4 ms·

Anyone happen to know what the time dilation factor would be for the Milky Way at somewhere near the outside edge and somewhere nearish to the center (say at 5%

by floxy 2y ago

Anyone happen to know what the time dilation factor would be for the Milky Way at somewhere near the outside edge and somewhere nearish to the center (say at 5% of the radius, so that we're not inside the central black hole)? The article says: "...in fact, an atomic clock located in a galaxy could tick up to a third slower than the same clock in the middle of a void." Maybe there is an nice simple formula assuming a uniform disk relating the time dilation to radius? I've always been under the impression that general relativity simulations needed supercomputers, but maybe current desktop machines are sufficiently "super" compared to 20 years ago that we could do simple calculations? Anyone have a recommendation on where to start learning enough general relativity to make a computer program? Assuming you've have a couple semesters of calculus-based college physics 101 under your belt as the prerequisite?

pantulis 2y ago
Yes, the formula is the Einstein's Field Equation. Alas, I don't think it's generally considered as nice and simple. But it's the tool for that: you would basically set an energy density distribution and the software would calculate the metric of spacetime (and that would include time distortion) I guess you can find repos on Github (first Google result for me was this: https://github.com/spacetimeengineer/spacetimeengine https://github.com/spacetimeengineer/spacetimeengine) and try your hand. But the theoretical aspect is intimidating, for sure. As for getting up to speed in the theory, I cannot but recommend Leonard Susskind lectures on GR that are available on YouTube. Susskind may be best known as one of the founding fathers of String Theory but the man is an exceptionally great teacher on this subject.
ben_w 2y ago
The formula themselves aren't that complicated, as software goes: https://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form https://en.wikipedia.org/wiki/Einstein_field_equations#Mathe... The main problem is that you need to have some intuition about what the formula is supposed do in order to test your code to make sure you've not made a silly mistake, and that relativity is very unintuitive. There's probably some existing python library that already does all of it, may be best to start with one of them.
- floxy 2y ago
  >in order to test your code to make sure you've not made a silly mistake Precession of Mercury around the Sun?
zifk 2y ago
Simulations require supercomputers for doing large scale, detailed calculations, but simple situations can be solved completely analytically. For example, gravitational time dilation can be calculated somewhat simply for a central gravitational potential: https://en.wikipedia.org/wiki/Gravitational_time_dilation#Outside_a_non-rotating_sphere https://en.wikipedia.org/wiki/Gravitational_time_dilation#Ou... General Relativity is incredibly math heavy but fundamentally the numerical methods involved are standard methods for differential equations. The hard part is going from the math to a solvable form. See https://arxiv.org/pdf/2008.12931 https://arxiv.org/pdf/2008.12931 for a broad overview. This will of course probably not make sense without an introduction to differential geometry, a beast of a topic itself. See some big textbook like https://arxiv.org/pdf/2412.08026 https://arxiv.org/pdf/2412.08026 or find yourself a copy of Gravitation by Misner, Thorne and Wheeler.
- floxy 2y ago
  >an introduction to differential geometry I did find "Functional Differential Geometry": https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/9580/9580.pdf?dl=1 https://mitp-content-server.mit.edu/books/content/sectbyfn/b... ...which uses Scheme to teach differential geometry. I would need to learn quite a bit more before tackling that book. Maybe something like: "Structure and Interpretation of Classical Mechanics"? https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/9579/sicm_edition_2.zip/book.html https://mitp-content-server.mit.edu/books/content/sectbyfn/b... ...but even there, it looks like I would need to start with something more basic.
cryptonector 2y ago
Given an estimate of the mass and mass distribution of the Milky Way we can compute the time dilation factor we expect assuming that the intergalactic space beyond the galaxy has zero time dilation. However, I think we cannot easily measure or infer the absolute time dilation we experience due to the rest of the universe. But relative time dilation factors are enough for the sort of analysis that is done in timescapes.
cryptonector 2y ago
Yes: Schwarszchild time dilation, while a solution to a one-massive body model, is a decent approximation if you think of M/r as an integration of the effects of all matter around you: \sqrt{1-\frac{2GM}_{rc^2}} Of course you can just use the Einstein field equations, but that's more work, though not an approximation. You'll need a good model of the mass distribution in the universe in order to actually get good answers.

by floxy 2y ago

pantulis 2y ago

Yes, the formula is the Einstein's Field Equation. Alas, I don't think it's generally considered as nice and simple. But it's the tool for that: you would basically set an energy density distribution and the software would calculate the metric of spacetime (and that would include time distortion) I guess you can find repos on Github (first Google result for me was this: https://github.com/spacetimeengineer/spacetimeengine https://github.com/spacetimeengineer/spacetimeengine) and try your hand. But the theoretical aspect is intimidating, for sure. As for getting up to speed in the theory, I cannot but recommend Leonard Susskind lectures on GR that are available on YouTube. Susskind may be best known as one of the founding fathers of String Theory but the man is an exceptionally great teacher on this subject.

ben_w 2y ago

The formula themselves aren't that complicated, as software goes: https://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form https://en.wikipedia.org/wiki/Einstein_field_equations#Mathe... The main problem is that you need to have some intuition about what the formula is supposed do in order to test your code to make sure you've not made a silly mistake, and that relativity is very unintuitive. There's probably some existing python library that already does all of it, may be best to start with one of them.

floxy 2y ago
>in order to test your code to make sure you've not made a silly mistake Precession of Mercury around the Sun?

zifk 2y ago

Simulations require supercomputers for doing large scale, detailed calculations, but simple situations can be solved completely analytically. For example, gravitational time dilation can be calculated somewhat simply for a central gravitational potential: https://en.wikipedia.org/wiki/Gravitational_time_dilation#Outside_a_non-rotating_sphere https://en.wikipedia.org/wiki/Gravitational_time_dilation#Ou... General Relativity is incredibly math heavy but fundamentally the numerical methods involved are standard methods for differential equations. The hard part is going from the math to a solvable form. See https://arxiv.org/pdf/2008.12931 https://arxiv.org/pdf/2008.12931 for a broad overview. This will of course probably not make sense without an introduction to differential geometry, a beast of a topic itself. See some big textbook like https://arxiv.org/pdf/2412.08026 https://arxiv.org/pdf/2412.08026 or find yourself a copy of Gravitation by Misner, Thorne and Wheeler.

floxy 2y ago
>an introduction to differential geometry I did find "Functional Differential Geometry": https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/9580/9580.pdf?dl=1 https://mitp-content-server.mit.edu/books/content/sectbyfn/b... ...which uses Scheme to teach differential geometry. I would need to learn quite a bit more before tackling that book. Maybe something like: "Structure and Interpretation of Classical Mechanics"? https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/9579/sicm_edition_2.zip/book.html https://mitp-content-server.mit.edu/books/content/sectbyfn/b... ...but even there, it looks like I would need to start with something more basic.

cryptonector 2y ago

Given an estimate of the mass and mass distribution of the Milky Way we can compute the time dilation factor we expect assuming that the intergalactic space beyond the galaxy has zero time dilation. However, I think we cannot easily measure or infer the absolute time dilation we experience due to the rest of the universe. But relative time dilation factors are enough for the sort of analysis that is done in timescapes.

cryptonector 2y ago

Yes: Schwarszchild time dilation, while a solution to a one-massive body model, is a decent approximation if you think of M/r as an integration of the effects of all matter around you: \sqrt{1-\frac{2GM}_{rc^2}} Of course you can just use the Einstein field equations, but that's more work, though not an approximation. You'll need a good model of the mass distribution in the universe in order to actually get good answers.