6 ms·

I put in the work to study GR from Susskind' theoretical minimum (fantastic lecture series: https://theoreticalminimum.com/courses/general-relativity/2012/fall

by bollu 6y ago

I put in the work to study GR from Susskind' theoretical minimum (fantastic lecture series: https://theoreticalminimum.com/courses/general-relativity/2012/fall https://theoreticalminimum.com/courses/general-relativity/20... ). I still don't understand GR. Yes I can now say "oh gravity is '''just''' curvature on a lorentizian manifold", but what in the world does it _mean_? My current picture is something like this: A Lorentizian manifold looks locally hyperbolic, so we need to imagine a light cone. Now the curvature of this manifold tells us how we need to distort light cones at different points to build up the full space. How the fuck do I visualize this? Is there some way to embed this in low dimension to play with it? For example, to understand regular curvature, you can play around with a sphere and a cone to get a feel for curvature ~= angle deficit ~= loopy deficit ~= holonomy. I don't know how to visualize the curvature of space-time, and I'm completely unconvinced I actually understand it beyond squiggles on paper. I'd love a way to embed and visualize this. Please tell me if you know how! As an aside: It is definitely possible to get a feel for _hyperbolic geometry_ by playing HyperRogue: (https://roguetemple.com/z/hyper/ https://roguetemple.com/z/hyper/) --- A free roguelike set in hyperbolic space. What I don't grok at all is the _curvature_ across hyperbolic spaces, analogous to how a sphere has _curvature_ across flat spaces. I'd love to know how to get a feel for this.

Ono-Sendai 6y ago
Sure, I can explain how to visualise it (I think :) ) First off, a note: lots of people think they know how to visualise GR, but actually don't. For example, explaining with marbles falling down curved rubber sheets doesn't help, because you're using gravity (the marble being pulled down) to explain gravity. And if you think you can just explain it with geodesics along a curved rubber sheet with a heavy weight in the middle, then think about how to explain why a stationary object falls towards the heavy weight. Anyway, here is how to visualise it: Imagine you are driving a car along some surface. We require that each wheel rotates at exactly the same speed with respect to the surface. Now imagine there is a decreasing curvature gradient from left to right across the path of your car - in other words the surface below the left wheel is curving faster than under the right wheel. Seen from above, the left wheel has extra distance to travel in the vertical direction than the right wheel, to achieve a given horizontal distance. This causes the direction of the car to turn slowly to the left - in other words the car is 'gravitationally attracted' to the left - i.e. towards the region of greater curvature, i.e. towards the attracting mass. To explain why static objects are gravitationally attracted to a mass, you just have to realise that according to Einstein, there are no such things as stationary objects (in some sense). Rather, even a 'stationary' object is travelling rapidly through spacetime. To go back to the car metaphor - the car is not stationary, rather it is always driving forwards (through spacetime). Some notes on this - think about what a bizarre, weirdly-shaped surface the car must be driving over. You can also think of the car front axle as being a wavefront of light. And finally - if you think this explanation is too bizarre, don't blame me, blame Einstein. Personally I don't actually like GR as a theory due to how weird it is.
- vagab0nd 6y ago
  > For example, explaining with marbles falling down curved rubber sheets doesn't help, because you're using gravity (the marble being pulled down) to explain gravity. Exactly, and it drives me nuts. There are many similar examples in SR and GR that maybe help a layperson intuitively, but creates confusion and frustration when you actually try to learn the theory. I can think of 2: The "escape velocity" explanation of black hole event horizon. Yes, the number comes out correct. No, it does not make sense and does not help me understand how black hole works. "Mass increases when object moves". No it doesn't. And no it doesn't explain why massive particles can't reach speed of light, because in the frame of the object, its speed is zero. I can't tell you how much time I've spent searching for the answers, and drowned by these answers that seemingly make sense.
  - ISL 6y ago
    For the latter concern, you may find this Physics Today article from Okun helpful: http://www.hysafe.org/science/KareemChin/PhysicsToday_v42_p31to36.pdf http://www.hysafe.org/science/KareemChin/PhysicsToday_v42_p3...
- steve76 6y ago
  Special Relativity: It's easier to win a race with a slow stopwatch. You run a 5 minute mile. Other person's fast watch shows 7. Nature doesn't care about your watch. Like a hot water tank it's going to press out everywhere. Nature efficiently takes some energy, slows down it's own watch because it's the easiest way to move. With heat being transferred through the vacuum of empty space between the sun and the earth, it will be really nice to say changing clocks is it's own thermal medium alongside conduction and convention. Now you just need to make it a general principle. Einstein and Brownian Motion: Brownian motion reverses the second law. A bath of particles causing each one to move instead of each particle having it's own little internal thermodynamic engine, like dust in the sunlight. That allows you to apply subatomic principles outside the atom. Nature doesn't care about what we call an atom. Atoms exist because all of the light and heat being radiated already sees them that way. Boltzmann and the Kinetic Theory of Gasses: Modeling an atom, focusing a lens, connecting conductive surfaces, I think of as little soup cans. Ideally, two particles either collide with each other, the walls of the soup can, or nothing else. That way you can just talk in probabilities, 99%, 10%, and put an end for a building block collect-a-thon. Planck and Photons: If you just had thermal transfer on temperature difference alone, closing the door on cupboard full of cast iron pots and pans causes on to be white hot and all the other ones turn as dark as they could be. Planck calculated if it exists, it needs to increase energy before it radiates higher energy. Einstein said you have a start, you have an end, you have a particle. That particle isn't a little cannonball shot out of the atom. More a photo of the entire object at once. Regardless of the intensity focused on one spot, the photon exchange, using time as it's medium, hits the entire photoelectric conductive surface at once, so frequency, the weight of the photon matters, not the intensity or amount per second. Gravity: I think of a film project strip at two different frame rates. Since nature is in the terms of photons and photoelectric effects, I can have one really long frame rate, and one really short frame rate. The Earth's on the slow one. It has to tick, so it orbits. Sun's on the long one, one tick for every 365 of the Earth.
pa7x1 6y ago
Some corrections to hopefully solve some misunderstandings. > A Lorentizian manifold looks locally hyperbolic, so we need to imagine a light cone. Now the curvature of this manifold tells us how we need to distort light cones at different points to build up the full space. A Lorentzian manifold is not necessarily locally hyperbolic and certainly the existence of light-cones do not follow from its hyperbolic nature. A Lorentzian manifold may be flat or curved, if it's flat we usually call it also a Minkowski space. Curved spaces come in many shapes of forms depending on the curvature which is a complicated geometric object. One particular case of curved space is a hyperbolic space which is one with constant negative curvature. A hyperbolic (constant negative curvature) Lorentzian (space+time) manifold is also called an Anti-de Sitter (AdS) space and plays a very important role in physics (see AdS/CFT correspondence). > How the fuck do I visualize this? Is there some way to embed this in low dimension to play with it? For example, to understand regular curvature, you can play around with a sphere and a cone to get a feel for curvature ~= angle deficit ~= loopy deficit ~= holonomy. Well, we humans suck at visualizing dimensions greater than 3 and projecting higher dimensional spaces into lower dimensional spaces (like we usually do representing a hypercube) is very hard to grasp unless you already know what you should be seeing. So your best bet is to choose problems with high degree of symmetry and visualizing only the interesting dimensions. For example, a stationary (i.e. doesn't change with time -> 1 dimension less to visualize) and spherical solution (2 more symmetries -> 2 dimensions less to visualize) can be easily visualized, in fact only has one interesting dimension! This solution is usually called Schwarzschild's solution. And because it has so high degree of symmetry it can be embedded a low dimensional space we can see, we usually call that Flamm's paraboloid and you can find a pretty picture here: https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s... > I don't know how to visualize the curvature of space-time, and I'm completely unconvinced I actually understand it beyond squiggles on paper. I'd love a way to embed and visualize this. Please tell me if you know how! Unless you develop the ability the visualize higher dimensional spaces your best bet is to select highly symmetrical problems and visualize the interesting dimensions of those, the way we generalize to higher dimensions is through math not through the visual cortex. The brain is a highly plastic structure so maybe with training you could develop the ability to work visually with higher dimensional spaces but I also wouldn't be surprised if the visual cortex has been evolved and finely tuned to work with the macroscopic dimensions it had available and this exercise would turn out to be futile. After all, the ability to generalize to arbitrary high dimensions may have some cost and animals would have benefited very little from it.
raattgift 6y ago
Test particles are super-useful. Synge in his textbook [1] discusses a five-point curvature detector, which is the formal treatment for a small cloud of non-interacting dust that reveals the Riemann curvature tensor. Baez does this succinctly and less formally at [2], using a decomposition of the Riemann curvature tensor into the Weyl tensor and the Ricci tensor. One could substitute ticking clocks for coffee grounds: in e.g. exterior Schwarzschild, when the flat-spacetime round-ball is instead a Weyl-curved spacetime American-football, the long axis of the football is radial to M, so the clock closest to M ticks slower than the others (cf. the Riemann tensor and especially the last sentence at [3]). One may compare with [4] (and the focusing theorem on the same page). - -- [1] Synge, J.L., "Gravitation: The general theory". (1964) Ch. XI, §8 [2] http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html [3] https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvatures https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvature... [4] https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitive_significance https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitiv...

by bollu 6y ago

Ono-Sendai 6y ago

Sure, I can explain how to visualise it (I think :) ) First off, a note: lots of people think they know how to visualise GR, but actually don't. For example, explaining with marbles falling down curved rubber sheets doesn't help, because you're using gravity (the marble being pulled down) to explain gravity. And if you think you can just explain it with geodesics along a curved rubber sheet with a heavy weight in the middle, then think about how to explain why a stationary object falls towards the heavy weight. Anyway, here is how to visualise it: Imagine you are driving a car along some surface. We require that each wheel rotates at exactly the same speed with respect to the surface. Now imagine there is a decreasing curvature gradient from left to right across the path of your car - in other words the surface below the left wheel is curving faster than under the right wheel. Seen from above, the left wheel has extra distance to travel in the vertical direction than the right wheel, to achieve a given horizontal distance. This causes the direction of the car to turn slowly to the left - in other words the car is 'gravitationally attracted' to the left - i.e. towards the region of greater curvature, i.e. towards the attracting mass. To explain why static objects are gravitationally attracted to a mass, you just have to realise that according to Einstein, there are no such things as stationary objects (in some sense). Rather, even a 'stationary' object is travelling rapidly through spacetime. To go back to the car metaphor - the car is not stationary, rather it is always driving forwards (through spacetime). Some notes on this - think about what a bizarre, weirdly-shaped surface the car must be driving over. You can also think of the car front axle as being a wavefront of light. And finally - if you think this explanation is too bizarre, don't blame me, blame Einstein. Personally I don't actually like GR as a theory due to how weird it is.

vagab0nd 6y ago
> For example, explaining with marbles falling down curved rubber sheets doesn't help, because you're using gravity (the marble being pulled down) to explain gravity. Exactly, and it drives me nuts. There are many similar examples in SR and GR that maybe help a layperson intuitively, but creates confusion and frustration when you actually try to learn the theory. I can think of 2: The "escape velocity" explanation of black hole event horizon. Yes, the number comes out correct. No, it does not make sense and does not help me understand how black hole works. "Mass increases when object moves". No it doesn't. And no it doesn't explain why massive particles can't reach speed of light, because in the frame of the object, its speed is zero. I can't tell you how much time I've spent searching for the answers, and drowned by these answers that seemingly make sense.
- ISL 6y ago
  For the latter concern, you may find this Physics Today article from Okun helpful: http://www.hysafe.org/science/KareemChin/PhysicsToday_v42_p31to36.pdf http://www.hysafe.org/science/KareemChin/PhysicsToday_v42_p3...
steve76 6y ago
Special Relativity: It's easier to win a race with a slow stopwatch. You run a 5 minute mile. Other person's fast watch shows 7. Nature doesn't care about your watch. Like a hot water tank it's going to press out everywhere. Nature efficiently takes some energy, slows down it's own watch because it's the easiest way to move. With heat being transferred through the vacuum of empty space between the sun and the earth, it will be really nice to say changing clocks is it's own thermal medium alongside conduction and convention. Now you just need to make it a general principle. Einstein and Brownian Motion: Brownian motion reverses the second law. A bath of particles causing each one to move instead of each particle having it's own little internal thermodynamic engine, like dust in the sunlight. That allows you to apply subatomic principles outside the atom. Nature doesn't care about what we call an atom. Atoms exist because all of the light and heat being radiated already sees them that way. Boltzmann and the Kinetic Theory of Gasses: Modeling an atom, focusing a lens, connecting conductive surfaces, I think of as little soup cans. Ideally, two particles either collide with each other, the walls of the soup can, or nothing else. That way you can just talk in probabilities, 99%, 10%, and put an end for a building block collect-a-thon. Planck and Photons: If you just had thermal transfer on temperature difference alone, closing the door on cupboard full of cast iron pots and pans causes on to be white hot and all the other ones turn as dark as they could be. Planck calculated if it exists, it needs to increase energy before it radiates higher energy. Einstein said you have a start, you have an end, you have a particle. That particle isn't a little cannonball shot out of the atom. More a photo of the entire object at once. Regardless of the intensity focused on one spot, the photon exchange, using time as it's medium, hits the entire photoelectric conductive surface at once, so frequency, the weight of the photon matters, not the intensity or amount per second. Gravity: I think of a film project strip at two different frame rates. Since nature is in the terms of photons and photoelectric effects, I can have one really long frame rate, and one really short frame rate. The Earth's on the slow one. It has to tick, so it orbits. Sun's on the long one, one tick for every 365 of the Earth.

pa7x1 6y ago

Some corrections to hopefully solve some misunderstandings. > A Lorentizian manifold looks locally hyperbolic, so we need to imagine a light cone. Now the curvature of this manifold tells us how we need to distort light cones at different points to build up the full space. A Lorentzian manifold is not necessarily locally hyperbolic and certainly the existence of light-cones do not follow from its hyperbolic nature. A Lorentzian manifold may be flat or curved, if it's flat we usually call it also a Minkowski space. Curved spaces come in many shapes of forms depending on the curvature which is a complicated geometric object. One particular case of curved space is a hyperbolic space which is one with constant negative curvature. A hyperbolic (constant negative curvature) Lorentzian (space+time) manifold is also called an Anti-de Sitter (AdS) space and plays a very important role in physics (see AdS/CFT correspondence). > How the fuck do I visualize this? Is there some way to embed this in low dimension to play with it? For example, to understand regular curvature, you can play around with a sphere and a cone to get a feel for curvature ~= angle deficit ~= loopy deficit ~= holonomy. Well, we humans suck at visualizing dimensions greater than 3 and projecting higher dimensional spaces into lower dimensional spaces (like we usually do representing a hypercube) is very hard to grasp unless you already know what you should be seeing. So your best bet is to choose problems with high degree of symmetry and visualizing only the interesting dimensions. For example, a stationary (i.e. doesn't change with time -> 1 dimension less to visualize) and spherical solution (2 more symmetries -> 2 dimensions less to visualize) can be easily visualized, in fact only has one interesting dimension! This solution is usually called Schwarzschild's solution. And because it has so high degree of symmetry it can be embedded a low dimensional space we can see, we usually call that Flamm's paraboloid and you can find a pretty picture here: https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s... > I don't know how to visualize the curvature of space-time, and I'm completely unconvinced I actually understand it beyond squiggles on paper. I'd love a way to embed and visualize this. Please tell me if you know how! Unless you develop the ability the visualize higher dimensional spaces your best bet is to select highly symmetrical problems and visualize the interesting dimensions of those, the way we generalize to higher dimensions is through math not through the visual cortex. The brain is a highly plastic structure so maybe with training you could develop the ability to work visually with higher dimensional spaces but I also wouldn't be surprised if the visual cortex has been evolved and finely tuned to work with the macroscopic dimensions it had available and this exercise would turn out to be futile. After all, the ability to generalize to arbitrary high dimensions may have some cost and animals would have benefited very little from it.

raattgift 6y ago

Test particles are super-useful. Synge in his textbook [1] discusses a five-point curvature detector, which is the formal treatment for a small cloud of non-interacting dust that reveals the Riemann curvature tensor. Baez does this succinctly and less formally at [2], using a decomposition of the Riemann curvature tensor into the Weyl tensor and the Ricci tensor. One could substitute ticking clocks for coffee grounds: in e.g. exterior Schwarzschild, when the flat-spacetime round-ball is instead a Weyl-curved spacetime American-football, the long axis of the football is radial to M, so the clock closest to M ticks slower than the others (cf. the Riemann tensor and especially the last sentence at [3]). One may compare with [4] (and the focusing theorem on the same page). - -- [1] Synge, J.L., "Gravitation: The general theory". (1964) Ch. XI, §8 [2] http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html [3] https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvatures https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvature... [4] https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitive_significance https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitiv...