3 ms·

Ideally, we'd understand space completely in terms of the relationships between things, and our coordinate geometry would not ascribe any "positions" to events,

by Retra 8y ago

Ideally, we'd understand space completely in terms of the relationships between things, and our coordinate geometry would not ascribe any "positions" to events, but only "relations" between events. Singularities are places where our improperly-formed coordinate maps break down. Which means we'd rather like to look at black holes as holes in the coordinate mapping, rather than as singularities, but we don't have the mathematical prowess to do that. Or good enough scientific understanding of the phenomena to know that we've done it correctly.

danbruc 8y ago
Coordinate systems are not the primary issue. It's true that coordinates can be problematic but the laws of physics are independent of coordinates because those are of course only a tool we use to perform calculations. But take for example the classical law of gravitation which requires dividing by the square of the distance between the two bodies. The distance is independent of the coordinates you use to describe the positions of the bodies but you still get a divergent force as the distance approaches zero. So the breakdown of the law is not due to a bad choice of coordinates but inherent in the equation and a strong indication that the law is probably only approximate for short distances.
- Retra 8y ago
  I believe you're just saying the same thing I am in different terms. The fact that zero is even a 'distance' you can divide by is consequence of the fact that you're deriving distance from the coordinates of the bodies. If distance was instead fundamental (and not a function of coordinates), you'd not even have a zero distance, and you'd have no inclination to attempt dividing by it. (But the math for doing that is basically to always account for every part of every body, and that's just entirely intractable.) This is assuming mostly Newtonian physics, of course. Once you start dealing with fields and quantum mechanics, none of this makes as much sense.
  - danbruc 8y ago
    Coordinates and distances are independent concepts. They are function from the space you are working with into the reals assigning each respectively each pair of elements a real number for each coordinate respectively one real number for the distance. You can for various reasons choose coordinates and distances in a way such that they have simple mathematical relationships but that is just an arbitrary choice. The classical law of gravity is only dependent on distances, which is a physical meaningful concept, but not on coordinates, which are not physically meaningful. In that sense distances are fundamental in a way in which coordinates are not and a distance of zero is a meaningful thing, namely two things are at the very same place.
- raattgift 8y ago
  I think what you're aiming for here is the statement: somewhere in a black hole spacetime the Kretschmann scalar goes to infinity. The Kretschmann scalar is the square of the Riemann tensor, or in notation R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}. There are no forces or coordinates or anything like that to even consider; it's a curvature invariant. ( https://en.wikipedia.org/wiki/Curvature_invariant_(general_relativity) https://en.wikipedia.org/wiki/Curvature_invariant_(general_r... ) If the Kretschmann scalar is non-infinite at a point then there is no gravitational singularity there. If you really want to think about a black hole spacetime in a coordinate system, then in Schwarzschild coordinates for a Schwarzschild black hole, K = \frac{48M^2}{r^6} where K is the Kretschmann scalar. You can see that when r=0, K is undefined. Other coordinate systems and black hole spacetimes are available. However, generically, one can distinguish between a coordinate singularity and a gravitational singularity by calculating the Kretschmann scalar at a point. If the Kretschmann scalar is non-infinite then it cannot be the latter, whatever the coordinate basis is. Continuing to think generically and without choosing coordinates: "distance" -> it's still interesting for considering non-gravitational body forces of objects as we take the Kretschmann scalar around the objects towards infinity. Essentially, tossing in one test particle just causes it to free-fall forever; tossing in two classical objects simultaneously (in Schwarzschild coordinates on Schwarzschild black hole spacetime)[1] means at some object- proper-time each object will collide with the other. They will still free-fall forever; there is a notional "down" if we have a gradient in the Kretschmann scalar field. However, the free-falling trajectories available narrow even after contact. For classical objects that are mutually repulsive, pressure[2] builds as they continue on converging free-falling trajectories and eventually classical forces cannot keep the two objects separated; they are forced into each other. The resulting compacted matter, whatever its nature, continues to free-fall forever. Quantum-mechanically, who knows? Firstly we're way into the ultraviolet completion of the Standard Model, and secondly there's no reason to expect quantum mechanical effects to contribute to the stress-energy tensor in ways that do not look sufficiently classical under averaging or other tricks that let one continue using the geodesics of the black hole background metric or make only small perturbations to them. And we're ignoring quantum-mechanical effects outside the black hole, like Hawking radiation. - -- [1] The simultaneity condition is a handwavy way of absorbing messy behaviour as we drop two identical charges into a black hole. See Cohen & Wald (1971), http://adsabs.harvard.edu/abs/1971JMP....12.1845C http://adsabs.harvard.edu/abs/1971JMP....12.1845C [2] Pressure here is encoded in the stress-energy tensor T as T^{ii}, i \neq 0. Since the Einstein Field Equations say essentially G = T, we have to deal with pressure as it becomes non-negligible, because as a component of T it must be reflected in the curvature represented here as the Einstein tensor G. This is a practical example of where General Relativity is a dynamical non-linear theory. Two classical objects with identical electric charge within the horizon of a black hole makes a surprising bookkeeping mess! (compare the Schwarzschild metric with the https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6... and there we are talking about what's outside the black hole and ignoring quantum mechanical effects altogether).

by Retra 8y ago

danbruc 8y ago

Coordinate systems are not the primary issue. It's true that coordinates can be problematic but the laws of physics are independent of coordinates because those are of course only a tool we use to perform calculations. But take for example the classical law of gravitation which requires dividing by the square of the distance between the two bodies. The distance is independent of the coordinates you use to describe the positions of the bodies but you still get a divergent force as the distance approaches zero. So the breakdown of the law is not due to a bad choice of coordinates but inherent in the equation and a strong indication that the law is probably only approximate for short distances.

Retra 8y ago
I believe you're just saying the same thing I am in different terms. The fact that zero is even a 'distance' you can divide by is consequence of the fact that you're deriving distance from the coordinates of the bodies. If distance was instead fundamental (and not a function of coordinates), you'd not even have a zero distance, and you'd have no inclination to attempt dividing by it. (But the math for doing that is basically to always account for every part of every body, and that's just entirely intractable.) This is assuming mostly Newtonian physics, of course. Once you start dealing with fields and quantum mechanics, none of this makes as much sense.
- danbruc 8y ago
  Coordinates and distances are independent concepts. They are function from the space you are working with into the reals assigning each respectively each pair of elements a real number for each coordinate respectively one real number for the distance. You can for various reasons choose coordinates and distances in a way such that they have simple mathematical relationships but that is just an arbitrary choice. The classical law of gravity is only dependent on distances, which is a physical meaningful concept, but not on coordinates, which are not physically meaningful. In that sense distances are fundamental in a way in which coordinates are not and a distance of zero is a meaningful thing, namely two things are at the very same place.
raattgift 8y ago
I think what you're aiming for here is the statement: somewhere in a black hole spacetime the Kretschmann scalar goes to infinity. The Kretschmann scalar is the square of the Riemann tensor, or in notation R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}. There are no forces or coordinates or anything like that to even consider; it's a curvature invariant. ( https://en.wikipedia.org/wiki/Curvature_invariant_(general_relativity) https://en.wikipedia.org/wiki/Curvature_invariant_(general_r... ) If the Kretschmann scalar is non-infinite at a point then there is no gravitational singularity there. If you really want to think about a black hole spacetime in a coordinate system, then in Schwarzschild coordinates for a Schwarzschild black hole, K = \frac{48M^2}{r^6} where K is the Kretschmann scalar. You can see that when r=0, K is undefined. Other coordinate systems and black hole spacetimes are available. However, generically, one can distinguish between a coordinate singularity and a gravitational singularity by calculating the Kretschmann scalar at a point. If the Kretschmann scalar is non-infinite then it cannot be the latter, whatever the coordinate basis is. Continuing to think generically and without choosing coordinates: "distance" -> it's still interesting for considering non-gravitational body forces of objects as we take the Kretschmann scalar around the objects towards infinity. Essentially, tossing in one test particle just causes it to free-fall forever; tossing in two classical objects simultaneously (in Schwarzschild coordinates on Schwarzschild black hole spacetime)[1] means at some object- proper-time each object will collide with the other. They will still free-fall forever; there is a notional "down" if we have a gradient in the Kretschmann scalar field. However, the free-falling trajectories available narrow even after contact. For classical objects that are mutually repulsive, pressure[2] builds as they continue on converging free-falling trajectories and eventually classical forces cannot keep the two objects separated; they are forced into each other. The resulting compacted matter, whatever its nature, continues to free-fall forever. Quantum-mechanically, who knows? Firstly we're way into the ultraviolet completion of the Standard Model, and secondly there's no reason to expect quantum mechanical effects to contribute to the stress-energy tensor in ways that do not look sufficiently classical under averaging or other tricks that let one continue using the geodesics of the black hole background metric or make only small perturbations to them. And we're ignoring quantum-mechanical effects outside the black hole, like Hawking radiation. - -- [1] The simultaneity condition is a handwavy way of absorbing messy behaviour as we drop two identical charges into a black hole. See Cohen & Wald (1971), http://adsabs.harvard.edu/abs/1971JMP....12.1845C http://adsabs.harvard.edu/abs/1971JMP....12.1845C [2] Pressure here is encoded in the stress-energy tensor T as T^{ii}, i \neq 0. Since the Einstein Field Equations say essentially G = T, we have to deal with pressure as it becomes non-negligible, because as a component of T it must be reflected in the curvature represented here as the Einstein tensor G. This is a practical example of where General Relativity is a dynamical non-linear theory. Two classical objects with identical electric charge within the horizon of a black hole makes a surprising bookkeeping mess! (compare the Schwarzschild metric with the https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6... and there we are talking about what's outside the black hole and ignoring quantum mechanical effects altogether).