Model for Gravity -- What mechanism distorts space in the real case?

[gmax137]

When I hear a call for the "mechanism" I think of the angels flapping their wings to keep the planets in their orbits. Physics has been done with such "mechanisms" for longer than any of us can remember.

 

[Vanadium 50]

I think of the angels flapping their wings to keep the planets in their orbits.

In modern times, this has become an animation or angels flapping their wings to keep the planets in orbit.

 

[mitchell porter]

Just what mechanism distorts space in the real case? General Relativity doesn't provide a mechanism.

In your opinion, do Newton's theory of gravity or Maxwell's theory of electromagnetism provide mechanisms for the physics they describe? Because General Relativity isn't that different.

 

[pervect]

The general relativity equation just shows an equalness between curved spacetime and the stress energy tensor, not a mechanism how the curvature arises when mass moves into a new position or new mass gets produced. It does not describe this process.

The "moving mass" changes the stress energy tensor, and the stress energy tensor can be more or less thought of as the "source of the gravitational field", i.e the source of curvature".

So, it does, IMO, answer the question. One major difficulty is in explaining what the stress energy tensor is, and why we need it. At least equally daunting - actually, probably more daunting - is a general explanation of what we mean by curvature, but I will just mention it as a problem and not talk about it in this post. IF we understand at least the right-hand side of Einstein's field equations (the stress-energy tensor), we'll be a step ahead of understanding neither side. Which at least alleviates the issue of GR being "the thing on the left side that you don't know what it is is equal to the thing on the right side that you don't know what it is, either". With enough work, we can understand first one side, and with more work, we can (potentially) understand both!

In the end, you still can't really answer "why" the stress energy tensor causes curvature, as in general why questions have no fixed endpoint. You can always ask "why" about any explanation. You can point out that the representation of curvature is always equal to the representation of the stress-energy tensor, and with enough background you can use this to perform actual calcuations, to compare with measurements, which is the ultimate goal of any theory.

The probably too-technical explanation of why we use the stress energy tensor is that it's covariant in special relativity. This basically means that we can specify how it transforms when we change coordinates. This is important and useful - but it requires a shared understanding of the concept of covariance and it's importance. That's something I started to talk about, but I gave up because it was becoming too long and muddled. It starts with the idea that coordinates are just labels, which also seems to not generally be appreciated by the larger audience.

So, lets step back away from covariance. The stress energy tensor incorporates both the effects of what we traditionally think of mass, and other things that are not usually thought of as having mass, such as radiation (important in cosmology), and fields in general. This is hardly a complete description, but it provides at least an overview of why we use the stress-energy tensor as the source of gravity in general relativity, and not "mass". We used to use "mass" in Newton's theory of gravity, but when we move beyond Newton, the concept of mass turns out to be basically inadequate. I will point out that even before general relativity, the "mass" of electromagnetic fields became a problem, in such notions as the "electromagnetic mass" problem. Wiki has a short description of this issue, see https://en.wikipedia.org/wiki/Electromagnetic_mass. Max Jammer has a couple of semi-popular books on the concept of Mass in classical and modern physics, of which I've read the one on classical physics (though I don't have it handy). One gets into general unpleasantness, with the mass of a truly classical point charge due to it's field becoming infinite :(. This is just a short overview - as I've said, people have literally written books about the issue.

The logical endpoint of the process of extending the concept of mass turns out to be (with our current understanding) to replace the whole idea of mass with the stress energy tensor, and things work well enough with this approach where we can calculate things.

I'm not sure if I've totally succeeded in my basic goal, which is to motivate the idea of the stress energy tensor. It's a hard problem - in the end , most texts that I've seen introduce it, without much attempt to motivate it, and leave the student to - perhaps - figure it out on their own.

This is a bit of an afterthought - but if the idea of point particles with point masses is just too deeply ingrained to avoid, one can model a swarm of point like particles bouncing around and interacting only when they directly "touch" as having a stress energy tensor. Less discussed (at least clasically) is the idea of how such a swarm of point particles can be used to represent a field, with possible complications such as the point particles having negative masses to generate attractive forces. But it may be helpful in motaivation, at least, to realize that such a swarm of particles CAN be represented by a stress-energy tensor, with any piece of the swarm having an energy density (energy / unit volume), a momentum density, and a pressure. It's not perfect (for instance, it doesn't help much with the problem of the infinite electromagnetic mass of a point charge I alluded to earlier), but it might be a helpful step up to understanding the stress energy tensor. Realizing that a swarm of particles has such a stress energy tensor, one is then just left with the challenge of thinking of an extended object as being able to be represented by such a swarm of idealized point-like particles - with only contact interacations, no interactions "at range".

Next up for a serious treatment would go back to the issue of how this entity that we've come up with (the stress energy tensor) transforms, so that we can work in different frames of reference (say, a stationary frame, and a moving frame, for example). This is complicated somewhat by the details of special relativity, in which things transform differently than they did before.

I'm not going to attempt this - but I wanted to mention it, as one of the goals of the whole "covariance" problem I mentioned earlier is to represent physics in such a way that we can use multiple coordinate systems interchangeably, and have all the results be self consistent. This basically consists of mathematical entities that we can use to represent a system in some frame or coordiante system A with transformations back and forth to coordinate system B. The idea of covariance is to insure that a solution in A (which is assumed to exist and be unique for the time being) is also a solution in B, and vica versa. This way we are just able to talk about the solution to a physical problem without having to specify the minutae such as having to specify a specific coordinate system. We simply talk about "the" solution, and it is understood that with the aid of the principle of covariance, we know how to transform "the solution" into whatever coordinates or frame we desire.

 

[cyboman]

Homework Statement:: Model for Gravity
Relevant Equations:: Rg -Rg = G/(8pi*c^4)T

The rubber table model for gravity can't quite translate to reality. One sees that a ball placed on the table distorts it, but this is due to it being in a uniform gravity field. Just what mechanism distorts space in the real case? General Relativity doesn't provide a mechanism.

My laymen instinct would be to say the mass of the entity is what distorts space-time, by mechanism I assume you would mean the math developed by Einstein to describe that phenomenon. If you're looking for something further you're probably chasing questioning the nature and description of matter and space itself.

 

[PeterDonis]

My laymen instinct would be to say the mass of the entity is what distorts space-time

More precisely, the stress-energy of the entity. (There are other subtleties involved but that's a good enough first answer.)

 

[pervect]

My laymen instinct would be to say the mass of the entity is what distorts space-time, by mechanism I assume you would mean the math developed by Einstein to describe that phenomenon. If you're looking for something further you're probably chasing questioning the nature and description of matter and space itself.

I've missed most of this thread, but the closest thing to a "'mechanism" that General Relativity has Einstein's field equations, henceforth called the "EFE". The mathematical statement of the EFE is the core of the theory. Other descriptions of it are intended to present some of the ideas while attempting to avoid the mathematics. Generally, such attempts do not succede well , unfortunately.

The curvature, the so-called distortion, of space-time would be described by a mathematical entity, the Riemann tensor, $R^{a}_{bcd}$. After some computation, this yields the left-hand side of Einstein's field equations, the Einstein curvature tensor, $G_{uv}$. The Einstien tensor G is not a full description of the curvature, the einstein tensor G doesn't contain all the information that the Riemann tensor R does.

Einstein's field equation says that the Einstein field tensor is proportional to the stress-energy tensor, already mentioned. Or, to be precise.

$$G_{uv} = \frac{8 \pi G}{c^4} T_{uv}$$

where $G_{uv}$ is the Einstein tensor and $T_{uv}$ is the stress energy tensor. The thing on the left is related to curavature, the thing on the right is related to energy (mass contributes to this by the famous equation E=mc^2), momentum, and stress. Usually in astronomy, we can replace "stress" with pressure, because in practice, the simpler concept of pressure is sufficient to describe the more complex notion of stress. So one will often see statements that the stress-energy tensor consists of energy, momentum, and pressure, which is a reasonable simplification for astronomical bodies.

So the "mechanism" can be roughly thought of as the process of solving the field equations, with the input to the mechanism being the stress-energy tensor, and the output of the mechanism being the description of the non-Euclidean space-time. "Distorted" is a popularization, as non-Euclidean geometry is not familiar to most people. However, a common example of a non-Euclidean geometry is the surface of a curved object, which has become popularized as distorting a flat background space into a curved one.

There are several caveats. The EFE are highly non-linear, so to fully solve them, you need to specify the stress-energy tensor everywhere, though in many cases one can use linear approximations which relax this constraint.

The stress-energy tensor exists in "undistorted" space-time, so it doesn't need non-Euclidean geometry to specify. Mathematically we call the undistored space-time the "tangent space". Using the curved surface model, every point on the curved surface has a flat plane tangent to the surface, and the stress-energy tensor can be thought of is existing in said "tangent space". The stress-energy tensor could be and occasionally is introduced in special relativity.

The actual mathematical process of solving the EFE usually involves specifying a metric. The metric gives both the stress-energy tensor, and the curvature tensor. So the rough conceptual model I gave of specifying the stress-energy tensor first and then deriving the Riemann tensor isn't quite the way things are actually done.

The challenge of popularizing the EFE is that neither the left hand side of the equation (the Einstein tensor), nor the right hand side of the equation (the stress-energy tensor) is familiar to most of the interested audience. One then winds up with the EFE being described as "the thing on the left", that you don't know what it is, being directly proportional to the thing on the right, which you also don't know what it is :(. Unfortunately, it takes quite a lot of study to appreciate and understand either the thing on the left (the Einstein tensor G), or the thing on the right (the stress energy tensor T). With enough effort, one can slowly come to some appreciation to what the thing on the left is, via a study of differential geometry, and "the thing on the right", by studying special relativity with an emphaisis on covariant formulations of the theory, using mathematical methods such as tensors that allow one to express physics in arbitarary coordinates.

 

[PeterDonis]

The actual mathematical process of solving the EFE usually involves specifying a metric.

Not entirely. It usually involves making some assumptions that impose constraints on the metric while leaving some unknown functions (i,.e,. the metric is not specified completely), making some other assumptions that impose constraints on the stress-energy tensor, computing the Einstein tensor in terms of the remaining unknown functions in the metric and their derivatives, and then solving the differential equations that result from the remaining independent components of the EFE, either in closed form (in the rare cases where that's possible) or numerically.

The two most commonly encountered solutions, Schwarzschild and FRW, are both examples of this:

To obtain the Schwarzschild solution, you first assume spherical symmetry, which constrains the metric to two unknown functions, and then assume vacuum (i.e., zero stress-energy tensor), which allows you to solve the two remaining independent components of the EFE for both of the unknown functions in the metric to get a closed form solution.

To obtain the family of FRW solutions, you assume homogeneity and isotropy, which constrains the metric to one unknown function of time and three possibilities for the spatial geometry of a slice of constant time. You then assume that the stress-energy takes the form of a perfect fluid. Computing the Einstein tensor from the metric and equating components then gives you a family of closed form solutions, usually expressed in terms of the Friedmann equations, which can be further narrowed down by specifying the equation of state of the perfect fluid.

 

[pervect]

A followup - I've recalled that in the past, another poster has recommended "sector models" as a good approach for an early introduction to the ideas of curved space-times at a more useful level than popularizations. Google finds a paper by Zahn and Kraus with about 50 citations, availble on arxiv. https://arxiv.org/pdf/1405.0323.pdf "Sector Models – A Toolkit for Teaching General Relativity. Part 1: Curved Spaces and Spacetimes" that might be of some interest. The same authors have another paper that mentions the field equations, the text of which is available at https://iopscience.iop.org/article/10.1088/1742-6596/1286/1/012025/pdf. I have only skimmed them, however, not read them in depth. But I thought it might be of some interest.

A quote from the abstract of the second paper:

The field equations of the general theory of relativity state the connection between
spacetime curvature and matter, often summarized as “matter curves spacetime”. In this
contribution we analyze the structure of the field equations from a pedagogical point of view and
describe a formulation that is accessible in secondary school and undergraduate physics. This
formulation of the field equations requires the concept of curvature in three dimensions and we
show how this notion can be introduced using sector models. The field equations are illustrated
using the black hole as an example.

I thought it was worth a mention.

 

[PeterDonis]

Sector Models

These are interesting, but I would make one caution: the "sector model" approach helps to visualize spatial curvature, but that is not the same as spacetime curvature. And the difference can often be quite stark.

For example, consider the black hole example. The "sector models" visualizations show that the spatial curvature around a black hole is positive radially but negative tangentially. However, the spacetime curvature is the opposite: radially it is negative (geodesics diverge) while tangentially it is positive (geodesics converge).

So I think one has to be very careful with such pedagogical methods.

 

[pervect]

These are interesting, but I would make one caution: the "sector model" approach helps to visualize spatial curvature, but that is not the same as spacetime curvature. And the difference can often be quite stark.

For example, consider the black hole example. The "sector models" visualizations show that the spatial curvature around a black hole is positive radially but negative tangentially. However, the spacetime curvature is the opposite: radially it is negative (geodesics diverge) while tangentially it is positive (geodesics converge).

So I think one has to be very careful with such pedagogical methods.

That is an issue, and while the author does explicitly address it, the solution does require a fairly advanced knowledge of special relativity.

To quote from the first paper: https://arxiv.org/pdf/1405.0323.pdf

In a spatial sector model a sector is rotated in order to lay it alongside the neighbouring sector. In the spatiotemporal case the rotation is replaced by a Lorentz transformation.

It's a step up from the proposal I often uses, saying that General relativity can be thought of as drawing space-time diagrams on "curved surfaces", going into much more detail about what is meant by a "curved surface". I always give that a very short shrift when I mention it, saying that a sphere is an example of what I mean by a curved surface and not attempting a more general defintion.

While the required knowledge of special relativity is a bit advanced, it's IMO more accessible and practical than trying to present differential geometry at the same intermediate level to the same audience.