3 ms·

The article comments on the site are thick with the autodidacts with an intuition that the author describes. They all seem to have missed a primary point made i

by g8oz 5y ago

The article comments on the site are thick with the autodidacts with an intuition that the author describes. They all seem to have missed a primary point made in the essay - if you can't describe it with math you have nothing.

paulpauper 5y ago
on the other hand, the curvature of space can be described visually (famous gravity well example)
- solveit 5y ago
  If that were the only way to describe the curvature of space then it would indeed be useless.
- jodrellblank 5y ago
  I have never found it comfortable to imagine gravity as a rubber sheet with a weight on it, where Earth's gravity is making the 'model' work by pulling the weight and stretching the sheet. Trying to turn that into a 3D Sun with a planet orbiting it when there is no 'down' separate to the system, trying to imagine gravitational lensing when there is no way to imagine a NASCAR style banked corner in a stretched void, trying to imagine why a black hole can be so dense that light cannot get away. Imagine a rubber sheet and light as a toy car climbing up the well, it's easy to imagine the well becoming so steep that the car wheels cannot grip it, or the car motor cannot lift the weight of the car. Try imagining light heading directly 'out' from a black hole, space is compressed so much that ... what? Light loses its grip on spacetime? Spacetime is not just stretched but is being pulled inwards like a conveyor belt at 3e8 meters/second so light is at an effective standstill? (yet all frames of reference see light moving away from them at the speed of light). It doesn't fit a rubber sheet for anything, really, even the basic planet orbiting the gravity well has no easy imagining in 3D. It's somehow ... Earth taking the path of least action? a straight line in curved spacetime?
  - ncmncm 5y ago
    It was never a productive metaphor, in any way, but it let people with no mathematics feel like they had a sense of things. If you want to reason correctly, it is a trap.
  - raattgift 5y ago
    The rubber sheet runs into problems when you try to introduce high inclination orbits, or non-inspiralling (or at least very very slowly inspiralling) nearly circular orbits. It also doesn't do well when the mass-to-mass ratios of the objects on the sheet are small. It also doesn't do well when you want the masses to spin. (How many of the large bodies in our solar system have nearly zero spin angular momentum? And the difference between perihelion and aphelion for Venus and Neptune are small, and there doesn't seem to be any danger of them inspiralling sunwards in the next several thousand of their respective solar orbits.)
- raattgift 5y ago
  Are you sure? Can you describe, visually using the famous gravity well example, the central massive object rotating on an axis with a small object in a stable (non-inspiralling over the course of thousands or more revolutions) polar (or indeed any arbitrary nonequatorial) orbit about it? One might compare your visualization with results from Gravity Probe B, which had an orbital inclination of 90.007 degrees, or Juno, which continues to have have this fun polar orbit (90.000 degree inclination) around Jupiter: https://en.wikipedia.org/wiki/Juno_(spacecraft)#/media/File:Animation_of_Juno_trajectory_around_Jupiter.gif https://en.wikipedia.org/wiki/Juno_(spacecraft)#/media/File:... (animated). Jupiter's mass and angular momentum ("J") parameters are, relevantly, greater than Earth's; additionally, GPB's was nearly circular, while Juno's is very highly elliptical. I'm also curious about how you visualize -- using the gravity well example -- an extremely elliptical orbit even with J=0 and orbital inclination=0 -- one that alllllmost grazes the surface of the central object at closest, and at farthest is light-seconds away in effectively flat spacetime. I'm not saying it can't be done, or even that you cannot do it. I just think it's hard -- I can't do it, or perhaps won't do it because I think it's easier to reach for an approximate solution of the Einstein Field Equations (and if one takes the mass of the small body to some unignorably higher level, through effective one-body methods). Others might reach for gravitoelectromagnetism. Others have some other favoured post-Newtonian or numerical method. Solve first, then grind out some useful visualization. But if you can successfully visualize first and extract a solution from that, with any sort of generality, then I'm honestly interested.

by g8oz 5y ago

paulpauper 5y ago

on the other hand, the curvature of space can be described visually (famous gravity well example)

solveit 5y ago
If that were the only way to describe the curvature of space then it would indeed be useless.
jodrellblank 5y ago
I have never found it comfortable to imagine gravity as a rubber sheet with a weight on it, where Earth's gravity is making the 'model' work by pulling the weight and stretching the sheet. Trying to turn that into a 3D Sun with a planet orbiting it when there is no 'down' separate to the system, trying to imagine gravitational lensing when there is no way to imagine a NASCAR style banked corner in a stretched void, trying to imagine why a black hole can be so dense that light cannot get away. Imagine a rubber sheet and light as a toy car climbing up the well, it's easy to imagine the well becoming so steep that the car wheels cannot grip it, or the car motor cannot lift the weight of the car. Try imagining light heading directly 'out' from a black hole, space is compressed so much that ... what? Light loses its grip on spacetime? Spacetime is not just stretched but is being pulled inwards like a conveyor belt at 3e8 meters/second so light is at an effective standstill? (yet all frames of reference see light moving away from them at the speed of light). It doesn't fit a rubber sheet for anything, really, even the basic planet orbiting the gravity well has no easy imagining in 3D. It's somehow ... Earth taking the path of least action? a straight line in curved spacetime?
- ncmncm 5y ago
  It was never a productive metaphor, in any way, but it let people with no mathematics feel like they had a sense of things. If you want to reason correctly, it is a trap.
- raattgift 5y ago
  The rubber sheet runs into problems when you try to introduce high inclination orbits, or non-inspiralling (or at least very very slowly inspiralling) nearly circular orbits. It also doesn't do well when the mass-to-mass ratios of the objects on the sheet are small. It also doesn't do well when you want the masses to spin. (How many of the large bodies in our solar system have nearly zero spin angular momentum? And the difference between perihelion and aphelion for Venus and Neptune are small, and there doesn't seem to be any danger of them inspiralling sunwards in the next several thousand of their respective solar orbits.)
raattgift 5y ago
Are you sure? Can you describe, visually using the famous gravity well example, the central massive object rotating on an axis with a small object in a stable (non-inspiralling over the course of thousands or more revolutions) polar (or indeed any arbitrary nonequatorial) orbit about it? One might compare your visualization with results from Gravity Probe B, which had an orbital inclination of 90.007 degrees, or Juno, which continues to have have this fun polar orbit (90.000 degree inclination) around Jupiter: https://en.wikipedia.org/wiki/Juno_(spacecraft)#/media/File:Animation_of_Juno_trajectory_around_Jupiter.gif https://en.wikipedia.org/wiki/Juno_(spacecraft)#/media/File:... (animated). Jupiter's mass and angular momentum ("J") parameters are, relevantly, greater than Earth's; additionally, GPB's was nearly circular, while Juno's is very highly elliptical. I'm also curious about how you visualize -- using the gravity well example -- an extremely elliptical orbit even with J=0 and orbital inclination=0 -- one that alllllmost grazes the surface of the central object at closest, and at farthest is light-seconds away in effectively flat spacetime. I'm not saying it can't be done, or even that you cannot do it. I just think it's hard -- I can't do it, or perhaps won't do it because I think it's easier to reach for an approximate solution of the Einstein Field Equations (and if one takes the mass of the small body to some unignorably higher level, through effective one-body methods). Others might reach for gravitoelectromagnetism. Others have some other favoured post-Newtonian or numerical method. Solve first, then grind out some useful visualization. But if you can successfully visualize first and extract a solution from that, with any sort of generality, then I'm honestly interested.