4 ms·

There are many ways for spacetime to warp, which can be put into two categories. The simpler kind, Ricci curvature, is the only kind of curvature in <4 dimensio

There are many ways for spacetime to warp, which can be put into two categories. The simpler kind, Ricci curvature, is the only kind of curvature in <4 dimensions and is produced by mass-energy, momentum, pressure, and shear stress, according to general relativity. The other kind, Weyl curvature, only exists in 4 or more spacetime dimensions and can exist in a vacuum. Gravitational waves are Weyl-curvature distortions of spacetime that propagate in a vacuum according to general relativity. (Also, gravitational waves do carry a little bit of energy, so they cause a small amount of Ricci curvature, but this is a secondary effect.)

raattgift 3y ago
I'm not sure why you want to draw attention to non-Lorentzian spacetimes in this context. > ... that propagate ... "propagate". That requires a decomposition of spacetime into space+time, and of course the decomposition of the Riemann curvature tensor, the setting of a background value for the Weyl tensor, and the use of perturbation theory. But if you're going down that path, why not use the metric tensor? g_munu = eta_munu + h_munu + h.o.t. is standard in post-Newtonian expansion approximations, and in particular https://en.wikipedia.org/wiki/Linearized_gravity https://en.wikipedia.org/wiki/Linearized_gravity (which doesn't track the higher-order terms). The Weyl curvature tensor C_abcd is useful in understanding that in a spherical region of space (not spacetime, so really we're in the land of extracting 3-Cotton-York C_ab) where a GW is incident suffers not from a volume deficit but from an ellipsoidal stretch-squash. But conceptual understanding of and calculation are... well, not really on speaking terms. Even theorists who take the full covariant theory seriously will decompose further, into e.g. an electrical and magnetic part, and add further structure to match the worldlines to the Raychaudhuri equation in shear and vorticity. > they cause a small amount of Ricci curvature ??? If nothing else, I think you'd need to choose between explaining this or explaining why "Ricci curvature is produced by [matter but] Weyl curvature ... can exist in a vacuum" (or choose neither). The sticky bead apparatus is a breaking of the T_munu = 0 vacuum condition.
- snarkconjecture 3y ago
  I was trying to give a conceptual understanding on the level of the person I was responding to, not instructions for doing actual calculations. I didn't mean to suggest euclidean metric, I just thought "3+1" would be extra jargon. The person asking questions was confused because they'd heard that "mass causes curvature" in GR, and that gravitational waves involve curvature. I figured it would help to explain that these are different kinds of curvature. I think I was totally wrong re: Ricci curvature. I was thinking "oh, GW carry energy and there's nonlinear evolution..." and got carried away, whoops. If you want to give a more accurate explanation (that makes sense to someone who's never taken a GR class) please do!
  - raattgift 3y ago
    There is no question that this stuff is hard (and has been cutting edge for decades) and that it is easy to make mistakes. > give a more accurate explanation (that makese sense to someone who's never taken a GR class) It's hard to know what level of understanding to aim for on HN. There are non-relativist working physicists here who somewhat casually read other areas of physics (and mathematics) here, for example, rather than e.g. physics SE or looking through literature reviews. I'm guessing that you have done GR but probably not much with approximations like GEM, nor looked into the history of gravitational waves (e.g. the 1950s-60s work by Bondi with collaborators like Pirani) before the wide availability of powerful computers and observational support for the linearized theory. My goal here is not to nitpick you, but rather to offer a couple of references that might interest you or anyone who is quietly reading along. > I think I was totally wrong re: Ricci curvature. R_munu = 0 for one (uncharged) BH, and also for two. Charged BHs are different (Reissner-Nordström's Ricci tensor is R_munu = +- g_munu r_{Q}^2 / r^4 where r_{Q}^2 = \frac{Q^2G}{4 \pi \varepsilon_0 c^4} and ε_0 is the electric constant, all thanks to the electromagnetic stress-energy tensor; the Ricci scalar remains 0). Superposing two Schwarzschild or Kerr solutions is messy [Krivan & Price 1998 Phys. Rev. D 58, 104003 was a nice overview arxiv html5[*] <https://ar5iv.labs.arxiv.org/html/gr-qc/9806017 https://ar5iv.labs.arxiv.org/html/gr-qc/9806017>, 'two locally Kerr holes, no matter how close they are, will not superpose into a single Kerr hole. Though [this] "failure" of the close limit is physically correct, it is inconvenient [for doing perturbation theory]'] but doesn't change Rmunu = 0. A charged binary is beyond the scope of this comment. > I was thinking "oh, GW carry energy ..." I can recommend two really good papers written at very different times. Weber & Wheeler 1957 <https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29.509 https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29....> (it's also on sci-hub) where they struggle with the question of the sense in which GWs are real ("the only well defined way there is to express the influence of [gravitational] radiation [is] in terms of its effect upon invariant space time intervals betwee two test bodies"). I enjoyed seeing the rare \dotequalsdot in eqns 28,30, "geometrically equal to". Also, they lead right into the next suggestion with, "Similarties between gravitational and elecgromagnetic waves thus make it simple to draw a number of reasonable inferences. The significance of these inferences has a much more subtle character in the gravitational case than in the electromagnetic case. Neither field densities nor test particle motions have a meaning independent of the choice of coordinate systems. The simple observable consequences of wave action are instead changes in the separation of nearby test particles -- changes that are related to the covariant components of the curvature tensor R_ijkl." Goswami & Ellis 2021 <https://iopscience.iop.org/article/10.1088/1361-6382/abdaf3 https://iopscience.iop.org/article/10.1088/1361-6382/abdaf3> arxiv html5[*] <https://ar5iv.labs.arxiv.org/html/1912.00591 https://ar5iv.labs.arxiv.org/html/1912.00591> with the somewhat stentorian title "Tidal forces are gravitational waves". It is really about the Weyl curvature in binaries. The authors explicity consider (for orbital motions of two massive bodies) a decomposition of the Weyl tensor C_abcd into an electric and magnetic part (for which see §III; and eqn 6 repeats the point on R_ab in vaccum), the former compared to Newtonian gravitation and the latter being general-relativistic and encoding the gravitational waves that do not arise in the Newtonian theory. Their focus is on the magnetic part of the Weyl tensor, since (they argue) that is where all the interesting stuff is encoded, and how energy gets from the orbital system to an observer and between the binary partners themselves. One highlight is a short paragraph summarizing some ~1960 work by Bondi and MacCrae [**], "Through a series of shape changing operations ... one sees that after each rotation, Tweedledum is gaining internal energy as the external tidal force is doing work on him, while Tweedledee is losing energy as she is doing work against the external force. This is an excellent example of how the internal energy of a system can be transferred to another system via gravitational induction, in Newtonian gravity." [emphasis mine]. - - [*] lost way downpage and heavily downvoted is a HN user pointing out that the page of cartoons linked at the top is not accessibilty-friendly. I accept the point, and make up for it a little with the html5 links here. One can straightforwardly paste the trailing element of the ar5iv into a search engine or whatnot in order to reach a PDF version of each arxived paper above. [**] The actual Bondi & McCrae paper(s) is(are) proving a bit elusive, but this gives a good overview: §§3-4, Tweedledum and Tweedledee and Energy Transfer (the latter starts with an excellent quote from Bondi in 1957), https://link.springer.com/article/10.1007/s10701-022-00660-z https://link.springer.com/article/10.1007/s10701-022-00660-z (open access). So this quickly became three papers, and I'll stop now, because I'm fairly sure you can find your way through them if you really want, and figure out how to get help if you get stuck.

raattgift 3y ago

I'm not sure why you want to draw attention to non-Lorentzian spacetimes in this context. > ... that propagate ... "propagate". That requires a decomposition of spacetime into space+time, and of course the decomposition of the Riemann curvature tensor, the setting of a background value for the Weyl tensor, and the use of perturbation theory. But if you're going down that path, why not use the metric tensor? g_munu = eta_munu + h_munu + h.o.t. is standard in post-Newtonian expansion approximations, and in particular https://en.wikipedia.org/wiki/Linearized_gravity https://en.wikipedia.org/wiki/Linearized_gravity (which doesn't track the higher-order terms). The Weyl curvature tensor C_abcd is useful in understanding that in a spherical region of space (not spacetime, so really we're in the land of extracting 3-Cotton-York C_ab) where a GW is incident suffers not from a volume deficit but from an ellipsoidal stretch-squash. But conceptual understanding of and calculation are... well, not really on speaking terms. Even theorists who take the full covariant theory seriously will decompose further, into e.g. an electrical and magnetic part, and add further structure to match the worldlines to the Raychaudhuri equation in shear and vorticity. > they cause a small amount of Ricci curvature ??? If nothing else, I think you'd need to choose between explaining this or explaining why "Ricci curvature is produced by [matter but] Weyl curvature ... can exist in a vacuum" (or choose neither). The sticky bead apparatus is a breaking of the T_munu = 0 vacuum condition.

snarkconjecture 3y ago
I was trying to give a conceptual understanding on the level of the person I was responding to, not instructions for doing actual calculations. I didn't mean to suggest euclidean metric, I just thought "3+1" would be extra jargon. The person asking questions was confused because they'd heard that "mass causes curvature" in GR, and that gravitational waves involve curvature. I figured it would help to explain that these are different kinds of curvature. I think I was totally wrong re: Ricci curvature. I was thinking "oh, GW carry energy and there's nonlinear evolution..." and got carried away, whoops. If you want to give a more accurate explanation (that makes sense to someone who's never taken a GR class) please do!
- raattgift 3y ago
  There is no question that this stuff is hard (and has been cutting edge for decades) and that it is easy to make mistakes. > give a more accurate explanation (that makese sense to someone who's never taken a GR class) It's hard to know what level of understanding to aim for on HN. There are non-relativist working physicists here who somewhat casually read other areas of physics (and mathematics) here, for example, rather than e.g. physics SE or looking through literature reviews. I'm guessing that you have done GR but probably not much with approximations like GEM, nor looked into the history of gravitational waves (e.g. the 1950s-60s work by Bondi with collaborators like Pirani) before the wide availability of powerful computers and observational support for the linearized theory. My goal here is not to nitpick you, but rather to offer a couple of references that might interest you or anyone who is quietly reading along. > I think I was totally wrong re: Ricci curvature. R_munu = 0 for one (uncharged) BH, and also for two. Charged BHs are different (Reissner-Nordström's Ricci tensor is R_munu = +- g_munu r_{Q}^2 / r^4 where r_{Q}^2 = \frac{Q^2G}{4 \pi \varepsilon_0 c^4} and ε_0 is the electric constant, all thanks to the electromagnetic stress-energy tensor; the Ricci scalar remains 0). Superposing two Schwarzschild or Kerr solutions is messy [Krivan & Price 1998 Phys. Rev. D 58, 104003 was a nice overview arxiv html5[*] <https://ar5iv.labs.arxiv.org/html/gr-qc/9806017 https://ar5iv.labs.arxiv.org/html/gr-qc/9806017>, 'two locally Kerr holes, no matter how close they are, will not superpose into a single Kerr hole. Though [this] "failure" of the close limit is physically correct, it is inconvenient [for doing perturbation theory]'] but doesn't change Rmunu = 0. A charged binary is beyond the scope of this comment. > I was thinking "oh, GW carry energy ..." I can recommend two really good papers written at very different times. Weber & Wheeler 1957 <https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29.509 https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29....> (it's also on sci-hub) where they struggle with the question of the sense in which GWs are real ("the only well defined way there is to express the influence of [gravitational] radiation [is] in terms of its effect upon invariant space time intervals betwee two test bodies"). I enjoyed seeing the rare \dotequalsdot in eqns 28,30, "geometrically equal to". Also, they lead right into the next suggestion with, "Similarties between gravitational and elecgromagnetic waves thus make it simple to draw a number of reasonable inferences. The significance of these inferences has a much more subtle character in the gravitational case than in the electromagnetic case. Neither field densities nor test particle motions have a meaning independent of the choice of coordinate systems. The simple observable consequences of wave action are instead changes in the separation of nearby test particles -- changes that are related to the covariant components of the curvature tensor R_ijkl." Goswami & Ellis 2021 <https://iopscience.iop.org/article/10.1088/1361-6382/abdaf3 https://iopscience.iop.org/article/10.1088/1361-6382/abdaf3> arxiv html5[*] <https://ar5iv.labs.arxiv.org/html/1912.00591 https://ar5iv.labs.arxiv.org/html/1912.00591> with the somewhat stentorian title "Tidal forces are gravitational waves". It is really about the Weyl curvature in binaries. The authors explicity consider (for orbital motions of two massive bodies) a decomposition of the Weyl tensor C_abcd into an electric and magnetic part (for which see §III; and eqn 6 repeats the point on R_ab in vaccum), the former compared to Newtonian gravitation and the latter being general-relativistic and encoding the gravitational waves that do not arise in the Newtonian theory. Their focus is on the magnetic part of the Weyl tensor, since (they argue) that is where all the interesting stuff is encoded, and how energy gets from the orbital system to an observer and between the binary partners themselves. One highlight is a short paragraph summarizing some ~1960 work by Bondi and MacCrae [**], "Through a series of shape changing operations ... one sees that after each rotation, Tweedledum is gaining internal energy as the external tidal force is doing work on him, while Tweedledee is losing energy as she is doing work against the external force. This is an excellent example of how the internal energy of a system can be transferred to another system via gravitational induction, in Newtonian gravity." [emphasis mine]. - - [*] lost way downpage and heavily downvoted is a HN user pointing out that the page of cartoons linked at the top is not accessibilty-friendly. I accept the point, and make up for it a little with the html5 links here. One can straightforwardly paste the trailing element of the ar5iv into a search engine or whatnot in order to reach a PDF version of each arxived paper above. [**] The actual Bondi & McCrae paper(s) is(are) proving a bit elusive, but this gives a good overview: §§3-4, Tweedledum and Tweedledee and Energy Transfer (the latter starts with an excellent quote from Bondi in 1957), https://link.springer.com/article/10.1007/s10701-022-00660-z https://link.springer.com/article/10.1007/s10701-022-00660-z (open access). So this quickly became three papers, and I'll stop now, because I'm fairly sure you can find your way through them if you really want, and figure out how to get help if you get stuck.