But what if we had something like "foam automata", discrete local functions that could alter the space they operate on? For example, if we represented the underlying space as graph, could we have functions that evolve as to change (reconnect) the edges and add/remove vertices of said graph?

Aside from 1D case, I was unable to figure out a set of rules that wouldn't quickly devolve into something that's either locally too dense or too sparse or disconnected. And making it follow some laws of conservation seems even harder.

Maybe some smarter readers of HN will figure it out, I would certainly be delighted.

What you're describing isn't too far from Causal Dynamical Triangulation.

https://en.wikipedia.org/wiki/Causal_dynamical_triangulation

Thanks, I was also confused, but according to some weird social customs enforced on HN, I cannot sincerely ask that question. On the other hand, I am not sure about the upvotes either.

I actually didn't mean to develop some new physics theory, I just wanted to play a mathematical game (like Game of Life, for example), which would alter its own space/geometry. And this seemed as a good place to mention it, and get some feedback if somebody was aware of something similar. Because I found it quite nontrivial (to make it work "decently", in a vague sense), despite my lack of ambitions to explain physics.

CDT looks interesting, but it's not quite what I had in mind.

(I upvoted because the unrigorous idea of some sort of "foam automata" is cool, even if a formal, universal version of it is impossible.)

It's been proven to be locally equivalent to General Relativity, but might differ globally (e.g. solving the bucket in an empty universe issue).

I found that article from here: https://en.wikipedia.org/wiki/Pregeometry_(physics)

Maybe the trick to stability is making less radical changes per generation, like changing edge weights, rather than wholesale creation and deletion?

Sounds familiar with someone else's post, specifically, this part: https://youtu.be/qTx98PUW6lE?t=4592

The Forgotten Mystery of Inertia https://news.ycombinator.com/item?id=15487928

Regarding the topic on hand -- Is it possible for the net statistical acceleration of the charged virtual particles of the quantum vacuum to be used as an absolute reference frame in order to gauge absolute velocity/intertial mass? Does anyone know if this idea or similar has ever been tested experimentally?

I very much doubt you could use that as an absolute reference frame. It would contradict the ideas of relativity and no one has observed such a thing as far as I know.

The reason if any, to think there might be something interesting here-- All objects as they move through the vacuum, leave a disturbance, a "wake" of sorts, in all quantum fields. These disturbances can appear as real particles to some observers, while being virtual particles for others. It may be possible then, for relativity to be satisfied considering this observer-dependent visibility. In fact, perhaps a binary statistical measure on the number of real particles may be the key to getting the absolute measurements.

When I think about the problem on an individual atom level, the rising up the bucket doesn't seem hard to explain at all. Friction from the bucket to the outer molecules makes the water molecules want to move in the same direction (slowing the bucket sightly, I'd imagine). Once those ones start moving, they too will pull along the nearby molecules via friction. Then momentum will cause them to move towards the outer edge of the bucket, pushing each other and pushing some up the walls of the bucket.

I'm not being facetious here- what's the problem? Have we tried the same experiment with extremely low-friction buckets?

Maybe this will help: http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Newton...

So being at rest you experience no forces in your local reference frame, rotating = acceleration in your local reference frame.

I'm not sure why the rest of the universe rotating around the bucket would impart acceleration upon it because the rest of the universe is outside the buckets local reference frame. What sort of mechanism can transfer acceleration through rotation from one reference frame to another? (genuine question)

After all, it's acceleration within the local reference frame that lets the object 'feel' the acceleration rotation - it lets the object 'feel' it's curved path through spacetime, so it knows it is rotating and not the universe around it.

Eh? You measure the rotation against the thing that performed the “rotate the rope” step in the experiment.

Thought experiments that contain the step “now magically remove X from the observable universe” should be treated with caution! :-)

If you change it to just “There exists a universe empty except for a rotating set of rocks connected by a rope”, rotation needs no meaning as it’s one of the axioms. Also, there is no observer in that universe, so you’re making a pretty dangerous assumption about the ability to observe a universe without being a part of it.

Now if you’re talking quantum vacuum thrusters, exploring exactly how that works should also give you a good idea of what space is! ;-)

The conundrum is: what is the rotation without the relative perspective of the stationary observer? What pulls the rope taut?

Ok, this seems to be a clash of intuitions explained by reference frames —

When you imagine an object in deep space or its own empty universe, you intuitively use an “inertial reference frame”, where “the physics of a system in an inertial frame have no causes external to the system”. I.e. you do not see any “fictitious” forces like the centrifugal force, so the rope would not be taut.

https://en.m.wikipedia.org/wiki/Inertial_frame_of_reference

If you are imagining the rope to be taut, you are imagining a force pulling it taut, but this force that you’re imagining is only explained by physics if you’re using a rotational (non-inertial) reference frame.

So if your system has unexplained “centrifugal” forces, those forces can be modelled as the system rotating within an inertial reference frame, without having any evidence of any “non-rotating” matter.

It just so happens that deep space is pretty close to an inertial reference frame, so your mind sort of makes that assumption.

Or, in other words, you could measure and perceive the rate of the rope’s rotation by analyzing the forces in system, without having any physical external point of reference.

As the article points out, the highest rise happens when the water is in sync with the bucket, so you can’t attribute the phenomenon to friction.

These questions won’t really stress me until we run out of ideas for physics experiments. :-)

To build on the atom-level view, imagine you take a snapshot in time of a cross-section of the bucket and the universe. Then you put debuggers on four particles. These debuggers tell us the linear velocities of each particle. Two of the particles, A and B, are part of the water in the bucket. The other two particles are somewhere out in the universe, but aligned in a line with C and D.

Here's the picture so far, with the size of the arrow indicating the linear velocity of each particle. The bucket+water has been rotating for a long time, so all the water is rotating normally.

  [center of bucket]
  A -->
  B ---->
  [edge of bucket]
  .
  .
  C
  D

  [center of bucket]
                A
                B
  [edge of bucket]
                .
                .
        <------ C
  <------------ D

This is a sketch, definitely not rigorous. But my main point is that you can't say "imagine the universe is rotating around the bucket" without introducing a lot of messiness. That's what it means to have a non-inertial/accelerating reference frame.

On another note, I do think the Nautilus article is conflating two different points of confusion. First, as I described above, rotating reference frames are weird. But despite that weirdness, we can fully explain it in terms of the underlying physical concepts (acceleration, inertia, space, etc). Second, we haven't fully explained the origin of those concepts yet. But that's a very different matter. It seems like a stretch to say that we can't explain the water vortex, or that we can't explain the bulging planet.

Hubris. Programmers believe they can solve anything. The belief is so strong that they can look at a centuries old problem, which has puzzled the greatest minds in science, and think they have the simple answer in minutes.

The belief that they have a simple answer is a result of applying their engineering skills to a domain that they understand poorly but think they know well; this misalignment is due to the use of subtly inaccurate metaphors and generalizations, e.g. classical mechanics.

Taking their simple answer and methodically explaining why it’s incorrect efficiently develops their knowledge.

So this hubris is adaptive — it allows them to build their knowledge in a domain quickly.

Some might even do it intentionally, since they know the Internet can’t bear to leave an error uncorrected. ;-)

And yet, it is now thought that spacetime itself can not be absolute, and must emerge from more fundamental structures, such as combinatorial geometries: https://youtu.be/qTx98PUW6lE?t=13m7s

Alternatively, you can choose a coordinate system where the particle is rotating, accelerating, or moving however you like. Your choice will be determined by the goals you have. (Without the laws of motion, there is no reason to prefer even intertial frames over nonintertial ones.)

There's a whole set of excellent reasons to prefer to work in inertial frames where possible, most of them having to do with calculational burden. Our case here is essentially flat electrovac with a test field - do you really want to work in Rindler or Born coordinates or worse if you don't have to? You're free to do so if you're feeling masochistic (or doing it as a learning exercise), just as you're free go further and solve Maxwell's equations in curved spacetime in arbitrary coordinates with the vector potentials in some arbitrary gauge (\partial_{\mu} A^{\nu} \neq 0). And with "proton?" then you might as well dive into QCD as well. Unnecessary extra work is surely a good reason to prefer to use the simpler maths where the results will be indistinguishable.

> Without the laws of motion

I don't understand this point. Are you ditching the action principle somehow?

There are no meaningful potentials in the one-particle universe that the OP described. However the particle is "really moving," you could just invent a coordinate system to make it "move" however you like.

Even introducing EM would bump you up to at least two particles! The only reasonable one particle universe would be one where the partice was coupled to nothing.

(It is under the "try to interpret it as they meant it" doctrine that I go from a proton to a particle that doesn't leave behind a disturbed field when it accelerates.)

Is the particle moving or is the coordinate system?

It seems, like you say, to be arbitrary. Without something else in the universe, the particle could have any rotation, velocity, inertia, etc... depending on which coordinate system you choose. The coordinate system seems to be layered on, from a universe where the idea of space and time were more useful.

No, according to the Copenhagen interpretation.

Note that in general relativity, which allows inertial and non-inertial reference frames as equally valid, there is some non-arbitrary motion of objects that is independent of the choice of frame. These are the so-called tidal gravitational effects that cannot be transformed away in any frame, and are easiest described as curvature of spacetime due to stress-energy.

A smaller point: "motion" and "tidal...effects" each implicitly chooses a splitting (in particular a 1+3 threading) of spacetime into space and time. That's fine; it's really hard for general relativity to confront observation without doing some splitting, and implicitly threading is standard behaviour (Synge does this in his textbook, for example, when discussing 3-velocity and spatial distance and offering up his world function biscalar; note that this is different from explicitly applying a 1+3 formalism).

Your central point is right though: coordinate transformations or arbitrary choices of splittings -- 1+3 threading or 3+1 slicings are examples -- cannot get rid of real curvature.

But to add to the first paragraph of their answer: if you set the 'T' tensor to the zero tensor in https://en.wikipedia.org/wiki/Einstein_field_equations#Mathe..., then you are effectively studying the vacuum. The left hand side describes the curvature of spacetime. This is a set of 10 non-linear partial differential equations which permit many solutions.

You can then add a "single particle" back to these solutions with varying degrees of complexity as outlined by raatgift.

It seems that the single particle may not be alone in the universe and that there is a vacuum entity that the particle could be depending upon. I'm not sure what forces or other properties the vacuum imbues the particle. Traveling along a curved spacetime wouldn't imply a force on a neutral particle, would it? I'm curious what properties of the particle the vacuum would change over time. I may have to dig deeper into the math to find out. I don't want to burden anyone with my questions.

My lay observation was that many of these properties are built up from relations (functions) between two points. The equations for distance have two points, velocity depends on distance over time, inertia depends on velocity. It's difficult to imagine what the other point would be in a single particle universe. It's why I wonder if there would be inertia in a universe with one particle. These concepts seem to spring forth once you add another particle, which causes me to think there isn't inertia per se, but a relation we observe that we call inertia under the condition of having a point moving relative to another with a certain mass at a certain speed. Useful no doubt, except perhaps in single particle universes. Similarly, if we have a party balloon and we take away all the gas and balloons in the universe except for one molecule of nitrogen, do we still have pressure? My answer here is not really, not practically, because these forces spring forth from the context of having multiple molecules. It's a helpful concept for use in the case balloons, tires and pipes, but there doesn't appear to be such a thing as pressure that exists in the universe independent of the particles within it.

> there doesn't appear to be such a thing as pressure that exists in the universe independent of the particles within it

This is the thing about fixing a gauge: you've decided here that the "standard" pressure is vacuum. That's perfectly fine. But it's also perfectly fine to set the 0 point higher, and talk about negative pressures.

We do that with temperature, for instance; we set the 0 point of our temperature gauges and talk about positive (or positive and negative for e.g. celsius and fahernheit) temperatures. We do that with calendars, for instance, talking about years before or after some 0 point. We do that unthinkingly when we talk about "a quarter past [some hour]" or "a quarter to [some hour]", for instance.

Gauging is done -- often implicitly -- on a huge range of physical quantities. We can even formalize the process into a gauge theory, wherein we encode such choices into a gauge field, which can have its own dynamics. (As an example, let's make a field throughout the Earth's atmosphere where every point has a value which would be reported by a calibrated barometric pressure sensor; we can choose one particular reading as a surface and call it "FL10" and fly airplanes in that surface/layer. Depending on latitude and local weather conditions, "FL10" will coincide or deviate from the 10000 feet above the WGS84 surface reported by GPS equipment at the same point. Further depending on surface features, a radar altimeter may report anything from ~ 10000 feet to "oops you've just crashed into a mountain". Moreover a pressure altimeter can slide "QNH" value around, essentially channging the zero point; so our pressure gauge field depends on how we measure, as well (different observers will report different values at the same point!). Moreover, very fast moving pressure altimeters (naively) may report different values compared to those that are slowly moving low in the subsonic regime. However, by and large, if two airplanes are near one another with their pressure altimeters corresponding to FL10, they have to be careful not to collide! So our pressure gauge field does have its uses in predicting physical events like collisions, even if there's nothing truly special, universal or even locally constant about FL10 or any similar surface.

Note that when we slide around zero points, the physical differences don't change. You're still suffering fever symptoms whether we tell you your temperature in degrees Rankine or kelvins-above-normal-body-temperature. The time in seconds from one northern winter solstice to the next is essentially the same whether you use the Gregorian, Julian, Chinese, or any other calendar to fix the day, month and year.

When we fix these zero positions and label the distances from them in degrees or years or volts or whatever, we are essentially laying down a set of coordinates that are at least locally valid. This lets us concentrate on local interactions without carrying around additional burdens from considering how how our local zero and gauge-markings might not be appropriate elsewhere in spacetime (or to a different culture, or whatever), and generally we can convert from one such fixing to another.

Although it is tempting in the pressure case to say that the zero is universal and so only the choice of units (pascals, pounds per square inch, whatever) is idiosyncratic, let's think about how to represent pressure without making any such choices.

In General Relativity pressure is encoded in the stress-energy tensor as the doubly-spatial components (T_{11} = T_{22} = T_{33} = p) where 1 2 and 3 are the three space-like axes and p is the total pressure. T_{12} is the flux of 1-momentum in the 2 direction, T_{13} is the flux of 1-momentum in the 3 direction, so T_{11} is the flux of 1-momentum in the 1 direction. If we make the 1 axis "left" and "right", then we can think of T_11 as "the flux of momentum from the left going towards the right through a point". That's just pressure pushing on something to the right (or a tension pulling on something to the left!).

We can assign any sort of label to the 1 2 and 3 directions as long as they're orthogonal. If 1 is left-right then 2 could be forward-backward and 3 could be above-below. Or we can say 1 is x-axis, 2 is y-axis, 3 is z-axis in Cartesian coordinates. Or we can say 1 is radial distance, 2 is azimuthal angle and 3 is polar angle in spherical coordinates. And so on. We can also set down units, whether they're metres or feet or light-seconds or anything. And finally, we can set down some origin for this system of coordinates. Whatever such choices we make, we quantify the relative momentum flux, but we don't actually change it.

That there is a flux at all is invariant in the face of these choices of coordinate and gauge. (The more precise statement is that the divergence of the total non-gravitational stress-energy vanishes, so one could in principle concoct an observer that sees the pressure-tension as some other form of stress-energy).

But your question is essentially, "what if there is nothing to the right to push on"? Well, if there is a flux of left-right momentum in the right direction, but nothing to the right, then for a truly pointlike object, there's now less stress-energy in that object and a bit of stress-energy to its right. Conversely, there is now some stress-energy to the right of the pointlike particle that has our test particle as a neighbour to its left. Depending on the gauge theory of the particle, we can now consider what becomes of the two neighbours: do they drift apart? Do they rejoin? In the former case, our gauge field can carry away the momentum to infinity; in fact, that's exactly what photons do: they are gauge bosons, massless, but not momentum-less. In the latter case, our gauge field might pull the momentum back inside the particle, essentially how asymptotic freedom works for objects that feel the strong nuclear force. And indeed the strong nuclear force is again representable as a gauge field: the gluon (massless, but not momentum-free). Photon pressure is even testable with e.g. solar sails. Gluon tension (tension being the inverse of pressure) is seen in hadronization.

One molecule of nitrogen is surprisingly busy with tension and pressure being carried around locally by gauge bosons!

Finally, depending on how deep you mean by "dig deeper into the math", you (or someone else reading this, including me at some point in the future :-) ) might like Terry Tao's take on this sort of thing: https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/ If nothing else, his approach to "explain[ing] what a gauge theory [is]" there is a bit different from mine here. :-)

> Travelling along a curved spacetime wouldn't imply a force on a neutral particle, would it?

There are exact solutions of the Einstein Field Equations (although typically this is done with the linearized version of them, the LEFEs, which is in the "almost certainly a good physical approximation" bucket) in which a change in the background metric propagates through vacuum in some direction as a plane wave.

The background metric could in principle be, for example, flat spacetime, or Schwarzschild, or any other vacuum solution of the EFEs. When one chooses a splitting of spacetime into space and time, the wave propagates through space.

A small set of test particles reacts to the wave as it passes over them, changing their respective "radar distances" for example. This view lends itself well to perturbation theory, where the wavelike perturbation of the background metric can (if weak) be quantized; those quanta are gravitons in perturbatively quantized gravity. That's a real theory that exists and is good everywhere where there is no strong gravity. "Strong gravity" can even be defined in terms of perturbatively quantized gravity as at least one loop of gravitons on a Feynman diagram. You generally don't find that until very close to a gravitational singularity (i.e., perturbatively quantized gravity as an effective field theory works inside stellar BH horizons too, and works even better inside even more massive BH horizons because there you are further away from the strong gravity near the centre.

In the spacetime view, the particles are instead worldlines that are mostly in regions described exactly by the background metric, but a small section of the worldlines are found in a small region where the metric is different. One can consider the spacetime intervals of points on the various worldlines and look for some minimum which will ought to be found in the region with that different metric. Exactly oppositely from Euclidean geometry, curved lines are shorter than straight lines, so a(n inertial) worldline which is found only in flat spacetime regions will be longer than one which is (even briefly) also found in a region with the slightly different metric.

Are such worldines travelling along? Nothing really travels at all if it's a worldline in a "block universe" spacetime picture like in the paragraph immediately above. But if one foliates spacetime into spaces with a time coordinate, movement emerges. You can choose a foliation and set of coordinates in which the different metric's effects are shoved into time-like effects (if the background is flat spacetime, a particle exposed to the gravitational wave is younger than a particle that isn't; c.f. the twin paradox); or you can choose a foliation and set of coordinates in which the different metric's effects are shoved into space-like ones ("radar" showing that the particle exposed to the gravitational wave is dragged or bobbed by its passage).

With some of these equivalent and (mostly) convertible views of the same physical picture, it would be perfectly reasonable to say that a force has been exerted. After all, one is interested in what causes a small piece of a long stable worldline to look different. The force acting on that section of worldline (or on the particle during that brief period of time), however, depends on one's choices of foliation, coordinates, and a few other things. If you can make some set of such choices such that the force vanishes, we call it an inertial force, d'Alembert force, or frame-dependent/fictitious force. Real forces cannot be made to vanish by any choice of coordinates etc.; real forces' "fingerprints" must do more than change an object's position or orientation in spacetime.

The tendency of objects -- particles, worldines, extended (many-particle) objects, worldtubes, or the full fields in which these worldtubes are just clusters of correlated numers -- to look mostly the same is one way of thinking about inertia: you'd want to attribute any unusual change to some force acting at the point of the change.

If you remove everything but your test object, so that no other object or field can impart any non-d'Alembertian force on it, then a magic observer at an enormous distance watching its behaviour could say that any changes in radar distance or (if the test object is a clock) ticking rate is the result of coupling with curved spacetime. But a magic observer who can see the whole worldline of the object and knows the layout of spacetime curvature will note that it matches a longest-possible solution of the geodesic equations.

> I don't want to burden anyone with my questions.

It's not a burden!

A test particle is one with negligible mass and extent, and thus does not itself perturb the curvature of the background vacuum spacetime.

If a real particle is electrically neutral, extremely low mass, and nearly pointlike, then in a vacuum spacetime it should behave very much like a test particle. Moreover, for microscopic real objects, which aren't truly pointlike and which may not be wholly chargeless, it is impractically hard to tell at a large enough spatial distance that it's not a test particle.

So a test particle or test field does not source the spacetime curvature -- it just rides along the background spacetime, which is arbitrarily curved by its nature.

We can extend the notion to a test field that is almost wholly massless, like a single-electron quantum field. The test field sits on the background, but doesn't influence the background's dynamics.

Real quantum fields, like test fields, permeate the whole of spacetime. However they tend to have abundant stress-energy in them, so source some of the curvature of the whole spacetime, and typically not uniformly. One can fix a gauge (like how one calibrates an air-pressure altimeter) that lets one talk about the inherent curvature of the background spacetime and the contribution to the total curvature of the lowest density of the quantum field. Typically the background is chosen to be some exact vacuum solution of the EFEs with corrections added via something like perturbation theory; flat spacetime, Schwarzschild and Robertson-Walker expanding spacetimes are popular backgrounds, depending on what one is studying.

Sometimes, though one wants to think about an electron rather than some characteristic set of numbers in some all-over-spacetime fields. In that case one (probably implicitly) fixes a gauge that lets one talk about the electron as an independent object that generates an electromagnetic field which, because that has some nonzero stress-energy (zero being set by our gauging choices and being found sufficiently far from our electron) it influences the background curvature of the spacetime (the background again being set by our gauging choices).

In either paradigm, one might want to think about, in reverse order, the gravitational back-reaction of the electron (and its EM field)'s gravitational footprint on itself, or the gravitational back-reaction of the overdensities in the all-over-spacetime (classical or quantum) fields on the regions of spacetime they are found in. "Gravity gravitates" is the problem: this arises from the EFEs' non-linearity. Fortunately, for a single electron these back-reactions will be very small, because electrons in either paradigm have very little stress-energy, so their contribution to any curvature will thus be so extremely tiny that it will be practically immeasurable, so there is basically no hope of measuring higher-order contributions (that is, the electron's gravitational influence's own influence on the curvature). Negiligibility saves a lot of headaches, because working with gravitational backreaction is hard.

Also, a dimension has some degree of continuity or, at least, regular discrete nature that I'm not sure your idea for the 5th dimension really fits naturally. Surely, we can relate to positions in your concept of a dimension, but does a relation require a dimension? I'm not sure. Maybe it does. Maybe my above thoughts are all locked in the concept of a euclidean geometry. My topology studies were always in the context of function spaces, so it was a bit too abstract to think about it this way.

http://fqxi.org/data/essay-contest-files/Barbour_The_Nature_...

Why doesn't the higgs field slow down objects moving in space?

2. If gravity is simply the curvature of spacetime itself, the higgs field gives particles mass, and mass curves spacetime, what is the relationship between the curvature of spacetime and the higgs field exactly, is it special somehow? (It seems somehow the way the higgs "offers resistance" is related to the curvature of spacetime)

This is not an unsolved problem. The key is to understand how inertial motion and inertial forces are explained in Newtonian mechanics vs. general relativity.

Let’s take two friends, Charlie and Isidor. Charlie is sipping coffee while reading a book sitting on a chair in his home, while Isidor is reading the same book while floating freely on the International Space Station. At first glance, it may seem that Charlie is experiencing inertial motion (with velocity 0, since he is at rest) and Isidor is in accelerated motion around the earth at a very high speed, since he is moving in a circular orbit. (Recall that in Newtonian mechanics, circular motion is always accelerated motion since the velocity vector varies with time, even though only in its direction.)

However, this is backwards. In fact, Charlie feels the constant force of the chair pushing on his bum and lower back, which is why he keeps changing his position when it gets uncomfortable, while Isidor is weightless, experiencing no forces acting on his body whatsoever. Also, absent any collision or explicit application of rocket thrust, the ISS and Isidor inside will maintain their current trajectory forever, while maintaining Charlie’s ‘at-rest’ position requires the constant application of force on his body, which is why when the chair eventually breaks (Charlie is a heavy dude), Charlie moves quickly towards the ground, spilling his coffee.

This observation is at the core of general relativity. Einstein remarked that, if we surround Charlie with a tent, he cannot possibly tell whether he is sitting on the beach or whether his chair is actually on the floor of a rocket accelerating constantly at 9.81 m/s^2 towards Alpha Centauri. At the same time, if the ISS had no windows, there is no experiment Isidor can conduct that would tell him whether the ship is orbiting the earth or is hurtling unaccelerated in a straight line in deep space. IOW, Charlie is in fact the one experiencing accelerated motion, while Isidor is in inertial motion.

So what, then, is inertial motion? In general relativity, a body is in inertial motion if its trajectory in spacetime follows a local spacetime geodesic. Earth’s gravity distorts space time around it to a shape in which the trajectory of the ISS at that altitude and velocity is actually a straight line according to the local metric imposed by the Earth’s gravitational field. Conversely, if we plot Charlie’s at-rest position on the beach through the same spacetime according to the same metric, it corresponds to a trajectory that deviates from the geodesic right up until the moment when the chair breaks, when for a very brief duration, his motion will actually follow the local geodesic.

What does this have to do with the water in the bucket? Once the water molecules rotate in sync with the bucket, their trajectories all deviate from the local geodesic lines. We can see that by imagining the bucket wall instantly disappearing; the water molecules would cease following the previously enforced circular trajectories, and they would start traveling in inertial motion away from the bucket, freely falling towards the ground (i.e., inertial motion in the Earth’s gravitational field). The sum of the forces required to maintain the water molecules on circular trajectories happen to lead to a curved water surface. The water molecules at rest in the non-rotating bucket are still in non-inertial motion similar to Charlie’s, but the forces acting on them to maintain those particular trajectories simply lead to a flat surface.

The point is that every time we want a body to deviate from its free, inertial trajectory in its local spacetime, we must apply a force. The water curvature is a direct result of all the forces which are necessary to steer the water molecules away from the local spacetime geodesics.

[1] https://physics.stackexchange.com/a/3988

We really have no idea?? Wow, it shows how little we really know about anything.