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I understand that on galactic scales, the expansion of space time has no appreciable effect, gravity being dominant and thus distance between stars remains fixed despite universal expansion.

Can anyone help me see this mathematically? I have been unable to find equations to show me both gravity and expansion together. I am interested in whether individual mass objects far from the virial mass centre would start to be caught in the space time expansion and thus end up farther from the galactic centre, i.e. some boundary condition where expansion does overcome gravity due to gravitational effects being weaker at the furthest fringes of matter attracted by the central masses.

ACuriousMind
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Mentasm
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1 Answers1

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One way to see this is simply to note the value of the Hubble constant, about $70\rm\,km/(s\cdot Mpc)$. The Milky Way galaxy is about 0.030 Mpc in diameter, so the Hubble "flow" from one end of the Milky Way to the other is only about 2 km/s. This is much smaller than the "peculiar" motions of objects within the galaxy. Heck, the eastward motion of Earth's surface near the equator is about 0.5 km/s.

Compare this to low-temperature phenomena in other areas of physics. If you're interested in milli-electronvolt physics, you can't explore it at room temperature ($kT = \rm25\,meV$); you have cool your material off to get rid of the random thermal motions.

Extending to the Local Group: the Andromeda galaxy and its satellites are approaching us at about 100 km/s, from sort-of a megaparsec away. So the "thermal motion" of galaxies in the Local Group is comparable to the Hubble flow, but still "hot" enough that the directions are more or less random.

Thriveth points out a subtle corollary in comment below: since there is no real distinction in GR between "expansion of matter within spacetime" and "expansion of spacetime carrying matter along," it is reasonable to conclude that within a galaxy, where motions are rapid and randomly oriented, there is no spacetime expansion at all. I admit to being somewhat out of my depth here.

rob
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  • "However, it doesn't explain this presumed mechanism of gravity stronger than the expansion mechanism" ... I think it does. Within the galaxy, you have gravity requiring objects to move at hundreds of kilometers per second, while Hubble flow would induce speeds orders of magnitude smaller. Gravity wins. The expansion beyond the galaxy is a little bit iffier. – rob Jan 13 '16 at 18:02
  • @rob this answer is very helpful, but i suppose i should explain the reason i am looking at the threshold between galactic gravitation and expansion...

    I was considering a thought experiment that took as a premise the notion most often explain to me that galactic and other masses 'roll' on, or spacetime expands 'under' (if using a 2D rubber sheet analogy of spacetime), the surface of expanding space time remaining bound together by their own gravitation, hence, the universe expands whilst locally, the interstellar distances in galaxies appear to remain remain effectively fixed.

     – Mentasm Jan 14 '16 at 03:51
  • ...It then occurred to me that if expansion was limited to flat(ish) spacetime, then this could lead to a model which could show a larger spacetime deformation than indicated by gravity alone. thus, lensed light would follow a path indicted by a hidden halo of mass, when in fact, the could be less mass than postulated due to expansion effects. – Mentasm Jan 14 '16 at 03:52
  • Lacking the math to derive all of this (its been 30 years since i did nay serious calculus), but having very strong programming skills, i planned to model the experiment in a computer simulation and tweaks the parameters until i could get observed effects without the need for (so much) hidden mass.

    Hence, i am seeking the right equations or a pointer to where i can find/derive them, so i can start doing some computer modelling.

     – Mentasm Jan 14 '16 at 03:52
  • @igael For framework, i use javascript / three.js, or c if computations become slow. i was planning to start with a simple model and later to introduce n-body methods for fringe stars if it look promising.

    I want to calculate spacetime curvature, so i presume Einstein field equations would be used. Next i would like to be able to add in a term to furthre stretch curvature already caused by gravity to have it further deform at low curvatures due to expansion. So, I need a additional term for expansion that only operates at a curvature threshold. A cosmological variable (and not constant!). J

     – Mentasm Jan 14 '16 at 06:39
  • So true @igael, I started on database project 25 years ago and now, thanks to webGL, it is getting finished. I figure I have another 25 years life left and never give up, so 3 years sounds ok. If I could see the math in the firm of an equation, then the model would be worked on principles, there would be no need to accurately model real structures. My interest is to postulate a credible alternative to the conclusion missing mass explains observations. There are so many supporting observations for concordance that dark matter probably is the answer, but I don't like it and have time to think.. – Mentasm Jan 14 '16 at 15:10
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    I am not sure this answer is actually correct. It is not that gravtiational/thermal motion dominates over cosmological expansion in gravitationally bound systems, it simply turns it off. I suppose we are used to thinking of it like "expansion of Space drags galaxies along", but it is equivalent and equally valid to say "Motion of galaxies drags Space along". If you think of it like that, it is easy to see that within a system in which all peculiar motion averages to zero, there is no expansion of Space.

    Spacetime tells matter how to move, but matter tells Spacetime how to curve.

     – Thriveth Jan 14 '16 at 16:37
  • @Thriveth Excellent point; edited. – rob Jan 14 '16 at 18:08
  • thanks @Thriveth. I am interested in how gravitation "turns it off" (expansion). 'Dominant' was a way to phrase. Nature is so rarely on/off and more fuzzy at transitions. My question was relating to whether this 'fuzziness' is an area worth investigation. In the terms of your comment, my thought experiment would be phrased: . "Spacetime tells matter how to move, but matter tells spacetime how to curve, and could expansion further expand flatter regions of curved spacetime near masses to give false indication of missing mass to an observers seeing Einstein rings etc.". – Mentasm Jan 19 '16 at 09:49
  • Thanks to all for the erudite commentary. My original question was answered by @rob, and while I am no closer to my goal, it is now plain to me now that if I am considering a thought experiment which is not 100% in line or respectful to the GR paradigm, then I will have to do a lot more groundwork to develop the idea. I will keep digging and perhaps when I am capable to ask my question in a better and more specific manner, with a starting mathematical premise, I will ask in a better way... – Mentasm Jan 21 '16 at 07:12