What is the mechanism by which energy/stress curves spacetime

[Sorcerer]

Ok, I can accept that as a reasonable criterion. Basically, you don’t accept instantaneous action at a distance as a mechanism. So the EFE would qualify as a mechanism (or at least would not be disqualified by this criterion) since it doesn’t have instantaneous action at a distance.

Yeah, it’s a serious upgrade to me. “Energy (etc) warps spacetime by virtue of its presence” makes a lot more sense than “invisible, immaterial magical ray pulls two things that have the same positive ‘charge’ (mass) together as if it were electrical attraction, except instantaneously.”

I suppose one could always ask more “why’s.” But how about this:

Energy density plays a role, right? Momentum density? Stress and pressure?

What does stress, for example, mean in the context of the EFEs? I believe getting a better understanding of what these components of the stress-energy tensor means will help me to see the EFEs as you do- AS the mechanism.

 

[Dale]

What does stress, for example, mean in the context of the EFEs?

Stress is the space-space component of the stress energy tensor. For example it describes the flux of the y component of momentum (a force in the y direction) across a surface normal to x. This is shear stress.

In the EFE this is related to the space-space components of the curvature. The specific type of curvature is a little strange, but basically you can think of the amount that a vector in the x direction changes as you parallel transport it in a plane normal to y.

 

[pervect]

Yeah, it’s a serious upgrade to me. “Energy (etc) warps spacetime by virtue of its presence” makes a lot more sense than “invisible, immaterial magical ray pulls two things that have the same positive ‘charge’ (mass) together as if it were electrical attraction, except instantaneously.”

I suppose one could always ask more “why’s.” But how about this:

Energy density plays a role, right? Momentum density? Stress and pressure?

What does stress, for example, mean in the context of the EFEs?I believe getting a better understanding of what these components of the stress-energy tensor means will help me to see the EFEs as you do- AS the mechanism.

Stress fundamentally means the same thing in physics as it does in mechanical engineering. Though I can't guarantee that engineers talk about it usually exactly the same language as physicists, unfortuately. Hence, for perfect understanding, it's a good idea to look at the math.

For genearl overview purposes, though, if one considers a cantilever beam with a load on the end, the beam is stressed.

Pressure is a form of stress, a form of stress that's the same in all directions (isotropic). If one has a spherical pressure vessel filled with an ideal gas, we'd say that there's stress in the walls of the pressure vessel (specifically, tension), and pressure in the gas. The stress energy tensor is a very general model that can model isotropic stresses (pressures), and non-isotropic ones as well.

It's not particular obvious without going through the math why one needs to include stress in the stress energy tensor, but it wouldn't be a tensor if one didn't include the stress components. Fundamentally, I'd say it's an unexpected consequence of the relativity of simultaneity, so it's not very intuitive.

 

[m4r35n357]

I believe getting a better understanding of what these components of the stress-energy tensor means will help me to see the EFEs as you do- AS the mechanism.

This and this helped me a lot when I first got interested, I think the first one is the closest you will get to a mechanism, maybe you will find it useful.

 

[pervect]

A few more things I can mention. The stress-energy tensor first makes its appearance in special relativity (henceforth SR), so it's needed to understand mechanical problems in SR. A typical SR problem that we've seen on PF that requires the stress-energy tensor would be calculating the angular momentum of an idealized rotating hoop. Be warned, we've also seen a lot of confusion on the part by people not familiar with tensors who try to understand this problem - it's not a particular easy problem to get right without the right methods.

Pedagogically, I would suggest learning about tensors first, before attacking general relativity. It's possible to understand about tensors in isolation, I suppose, but I'd generally recommend a physical application. Applied tensors are usually introduced in electromagnetism. Understanding why the electric field and the magnetic field are combined to form one larger tensor, the electromagnetic field tensor or Faraday tensor, is probably a good first introduction to the use of tensors in physics, and aids in understanding the interrelationships between seemingly different ideas. Understanding this point will aid one in understanding why stress (including pressure) and density are related the way they are in special relativity.

The relationship between stress (including pressure) and density is unexpected, but the principles unifying them into a lager entity are similar to the principles that unify the electric and magnetic fields in electromagnetism. The tensor methods unify the electric and magnetic field into the Faraday tensor - similarly they unify stress, energy, and momentum into the stress-energy tensor.
electromagnetism (EM).

A prerequisite of understanding tensors in general is partial differential equations, henceforth PDE's. PDE's can also be taught in isolation, but are frequently bundled in an applied context, typically Maxwell's equations. The first glimpse one has of Maxwell's equations is presented without tensors. At the graduate level, one introduces tensors and revisits the problem of Maxwell's equations, learning new and powerful techniques.

As a side bonus, one will gain a better understanding of covariance. Tensors transform partial differential equations in a manner that's by definition unaffected by coordinate transformations and/or changes in "frames of reference". I this is a unaswered question for a lot of people, though the mathematical requirements to appreciate how tensors answer this question can be daunting. Tensor methods also play well with Lagrangian methods, as those methods are also about expressing physics in "generalized coordinates". Lagrangian methods are also stressed at the graduate level, though they may be first introduced earlier. It is not essential to learn about Lagrangian mechanics before tackling tensors, however, it can be left until later.