I Question about geometrical engineering (placeholder name)

[Amazium]

TL;DR Summary

When two objects move apart due to an applied force, how does the surrounding space behave? Does it compress, expand, or take on a specific curvature? And if we understand this field distortion, could we replicate it artificially—causing objects to repel without direct propulsion?

Imagine a hypothetical 2D vector field wherein two circular objects are positioned near each other but not touching. If we could see the curvature of the soace surrounding each object, we’d see the vectors shrink and point inward towards the center of each object. I want to examine the backdrop as the object though, not the two material objects. If we set the total energetic potential of this undisturbed system to zero and make it a rule that the field seeks to minimize the total energy at every frame, it will naturally resolve any asymmetrical disturbance whenever one exists. In the case of two massive objects, this causes them to move toward eachother.

So one object will still cause the surrounding space to seek resolution, only it will do so equally in all directions, resulting in no net change. The curvature around the object will remain the same over time. If the object is moving, not accelerating, the surrounding curvature will remain constant, theoretically forever.

With two objects, there are now two disturbances AND an asymmetrical configuration of the space between the two. The space between the two will have a specific quality, whether it be that it is compressed, stretched, etc whatever, it will have specific characteristics and properties. The opposite of this property should therefore cause a repulsion between the two objects, also to minimize the total energetic potential of the system.

This is often viewed as impossible as gravity is always assumed to be attractive, but I’d argue that it isn’t, and it’s actually obvious that it’s not.

Examining two objects that move apart, instead of focusing on the objects themselves, we examine the space between them. Imagine any conceivable obejct moving away from another. Let’s say a motorcycle moving away from a person. What does the space between the motorcycle and the person look like when they are at rest relative to each other, when the motorcycle accelerates away from the person, and when the motorcycle approaches the person?

I’m trying to wrap my head around this concept of only examining space as the object rather than the objects themselves, and would like to do so more mathematically than philosophically. Can anyone point me in the right direction or refer me to already existing maths that cover this topic? I know that Einstein’s field equations cover this to some extent, but gravity is only ever discussed as an attractive curvature from what I can tell. I’d like to examine it as a more polarized distortion which can cause repulsion just as often as it causes attraction, inseparably so, such that any time there is attractive curvature, repulsive curvature manifests as well, just not as obviously.

Thanks!

 

[Nugatory]

In general relativity gravity is not a force, so it is a mistake to think in terms of “attractive” or “repulsive” forces. Objects moving under the influence of gravity are in free fall, subject to no force, and what appears to be attraction or repulsion is their straight-line paths through curved spacetime either converging or diverging.
Consider two people starting one meter apart at the equator and both walking due north on the curved surface of the earth; they would draw closer to one another and collide at the North Pole. Gravitational “attraction” is the same thing, except moving through four-dimensional curved spacetime instead of two-dimensional curved space.

Curvature effects are not always attractive. Consider two masses, one meter apart. Orient them parallel to the surface of the earth and release them, and they will move towards one another as they fall; but orient them vertically relative to one another and they will move apart.

I’m trying to wrap my head around this concept of only examining space as the object rather than the objects themselves, and would like to do so more mathematically than philosophically. Can anyone point me in the right direction or refer me to already existing maths that cover this topic? I know that Einstein’s field equations cover this to some extent

That is exactly what the Einstein field equations do. We put everything - all the masses, all the energies, all the radiation, amounts and positions changing with time - into the stress-energy tensor on the right-hand side of the EFE, and what emerges (after we’ve solved this fiendishly difficult, usually intractable set of non-linear differential equations) is the metric tensor which exactly specifies the curvature of spacetime at every point. That’s how we answer the first part of your question:

When two objects move apart due to an applied force, how does the surrounding space behave? Does it compress, expand, or take on a specific curvature?

As for the second part “if we understand this field distortion, could we replicate it artificially—causing objects to repel without direct propulsion?” As it said above, it’s a mistake to think in terms of attraction and repulsion. But we can work the EFE in the other direction: what spacetime curvature would lead the two objects to follow diverging paths, plug that into the left-hand side of the EFE, see if there is a stress-energy tensor that works, then arrange things according to that SE tensor. Usually there is no such solution, and even if there is it may involve impossible things like negative masses.

[edit: I hit the “reply” button to soon, ended up editing in the second half of this post after the first half was visible. Apologies to everyone who saw the incomplete version]

 

[Amazium]

In general relativity gravity is not a force, so it is a mistake to think in terms of “attractive” or “repulsive” forces. Objects moving under the influence of gravity are in free fall, subject to no force, and what appears to be attraction or repulsion is their straight-line paths through curved spacetime either converging or diverging.
Consider two people starting one meter apart at the equator and both walking due north on the curved surface of the earth; they would draw closer to one another and collide at the North Pole. Gravitational “attraction” is the same thing, except moving through four-dimensional curved spacetime instead of two-dimensional curved space.

Curvature effects are not always attractive. Consider two masses, one meter apart. Orient them parallel to the surface of the earth and release them, and they will move towards one another as they fall; but orient them vertically relative to one another and they will move apart.
That is exactly what the Einstein field equations do. We put everything - all the masses, all the energies, all the radiation, amounts and positions changing with time - into the stress-energy tensor on the right-hand side of the EFE, and what emerges (after we’ve solved this fiendishly difficult, usually intractable set of non-linear differential equations) is the metric tensor which exactly specifies the curvature of spacetime at every point.

Nugatory, I understand that the “force” is a result of the curvature of space. My question is asking how the curvature of space between two objects moving apart is different than two objects moving together. When an object moves toward another object, the space between the two has to be fundamentally different than when they move away from each other, otherwise they would not behave the way they do, because the space they occupy together would not accommodate their combined motion in that direction.

Even if you assume the object contains an intrinsic attribute different than its opposite such as with magnets and electric charges, the way that those opposing objects distort the space between them must be fundamentally opposite or different depending on whether or not you get attraction or repulsion to occur. The space and the objects are not separate entities, they are continuous expressions of eachother, where a property change of one results in an equal property change in the other. So if you delete the objects entirely from your examination, you should see space acting in a way that implies the exact objects which are now absent, and likewise, if you assume there is no space, and there is only the material system of the two objects, you should be able to derive the exact behavior of the space between them.

I’m not a physicist or a mathematician, I admit, and this could be exactly what the Einstein field equations describe. But it seems to be very unidirectional in terms of its polarity. If curvature favors motion in any direction, then the opposite curvature should be present in the opposite direction, such that motion is favored from both perspectives. Like as an object falls toward earth, the curvature of space behind the object will favor repelling the object away from that point in space, while the space in front of it (in terms of its trajectory of motion) will favor attraction. Both are happening simultaneously. And so it should be possible to replicate this effect by intentionally distorting space such that the space in front of an object favors its motion into or away from that point in space.

This is not at odds with GR from my understanding, as GR allows us to curve space with mass, energy density, momentum, pressure, etc. so I don’t mean to invalidate anything here, I only wish to expand upon it, if possible. Unless there is an error in my logic which I can’t see, which is also very possible.

 

[cianfa72]

Nugatory, I understand that the “force” is a result of the curvature of space. My question is asking how the curvature of space between two objects moving apart is different than two objects moving together.

This is not at odds with GR from my understanding, as GR allows us to curve space with mass, energy density, momentum, pressure, etc. so I don’t mean to invalidate anything here, I only wish to expand upon it, if possible. Unless there is an error in my logic which I can’t see, which is also very possible.

The curvature we're talking about is the spacetime curvature (i.e. geodesic deviation) and it isn't the curvature of just "space". Note that "space" let me say is "ill-defined" in GR since it depends on the specific spacetime's spacelike foliation you pick (i.e. it depends from your notion of "space at given point in time").

 

[Amazium]

The curvature we're talking about is the spacetime curvature (i.e. geodesic deviation) and it isn't the curvature of just "space". Note that "space" let me say is "ill-defined" in GR since it depends on the specific spacetime's spacelike foliation you pick (i.e. it depends from your notion of "space at given point in time").

cianfa72, ok same question just replace “space” with “spacetime” wherever I used it incorrectly.

 

[Amazium]

In general relativity gravity is not a force, so it is a mistake to think in terms of “attractive” or “repulsive” forces. Objects moving under the influence of gravity are in free fall, subject to no force, and what appears to be attraction or repulsion is their straight-line paths through curved spacetime either converging or diverging.
Consider two people starting one meter apart at the equator and both walking due north on the curved surface of the earth; they would draw closer to one another and collide at the North Pole. Gravitational “attraction” is the same thing, except moving through four-dimensional curved spacetime instead of two-dimensional curved space.

Curvature effects are not always attractive. Consider two masses, one meter apart. Orient them parallel to the surface of the earth and release them, and they will move towards one another as they fall; but orient them vertically relative to one another and they will move apart.
That is exactly what the Einstein field equations do. We put everything - all the masses, all the energies, all the radiation, amounts and positions changing with time - into the stress-energy tensor on the right-hand side of the EFE, and what emerges (after we’ve solved this fiendishly difficult, usually intractable set of non-linear differential equations) is the metric tensor which exactly specifies the curvature of spacetime at every point. That’s how we answer the first part of your question:

As for the second part “if we understand this field distortion, could we replicate it artificially—causing objects to repel without direct propulsion?” As it said above, it’s a mistake to think in terms of attraction and repulsion. But we can work the EFE in the other direction: what spacetime curvature would lead the two objects to follow diverging paths, plug that into the left-hand side of the EFE, see if there is a stress-energy tensor that works, then arrange things according to that SE tensor. Usually there is no such solution, and even if there is it may involve impossible things like negative masses.

[edit: I hit the “reply” button to soon, ended up editing in the second half of this post after the first half was visible. Apologies to everyone who saw the incomplete version]

But even if there exists no solution in GR, there still exists an actual solution to my problem, because if you position three objects in space in a straight line, with respective masses 1, 1, and 3 kg, the middle object (1 kg) will move away from the left object and toward the right object. The left object will gradually come together with the other two, but momentarily the space between the left and middle object will cause the distance between them to temporarily increase, which I’m calling repulsion in an informal way, but maybe there is a better more rigorously accurate term.

If we knew nothing about these objects or the space between them, we would note that the left most object appeared to repel the middle object, even if only temporarily. Can spacetime be manipulated between two objects to favor relative motion in a seemingly unnatural direction? And if this is the case, what math do I need to learn to describe the medium I’m referring to? Tensor calculus?

 

[Ibix]

If you are trying to invent repelling gravitational fields the short answer is you can't - it would violate the equivalence principle.

Tidal forces due to some third body may cause two bodies to move together or apart, as @Nugatory has already stated.

 

[Nugatory]

My question is asking how the curvature of space between two objects moving apart is different than two objects moving together. When an object moves toward another object, the space between the two has to be fundamentally different than when they move away from each other, otherwise they would not behave the way they do,

There is indeed a difference: in one case the curvature is such that initially parallel straight lines in the forwards-in-time direction converge and eventually intersect; in the other the curvature is such that these lines diverge. (Of course if there were no curvature the initially parallel lines would remain parallel forever - for example two objects with negligible mass and at rest relative to one another in intergalactic space far from any external gravitational influence will neither draw clloser nor apart).

An important note: you should stop thinking in terms of points in space; we're working with spacetime and what you are calling a point in space is actually a line in spacetime. If you are not familiar with the two-dimensional Minkowski flat spacetime spacetime diagrams of special relativity (and the way you are talking suggests that you aren't) it is essential that you learn this way of thinking before yo can take on curved spacetimes.

 

[Amazium]

If you are trying to invent repelling gravitational fields the short answer is you can't - it would violate the equivalence principle.

Tidal forces due to some third body may cause two bodies to move together or apart, as @Nugatory has already stated.

Can you elaborate on how it would violate the equivalence principle?

 

[Ibix]

Can you elaborate on how it would violate the equivalence principle?

Two test particles released under the same conditions must fall at the same rate, or else you could tell a gravitational field apart from an accelerating coordinate system in flat spacetime. If you have one that flies in the opposite direction, that's not the same rate.

 

[Amazium]

Two test particles must fall at the same rate. If you have one that flies in the opposite direction, that's not the same rate.

As in the same unit of distance per unit of time, sure, but curved spacetime has varying units of distance and time, so two objects could theoretically fall at different rates of speed if one existed closer to a curved volume of spacetime than the other, even if they are next to eachother, no? Two photons passing a blackhole in close proximity for example. One, slightly closer might get trapped in the event horizon while the other just bends towards the black hole slightly but escapes out into space. The distance between the photons increases forever in that case, which mirrors the concept of repulsion.

 

[Ibix]

As in the same unit of distance per unit of time, sure, but curved spacetime has varying units of distance and time, so two objects could theoretically fall at different rates of speed if one existed closer to a curved volume of spacetime than the other, even if they are next to eachother, no? T

That's a very long-winded way of saying that if you start things on different paths they follow different paths.

You are talking about tidal gravity here, which can certainly be exploited during slingshot maneuvers and the like. It's not repelling gravity, though. Fundamentally, it's not really different to the fact that two straight lines that aren't parallel get further apart. Curved spacetime just gives you more complicated free-fall paths than just straight lines.

 

[Amazium]

That's a very long-winded way of saying that if you start things on different paths they follow different paths.

You are talking about tidal gravity here, which can certainly be exploited during slingshot maneuvers and the like. It's not repelling gravity, though. Fundamentally, it's not really different to the fact that two straight lines that aren't parallel get further apart. Curved spacetime just gives you more complicated free-fall paths than just straight lines.

Exactly, so it’s in no violation of the laws of physics to induce a specific manner of curvature between two objects which causes them to diverge in time, geodesically, from the desired frame of reference.

 

[Ibix]

Ok. But then there's no difference between the geometry that leads to separating geodesics and converging geodesics. That's entirely a matter of launch parameters - position, velocity and timing.

 

[Amazium]

I disagree. My hypothesis which I’m willing to accept as incorrect if it is, is that if we could replicate the conditions of spacetime between the two photons, we could cause divergence of the two. In other words, I’m assuming that there is more than one route to achieve the effects of tidally slingshotting something away from a source of gravity. If the spacetime between the objects was indistinguishable in both cases, the motion would also be indistinguishable. So it becomes more a question of distorting space in just the right way to induce the effect rather than focusing on the material objects themselves so much.

 

[Ibix]

I disagree

You disagree with what? That things in free fall in the same region can get closer to each other and further away from each other?

So it becomes more a question of distorting space in just the right way to induce the effect

Sure. You just need to custom-build a planet.

You might want to look up Alcubierre's warp drive solution. It's impossible because it requires matter that violates the energy conditions, but it's along the lines of a gravitational drive. It doesn't depend on repulsive gravity, though, nor misinterpreting geodesic deviation as repulsion.

 

[Amazium]

You disagree with what? That things in free fall in the same region can get closer to each other and further away from each other?

Sure. You just need to custom-build a planet.

You might want to look up Alcubierre's warp drive solution. It's impossible because it requires matter that violates the energy conditions, but it's along the lines of a gravitational drive. It doesn't depend on repulsive gravity, though, nor misinterpreting geodesic deviation as repulsion.

I disagree with more than one of your assumptions. Your focus being on the motion of the object rather than the geometry of space. You assume that divergent conditions depend on classical motion of the objects within the medium, rather than on the medium itself. I think both can produce indistinguishable results. A fast moving object curves spacetime, but curved spacetime can also produce a fast moving object.

I also disagree that alcubierres drive is the ONLY way to produce the intended effect. It might be the ONLY KNOWN way, or the ONLY ASSUMED way, sure. Not the only way. And to say you’d need a planet to generate useful curvature is an assumption with insufficient evidence also. It’s the route you’d take if you were focused primarily on the object in space rather than the space. You can produce enough curvature to propel an object into space right now. It’s called thrust, and whatever it does to the space around the thrusted object is again virtually indistinguishable from curved space that causes the object to rapidly diverge from the earth in time.

And I am not extensively educated in physics, so I may use terminology which isn’t part of the formally established vocabulary, like repulsion, when what I mean is apparent repulsion, which I mean to describe geodesic divergence. I would argue that all repulsion/attraction must also be the result of locally curved space also, so that even magnetic and electric charge attraction/repulsion are erroneous terms which we think are intrinsic properties, but are instead a result of curvature in the field they occupy.

If it helps you can imagine me as someone who speaks broken english. Sure you could keep getting frustrated and correcting me semantically, or you could just realize that I speak broken physics and try to understand what it is I’m actually trying to say.

 

[Ibix]

A fast moving object curves spacetime

No differently from a stationary one.

I think both can produce indistinguishable results

And I think you don't understand relativity. You will need to learn it if you expect to produce anything of value.

I would recommend Spacetime Physics by Taylor and Wheeler or Morin's Special Relativity for the Enthusiastic Beginner. The former is free from Taylor's website, and the latter is a £10 download or something like that. Once you've learned special relativity you can start on general relativity. I mostly used Sean Carroll's lecture notes, which can be found free online, and I also found @bcrowell's GR book very useful (it can be found at his website, lightandmatter.com/genrel/). Depending on your level of physics education you may need to review classical mechanics and maybe electromagnetism first. We can help you with things you don't understand as you work through that, but we can't teach it all in forum posts.

 

[cianfa72]

Ok. But then there's no difference between the geometry that leads to separating geodesics and converging geodesics. That's entirely a matter of launch parameters - position, velocity and timing.

Exactly. Note that in this example we are neglecting the masses of the two particles involved as "sources" of the "underlying" field. They are initially left in an existing gravitational field, i.e. they do not change/alter with their masses the spacetime curvature.

 

[Nugatory]

but curved spacetime has varying units of distance and time, so two objects could theoretically fall at different rates of speed if one existed closer to a curved volume of spacetime than the other,

The equivalence principle applies only within a region of spacetime (and note that that is spacetime not space) small enough that tidal effects can be ignored. This is equivalent to saying that the curvature is, to the limits of accuracy of our measuring devices, the same across the region that we are considering.

If you have not already done so, this would be a really good time to google for "Einstein's elevator". Why objects fall at the same rate inside the elevator and why no mater where the are dropped from their trajectory intersects the floor of the elevator at the a right angle. Any viable theory of gravity has to produce the same results if the elevator is being held at rest within a gravitational field, as long as the elevator is sufficiently small.

 

[Nugatory]

My hypothesis which I’m willing to accept as incorrect if it is, is that if we could replicate the conditions of spacetime between the two photons, we could cause divergence of the two. In other words, I’m assuming that there is more than one route to achieve the effects of tidally slingshotting something away from a source of gravity. If the spacetime between the objects was indistinguishable in both cases, the motion would also be indistinguishable. So it becomes more a question of distorting space in just the right way to induce the effect rather than focusing on the material objects themselves so much.

Sure, you're just describing the "work the EFE in the other direction" process I described at the end of post #2 of this thread. But if the spacetime geometry is indistinguishable then the stress-energy tensor must be the same, and we already know what SE tensor produces that geometry: the one describing a nearby massive object.

 

[Nugatory]

cianfa72, ok same question just replace “space” with “spacetime” wherever I used it incorrectly.

That doesn't work - the substitution generally produces nonsense. Among the difficulties:
- "Point in space" would turn into "point in spacetime", but these are different things - so different that even in the intro textbooks we use the word "event" instead of "point" for the latter. A point in space is a line in spacetime.
- Ideas like "near to", "in between", "on the other side of" do not make sense for points in spacetime, aka events, until we've applied a lot more rigor to the definitions.

 

[PeterDonis]

If we set the total energetic potential of this undisturbed system to zero and make it a rule that the field seeks to minimize the total energy at every frame, it will naturally resolve any asymmetrical disturbance whenever one exists.

one object will still cause the surrounding space to seek resolution, only it will do so equally in all directions, resulting in no net change.

With two objects, there are now two disturbances AND an asymmetrical configuration of the space between the two. The space between the two will have a specific quality, whether it be that it is compressed, stretched, etc whatever, it will have specific characteristics and properties. The opposite of this property should therefore cause a repulsion between the two objects, also to minimize the total energetic potential of the system.

I don't know where you're getting all this from, but it's not from standard General Relativity.

gravity is always assumed to be attractive

No, it's not. In General Relativity, we derive the behavior of "gravity" by solving the Einstein Field Equation. There are solutions in which "gravity" is not always attractive.

I know that Einstein’s field equations cover this to some extent

Not "to some extent". The Einstein Field Equation covers all of it.

gravity is only ever discussed as an attractive curvature from what I can tell.

You are mistaken. See above.

 

[Ibix]

There are solutions in which "gravity" is not always attractive.

Can you expand on what you have in mind here?

 

[ohwilleke]

[edit: I hit the “reply” button to soon, ended up editing in the second half of this post after the first half was visible. Apologies to everyone who saw the incomplete version]

There's nothing wrong with editing and improving upon a reply after you originally post it (especially if the edits are made not long after the original post). The system allows this and I do that to some degree in most of my posts.

PF posts are frequently highly technical and it is easy to miss some subtle inaccuracy in your wording or formatting, or some overlooked aspect of an answer missed the first time around that becomes obvious once you've posted it and reread it.

The goal is to make the final product useful to not just the person asking the original question and seeing it in the moment, but also to future readers who will appreciate a more polished response.

 

[cianfa72]

Ideas like "near to", "in between", "on the other side of" do not make sense for points in spacetime, aka events, until we've applied a lot more rigor to the definitions.

You mean define/establish a topology for the spacetime as (at least) a 4D topological manifold.

 

[Nugatory]

You mean define/establish a topology for the spacetime as (at least) a 4D topological manifold.

Paricipant hat off, mentor hat on
Yes, but.... Did you look at the thread level tag?

 

[PeterDonis]

Can you expand on what you have in mind here?

Dark energy.

 

[Jaime Rudas]

Dark energy.

Could the expansion of the universe without dark energy also be considered as such (i.e., as a solution in which "gravity" isn't attractive)?

 

[PeterDonis]

Could the expansion of the universe without dark energy also be considered as such (i.e., as a solution in which "gravity" isn't attractive)?

No. Without dark energy the expansion decelerates, which is an indication of attractive gravity.

 

[pervect]

TL;DR Summary: When two objects move apart due to an applied force, how does the surrounding space behave? Does it compress, expand, or take on a specific curvature? And if we understand this field distortion, could we replicate it artificially—causing objects to repel without direct propulsion?

Imagine a hypothetical 2D vector field wherein two circular objects are positioned near each other but not touching. If we could see the curvature of the soace surrounding each object, we’d see the vectors shrink and point inward towards the center of each object.

This is a somewhat late post, but I'll put in my 2 cents.

I'm not aware of any description of curvature as a 2d vector field, so I don't know what you are imagining, but I don't recognize it as any conventional description of curvature. I suspect you have some personal idea here that may not be correct about what curvature is.

I don't know if you are familiar with tensors, but the basic description of curvature is a rank 4 tensor, called the Riemann curvature tensor. If you are serious about developing a concept of 'geometrical engineering', you'll need to eventually understand the Riemann tensor fully. But that is a fairly high bar. It starts with studying tensors, and differential geometry. Vector spaces are the start of the journey, you seem to have at least some familiarity with those to talk about vector fields. If you also happen to know about dual vectors, then tensors are just a short step away, so perhaps they are not as distant as I've made them out to be. A rank m+n tensor can be thought of as a map from m vectors and n dual vectors to a scalar (an observer-independent number). But I don't know if you know about dual vectors, if you have an interest in linear algebra, it's not a bad thing to learn about. It'll be useful in understanding tensors. And you'll need to understand at least two important tensors to understand General relativity - the stress-energy tensor and the Riemann curvature tensor. You'll probably need to understand a few other tensors as well, such as the Ricci tensor, but the Riemann and stress energy tensors are fundamental. So the proposed goal is - step one, understand what a tensor is. Step two, understand specifically what the Riemann tensor and the Stress energy tensor specifically are.

I can give you my usual "grab bag" of less mathematically demanding descriptions of curvature and GR than jumping directly to the Riemann curvature tensor, but in general you get what you put into the topic, so you won't get as much from the below approaches as you would from a more thorough study - but it will also be less work and less demanding on your background.

One choice that appears to be somewhat relevant to your interests is "The Meaning of Einstein's Equation" by Baez and Bunn. See for instance https://arxiv.org/abs/gr-qc/0103044. This is a link to the abstract, to get the full paper you'll need to look at the PDF. I'll quote from the abstract to attempt to give you an idea of what this is about.

This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of its consequences, and explain how the formulation given here is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity.

Baez and Bunn's approach will give you the insight that gravity from normal matter, with positive energy densities and positive pressures, is attractive, not repulsive. There is a concept of "repulsive gravity" in there, but it is not related to normal matter, but to cosmological ideas like "dark energy". However, dark energy is not directly related to the expansion of the universe, , but the rate of acceleration of the expansion, so it's not a perfect fit for what you're looking for, even though I think it is relevant.

One other note before I continue with the main theme. Baez and Bunn mention that you need to understand SR before you understand GR. My experience is that a lot of people interested in GR have not yet mastered SR. I have no idea where you are at. But it is good l advice to study SR first, before GR, so I'll mention that.

One of the key takeaways from Baez & Bunn is the idea that pressure, as well as density, causes gravity. Most textbooks I'm aware of do not directly mention that pressure causes gravity - the best you'll find is a mention that the stress-energy tensor (there we go with tensors again, you'll really need to have some understanding of them to understand GR fully) causes gravity, and they'll also mention that pressure is part of the stress energy tensor. Therefore one can conclude that pressure has a gravitational effect, which to be honest is a rather strange idea, but that's the way it turns out to work. Normally, matter has a positive energy density, and if we consider it as composed of particles such as atoms, the pressure is zero when the atoms don't move, and positive when the atoms do move. And - it never goes negative, normally. GR then says that the gravitational effect of the moving swarms of matter particles is different (and in some sense, stronger) than the gravitational effect would be of all the particles if they were not moving. Quantitatively, this relates to the expression rho+3P that Baez and Bunn mention. The stationary matter has some density rho and no pressure. The moving matter has the same density rho, but a positive pressure. Thus when we have a swarm of moving particles, P is positive, and rho+3P is greater than rho because P is positive. This probably won't make sense unless you read the paper to understand why rho+3P is a significant quantity.

The effects of dark energy, which is believed to cause an anti-gravity like effect that propels the accelerating expansion of the universe, are thought to involve negative pressures. This is weird, and can't be explained by moving particles of positive density. Quantum mechanics does predict the possibility of negative pressures, though, such as the Casimir effect however. Dark energy is still a mystery - the main reason we think it exists is the observed acceleration of the expansion of the universe. We haven't actually seen dark energy directly in the laboratory.

One way of imagining negative pressures (and the way I secretly use - well it used to be secret) is to imagine a bunch of particles of positive mass that are stationary, combined with a bunch of moving particles with a negative mass. You can expect most people to think that this is a rather silly way of looking at things, I would guess. Defintiely don't imagine that this is a peer-reviewed idea you'll find in textbooks.

Going back to Baez & Bunn's paper - the Raychaudhuri equation gives some specific significance to the sum of the matter/energy density (usually called rho) and the sum of the pressures along the three principal axes. In the spherically symmetrical case, the pressure is the same along all three axes, and we get the quantity rho+3P that Baez and Bunn talk about a lot in their paper. Baez and Bunn do not link this quantity to expansion, however - they link it to the acceleration of expansion. Specifically, the second derivative of the volume of their "ball of coffee ground". (I won't explain the ball of coffee grounds here, you'll need to read the paper to see what I'm talking about. Sorry, but this post is already too long.).

The second derivative makes it an acceleration, the first derivative of the volume would be related to expansion.

Onto some other approaches I like I'll mention briefly. Secotor models don't get a great deal of discussion, but they are an actual way to understand GR well enough to make some predictions that has been presented in the literature by a few authors. I'll refer you to https://www.physicsforums.com/threa...es-in-general-relativity.1066400/post-7128432 for some references there by another poster, robphy. I don't think there is neceessarily any direct relevance to your propulsive interests with the sector model, though. To me the approach is interesting but less powerful than the usual exposition in terms of the Riemann tensor, but I feel it may be more accessible.

 

[cianfa72]

In General Relativity, we derive the behavior of "gravity" by solving the Einstein Field Equation. There are solutions in which "gravity" is not always attractive.

As pointed out by @Nugatory, consider two free-falling particles oriented vertically relative to one another near the Earth. They will move apart.

However, I believe the above doesn't count as an example of not attractive "gravity", though.

 

[PeterDonis]

As pointed out by @Nugatory, consider two free-falling particles oriented vertically relative to one another near the Earth. They will move apart.

However, I believe the above doesn't count as an example of not attractive "gravity", though.

The example you give is an example of tidal gravity, which is a different thing from the "gravity" that the OP is talking about. It's unfortunate that the same word, "gravity", is used to describe different things, but we are stuck with that terminology.

In more technical language, the tidal gravity you describe is due to the Weyl tensor, but the "gravity" the OP is asking about with regard to "attractive" vs. "repulsive" is due to the Ricci tensor.