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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 sma

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.
- raattgift 9y ago
  The flexibility of General Relativity lets one consider the same physical picture in different ways, which can aid in understanding. It can also serve to confuse too! :-) A fair amount of research in General Relativity is in laying down some rules which lets one reject a mathematically consistent and complete picture as unphysical, for example, or alternatively which lets a picture that is only partial -- or which can't be proven to be complete or wholly consistent -- as a good physical approximation. > Travelling along a curved spacetime wouldn't imply a force on a neutral particle, would it? There are exact solutions of the Einstein Field Equations (although typically this is done with the linearized version of them, the LEFEs, which is in the "almost certainly a good physical approximation" bucket) in which a change in the background metric propagates through vacuum in some direction as a plane wave. The background metric could in principle be, for example, flat spacetime, or Schwarzschild, or any other vacuum solution of the EFEs. When one chooses a splitting of spacetime into space and time, the wave propagates through space. A small set of test particles reacts to the wave as it passes over them, changing their respective "radar distances" for example. This view lends itself well to perturbation theory, where the wavelike perturbation of the background metric can (if weak) be quantized; those quanta are gravitons in perturbatively quantized gravity. That's a real theory that exists and is good everywhere where there is no strong gravity. "Strong gravity" can even be defined in terms of perturbatively quantized gravity as at least one loop of gravitons on a Feynman diagram. You generally don't find that until very close to a gravitational singularity (i.e., perturbatively quantized gravity as an effective field theory works inside stellar BH horizons too, and works even better inside even more massive BH horizons because there you are further away from the strong gravity near the centre. In the spacetime view, the particles are instead worldlines that are mostly in regions described exactly by the background metric, but a small section of the worldlines are found in a small region where the metric is different. One can consider the spacetime intervals of points on the various worldlines and look for some minimum which will ought to be found in the region with that different metric. Exactly oppositely from Euclidean geometry, curved lines are shorter than straight lines, so a(n inertial) worldline which is found only in flat spacetime regions will be longer than one which is (even briefly) also found in a region with the slightly different metric. Are such worldines travelling along? Nothing really travels at all if it's a worldline in a "block universe" spacetime picture like in the paragraph immediately above. But if one foliates spacetime into spaces with a time coordinate, movement emerges. You can choose a foliation and set of coordinates in which the different metric's effects are shoved into time-like effects (if the background is flat spacetime, a particle exposed to the gravitational wave is younger than a particle that isn't; c.f. the twin paradox); or you can choose a foliation and set of coordinates in which the different metric's effects are shoved into space-like ones ("radar" showing that the particle exposed to the gravitational wave is dragged or bobbed by its passage). With some of these equivalent and (mostly) convertible views of the same physical picture, it would be perfectly reasonable to say that a force has been exerted. After all, one is interested in what causes a small piece of a long stable worldline to look different. The force acting on that section of worldline (or on the particle during that brief period of time), however, depends on one's choices of foliation, coordinates, and a few other things. If you can make some set of such choices such that the force vanishes, we call it an inertial force, d'Alembert force, or frame-dependent/fictitious force. Real forces cannot be made to vanish by any choice of coordinates etc.; real forces' "fingerprints" must do more than change an object's position or orientation in spacetime. The tendency of objects -- particles, worldines, extended (many-particle) objects, worldtubes, or the full fields in which these worldtubes are just clusters of correlated numers -- to look mostly the same is one way of thinking about inertia: you'd want to attribute any unusual change to some force acting at the point of the change. If you remove everything but your test object, so that no other object or field can impart any non-d'Alembertian force on it, then a magic observer at an enormous distance watching its behaviour could say that any changes in radar distance or (if the test object is a clock) ticking rate is the result of coupling with curved spacetime. But a magic observer who can see the whole worldline of the object and knows the layout of spacetime curvature will note that it matches a longest-possible solution of the geodesic equations. > I don't want to burden anyone with my questions. It's not a burden!

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.

raattgift 9y ago
The flexibility of General Relativity lets one consider the same physical picture in different ways, which can aid in understanding. It can also serve to confuse too! :-) A fair amount of research in General Relativity is in laying down some rules which lets one reject a mathematically consistent and complete picture as unphysical, for example, or alternatively which lets a picture that is only partial -- or which can't be proven to be complete or wholly consistent -- as a good physical approximation. > Travelling along a curved spacetime wouldn't imply a force on a neutral particle, would it? There are exact solutions of the Einstein Field Equations (although typically this is done with the linearized version of them, the LEFEs, which is in the "almost certainly a good physical approximation" bucket) in which a change in the background metric propagates through vacuum in some direction as a plane wave. The background metric could in principle be, for example, flat spacetime, or Schwarzschild, or any other vacuum solution of the EFEs. When one chooses a splitting of spacetime into space and time, the wave propagates through space. A small set of test particles reacts to the wave as it passes over them, changing their respective "radar distances" for example. This view lends itself well to perturbation theory, where the wavelike perturbation of the background metric can (if weak) be quantized; those quanta are gravitons in perturbatively quantized gravity. That's a real theory that exists and is good everywhere where there is no strong gravity. "Strong gravity" can even be defined in terms of perturbatively quantized gravity as at least one loop of gravitons on a Feynman diagram. You generally don't find that until very close to a gravitational singularity (i.e., perturbatively quantized gravity as an effective field theory works inside stellar BH horizons too, and works even better inside even more massive BH horizons because there you are further away from the strong gravity near the centre. In the spacetime view, the particles are instead worldlines that are mostly in regions described exactly by the background metric, but a small section of the worldlines are found in a small region where the metric is different. One can consider the spacetime intervals of points on the various worldlines and look for some minimum which will ought to be found in the region with that different metric. Exactly oppositely from Euclidean geometry, curved lines are shorter than straight lines, so a(n inertial) worldline which is found only in flat spacetime regions will be longer than one which is (even briefly) also found in a region with the slightly different metric. Are such worldines travelling along? Nothing really travels at all if it's a worldline in a "block universe" spacetime picture like in the paragraph immediately above. But if one foliates spacetime into spaces with a time coordinate, movement emerges. You can choose a foliation and set of coordinates in which the different metric's effects are shoved into time-like effects (if the background is flat spacetime, a particle exposed to the gravitational wave is younger than a particle that isn't; c.f. the twin paradox); or you can choose a foliation and set of coordinates in which the different metric's effects are shoved into space-like ones ("radar" showing that the particle exposed to the gravitational wave is dragged or bobbed by its passage). With some of these equivalent and (mostly) convertible views of the same physical picture, it would be perfectly reasonable to say that a force has been exerted. After all, one is interested in what causes a small piece of a long stable worldline to look different. The force acting on that section of worldline (or on the particle during that brief period of time), however, depends on one's choices of foliation, coordinates, and a few other things. If you can make some set of such choices such that the force vanishes, we call it an inertial force, d'Alembert force, or frame-dependent/fictitious force. Real forces cannot be made to vanish by any choice of coordinates etc.; real forces' "fingerprints" must do more than change an object's position or orientation in spacetime. The tendency of objects -- particles, worldines, extended (many-particle) objects, worldtubes, or the full fields in which these worldtubes are just clusters of correlated numers -- to look mostly the same is one way of thinking about inertia: you'd want to attribute any unusual change to some force acting at the point of the change. If you remove everything but your test object, so that no other object or field can impart any non-d'Alembertian force on it, then a magic observer at an enormous distance watching its behaviour could say that any changes in radar distance or (if the test object is a clock) ticking rate is the result of coupling with curved spacetime. But a magic observer who can see the whole worldline of the object and knows the layout of spacetime curvature will note that it matches a longest-possible solution of the geodesic equations. > I don't want to burden anyone with my questions. It's not a burden!