Markus Hanke

I think it is important that you actually read the replies you get on here, because otherwise you will just keep going in circles. As already explained, there is no force acting on the galaxies. There are no forces involved in any of this at all. Gravity is not a force - it’s a geometric property of spacetime. If you drop an accelerometer, it will read exactly zero at all times (you can try that out yourself at home), so as per F=ma, with a=0, there is no force. Yet it will still fall under the influence of gravity, and according to its rules. Also, the weak/strong/EM interactions are not forces in the Newtonian sense either - they are interacting quantum fields. There are no mechanical forces involved anywhere in this. I think your basic problem is that you assume the universe and everything that is in it to be Newtonian (essentially the kind of physics you learn in high school) - but in reality it isn’t. Newtonian mechanics is just a highly simplified approximation that applies only under very limited circumstances. Even on comparatively small scales such as the solar system, Newtonian physics already fails miserably. Cosmology then is very far outside its domain of applicability. Trying to ask about what forces act on galaxies etc is hence largely meaningless, because the very concepts are essentially meaningless in the context of cosmology. There are other things at play here. All of this can very easily be understood in the framework of spacetime geometry - but if you do not acknowledge that as a valid concept (your own prerogative, of course), then there will be little point in this discussion.

This does not make any sense, I’m afraid.

Again, you need to remember that space is not any kind of mechanical medium, and that there is no motion involved, in the sense that no forces act on anything. If you were to attach an accelerometer to any of these galaxies, it would read exactly zero at all times, so there is no acceleration and hence no forces that act on anything. All that happens is that the distance between galaxies increases, because space there expands - so there is relative/apparent motion due to the increase in distances, but no local motion that involves forces or the transfer of energy. It’s purely a geometric phenomenon.

I don’t understand this question; can you explain a bit more? No, gravity is distinct from the weak, strong and EM interactions. They are not the same thing, and function in very different ways. Again, I am unsure what you mean by this. Space is not a separate entity in and of itself, and hence it does not “interact” with anything. It’s best understood as a background, a “stage” of you so will. I think a good way to look at space(time) is as a collection of events, and the relationships between these events are the geometry of spacetime. On very small scales, as on atomic and subatomic levels, gravity plays almost no role at all under normal circumstances, since its coupling strength is very much weaker than any of the other interactions by many, many orders of magnitude. There are, however, scenarios where gravity becomes so strong that it cannot be neglected even on small scales (e.g. the interior region of black holes, or the very earliest times in the evolution of the universe) - but we are not currently able to describe such domains, because unlike the other three fundamental interactions, gravity cannot be straightforwardly quantised, since its nature is very different from the one of the other interactions. This is currently an area of very active and ongoing research.

The mechanism would be the law of gravity itself. The macroscopic behaviour of galaxies etc across very large scales is taken to conform to the same law of gravity that also governs small scales, such as the motion of bodies in our solar system. We know from experiment and observation that this law (being the Einstein equations) is valid to a very high degree of accuracy on scales on the order of the solar system - from this, we extrapolate to larger scales, and the tendency to expand naturally emerges. So in essence, this tendency happens for the same reason why a rock falls towards the surface of the earth, when released somewhere high above. It’s a manifestation of gravity. It’s not so much an assumption as an extrapolation - it being that gravity works the same way on large scales as it does on small scales. If that extrapolation is accurate, then metric expansion naturally happens. Why is this a problem? We have no real reason to believe - as of yet - that gravity works differently on large scales than it does here in our neighbourhood. Of course, it is possible that gravity is scale-dependent - in fact, many alternative models of gravity have been developed over the past several decades that are based on precisely that possibility, so modern physics is definitely open to this idea. But the fact is also that none of these models have been able to match experiment and observation with the same degree of accuracy as General Relativity does. As “illogical” as standard cosmology may appear to the untrained eye (and I do grant you that it can appear that way), it is actually the simplest possible model to explain what we can observe. Most alternative models are very much more complicated, and require even more illogical assumptions.

Space is a 3D hypersurface of simultaneity, for some fixed instant in time. Spacetime is the collection of all such hypersurfaces. This question does not make sense, because there is no acceleration involved, so the galaxies do not “move” as a result of metric expansion. It’s only the empty space between the galaxies that expands. This phenomenon has to do with the geometry of spacetime, not with any mechanical forces. I don’t know what you mean by this - space is not a medium, it doesn’t “flow”. That’s because the question does not make any sense in the context of physics.

It’s space that expands, not spacetime. This is an important distinction in this context. The tendency for the spatial part of it to expand is intrinsic in this type of spacetime geometry; this means it’s a natural feature of this particular type of geometry. It requires no further cause or agent, other than the nonlinear law of gravity itself. This is somewhat analogous to shaving foam - once released from its spray can, it will have a natural tendency to expand, without the need for any external catalyst; this tendency is already intrinsic in its chemical composition and their physical properties. The function of dark energy is only to regulate the rate at which the expansion happens - you can accelerate or decelerate the expansion over time, or - if its distribution is chosen in just the right manner - bring it to a halt. You can again liken that to shaving foam - its rate of expansion depends (assuming normal atmospheric pressure) on ambient temperature, just as the rate of metric expansion depends on the distribution of dark energy. They aren’t. What is expanding is only the space between them, so it’s the distance between galaxies that increases over time. In general, expansion happens only in regions that are not (or only very weakly) gravitationally bound; also, this expansion is noticeable only across very large scales, on the order of dozens of MPc and above.

GR is not intended as a complete description of the universe, it’s specifically a model for gravity (which is what this thread is about), that is all. And as such, it does a remarkably good job.

That means the OP is approaching this subject from the wrong angle, because GR is the proper description of gravity; Newtonian physics are only the weak field limit. And issues about who observes what don’t arise if the proper formalism is used right from the start. Let’s think about this for a minute. First of all, “energy” - when taken in isolation - is an observer-dependent concept, so observers will in general not agree on its conservation, or lack thereof. It’s therefore unsuitable as the source of gravity. So before we can do anything else, we need to ask ourselves just what it is that is actually conserved. To answer this, we consider a small patch of spacetime, and apply Noether’s theorem - out of all the possible symmetries, we find that it is time translation invariance that gives us a conserved quantity related to energy; that is the stress-energy-momentum tensor. It is locally (!) conserved in the sense that [math]\displaystyle{\triangledown_{\mu}T^{\mu \nu}=0}[/math] Two things need to be noted here: first of all, the fundamental concept here is not energy-momentum, but spacetime. It’s from spacetime and its local symmetries that, via Noether’s theorem, the notion of energy-momentum arises. Therefore, saying that gravity is the result of the conservation of energy-momentum is missing the point - gravity is the result of the geometry of spacetime. Energy-momentum just forms the source term in the dynamical laws; that’s not the same thing. Secondly, energy-momentum is a purely local concept, and as a tensor it is a purely local mathematical object. Its conservation cannot be easily extended across regions of curved spacetime, because if you integrate the above expression across some volume of curved spacetime, you are left with non-covariant curvature terms that do not identically vanish. So physically, energy-momentum is conserved everywhere locally, but in general no conservation law exists for global regions. Since gravity is clearly a global concept, this will again not get us very far. This being said, the above expression is one of the constraints that must hold when we construct the field equations from the source term. If the energy-momentum tensor is conserved in the above manner, then so must be the entire other side of the field equation. When we bring together the entire list of mathematical and physical requirements needed to make everything self-consistent, we find that the only object that fulfills them all is a tensor constructed from the Ricci tensor, less its trace; this is called the Einstein tensor. Up to a proportionality constant (and a term of the form const*metric, which we ignore here), this fixes the field equations: [math]\displaystyle{G^{\mu \nu}=\kappa T^{\mu \nu}}[/math] And we find that the Einstein tensor is conserved just in the same way as the energy-momentum tensor: [math]\displaystyle{\triangledown_{\mu}G^{\mu \nu}=0}[/math] To summarise: the conservation of energy-momentum plays a role in determining the form of the field equations, but it isn’t in itself where gravity comes from. This should be rather obvious, because in vacuum the above equations (after trace-reversal) reduce to [math]\displaystyle{R^{\mu \nu}=0}[/math] There is no energy-momentum, but we still get non-trivial solutions (i.e. gravity) to these equations.

There is one fundamental issue in this thread that hasn’t even been acknowledged - “gravity” cannot be described as a scalar field; it can’t even consistently be modelled as a vector field. It’s a rank-2 tensor - and it has to be, in order to capture all relevant degrees of freedom. What’s more, real-world gravity is self-coupling, and hence non-linear, unlike any of the concepts bandied about in this discussion. It is therefore futile to play around with notions of acceleration and energy, since at the very best this would produce something akin to linear Newtonian gravity, which is not a complete description of what we see in the real world. So what is the point in all of this?

The top-down approach (trying to police what appears in the media) is never going to work, in my opinion. The best chance we have on this issue is going bottom-up - education is the key. We need to teach people the critical thinking skills that will enable them to distinguish between what is real and what is fake, or at the very least to question suspicious claims. To some of us these skills come naturally, but to the vast majority of the populace they do not. However, at least to some degree, this is a skill that can be taught and learned - I think it is high time we make engagement with media a subject in our school curriculums, instead of taking it for granted that people somehow just have the necessary skills. Is education a magic bullet? Of course not, but it would at least alleviate the problem.

While that is indeed the basic idea, I should clarify that the notion of “event” in GR is not the same as the term “event” in everyday speech. A GR event is simply a point in space at an instant in time, meaning a point on a 4-dimensional manifold. When you describe a real-world object that persists over a period of time, then this becomes a collection of events, i.e. a world line in spacetime. But yes, in GR terms, the “universe” is the collection of all points in space at all instances of time. GR is a purely classical model, meaning it does not account for any quantum effects, such as uncertainties in measurement outcomes. That is what I meant when I mentioned that everything is deterministic in GR. You can try to interpret QM in the context of spacetime, which basically leads you to the “many worlds” interpretation. That is indeed a gigantic collection of events!

It’s like one of those old style film projectors - you project a rapid succession of frames onto a screen, and as a result you get the illusion of motion. But in reality, there is no motion, because the reel of film itself is a completely static construct. All frames are there already, eternalised on the reel, and never change; in fact, the very notion of “change” is meaningless here, except as a relationship between static frames. But neither the frames themselves, nor the collection of all frames (the film reel) ever changes in any way. In the context of GR, there is also not really any such thing as motion, or the passage of time; those are only auxiliary concepts that are “left-overs” (so to speak) from the old Newtonian paradigm. For example, we usually think of the moon revolving around Earth - a dynamic process, described by an elliptic orbit in space, that repeats in time. But in GR, we no longer separate space and time, so the moon becomes a bundle of world lines in spacetime, that looks like a helix winding around another bundle of world lines that is the Earth. This geometric structure encompasses both space and time, and is itself completely static (just like the film reel), because it encompasses all locations in space of these objects at all instances in time throughout their history. Any notion of “past”, “present”, “future” and “motion” is now completely arbitrary, and not fundamental to the universe. Everything that was, is and will be, is there - and that collection of all points in space at all instances in time is not embedded in anything else, and hence the notion of “change” is meaningless if applied to it. As such, GR is a completely deterministic view of the world (ref “block universe”). The same is true for the spacetime geometry that accompanies the world lines (or for gravitational waves) - they are just continuous deformations at all points in space and all instances in time, and themselves completely static. But of course, one can always recover the notion of dynamics, by “slicing up” spacetime into hypersurfaces of simultaneity. You then have slices of 3D space, which are ordered in an oriented way (i.e. from past to future), and which are causally linked to each other by dynamic laws. This is called the “ADM formalism”, and is analogous to the film reel being projected onto a screen. However, such a “slicing-up” of spacetime into hypersurfaces is a completely arbitrary procedure, and not in any way fundamental to nature. One must remember that - in the context of GR - the passage of time is merely an artefact of human perception of consciousness. There is no fundamental mechanism that picks out a particular hypersurface of simultaneity, and gives it any kind of physical property that somehow distinguishes it from any other hypersurface (“the present” as opposed to past and future), or “advances” that hypersurface selection in a linear manner. GR simply treats all points in space and all instances in time in the exact same way. One can impose other structures on top of that, but those will always be arbitrary, and have no real physical significance. To give a very practical example - the process of free fall is not described as a dynamic process. Instead, it becomes a purely geometric problem - we find that world line between two given events on a static manifold, which forms a geodesic of that manifold, given a connection and a metric. Mathematically this is done by finding that world line, which parallel-transports its own tangent vector at all points; so the “process” of free fall becomes a simple concept in static geometry. If you write this statement down in mathematical form, you end up with the geodesic equation. That’s all there is to it. As mentioned before, it can be a useful analogy to explain certain situations in a visually appealing manner. But that’s all it is - an analogy.

The gravity/fluid duality is not the same thing as the “flow of spacetime into the earth” comment I responded to. If you think about it carefully, you will realise quickly that spacetime is a completely static construct, in the sense that it is not embedded in any higher-dimensional manifold with additional dimensions of time. Spacetime is hence the totality of all events at all times, and therefore cannot in itself exhibit any dynamics. You can, however, use the idea of dynamics as an illustrative analogy in some situations, similar to the rubber sheet analogy - but it is always important to bear in mind that an analogy is not the same as the actual model. I should also mention that the gravity/fluid duality never really made it into the mainstream, because the expected experimental evidence of gravitational turbulence has not been observed.

Such a concept does not make any physical or mathematical sense.

There is not really any such thing. There is a waterfall analogy that is sometimes used to illustrate certain concepts of GR that would otherwise be difficult to explain without going deeply into the maths; but that is just an illustrative analogy. Unlike is arguably the case in quantum mechanics, there is no space for differing “interpretations” in GR.

Precisely. Thanks for your efforts in putting together that excellent post.

To the best of my knowledge, the time dilation in the interior of a shell cavity wrt some clock at infinity is already a well established fact. I never mentioned anything about singularities, nor do I claim that GR breaks down anywhere in this type of spacetime. Obviously it doesn’t, since it is globally geodesically complete. Time dilation is directly related to differences in gravitational potential, so the Newtonian limit both in the cavity and at infinity are very relevant here. In fact, it is all we need to know - if there is a difference in gravitational potential between these regions, then there will necessarily be time dilation. And we know that there is a difference, even in the weak field Newtonian limit. This difference will increase, the more massive we make the shell.

No, it’s Minkowskian. That means that - within the metric - the time and space parts have opposite sign. This gives spacetime a type of hyperbolic geometry. In Euclidean geometry, all parts of the metric have the same sign. No, this cannot be done in a consistent manner. In Euclidean geometry, for example, speeds add linearly - if you ride on a very fast rocket, and shine a torch light into the direction of motion, this would give you a superluminal ray of light. Obviously that is not what happens in the real world.

The boundary conditions in this case are simply that the spacetime must be asymptotically flat at infinity (i.e. reduces to Newtonian gravity), and smooth and continuous everywhere else, including at the inner and outer boundaries of the shell. That is what it means to have a global spacetime manifold. This continuity condition is crucial - if you have points where spacetime is not continuous and differentiable, then the field equations do not apply there, and the whole thing becomes internally inconsistent. When you account for the continuity condition, the metric constants become fixed automatically. For an example of how boundary conditions are used in GR to match solutions and ensure continuity, see §23 of Misner/Thorne/Wheeler, which deals with the interior Schwarzschild metric. This way is how I learned to do it, and I don’t see how it could possibly be “wrong”, so long as the result matches the situation in the Newtonian limit, and reproduces the Birkhoff theorem outcome, which it does. My reference would be the well-understood Newtonian limit. We already know that in the Newtonian case, the gravitational potential in the interior cavity is not the same as the one at infinity. The GR solution must reproduce this boundary case, since the g{tt} component of the metric tensor is directly related to gravitational potential in the Newtonian limit. If time dilation was zero in the cavity, as compared to a reference clock at infinity, you would end up with a contradiction in the Newtonian limit. Therefore, globally speaking, g{tt} cannot be the same inside the cavity and at infinity. For a more direct confirmation of what the consensus on this is, physicsforums.com would be a good place to ask, since that is where the actual experts in the field go. The general argument though is along these lines: https://www.quora.com/Does-a-hollow-sphere-of-mass-still-cause-GR-time-dilation-inside-it-even-though-there-is-no-net-gravitational-field.

The difference is the geometric length of the observers’ world lines, which connect the two events - and that length physically corresponds to what a clock travelling along that world line records. The longest possible world line between two given events is always a geodesic in spacetime - which physically corresponds to an inertial frame. However, an accelerating rocket does not trace out a geodesic, because it isn’t an inertial frame, due to the presence of proper acceleration. Therefore, the world line of the rocket is shorter, meaning less time is recorded by a co-moving clock; that is your time dilation. No complications need to come into this, just simply compare the geometric length of the world lines traced out by the two observers. You’ll find they are not equal. It’s that simple.

As a matter of fact, yes, I have (though it was years ago) - and my result was precisely the one I had described above, which also corresponds to the scientific consensus on this matter. It certainly did not involve any step functions though, I don’t know how you got that. To me that’s a red flag right there, because in general, step functions are not smooth and continuous (and hence not differentiable) at the points where the steps occur. Then you must have done something wrong, because your result would mean that the energy-momentum of the massive shell has a gravitational effect in one radial direction, but not the other, as seen from the shell. That is obviously unphysical. I’m also pretty sure (but would need to check in more detail) that it would be mathematically inconsistent, since it can’t be asymptotically flat at both infinity and some other point that is not at infinity, while remaining smooth and continuous everywhere in between, in a spacetime that is not globally empty. Surely you wouldn’t seriously expect to have a mass-energy distribution without any gravitational effect in some region just outside it, would you? Again, this would mean you did something wrong, because if you have a region of flat spacetime that is surrounded by a thin massive shell, then you are guaranteed to have a Schwarzschild spacetime in the exterior, due to Birkhoff’s theorem. If you don’t get this, then there is an error in your maths somewhere. I would say check your boundary conditions at the shell - from what I remember (again, this was years ago, and I don’t know where my notes from that time are gone), these were quite tricky to get right, and I got stuck at that point too. Do remember that the overall global metric has to be both smooth and continuous everywhere, including at the two boundaries, as well as the non-vacuum interior of the thin shell itself. In your case it looks like you have no gravity whatsoever in the interior cavity, and then suddenly gravity in the non-vacuum shell itself, meaning there is a discontinuity in your solution at the interior boundary of the shell. Perhaps that is why you ended up with a step function...? Check your maths again.

Yes, I did indeed say that the universe is fundamentally quantum. Nonetheless, in this thread I am only talking about classical gravity, being the kind we know about and can mathematically describe. Whether aforementioned topological principle plays a role in quantum gravity is not a question I can answer, since we don’t have a model for quantum gravity just yet. It remains to be seen.

In some sense, gravity is indeed the result of a conservation law of sorts, but it has nothing to do with energy. The law in question is the topological principle that the boundary of a boundary is zero. Misner/Thorne/Wheeler have described this very nicely in “Gravitation”. So far as energy is concerned though, it should be remembered that the energy associated with a region of curved spacetime is a difficult to define concept. It is also not localisable.

I can fluently switch without having to consciously think about it. I sometimes also mix languages, so I could have more than one language in a single thought. I’m nonetheless aware of myself using different languages, mostly because I am also a synaesthetic, and the German/English languages have very different patterns, colours, textures, and general “feel” to them (for me). There’s no mistaking their differences.