4 ms·

The classic GR line is "the stress-energy tensor tells spacetime (i.e. the metric tensor) how to bend and spacetime tells the stress-energy tensor how to move".

by klank 1y ago

raattgift 1y ago
tl;dr: drop the use of tensors in the wording and the slogan works better. This sort of "classic" line only applies to a 3+1 slicing or 1+3 threading of the fully-solved Einstein Field Equations, and only in a certain limit. A complete solution of the Einstein Field Equations relates the field-values (the curvature and matter tensors) at every point in the manifold, and therefore there's no moving or bending to speak of in the so-called block universe - there's just tensor-values at every point, and the tensor-values of each tensor have a strict relationship. The choice of any curve along which to slice is totally arbitrary (it doesn't have to be a timelike geodesic, even!), although it certainly helps if for every slice there is no everywhere-non-spacelike curve which passes through the slice more than once, and (where the full solution allows) every point in the manifold is in exactly one slice. Once you've sliced spacetime, you can consider the evolution of a field quantity from one slice to its neighbours - this is 3+1 general relativity (3-dimensional spacetime-filling fields exist on every spatial (or really "constant time coordinate") slice, the slices ordered by some choice of time axis & given coordinates: this is called a foliation), of which there are some formalisms. Taken along an infinite timelike geodesic, this approach gives you quantities that "look like" 3-d objects evolving in time, moving from point to point in spacetime, and interacting (feeling and generating) the gravitational interaction. Threading is similar, and is often done in a 1+3 formalism, but decomposes the tensors in terms of the "radar distance" among sets of initially parallel and initially close-to-each-other curves. Threading is a lot less famous; Landau & Lifshitz's Classical theory of fields goes into some detail. For the purposes of this HN comment, threading also lets one see the evolution of 3-d objects' trajectories as they interact with each other, even though it is just applying "radar coordinates" among a set of curves traced through the fully solved block universe. (Arguably one can trace actual physical objects that are coupled to these curves, and the physical objects could beam light to one another, so it could be a little less arbitrary than selecting one curve (with an object on it or not) as the basis for of a foliation, but then one can get into ratholes involving real interacting matter fields where local interactions can delay "radar" signals... so one probably wants to add back in some arbitrariness by e.g. gauge fixing). Perplexingly humans do not normally think of themselves or their parts (like those carbon atoms you ingested at lunch the other day but just exhaled) coupled to curves through a 4-d manifold, and struggle with the consequences of taking constant-time slices based on e.g. their wristwatches (c.f. practically any HN discussion dealing with recent detections of astronomical events, especially things like stellar deaths). Of course also perplexingly humans do not remember their future as well as their past (although they're honestly not very good at remembering their past in detail, and tend to reconstruct it seemingly similarly to how they predict their future). Finally I guess you could think of the decomposition of the tensors in a 3+1 or 1+3 as producing other tensors, just like you could think of scalar or vector quantities being tensor quantities, but then they are not the metric tensor and the stress-energy tensor of the Einstein Field Equations. Also of course there are other tensors and scalars on the LHS of the EFEs, and they inevitably enter into equations of motion (dissipation from tides raised on partners in relativistic few-body relativistic systems explain certain orbital characteristics!), so the metric and its encoding of changes in angles and displacements isn't the full story in the slogan anyway. That's the problem with slogans. And there are rather a lot of them that get thrown about wrt Einsteinian gravitation. P.S.: I brought up threading (which few working scientists will ever really run into) principally because presently the top comment in this discussion essentially wonders -- somewhat off topically -- how the expansion of the universe could happen instead of everything collapsing into black holes. I was tempted to answer that in terms of the Raychaudhuri equations (two pre-Standard-Model particles of matter initially close together are taken away from each other by e-folds of inflation, and other factors (angular momentum and inelasticities from phase changes / symmetry breaking that gives rise to electrons, photons, protons, and so forth) also tend to keep their trajectories from ever "focusing"). However, I didn't have the time or energy to ELI5 it and scrolled along until deciding to think about your unusual wording of a usual slogan that I have repeated myself.

The classic GR line is "the stress-energy tensor tells spacetime (i.e. the metric tensor) how to bend and spacetime tells the stress-energy tensor how to move".

by klank 1y ago

The classic GR line is "the stress-energy tensor tells spacetime (i.e. the metric tensor) how to bend and spacetime tells the stress-energy tensor how to move".

raattgift 1y ago

tl;dr: drop the use of tensors in the wording and the slogan works better. This sort of "classic" line only applies to a 3+1 slicing or 1+3 threading of the fully-solved Einstein Field Equations, and only in a certain limit. A complete solution of the Einstein Field Equations relates the field-values (the curvature and matter tensors) at every point in the manifold, and therefore there's no moving or bending to speak of in the so-called block universe - there's just tensor-values at every point, and the tensor-values of each tensor have a strict relationship. The choice of any curve along which to slice is totally arbitrary (it doesn't have to be a timelike geodesic, even!), although it certainly helps if for every slice there is no everywhere-non-spacelike curve which passes through the slice more than once, and (where the full solution allows) every point in the manifold is in exactly one slice. Once you've sliced spacetime, you can consider the evolution of a field quantity from one slice to its neighbours - this is 3+1 general relativity (3-dimensional spacetime-filling fields exist on every spatial (or really "constant time coordinate") slice, the slices ordered by some choice of time axis & given coordinates: this is called a foliation), of which there are some formalisms. Taken along an infinite timelike geodesic, this approach gives you quantities that "look like" 3-d objects evolving in time, moving from point to point in spacetime, and interacting (feeling and generating) the gravitational interaction. Threading is similar, and is often done in a 1+3 formalism, but decomposes the tensors in terms of the "radar distance" among sets of initially parallel and initially close-to-each-other curves. Threading is a lot less famous; Landau & Lifshitz's Classical theory of fields goes into some detail. For the purposes of this HN comment, threading also lets one see the evolution of 3-d objects' trajectories as they interact with each other, even though it is just applying "radar coordinates" among a set of curves traced through the fully solved block universe. (Arguably one can trace actual physical objects that are coupled to these curves, and the physical objects could beam light to one another, so it could be a little less arbitrary than selecting one curve (with an object on it or not) as the basis for of a foliation, but then one can get into ratholes involving real interacting matter fields where local interactions can delay "radar" signals... so one probably wants to add back in some arbitrariness by e.g. gauge fixing). Perplexingly humans do not normally think of themselves or their parts (like those carbon atoms you ingested at lunch the other day but just exhaled) coupled to curves through a 4-d manifold, and struggle with the consequences of taking constant-time slices based on e.g. their wristwatches (c.f. practically any HN discussion dealing with recent detections of astronomical events, especially things like stellar deaths). Of course also perplexingly humans do not remember their future as well as their past (although they're honestly not very good at remembering their past in detail, and tend to reconstruct it seemingly similarly to how they predict their future). Finally I guess you could think of the decomposition of the tensors in a 3+1 or 1+3 as producing other tensors, just like you could think of scalar or vector quantities being tensor quantities, but then they are not the metric tensor and the stress-energy tensor of the Einstein Field Equations. Also of course there are other tensors and scalars on the LHS of the EFEs, and they inevitably enter into equations of motion (dissipation from tides raised on partners in relativistic few-body relativistic systems explain certain orbital characteristics!), so the metric and its encoding of changes in angles and displacements isn't the full story in the slogan anyway. That's the problem with slogans. And there are rather a lot of them that get thrown about wrt Einsteinian gravitation. P.S.: I brought up threading (which few working scientists will ever really run into) principally because presently the top comment in this discussion essentially wonders -- somewhat off topically -- how the expansion of the universe could happen instead of everything collapsing into black holes. I was tempted to answer that in terms of the Raychaudhuri equations (two pre-Standard-Model particles of matter initially close together are taken away from each other by e-folds of inflation, and other factors (angular momentum and inelasticities from phase changes / symmetry breaking that gives rise to electrons, photons, protons, and so forth) also tend to keep their trajectories from ever "focusing"). However, I didn't have the time or energy to ELI5 it and scrolled along until deciding to think about your unusual wording of a usual slogan that I have repeated myself.