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Out of curiosity, where is the spacetime curvature coming from in the single particle universe? And if the particle curves spacetime, what effect does it have

Out of curiosity, where is the spacetime curvature coming from in the single particle universe? And if the particle curves spacetime, what effect does it have on itself?

raattgift 9y ago
In General Relativity, spacetime itself can be curved by its very nature. Vacuum solutions which are exact solutions of the Einstein Field Equations (the core set of partial differential equations of General Relativity; the EFEs), have been the subject of extensive research since the 1920s. A test particle is one with negligible mass and extent, and thus does not itself perturb the curvature of the background vacuum spacetime. If a real particle is electrically neutral, extremely low mass, and nearly pointlike, then in a vacuum spacetime it should behave very much like a test particle. Moreover, for microscopic real objects, which aren't truly pointlike and which may not be wholly chargeless, it is impractically hard to tell at a large enough spatial distance that it's not a test particle. So a test particle or test field does not source the spacetime curvature -- it just rides along the background spacetime, which is arbitrarily curved by its nature. We can extend the notion to a test field that is almost wholly massless, like a single-electron quantum field. The test field sits on the background, but doesn't influence the background's dynamics. Real quantum fields, like test fields, permeate the whole of spacetime. However they tend to have abundant stress-energy in them, so source some of the curvature of the whole spacetime, and typically not uniformly. One can fix a gauge (like how one calibrates an air-pressure altimeter) that lets one talk about the inherent curvature of the background spacetime and the contribution to the total curvature of the lowest density of the quantum field. Typically the background is chosen to be some exact vacuum solution of the EFEs with corrections added via something like perturbation theory; flat spacetime, Schwarzschild and Robertson-Walker expanding spacetimes are popular backgrounds, depending on what one is studying. Sometimes, though one wants to think about an electron rather than some characteristic set of numbers in some all-over-spacetime fields. In that case one (probably implicitly) fixes a gauge that lets one talk about the electron as an independent object that generates an electromagnetic field which, because that has some nonzero stress-energy (zero being set by our gauging choices and being found sufficiently far from our electron) it influences the background curvature of the spacetime (the background again being set by our gauging choices). In either paradigm, one might want to think about, in reverse order, the gravitational back-reaction of the electron (and its EM field)'s gravitational footprint on itself, or the gravitational back-reaction of the overdensities in the all-over-spacetime (classical or quantum) fields on the regions of spacetime they are found in. "Gravity gravitates" is the problem: this arises from the EFEs' non-linearity. Fortunately, for a single electron these back-reactions will be very small, because electrons in either paradigm have very little stress-energy, so their contribution to any curvature will thus be so extremely tiny that it will be practically immeasurable, so there is basically no hope of measuring higher-order contributions (that is, the electron's gravitational influence's own influence on the curvature). Negiligibility saves a lot of headaches, because working with gravitational backreaction is hard.
frutiger 9y ago
raatgift develops an excellent answer describing the many options one has when attempting to model a "single particle" (and goes into detail as to what that small phrase can mean). But to add to the first paragraph of their answer: if you set the 'T' tensor to the zero tensor in https://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form https://en.wikipedia.org/wiki/Einstein_field_equations#Mathe..., then you are effectively studying the vacuum. The left hand side describes the curvature of spacetime. This is a set of 10 non-linear partial differential equations which permit many solutions. You can then add a "single particle" back to these solutions with varying degrees of complexity as outlined by raatgift.
- state_less 9y ago
  Thanks raatgift and frutiger. It seems that the single particle may not be alone in the universe and that there is a vacuum entity that the particle could be depending upon. I'm not sure what forces or other properties the vacuum imbues the particle. Traveling along a curved spacetime wouldn't imply a force on a neutral particle, would it? I'm curious what properties of the particle the vacuum would change over time. I may have to dig deeper into the math to find out. I don't want to burden anyone with my questions. My lay observation was that many of these properties are built up from relations (functions) between two points. The equations for distance have two points, velocity depends on distance over time, inertia depends on velocity. It's difficult to imagine what the other point would be in a single particle universe. It's why I wonder if there would be inertia in a universe with one particle. These concepts seem to spring forth once you add another particle, which causes me to think there isn't inertia per se, but a relation we observe that we call inertia under the condition of having a point moving relative to another with a certain mass at a certain speed. Useful no doubt, except perhaps in single particle universes. Similarly, if we have a party balloon and we take away all the gas and balloons in the universe except for one molecule of nitrogen, do we still have pressure? My answer here is not really, not practically, because these forces spring forth from the context of having multiple molecules. It's a helpful concept for use in the case balloons, tires and pipes, but there doesn't appear to be such a thing as pressure that exists in the universe independent of the particles within it.

Out of curiosity, where is the spacetime curvature coming from in the single particle universe? And if the particle curves spacetime, what effect does it have on itself?

raattgift 9y ago

In General Relativity, spacetime itself can be curved by its very nature. Vacuum solutions which are exact solutions of the Einstein Field Equations (the core set of partial differential equations of General Relativity; the EFEs), have been the subject of extensive research since the 1920s. A test particle is one with negligible mass and extent, and thus does not itself perturb the curvature of the background vacuum spacetime. If a real particle is electrically neutral, extremely low mass, and nearly pointlike, then in a vacuum spacetime it should behave very much like a test particle. Moreover, for microscopic real objects, which aren't truly pointlike and which may not be wholly chargeless, it is impractically hard to tell at a large enough spatial distance that it's not a test particle. So a test particle or test field does not source the spacetime curvature -- it just rides along the background spacetime, which is arbitrarily curved by its nature. We can extend the notion to a test field that is almost wholly massless, like a single-electron quantum field. The test field sits on the background, but doesn't influence the background's dynamics. Real quantum fields, like test fields, permeate the whole of spacetime. However they tend to have abundant stress-energy in them, so source some of the curvature of the whole spacetime, and typically not uniformly. One can fix a gauge (like how one calibrates an air-pressure altimeter) that lets one talk about the inherent curvature of the background spacetime and the contribution to the total curvature of the lowest density of the quantum field. Typically the background is chosen to be some exact vacuum solution of the EFEs with corrections added via something like perturbation theory; flat spacetime, Schwarzschild and Robertson-Walker expanding spacetimes are popular backgrounds, depending on what one is studying. Sometimes, though one wants to think about an electron rather than some characteristic set of numbers in some all-over-spacetime fields. In that case one (probably implicitly) fixes a gauge that lets one talk about the electron as an independent object that generates an electromagnetic field which, because that has some nonzero stress-energy (zero being set by our gauging choices and being found sufficiently far from our electron) it influences the background curvature of the spacetime (the background again being set by our gauging choices). In either paradigm, one might want to think about, in reverse order, the gravitational back-reaction of the electron (and its EM field)'s gravitational footprint on itself, or the gravitational back-reaction of the overdensities in the all-over-spacetime (classical or quantum) fields on the regions of spacetime they are found in. "Gravity gravitates" is the problem: this arises from the EFEs' non-linearity. Fortunately, for a single electron these back-reactions will be very small, because electrons in either paradigm have very little stress-energy, so their contribution to any curvature will thus be so extremely tiny that it will be practically immeasurable, so there is basically no hope of measuring higher-order contributions (that is, the electron's gravitational influence's own influence on the curvature). Negiligibility saves a lot of headaches, because working with gravitational backreaction is hard.

frutiger 9y ago

raatgift develops an excellent answer describing the many options one has when attempting to model a "single particle" (and goes into detail as to what that small phrase can mean). But to add to the first paragraph of their answer: if you set the 'T' tensor to the zero tensor in https://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form https://en.wikipedia.org/wiki/Einstein_field_equations#Mathe..., then you are effectively studying the vacuum. The left hand side describes the curvature of spacetime. This is a set of 10 non-linear partial differential equations which permit many solutions. You can then add a "single particle" back to these solutions with varying degrees of complexity as outlined by raatgift.

state_less 9y ago
Thanks raatgift and frutiger. It seems that the single particle may not be alone in the universe and that there is a vacuum entity that the particle could be depending upon. I'm not sure what forces or other properties the vacuum imbues the particle. Traveling along a curved spacetime wouldn't imply a force on a neutral particle, would it? I'm curious what properties of the particle the vacuum would change over time. I may have to dig deeper into the math to find out. I don't want to burden anyone with my questions. My lay observation was that many of these properties are built up from relations (functions) between two points. The equations for distance have two points, velocity depends on distance over time, inertia depends on velocity. It's difficult to imagine what the other point would be in a single particle universe. It's why I wonder if there would be inertia in a universe with one particle. These concepts seem to spring forth once you add another particle, which causes me to think there isn't inertia per se, but a relation we observe that we call inertia under the condition of having a point moving relative to another with a certain mass at a certain speed. Useful no doubt, except perhaps in single particle universes. Similarly, if we have a party balloon and we take away all the gas and balloons in the universe except for one molecule of nitrogen, do we still have pressure? My answer here is not really, not practically, because these forces spring forth from the context of having multiple molecules. It's a helpful concept for use in the case balloons, tires and pipes, but there doesn't appear to be such a thing as pressure that exists in the universe independent of the particles within it.