I Question about geometrical engineering (placeholder name)

[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.