzschild geometry. Note though that, even though both of these distant regions are flat, if you were to place a clock into the interior hollow of the shell, you would find that it is gravitationally time-dilated with respect to a reference clock in the flat region for away.

Yes, but you need to look at it in geometric terms. You start out at some point P, and make that the origin of your coordinate system. Velocity is a vector, so you draw that onto your chart, which gives you an arrow that points from your origin to some new point P’: P’=0+v. The factor t is a scalar, so what the multiplication v*t means is that it lengthens the vector v, keeping its origin and direction unchanged: |s|=(0+v)*t=v*t That is the meaning of this kind of multiplication - it lengthens a vector along the direction it points in. The resulting quantity s is then just the total length of that new vector, which physically speaking is the point you get to after travelling for a time t with velocity v, being the tip of the new arrow in your drawing.

I am honestly not sure if I follow your thought process correctly, since such a notion as “time dilation gradient” does not make much sense to me. But nonetheless, the aforementioned case of an orbit around a rotating mass should be an example. Another scenario that immediately comes to mind would be two parallel beams of light (or any other pp-wave spacetime, for that matter) - if you fire two parallel beams of light in the same direction, there will be no gravitational attraction between them, even though they carry energy. But if you fire the same two beams of light so that they are initially parallel, but travel in opposite directions (i.e. you let emitter and receiver trade places for one of the beams), then they will indeed experience a gravitational attraction. There will of course be time dilation between a clock inside the volume of dust, and some other reference clock outside of the dust cloud; but there is no time dilation between two clocks that are both located inside the dust cloud. I should have been more clear on this, as I was initially thinking of the cosmological case, where there is no “outside”. As I said, I don’t immediately recall where I saw that proof, it was a few years back when I came across it, and it was in a printed textbook. However, I can offer an outline (!) of my own attempt at proving this, for whatever it is worth. For this, allow me to go back to the basics, and consider what it actually means for a manifold (such as spacetime) to have curvature, and how to capture this mathematically. Imagine you choose some arbitrary point P on your manifold, and pick out an arbitrary tangent vector attached to that point. Now you parallel-transport that tangent vector around a small (i.e. infinitesimal) loop that starts and ends at your point P. The question is - will the initial vector before the parallel transport operation coincide with the final vector at the end of the procedure, regardless of the specific curve the loop describes, and what direction I travel on that loop? On a flat manifold, using standard calculus, the answer is obviously yes (I use single bars “|” to denote ordinary derivatives), since ordinary derivatives commute: \[A_{\mu |\nu \gamma } -A_{\mu |\gamma \nu } =0\] However, if we allow the manifold to not be flat, then the situation changes; following the standard prescription for this (refer to any textbook on differential geometry), we must now replace ordinary with covariant derivatives, which do not in general commute. The degree to which they fail to commute is (I use double bars “||” to denote covariant derivatives): \[A_{\mu ||\nu \gamma } -A_{\mu ||\gamma \nu } =R{^{\delta }}{_{\mu \nu \gamma }} A_{\delta }\] The object \(R_{\mu \nu \gamma \delta}\) is called the Riemann curvature tensor, and it uniquely specifies all aspects of the geometry of a given manifold. The question then becomes how you explicitly calculate the components of the Riemann tensor, i.e. what kind of object is it a function of? For this you need to only remember that GR uses the Levi-Civita connection, which is torsion free; this implies symmetry in the lower indices of the Christoffel symbols: \[\Gamma {^{\gamma }}{_{\mu \nu }} -\Gamma {^{\gamma }}{_{\nu \mu }} =0\] This being the case, you can then work out an explicit coordinate expression for the Riemann tensor from the above equations. I won’t typeset it here now since it is tedious to write in LaTeX notation (you can easily Google it, if you are interested) - I will simply point out that the Riemann tensor turns out to be a function of the connection coefficients and their derivatives only, which in turn are functions of the metric tensor and its derivatives only. So in other words, and that is the point of this whole exercise, given the fact that GR uses the Levi-Civita connection to describe parallel transport, a unique description of all relevant aspects of a manifold’s geometry under GR (i.e. the Riemann curvature tensor) arises from a rank-2 tensor, being the metric tensor. A simple accounting of the indices in the above expressions show that there is no mathematical possibility of any lower rank object (such as a scalar or vector) doing the same job. Which is what we wanted to show. The above is obviously only an outline - you could fill in the details and actual calculations yourself, using any standard textbook on differential geometry. I don’t know how rigorous the above really is, but that’s how I would approach such a proof - quantify the failure of derivatives to commute on curved manifolds; then, given a connection, check what kind of object the coordinate expression for the curvature tensor depends on. It seems pretty simple and logical to me. But if someone here who is actually an expert in the area can think of a better, more rigorous way, or can point out an error in the above reasoning, then I would definitely be interested in seeing it!

Well, you can consider a hollow sphere made from a thin shell of matter. The exterior of the shell looks like any other spherically symmetric body, so it is described by the usual Schwarzschild metric. The hollow interior of the shell however is a different story - no tidal gravity is detected therein, meaning a test particle placed anywhere into the interior remains at rest. At the same time though, if you place a clock into the interior, and somehow compare its tick rate against a reference clock far away on the outside, you will find that it is time dilated, even though no forces (which would cause it to move) are detected locally where the clock is. So this would be an example of time dilation, but no tidal forces. Another even simpler example would be a uniformly accelerated frame in an otherwise empty region of spacetime; again, an accelerated clock is dilated, but there is no tidal gravity. A real-world example of a case where you have tidal gravity in the spatial part of the metric, but no time dilation, would be a region of spacetime that is uniformly filled with dust, in a way that ensures homogeneity and isotropy. The FLRW metric - on which our current understanding of cosmology, the Lambda-CDM model, is based - is an example of this. In this metric the temporal part is constant, but the spatial part is not. I should also mention here that within metrics, each coordinate coefficient can depend on all coordinates, including time. So not only can things vary as you move in space, they can also vary with time, and with any possible combination of the two. So you can get quite complicated spacetimes that are neither static nor stationary, with highly non-intuitive geometries. And if that wasn’t enough, then it needs mentioning that the dynamics of GR are highly non-linear, meaning gravity self-interacts; hence (at least in principle) you can have topological constructs that are formed and held together purely by their own gravitational self-energy, in the complete absence of any “traditional” sources. I think you are beginning to see now that the dynamics of spacetime are very rich and varied - they can’t be captured by just assigning some scalar field. I actually seem to remember having once seen a formal proof that a rank-2 tensor is the lowest rank object required to capture all dynamics of GR, I just can’t remember where I have seen it. If I come across it, I will post it here.