How exactly does gravity work?

[A.T.]

Why can an object in the center of the flat spacetime grid sit still, moving only in the dimension of time, yet it must move in the dimensions of space in the curved spacetime grid?
View attachment 219229 View attachment 219230

1) Your second grid is not distorted the right way, because x (time) and z (vertical space) are not orthogonal at all points. See the video in post #28.

2) You need a better grasp of the concept of geodesics (locally straight worldines of freefallers with zero net force), which easier on a 2D surface (your y is not needed). Check out this:

http://www.relativitet.se/spacetime1.htmlAnd this:

 

[bhobba]

The basic answer to this is "that's what they do". Why do things travel in straight lines in flat spacetime? It's just what they do.

Well one can get the Principle of Least Action from QM. That determines how particles travel from one point to another. Suppose we have a free particle in flat space-time and we apply the principle of relativity. Well the action in Lagrangian form by definition is ∫L dt L is the Lagrangian. From the POR we want L dt to be invariant. dt is not invariant - but dτ where τ is the proper time is. L is a scaler we will call c. So we have the Lagrangian ∫c dτ. -c by definition is called the mass (in units the speed of light c = 1). From that we get the the principle of inertia. Since c is a constant you can also get the motion by extremizing ∫dτ

Now in GR dτ^2 = guv dxu dxv. We apply the same rule - the path is extremizing ∫dτ - but in curved space time that is the definition of geodesics - so particles in GR move along geodesics.

We do not know the answer to everything, and never will, but we sometimes know the why for things that are thought fundamental - but for some reason is not explained in most textbooks. That's why its important to (occasionally anyway) study textbooks that explain things as complete as possible - even if a bit different.

For mechanics the textbook I have found that does that best is - Landau - Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

It takes an entirely different approach that emphasizes symmetry which is not the usual approach - but IMHO gets to the core much better.

Just a personal opinion - in a certain sense you are correct. I haven't really explained why QM - so it doesn't matter what you do its exactly as you say - It's just what they do. Every theory, every single one assumes some things. However I just wanted it out there - the real thing we do not know - the thing that is actually - that's just how nature is - is QM. We do know a few things about the why of QM but the QM sub-forum is the place to discuss it - not here. Still those things do not answer - why QM - or even - what the damn does it mean.

Thanks
Bill

 

[Dragon27]

What I *think* I want to ask is why do objects *have* to move in curved spacetime if they do not have to move in flat spacetime.

But objects have to move in flat space-time too. They never stand still in space-time. They may have zero velocity in certain frames of reference (and 'stand still' in space, not in space-time), but not in all of them at once. The same can be said about curved space-time. If you move along with the falling apple, it would appear to be standing still in you frame of reference.

Things can exist in curved spacetime without moving in the dimensions of space. The apple on the tree is doing so. It moves only in the dimension of time, because the tree is holding it still in the dimensions of space. But why MUST it begin to follow the curved spacetime when the tree let's go of the stem? Why does it not simply stay at the spot on the curve where it has been all along? It was not moving in space before. All of its lightspeed motion was in the dimension of time. Why does it lose some of its motion through time and gain motion in space?

The tree is 'holding it still' in its own frame of reference. For a person running nearby the tree would appear moving along with the apple. For a person falling through the air the tree (and the apple) would appear to be accelerating upwards. Which ones of this motions of the apple is 'the right one'? Is the apple hanging still on the tree (like an observer standing on the ground would say), or is the apple accelerating upwards (like an observer falling to the ground would say)?
The apple is constantly in motion dictated by the curvature of space-time and other forces acting on the apple (like the one from the tree branch). It's just that in some frames of reference this motion could be described as 'not moving in space'.

 

[PeroK]

I don't think that's it. I can understand that, if spacetime is curved, anything moving in that curved chunk of spacetime must follow the curve. A ball rolling down through a curved tube can only follow the path of the tube. What I *think* I want to ask is why do objects *have* to move in curved spacetime if they do not have to move in flat spacetime. Things can exist in curved spacetime without moving in the dimensions of space. The apple on the tree is doing so. It moves only in the dimension of time, because the tree is holding it still in the dimensions of space. But why MUST it begin to follow the curved spacetime when the tree let's go of the stem? Why does it not simply stay at the spot on the curve where it has been all along? It was not moving in space before. All of its lightspeed motion was in the dimension of time. Why does it lose some of its motion through time and gain motion in space?

The apple "wants" to move in both cases, but in one case a force is preventing it. That's perhaps the obvious thing you are overlooking. Gravity, even when described as spacetime curvature is like a force. Everything wants to move towards the centre of the Earth and - if there is no force holding it back, that's what it will do. It MUST, because that is the law of nature. In that sense it is no different from Newton's laws of nature.

When the force is removed the apple follows a natural path through spacetime, which is towards the centre of the Earth. The difference between Newton and GR is the reason that it wants to move down. In Newton's gravity there is a force pulling it; in GR down is the natural path that is determined by the Lagrangian principle.

You could also turn your question round. Suppose the apple is lying on the ground. It doesn't move. You come along and kick it. Why does it move? If it can remain at rest on the ground before you kicked it, why can't it remain at rest when you kick it?

The major conceptual hurdle for you is to see that there are more than just "forces" in the laws of nature. If you insist that only a force can move an object, then you are stuck in 18th Century physics. To study any modern physics, you must accept that the laws of nature are not necessarily those of Newton and that the laws of physics may take an alternative form.

 

[Patterner]

Apologies for the jpgs. They were not meant to be accurate, "to scale", or much of anything other than a general idea.

While still attached to the tree, the apple is not moving relative to the Earth or the curves in spacetime that the Earth creates. Why must it move relative to the curves when the tree releases it?

 

[PeroK]

Apologies for the jpgs. They were not meant to be accurate, "to scale", or much of anything other than a general idea.

While still attached to the tree, the apple is not moving relative to the Earth or the curves in spacetime that the Earth creates. Why must it move relative to the curves when the tree releases it?

You can't talk about "moving relative to the curves in spacetime". Curved spacetime tells a particle how to move. In the same way as a Newtonian force tells a particle how to move.

When the apple is attached to the tree, gravity (curved spacetime) is telling the particle to move down. And the force from the tree is telling the apple to move up. The two are balanced, cancel each other out and the apple stays where it is. If the tree no longer pushes the apple up, then curved spacetime has its way and the apple falls.

 

[Dale]

Why must it move relative to the curves when the tree releases it?

Brcause at that point it’s worldline changes from curved to straight. A straight worldline cannot follow a curved time axis!

 

[Nugatory]

While still attached to the tree, the apple is not moving relative to the Earth or the curves in spacetime that the Earth creates. Why must it move relative to the curves when the tree releases it?

When there is no force on the apple, it moves on a straight-line path through spacetime because there's no force to pull it off that path. The curvature of spacetime means that that straight-line path intersects the path through spacetime of the surface of the earth.

While the apple is attached to the tree, there is a force on it from its stem. This force pulls it off the straight-line path and holds it on a different path, one that does not intersect the path of the surface of the earth,

 

[A.T.]

Why must it move relative to the curves when the tree releases it?

Because that's how a geodesic path looks like in that case. Read the link and watch the video in post #36.

 

[Patterner]

But why does it HAVE to follow a geodesic path in the dimensions of space? Why can it not remain motionless in space, continuing to move only in the dimension of time?

Let me ask this. Let's say we can find an area of space, between galaxies, or wherever, where spacetime is as flat as possible. Let's say we, using our incredible scifi spacetime curvature detector, know it to be absolutely flat. Except for a baseball just sitting in the middle of this vast ocean of nothing.

If a comet goes through this area, and gets close enough to the baseball, and the areas of curved spacetime that each creates meet, the baseball, which is in relative motion, will move along the new geodesic.

But what if our crazy scifi gadgetry allows us to curve spacetime without introducing mass or energy. JUST curved spacetime. Shaped as though the planet was next to the baseball. Although there was no motion that set up this situation, the baseball is going to begin moving along the new geodesic, in the direction of where the planet would be, if the planet's presence was what was causing the curvature. Because, if spacetime is curved, an object MUST move along the curve.

Correct?

I've watched the video, A.T., and the other one in your first post of this thread, multiple times each, and now read the link, also. If they are telling me WHY the baseball would react the way it does in the scenario I just invented, I'm not aware of it. I guess, as Ibix said, that's just the way it is.

 

[Dale]

But why does it HAVE to follow a geodesic path in the dimensions of space?

Newton's first law describes inertia: an object in motion will continue to move in a straight line at a constant speed, etc. In other words, an inertial object's worldline is a straight line in spacetime. The technical word for a straight line is a geodesic.

So "why" it follows a geodesic is because of inertia. If an object is not experiencing a force then by the principle of inertia its worldline is a geodesic, i.e. a straight line. Conversely, an object which is experiencing a force will have a curved worldline, i.e. not a geodesic.

 

[PeterDonis]

why does it HAVE to follow a geodesic path in the dimensions of space? Why can it not remain motionless in space, continuing to move only in the dimension of time?

You are missing the point. The geodesic path is in spacetime, not space. You can always choose coordinates so that the space coordinates of the path stay the same and only the time coordinate changes. In the case of the apple, if it's freely falling, i.e., following a geodesic of spacetime, then to have its path be motionless in space, you would choose freely falling coordinates. But that's a matter of your choice of coordinates. There is no absolute sense in which an object is "moving in space" or "motionless in space". The only absolute is its worldline in spacetime.

 

[Nugatory]

Let's say we, using our incredible scifi spacetime curvature detector, know it to be absolutely flat. Except for a baseball just sitting in the middle of this vast ocean of nothing.

We don't need any magic sci-fi devices to test for flatness; all we need is two small objects. We let them float in empty space near one another and at rest relative to one another, and then measure the distance between them. If that distance doesn't change over time, then the spacetime is flat. Here's what going on:
- The paths through spacetime of the two objects are straight lines. We know this because they aren't subject to any force. This is just inertia, as @Dale has already mentioned.
- The two straight lines are initially parallel to one another. We know this because we started the two bjects at rest relative to one another. (Note, however, that the two objects may be moving relative to us).
- If there is no curvature, parallel straight lines remain parallel. Only if there is curvature present can they converge (the distance between the two objects is decreasing) or diverge (the distance between the two objects is increasing).

The generally accepted word for a path through spacetime is "worldline", so I'll start using that terminology).

It would be a good exercise to get a piece of graph paper and trying drawing the paths through spacetime (flat spacetime, because the paper is flat) of two nearby objects to see how they are parallel if they are at rest relative to one another. Use the y-axis for time and the x-axis for position, and try both the case in which the two objects are at rest relative to you (the lines go straight up the page) and moving relative to you (the lines are at an angle to the grid). Then try drawing a diagram for two objects that are moving relative to one another; their worldlines will intersect. These diagrams are called "spacetime diagrams" or "Minkowski diagrams", and being able to interpret them is absolutely essential to understanding relativity - that's why this is such a good exercise..

So if the distance between the two objects doesn't change, we have a flat spacetime. What would be an example of detecting curvature this way? Suppose we start our two objects at rest relative to one another, but in a region of spacetime that is not flat because the planet Earth is nearby: in fact, they start side by side and in free fall towards the surface of the earth. I'm freefalling alongside one of them, so neither I nor the objects are experiencing any forces, and as far as I am concerned, none of us are moving. (If I look down, I see the surface of the Earth rushing towards me, but that just means that the surface of the Earth is moving relative to me).

But if I have a very accurate laser rangefinder, I will measure that the two objects are drifting slowly towards one another. What's going on is that the spacetime is curved so the initially parallel worldlines are converging. If the surface of the Earth didn't get in the way, the worldlines of the two objects and the worldline of the point at the center of the Earth would all come together at a common point.

 

[Sorcerer]

after giving this some thought, i can not accept that time is the cause of this illusion of motion. time is a perception of the mind, i do not believe that time acts like a physical force, pushing objects through space in a strait direction. instead, doesn't it make a lot more since to say that the expansion of space with in our universe is the cause of the motion we perceive?

I know this Australian guy is probably gone for good, but I don't know if I can accept this. I've seen the math showing how intimately energy and time are tied together (Noether's theorem showing the connection between time symmetry and energy conservation). To me, if time is merely a perception of the mind, and so too must energy be. But that's madness, because energy is tied to motion, and if motion is just a perception of the mind, then by all means go stand in front of a moving bus.

Also it's pretty clear he didn't get what was said here, about maximizing proper time. Not that I am blaming the guy, as it's clear he hasn't really read much about special relativity, or physics in general.

I have never actually considered this concept, though, that minimizing the Lagrangian is the same as maximizing proper time. Of course it makes perfect sense: minimizing the Lagrangian is taking the shortest possible path, right? And the shortest possible path would be the one moving through space the least (in time-like intervals anyway, I suppose... maybe), which would mean maximal proper time, right?

But this concept of "minimizing the Lagrangian = maximizing proper time" is speaking in FOUR dimensions, rather than three, right? If anyone wants to go a bit deeper with that I'm willing to read it.

 

[Dragon27]

But why does it HAVE to follow a geodesic path in the dimensions of space? Why can it not remain motionless in space, continuing to move only in the dimension of time?

Take an object that doesn't move, change the frame of reference and BAM, it's moving. Why does it have to move? Why can't it stand still in all frames of references at once? Strange questions.

A free object will move along a geodesic in a curved space-time. It's a law, a postulate, it doesn't have to be explained. We have to start somewhere, have to have some assumptions or premises, don't we?

 

[A.T.]

But why does it HAVE to follow a geodesic path in the dimensions of space?

The worldline is a geodesic in spacetime (not space) because there is no force to bend the path, so it's locally straight. This is the same for Newton and Einsteins model.

 

[PAllen]

I know this Australian guy is probably gone for good, but I don't know if I can accept this. I've seen the math showing how intimately energy and time are tied together (Noether's theorem showing the connection between time symmetry and energy conservation). To me, if time is merely a perception of the mind, and so too must energy be. But that's madness, because energy is tied to motion, and if motion is just a perception of the mind, then by all means go stand in front of a moving bus.

Also it's pretty clear he didn't get what was said here, about maximizing proper time. Not that I am blaming the guy, as it's clear he hasn't really read much about special relativity, or physics in general.

I have never actually considered this concept, though, that minimizing the Lagrangian is the same as maximizing proper time. Of course it makes perfect sense: minimizing the Lagrangian is taking the shortest possible path, right? And the shortest possible path would be the one moving through space the least (in time-like intervals anyway, I suppose... maybe), which would mean maximal proper time, right?

But this concept of "minimizing the Lagrangian = maximizing proper time" is speaking in FOUR dimensions, rather than three, right? If anyone wants to go a bit deeper with that I'm willing to read it.

If the Lagrangian is proper time, a geodesic is found by maximizing the Lagrangian, not minimizing it (in GR, this is a local maximization rather than a global one). Literally, you extremize the the Lagrangian via the Euler Lagrange equations. They are analogous to finding zero derivative in simple calculus. Whether the result is a minimum or maximum depends on the Lagrangian, just as a zero derivative can be a minimum, maximum, or saddle point. It is just that all Lagrangians you’ve encountered before GR have the property that the extremum is a minimum (often globally, always locally).

[edit: as noted below, I meant before relativity, not GR per se. It is as true for SR as GR]

 

[Sorcerer]

If the Lagrangian is proper time, a geodesic is found by maximizing the Lagrangian, not minimizing it (in GR, this is a local maximization rather than a global one). Literally, you extremize the the Lagrangian via the Euler Lagrange equations. They are analogous to finding zero derivative in simple calculus. Whether the result is a minimum or maximum depends on the Lagrangian, just as a zero derivative can be a minimum, maximum, or saddle point. It is just that all Lagrangians you’ve encountered before GR have the property that the extremum is a minimum (often globally, always locally).

Now that’s interesting. Does it have something to do with the negative sign in the 4-D metric?

 

[PAllen]

Now that’s interesting. Does it have something to do with the negative sign in the 4-D metric?

Indeed it does.

 

[bhobba]

It is just that all Lagrangians you’ve encountered before GR have the property that the extremum is a minimum (often globally, always locally).

In an inertial frame the action of a free particle is ∫-mdτ, τ is the proper time and m the rest mass - I explained in post 37 why that is - its to do with QM implying the principle of least action (Feynman's sum over histories approach - most cancel except those of stationary action), but of course most just assume its true without explaining why, and the Principle of Relativity which means it must be the same in all frames otherwise you could tell one inertial frame from another.

Its a minimum, just as the PLA states. However because rest mass is a positive constant, and the negative sign in front of it, you get exactly the same answer maximizing ∫dτ. In GR dτ^2 = guv dxu dxv so the the geodesic is maximizing ∫√guv dxu dxv . This is the so called principle of maximal time. You would, intuitively expect it to be a minimum by analogy with Riemanian geometry - but this is pseudo Riemannian geometry and the metric is different, which leads to it being a maximum, exactly as you, correctly say. Strange hey.

Thanks
Bill

 

[Nugatory]

A free object will move along a geodesic in a curved space-time.

And the word "curved" is unnecessary in this sentence. A free object also moves along a geodesic in flat spacetime.

(I expect that @Dragon27 probably already understands this - the correction is for the benefit of other people reading this thread).

 

[bhobba]

Of course it makes perfect sense: minimizing the Lagrangian is taking the shortest possible path, right?

See my elaboration of what PAllen wrote which I will briefly recap.

Sure is and its correct - the principle of least action has that word LEAST in it - our physical laws would be wrong if it wasn't minimizing. It's just the Lagrangian in relativity contains that -m in front of it that changes maximizing proper time to minimizing the Lagrangian.

Why are Lagrangian's minimums? Feynman just shows it's a stationary parth when you infinitesimally change the path so it no longer cancels. Its because mass is positive - but I will leave you to investigate the why of that in - Landau - Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

Thanks
Bill

 

[bhobba]

But why does it HAVE to follow a geodesic path in the dimensions of space?

I have already done a number of posts above the B level in this thread. I did it in the hope even some B level people will get a gist. That is just a hope - you may not get the gist - which is absolutely nothing to worry about.

The answer has to do with mathematical elaboration of basic laws of nature, like Quantum Mechanics and things like the Principle of Relativity which applies to inertial frames. When you work through the math that is what it implies.

Thanks
Bill

 

[Sorcerer]

See my elaboration of what PAllen wrote which I will briefly recap.

Sure is and its correct - the principle of least action has that word LEAST in it - our physical laws would be wrong if it wasn't minimizing. It's just the Lagrangian in relativity contains that -m in front of it that changes maximizing proper time to minimizing the Lagrangian.

Why are Lagrangian's minimums? Feynman just shows it's a stationary parth when you infinitesimally change the path so it no longer cancels. Its because mass is positive - but I will leave you to investigate the why of that in - Landau - Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

Thanks
Bill

What if you use the (-,-,-,+) sign convention? (Where space gets the negative sign) Does they change anything or is it irrelevant? I would guess in this case you’d actualy maximize it.

 

[bhobba]

I know this Australian guy is probably gone for good, but I don't know if I can accept this..

Don't - what he wrote is rubbish - and I am an Australian guy to if that makes any difference .

Thanks
Bill

 

[Sorcerer]

Don't - what he wrote is rubbish - and I am an Australian guy to if that makes any difference .

Thanks
Bill

Lol. I did say “this” Australian guy, for the record. Interesting possibly slightly xenophobic side-note regarding an Australian physics professor I had- he called out the rest of the physics department for having an example problem using a “koala bear.” It was pretty darn funny.

 

[bhobba]

What if you use the (-,-,-,+) sign convention? (Where space gets the negative sign) Does they change anything or is it irrelevant? I would guess in this case you’d actualy maximize it.

No.

See my explanation - it does not involve the signature of the metric. It's to do with the free particle Lagrangian which is the same regardless of the metric you use.

Thanks
Bill

 

[Patterner]

Unfortunately, I don't know anything about the math of all this. It's been about 37 years since high school trig, and that's as far as I went. And the first time a heard or saw the words "geodesic" and "worldline" was four days ago.

But can't this, at least the initial concept I'm trying to wrap my head around, be put into the words of a fairly simple thought experiment? I've heard Einstein was fond of them. :D Some of you may well be answering my question, but in terms I don't understand. So let me try this wording. Trying to make everything as unambiguous as possible.

Let's say there's a planet sitting in the middle of a section of flat spacetime. The only point of reference is a distant galaxy that looks like a star to the naked eye. Its pretty far away. Relative to this galaxy, this lone point of reference, the planet is not moving in space. It is not even rotating. It is moving in the dimension of time, but not the dimensions of space. It is utterly still.

The planet causes a spacetime curvature, as planets tend to do. That curvature has a certain size and shape, which is the result of the planet's composition, size, and whatever else affects the size and shape of spacetime curvatures. If we could inject die into this Curved Hunk o' Spacetime, we would be able to see it. But we can't, so we're stuck with our imaginations. A planet, surrounded by this lattice; this matrix; this CHoS.

We can also, in our minds' eyes, remove the planet from this scenario. Now we have a shimmering CHoS, sitting absolutely still, relative to the only point of reference.

Now getting back to the baseball. The CHoS is not there yet. Only a baseball is there, sitting still in flat spacetime. And who comes along but Galactus! From the distant galaxy, he sees the baseball. And, with his Power Cosmic, he creates our our old friend, CHoS, so that the very edge of it touches the baseball. He creates it so that it does not move, relative to the baseball, the Galaxy, and himself. All four things are, relative to each other, dead in space. No motion, of anything relative to anything, lead to the baseball being on the edge of this section of curved spacetime. But there it is.

My only question is this: Will the baseball move toward the spot that would be the center of the planet, the center gravity, if the planet was there? (If so, I assume through it, than probably pulled back again, through toward the spot it started, back and forth.) That is, does the presence of spacetime curvature, by itself, without the help of relative motion, magnetism, or any other darned thing, cause an object to move in the dimensions of space?

This all seems pretty simple and clear enough to me. I would expect a Yes or No answer is possible. But, not knowing enough about this stuff, maybe it's not?

 

[Ibix]

It is moving in the dimension of time, but not the dimensions of space.

The point is that this is a choice you make. You are free to choose that the planet is moving in space, simply by choosing to work in another frame. This is why the theory is called relativity - there's no such thing as "at rest" in any absolute sense. It's always relative to something. You've chosen some galaxy, but you can choose something else. One of the stars in the galaxy, perhaps.

 

[pervect]

The planet causes a spacetime curvature, as planets tend to do. That curvature has a certain size and shape, which is the result of the planet's composition, size, and whatever else affects the size and shape of spacetime curvatures. If we could inject die into this

I'm not sure I understand your question, but as far as your image goes, you might like http://www.eftaylor.com/pub/chapter2.pdf, an excerpt from Taylor's book, "Exploring Black Holes".

Nothing is more distressing on first contact with the idea of curved space-
time than the fear that every simple means of measurement has lost its
power in this unfamiliar context. One thinks of oneself as confronted with
the task of measuring the shape of a gigantic and fantastically sculptured
iceberg as one stands with a meterstick in a tossing rowboat on the surface
of a heaving ocean.

Were it the rowboat itself whose shape were to be measured, the proce-
dure would be simple enough (Figure 1). Draw it up on shore, turn it
upside down, and lightly drive in nails at strategic points here and there
on the surface. The measurement of distances from nail to nail would
record and reveal the shape of the surface. Using only the table of these
distances between each nail and other nearby nails, someone else can
reconstruct the shape of the rowboat. The precision of reproduction can be
made arbitrarily great by making the number of nails arbitrarily large.

In space-time, the nails are replaced by events. I'm not sure if you're familiar with events, they aren't terribly complicated. Events have a location and a time of occurence. For example, if one was writing a police report about a crime, one might give the location (a street address, say), and a time and date of the crime. The street address is the location of the event, the crime, that tells where it occurred in space. Specifying the location of the event is not sufficient, however, we also need to know the time at which it occured. Events are one of the most fundamental elements of space-time, replacing the "nails" in Taylor's rowboat, which only have a location in space, but do not have a time of occurence.

The tricky part here is understanding what replaces the notion of the "distance between nails" on the rowboat. What is the equivalent "distantce" between events in space-time? The needed concept here is called the Lorentz interval. The Lorentz interval says that given two events in space-time, there is a single number that is simliar to a "distance", a number that is independent of the observer. A key part of why we need the Lorentz interval is a feature of special relativity called "the relativity of simultaneity". This feature says that whether or not two events occur "at the same time" depends on the observer, specifically the observer's state of motion, the observer's velocity. The end result is that we do not have a separate "spatial distance" and a "time distance" between two events that is the same for all observers. The spatial distance can change due to Lorentz contraction, the time "distance" can change due to the relativity of simultaneity. The Lorentz interval, however, does not change. It's the same for everyone, and it makes talking about what happens much, much easier.

I would guess offhand from what you write that you are not already familiar with these concepts (the Lorentz interval and the relativity of simultaneity). They both occurs in special relativity, which is much simpler mathematically than General relativity, though people still stumble over some of the needed concepts.

Myl recommendation to everyone is to understand special relativity first, before tackling General relativity. But I'd settle for people realizing that they should at least study special relativity in addition to general relativity.

 

[Nugatory]

That is, does the presence of spacetime curvature, by itself, without the help of relative motion, magnetism, or any other darned thing, cause an object to move in the dimensions of space?

I don't understand what you mean by "without the help of relative motion", because all movement in space is always relative motion - there is no other kind of motion.

But with that said, the answer to the rest of your question is that one thing and only one thing "causes" an object to move in the dimensions of space: and that one thing is which point in space you choose to define as not moving.

We've mentioned an object free-falling towards the surface of the Earth several times already in this thread. Choose the object to be at rest and the surface of the Earth is moving upwards through space; choose a point on the surface of the Earth to be at rest and the object is moving downwards through space. Either way the choice is purely a choice of point of view, with no physical significance.

In general the statement "X is moving through space" tells us only about the point of view of the person making that statement, and tells us nothing about what X is really doing. If you want to know that, you have to draw X's worldline, its path through spacetime. All observers, regardless of their point of view, will agree about which points in spacetime that worldline passes through, so there is no point-of-view problem in this description.

 

[Dale]

We can also, in our minds' eyes, remove the planet from this scenario.

You cannot remove the planet and keep the curvature of spacetime. The planet is the source of the curvature: no source -> no curvature. It is best to leave it.

our old friend, CHoS, so that the very edge of it

Hmm, there isn’t an edge. The curvature from a planet gets less and less the further away, but there is no edge where it suddenly goes to zero. I don’t think that the edge is helpful to your thought experiment.

That is, does the presence of spacetime curvature, by itself, without the help of relative motion, magnetism, or any other darned thing, cause an object to move in the dimensions of space?

This question is better answered by the following simplification of your scenario:

Consider a spherical non-rotating planet and a force-free ball initially at rest with respect to the planet in the vacuum some finite distance away from the planet. Neglect all other effects besides the curvature of spacetime due to the planet. Does the presence of spacetime curvature, by itself, cause the ball to move with respect to the planet?

The answer is: Yes

 

[Patterner]

You cannot remove the planet and keep the curvature of spacetime. The planet is the source of the curvature: no source -> no curvature. It is best to leave it.

Is there no hope of finding other means of curving spacetime, so we can put gravity wherever we want? Like on spaceships?

Hmm, there isn’t an edge. The curvature from a planet gets less and less the further away, but there is no edge where it suddenly goes to zero. I don’t think that the edge is helpful to your thought experiment.

One of the many less-than-fully-thought-out things I've posted in this thread. Still, there must be a distance from the Earth where it does not curve spacetime sufficiently to affect a baseball?

This question is better answered by the following simplification of your scenario:
Consider a spherical non-rotating planet and a force-free ball initially at rest with respect to the planet in the vacuum some finite distance away from the planet. Neglect all other effects besides the curvature of spacetime due to the planet. Does the presence of spacetime curvature, by itself, cause the ball to move with respect to the planet?
The answer is: Yes

Thank you!

 

[Patterner]

I would guess offhand from what you write that you are not already familiar with these concepts (the Lorentz interval and the relativity of simultaneity).

Lol. Indeed, I am not. Your post is the first I've ever heard of Lorentz. Thank you for the link. I'll see what I can make of it.

 

[Dale]

Is there no hope of finding other means of curving spacetime, so we can put gravity wherever we want? Like on spaceships?

You could always build planet sized spaceships.

One of the many less-than-fully-thought-out things I've posted in this thread. Still, there must be a distance from the Earth where it does not curve spacetime sufficiently to affect a baseball?

You could always specify some small speed and some large time where if over that large time the baseball has acquired less than the small speed, then you will call it “unaffected”. There would not be any edge, but at that point you would kind of shrug and say “who cares”.

However, setting a “who cares” radius and then asking about a baseball placed there will only lead to “who cares” answers. That is fairly uninteresting for us to write and uninformative for you to read, so I hope you don’t do it.