Hi all, Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know!
I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin.
Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense.
So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot?
Thank you for reading! Cheers,
Adam
Adam
The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task.
Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition.
Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE.
If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details?
-erik
On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst adamdrewherbst@gmail.com wrote:
Hi all, Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know!
I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin.
Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense.
So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot?
Thank you for reading! Cheers,
Adam _______________________________________________ Users mailing list Users@einsteintoolkit.org http://lists.einsteintoolkit.org/mailman/listinfo/users
Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it myself. I'll see if I can clear up the write-up I had and send it over. But I think you are right that I have not developed this enough to be tested numerically yet. After reading more, I think the Hilbert action approach doesn't make sense anyway. Also, as far as I can tell, the curvature singularity is unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which might be an elliptic PDE but it would be fourth-order in the metric. Could a 4th-order PDE be simulated?
Thank you kindly, Adam
On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter schnetter@cct.lsu.edu wrote:
Adam
The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task.
Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition.
Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE.
If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details?
-erik
On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst adamdrewherbst@gmail.com wrote:
Hi all, Before tackling the learning curve, I want to see if there's any chance
I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know!
I want to find a stationary spacetime, in which each time-slice has a
topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin.
Now the thing is, what I really want to do is start with the ansatz
described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense.
So I hope this was coherent. And if it is possible, can you let me know
which modules I should start getting familiar with in order to give it a shot?
Thank you for reading! Cheers,
Adam _______________________________________________ Users mailing list Users@einsteintoolkit.org http://lists.einsteintoolkit.org/mailman/listinfo/users
-- Erik Schnetter schnetter@cct.lsu.edu http://www.perimeterinstitute.ca/personal/eschnetter/
Hi Adam,
If it indeed turns out that your problem can be cast as a 4th order elliptical PDE, I don't see any reason why this could not be simulated. In fact in the thorn NoExcision, we actually use up to a 6th order ellitpical PDE to fill in the interior of a black hole with constraint violating data that smoothly matches the exterior data. In this thorn we implemented a conjugate gradient method to solve the equations and didn't see any issues with the fact the the equations involved 6th derivatives.
Cheers,
Peter
On Wednesday 2021-03-03 14:52, Adam Herbst wrote:
Date: Wed, 3 Mar 2021 14:52:53 From: Adam Herbst adamdrewherbst@gmail.com To: Erik Schnetter schnetter@cct.lsu.edu Cc: Einstein Toolkit Users users@einsteintoolkit.org Subject: Re: [Users] Can I simulate this exotic static topological spacetime with the ET?
Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it myself. I'll see if I can clear up the write-up I had and send it over. But I think you are right that I have not developed this enough to be tested numerically yet. After reading more, I think the Hilbert action approach doesn't make sense anyway. Also, as far as I can tell, the curvature singularity is unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which might be an elliptic PDE but it would be fourth-order in the metric. Could a 4th-order PDE be simulated?
Thank you kindly, Adam
On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter schnetter@cct.lsu.edu wrote: Adam
The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task. Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition. Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE. If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details? -erik On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst <adamdrewherbst@gmail.com> wrote: > > Hi all, > Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know! > > I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin. > > Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense. > > So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot? > > Thank you for reading! Cheers, > > Adam > _______________________________________________ > Users mailing list > Users@einsteintoolkit.org > http://lists.einsteintoolkit.org/mailman/listinfo/users -- Erik Schnetter <schnetter@cct.lsu.edu> http://www.perimeterinstitute.ca/personal/eschnetter/
Great to know, thanks Peter! I will continue to work on it and see if it comes to that point. I really appreciate all the information.
Adam
On Fri, Mar 5, 2021 at 8:39 AM Peter Diener diener@cct.lsu.edu wrote:
Hi Adam,
If it indeed turns out that your problem can be cast as a 4th order elliptical PDE, I don't see any reason why this could not be simulated. In fact in the thorn NoExcision, we actually use up to a 6th order ellitpical PDE to fill in the interior of a black hole with constraint violating data that smoothly matches the exterior data. In this thorn we implemented a conjugate gradient method to solve the equations and didn't see any issues with the fact the the equations involved 6th derivatives.
Cheers,
Peter
On Wednesday 2021-03-03 14:52, Adam Herbst wrote:
Date: Wed, 3 Mar 2021 14:52:53 From: Adam Herbst adamdrewherbst@gmail.com To: Erik Schnetter schnetter@cct.lsu.edu Cc: Einstein Toolkit Users users@einsteintoolkit.org Subject: Re: [Users] Can I simulate this exotic static topological
spacetime
with the ET?
Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it
myself.
I'll see if I can clear up the write-up I had and send it over. But I
think
you are right that I have not developed this enough to be tested
numerically
yet. After reading more, I think the Hilbert action approach doesn't make sense anyway. Also, as far as I can tell, the curvature singularity is unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which
might
be an elliptic PDE but it would be fourth-order in the metric. Could a 4th-order PDE be simulated?
Thank you kindly, Adam
On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter schnetter@cct.lsu.edu
wrote:
Adam The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task. Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition. Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE. If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details? -erik On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst <adamdrewherbst@gmail.com> wrote: > > Hi all, > Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know! > > I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin. > > Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense. > > So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot? > > Thank you for reading! Cheers, > > Adam > _______________________________________________ > Users mailing list > Users@einsteintoolkit.org > http://lists.einsteintoolkit.org/mailman/listinfo/users -- Erik Schnetter <schnetter@cct.lsu.edu> http://www.perimeterinstitute.ca/personal/eschnetter/
Hi Erik / Peter, Here is the write-up of the idea I'd like to simulate. I know it is pretty outlandish and not very likely to be true at the end of the day, but I can't shake the fact that it seems to explain the baryons so naturally. So I'd be ecstatic if you'd take a look and see if you think it would be possible to simulate this model of the electron. Even if I could just use the Toolkit for something like calculating the d'Alembertian of the Riemann tensor, so I could play with the metric and try to get it to converge to zero.
https://adamdrewherbst.pythonanywhere.com/welcome/spacetime/index?language=e...
But honestly, I would really appreciate it if any of you spacetime experts could tell me your reaction to the model as a whole, because it's hard to get that kind of feedback! If you see a multitude of reasons it should be dumped without further ado, well, that would be valuable too. But I understand you may not have the time for that. In any case, looking forward to a response!
Thank you, Adam
On Fri, Mar 5, 2021 at 5:10 PM Adam Herbst adamdrewherbst@gmail.com wrote:
Great to know, thanks Peter! I will continue to work on it and see if it comes to that point. I really appreciate all the information.
Adam
On Fri, Mar 5, 2021 at 8:39 AM Peter Diener diener@cct.lsu.edu wrote:
Hi Adam,
If it indeed turns out that your problem can be cast as a 4th order elliptical PDE, I don't see any reason why this could not be simulated. In fact in the thorn NoExcision, we actually use up to a 6th order ellitpical PDE to fill in the interior of a black hole with constraint violating data that smoothly matches the exterior data. In this thorn we implemented a conjugate gradient method to solve the equations and didn't see any issues with the fact the the equations involved 6th derivatives.
Cheers,
Peter
On Wednesday 2021-03-03 14:52, Adam Herbst wrote:
Date: Wed, 3 Mar 2021 14:52:53 From: Adam Herbst adamdrewherbst@gmail.com To: Erik Schnetter schnetter@cct.lsu.edu Cc: Einstein Toolkit Users users@einsteintoolkit.org Subject: Re: [Users] Can I simulate this exotic static topological
spacetime
with the ET?
Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it
myself.
I'll see if I can clear up the write-up I had and send it over. But I
think
you are right that I have not developed this enough to be tested
numerically
yet. After reading more, I think the Hilbert action approach doesn't
make
sense anyway. Also, as far as I can tell, the curvature singularity is unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which
might
be an elliptic PDE but it would be fourth-order in the metric. Could a 4th-order PDE be simulated?
Thank you kindly, Adam
On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter schnetter@cct.lsu.edu
wrote:
Adam The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task. Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition. Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE. If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details? -erik On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst <adamdrewherbst@gmail.com> wrote: > > Hi all, > Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know! > > I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin. > > Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense. > > So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot? > > Thank you for reading! Cheers, > > Adam > _______________________________________________ > Users mailing list > Users@einsteintoolkit.org > http://lists.einsteintoolkit.org/mailman/listinfo/users -- Erik Schnetter <schnetter@cct.lsu.edu> http://www.perimeterinstitute.ca/personal/eschnetter/
Adam
To solve a problem numerically, one must first have a well-posed formulation of the problem, and then choose a well-posed discretization. Both are difficult to obtain from the equations. If one just implements an equation, some boundary and/or initial conditions, and then runs a solver, most likely things won't work, and one won't have the slightest idea what is going wrong. In addition to the above, you'll need some intuition for length scales, time scales, curvature scales, etc. Starting with a 4d spacetime is probably hopeless.
Is there a way to simplify the problem to fewer dimensions? To simpler equations? Maybe to simpler physics even, solving a strawman problem?
For example, when learning how to solve the Einstein equations (which are nonlinear tensorial wave equations), we started with solving the linear scalar wave equation in one dimension. If there is no one-dimensional case, then maybe assuming axisymmetry or stationarity will help, or maybe one can study a linearization of the equations about some background, etc.
Even when simulating binary black holes (which is, by now, a well understood problem, since we have been simulating them for 15 years), it is difficult to get started from scratch. Most people start by taking an existing simulation and making small variations (masses, spins, initial velocities, etc.), or they take a known physical scenario and change numerical parameters (resolution, boundaries, numerical methods). The "original" black hole simulations were quite difficult to obtain and were based on years of experience, including experience from axisymmetric black hole simulations from many years earlier.
Since you are interested in studying a completely new set of equations, I suggest to consider first a much simplified problem. I wish that tools such as the Einstein Toolkit were black box solvers (similar to Mathematica's "Integrate" function), but in truth we're far away from that...
-erik
On Mon, Mar 8, 2021 at 5:19 PM Adam Herbst adamdrewherbst@gmail.com wrote:
Hi Erik / Peter, Here is the write-up of the idea I'd like to simulate. I know it is pretty outlandish and not very likely to be true at the end of the day, but I can't shake the fact that it seems to explain the baryons so naturally. So I'd be ecstatic if you'd take a look and see if you think it would be possible to simulate this model of the electron. Even if I could just use the Toolkit for something like calculating the d'Alembertian of the Riemann tensor, so I could play with the metric and try to get it to converge to zero.
https://adamdrewherbst.pythonanywhere.com/welcome/spacetime/index?language=e...
But honestly, I would really appreciate it if any of you spacetime experts could tell me your reaction to the model as a whole, because it's hard to get that kind of feedback! If you see a multitude of reasons it should be dumped without further ado, well, that would be valuable too. But I understand you may not have the time for that. In any case, looking forward to a response!
Thank you, Adam
On Fri, Mar 5, 2021 at 5:10 PM Adam Herbst adamdrewherbst@gmail.com wrote:
Great to know, thanks Peter! I will continue to work on it and see if it comes to that point. I really appreciate all the information.
Adam
On Fri, Mar 5, 2021 at 8:39 AM Peter Diener diener@cct.lsu.edu wrote:
Hi Adam,
If it indeed turns out that your problem can be cast as a 4th order elliptical PDE, I don't see any reason why this could not be simulated. In fact in the thorn NoExcision, we actually use up to a 6th order ellitpical PDE to fill in the interior of a black hole with constraint violating data that smoothly matches the exterior data. In this thorn we implemented a conjugate gradient method to solve the equations and didn't see any issues with the fact the the equations involved 6th derivatives.
Cheers,
Peter
On Wednesday 2021-03-03 14:52, Adam Herbst wrote:
Date: Wed, 3 Mar 2021 14:52:53 From: Adam Herbst adamdrewherbst@gmail.com To: Erik Schnetter schnetter@cct.lsu.edu Cc: Einstein Toolkit Users users@einsteintoolkit.org Subject: Re: [Users] Can I simulate this exotic static topological spacetime with the ET?
Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it myself. I'll see if I can clear up the write-up I had and send it over. But I think you are right that I have not developed this enough to be tested numerically yet. After reading more, I think the Hilbert action approach doesn't make sense anyway. Also, as far as I can tell, the curvature singularity is unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which might be an elliptic PDE but it would be fourth-order in the metric. Could a 4th-order PDE be simulated?
Thank you kindly, Adam
On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter schnetter@cct.lsu.edu wrote: Adam
The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task. Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition. Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE. If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details? -erik On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst <adamdrewherbst@gmail.com> wrote: > > Hi all, > Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know! > > I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin. > > Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense. > > So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot? > > Thank you for reading! Cheers, > > Adam > _______________________________________________ > Users mailing list > Users@einsteintoolkit.org > http://lists.einsteintoolkit.org/mailman/listinfo/users -- Erik Schnetter <schnetter@cct.lsu.edu> http://www.perimeterinstitute.ca/personal/eschnetter/
Hi Erik, This is quite sobering. I am very grateful for the in-depth response, and frankly in awe of all you folks are doing. Thank you so much!
Adam
On Mon, Mar 8, 2021 at 6:16 PM Erik Schnetter schnetter@cct.lsu.edu wrote:
Adam
To solve a problem numerically, one must first have a well-posed formulation of the problem, and then choose a well-posed discretization. Both are difficult to obtain from the equations. If one just implements an equation, some boundary and/or initial conditions, and then runs a solver, most likely things won't work, and one won't have the slightest idea what is going wrong. In addition to the above, you'll need some intuition for length scales, time scales, curvature scales, etc. Starting with a 4d spacetime is probably hopeless.
Is there a way to simplify the problem to fewer dimensions? To simpler equations? Maybe to simpler physics even, solving a strawman problem?
For example, when learning how to solve the Einstein equations (which are nonlinear tensorial wave equations), we started with solving the linear scalar wave equation in one dimension. If there is no one-dimensional case, then maybe assuming axisymmetry or stationarity will help, or maybe one can study a linearization of the equations about some background, etc.
Even when simulating binary black holes (which is, by now, a well understood problem, since we have been simulating them for 15 years), it is difficult to get started from scratch. Most people start by taking an existing simulation and making small variations (masses, spins, initial velocities, etc.), or they take a known physical scenario and change numerical parameters (resolution, boundaries, numerical methods). The "original" black hole simulations were quite difficult to obtain and were based on years of experience, including experience from axisymmetric black hole simulations from many years earlier.
Since you are interested in studying a completely new set of equations, I suggest to consider first a much simplified problem. I wish that tools such as the Einstein Toolkit were black box solvers (similar to Mathematica's "Integrate" function), but in truth we're far away from that...
-erik
On Mon, Mar 8, 2021 at 5:19 PM Adam Herbst adamdrewherbst@gmail.com wrote:
Hi Erik / Peter, Here is the write-up of the idea I'd like to simulate. I know it is
pretty outlandish and not very likely to be true at the end of the day, but I can't shake the fact that it seems to explain the baryons so naturally. So I'd be ecstatic if you'd take a look and see if you think it would be possible to simulate this model of the electron. Even if I could just use the Toolkit for something like calculating the d'Alembertian of the Riemann tensor, so I could play with the metric and try to get it to converge to zero.
https://adamdrewherbst.pythonanywhere.com/welcome/spacetime/index?language=e...
But honestly, I would really appreciate it if any of you spacetime
experts could tell me your reaction to the model as a whole, because it's hard to get that kind of feedback! If you see a multitude of reasons it should be dumped without further ado, well, that would be valuable too. But I understand you may not have the time for that. In any case, looking forward to a response!
Thank you, Adam
On Fri, Mar 5, 2021 at 5:10 PM Adam Herbst adamdrewherbst@gmail.com
wrote:
Great to know, thanks Peter! I will continue to work on it and see if
it comes to that point. I really appreciate all the information.
Adam
On Fri, Mar 5, 2021 at 8:39 AM Peter Diener diener@cct.lsu.edu wrote:
Hi Adam,
If it indeed turns out that your problem can be cast as a 4th order elliptical PDE, I don't see any reason why this could not be simulated. In fact in the thorn NoExcision, we actually use up to a 6th order ellitpical PDE to fill in the interior of a black hole with constraint violating data that smoothly matches the exterior data. In this thorn we implemented a conjugate gradient method to solve the equations and didn't see any issues with the fact the the equations involved 6th derivatives.
Cheers,
Peter
On Wednesday 2021-03-03 14:52, Adam Herbst wrote:
Date: Wed, 3 Mar 2021 14:52:53 From: Adam Herbst adamdrewherbst@gmail.com To: Erik Schnetter schnetter@cct.lsu.edu Cc: Einstein Toolkit Users users@einsteintoolkit.org Subject: Re: [Users] Can I simulate this exotic static topological
spacetime
with the ET?
Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it
myself.
I'll see if I can clear up the write-up I had and send it over. But
I think
you are right that I have not developed this enough to be tested
numerically
yet. After reading more, I think the Hilbert action approach doesn't
make
sense anyway. Also, as far as I can tell, the curvature singularity
is
unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which
might
be an elliptic PDE but it would be fourth-order in the metric. Could
a
4th-order PDE be simulated?
Thank you kindly, Adam
On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter schnetter@cct.lsu.edu
wrote:
Adam The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task. Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition. Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE. If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details? -erik On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst <adamdrewherbst@gmail.com> wrote: > > Hi all, > Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know! > > I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin. > > Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense. > > So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot? > > Thank you for reading! Cheers, > > Adam > _______________________________________________ > Users mailing list > Users@einsteintoolkit.org > http://lists.einsteintoolkit.org/mailman/listinfo/users -- Erik Schnetter <schnetter@cct.lsu.edu> http://www.perimeterinstitute.ca/personal/eschnetter/-- Erik Schnetter schnetter@cct.lsu.edu http://www.perimeterinstitute.ca/personal/eschnetter/
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