Timeline for Coordinates for FLRW metric
Current License: CC BY-SA 3.0
30 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2015 at 19:43 | history | edited | Shadumu | CC BY-SA 3.0 |
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| Jan 26, 2015 at 18:17 | answer | added | Zo the Relativist | timeline score: 7 | |
| Jan 26, 2015 at 17:37 | answer | added | Christoph | timeline score: 0 | |
| Jan 26, 2015 at 17:27 | history | tweeted | twitter.com/#!/StackPhysics/status/559764678096986112 | ||
| Jan 26, 2015 at 16:31 | history | reopened |
Christoph Kyle Kanos JamalS Hypnosifl Ali |
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| Jan 26, 2015 at 16:26 | comment | added | Shadumu | @Hypnosifl so indeed when we use another frame and talk about whether stars are getting further away from us, we need to make sure simultaneity when we measure, then we may obtain different conclusions depending on what frame we choose, just like the length contraction. What we observe is redshift and it should be coordinate independent, but when we propose the FRW model to fit the data, we are free to choose a coordinate to describe it. In some of them, we may say the universe is not expanding. | |
| Jan 26, 2015 at 16:24 | comment | added | Hypnosifl | By the way, I do know of a special case that might interest you--in the standard coordinate system for FLRW universes the curvature of space in each simultaneity surface is hyperbolic whenever the mass density is below a critical value, and this is true even when the density goes to zero. This is known as the Milne model, but it is really just a different coordinate system on flat Minkowski spacetime. The 2nd large diagram here shows how lines of simultaneity look when plotted in an inertial frame. | |
| Jan 26, 2015 at 16:21 | comment | added | Shadumu | @Jim think about the positive cosmological constant Lambdavacuum solution. In some coordinates, the universe seems static, whereas in some other, the universe is expanding. I beg similar things can happen to FRW | |
| Jan 26, 2015 at 16:17 | comment | added | Hypnosifl | @user3229471 - I would say that length contraction is dependent on one's choice of simultaneity convention which is coordinate-dependent, although given a spacelike curve between points on the worldlines of either end of an object (a curve that would be confined to a single simultaneity surface if it was intended as a measure of instantaneous "length" in some coordinate system), the proper distance along that spacelike curve is coordinate-independent (just like proper time along timelike curves is coordinate-independent). | |
| Jan 26, 2015 at 16:13 | comment | added | Jim | Ah, I understand now. I think that would be more a problem with the coordinate transformation. The system transformed to could be unphysical or in a complicated, non-inertial, and unsymmetric motion. But I think that in all cases over long time spans, the FLRW-metric always looks like it s expanding. I haven't got a math proof in mind to back up that claim, however | |
| Jan 26, 2015 at 16:04 | comment | added | Shadumu | @Jim Sorry, I can't give you the metric, I am just thinking that in principle you can do it. You are free to choose a coordinate to describe the same metric. It is just your interpretations could be different, just as the length of a rod will have different values in different frames in SR. | |
| Jan 26, 2015 at 15:56 | comment | added | Jim | Wouldn't that no longer be the FLRW-metric? Perhaps a demonstration is in order. Could you give an example of such a transformed FLRW-metric? or at least just the $g_{\mu\nu}$? | |
| Jan 26, 2015 at 15:40 | comment | added | Shadumu | @Jim you can do a general coordinate transformation in which the t and space coordinates mixed up, the new set of coordinates is equally good but the interpretation can be different, though physics being the same. | |
| Jan 26, 2015 at 15:33 | comment | added | Jim | I'm confused, the expansion of the universe is a conclusion that can be drawn in a coordinate-independent way. We can see the expansion of the metric without making a choice of coordinates. $ds^2=-dt^2+a(t)^2d\Sigma^2$. This shows that the spatial dimensions expand with the scale factor and $d\Sigma$ is coordinate independent. So perhaps I misunderstand the question | |
| Jan 26, 2015 at 15:10 | comment | added | Shadumu | @Hypnosifl what about length contraction in SR, do you think that is physical or just a matter of different coordinate distances? What if the FRW metric was first proposed in some other coordinates, we might change the way we think about the evolution of universe, space might be shrinking or static in some coordinates, though the metric/physics is still the same. | |
| Jan 25, 2015 at 23:42 | review | Reopen votes | |||
| Jan 26, 2015 at 16:31 | |||||
| Jan 25, 2015 at 23:22 | comment | added | Christoph | Am I the only one who finds this question perfectly clear? The answer I was going (actually started) to write would have shown how cosmological time and proper distance along associated spatial slicings are special and why FLRW spacetime is not Minkowski space, even though conformal time and comoving coordinates make it look that way | |
| Jan 25, 2015 at 23:17 | history | closed |
John Rennie ACuriousMind♦ Brandon Enright PhotonBoom ProfRob |
Needs details or clarity | |
| Jan 25, 2015 at 22:40 | comment | added | Hypnosifl | I'm sure you could use a different set of coordinates to express the metric, but it'd probably be more complicated mathematically. If you specifically want to see an example of the FLRW metric expressed in a different coordinate system (with a different definition of simultaneity do that each spatial hypersurface would have a different curvature than the hypersurfaces of simultaneity of the standard cosmological coordinate system), you might consider asking a new question about this, maybe someone here would know of some published example, or could construct one themselves. | |
| Jan 25, 2015 at 21:31 | answer | added | Qmechanic♦ | timeline score: 2 | |
| Jan 25, 2015 at 21:29 | comment | added | Shadumu | @Hypnosifl I see your point. But why this particular coordinate is used to describe the metric but not any other coordinates | |
| Jan 25, 2015 at 17:43 | comment | added | Hypnosifl | But what do you mean that the "universe is expanding as we observed"? What specific physical observations are you talking about? If you phrase things in terms of local observations like the relation between the redshift of a galaxy and the apparent brightness of standard candles in it (which can be used to deduce distance in the standard cosmological coordinate system), then all coordinate systems predict the same results for these local observations, even if the observations don't have the same meaning in terms of coordinate distances etc. | |
| Jan 25, 2015 at 17:25 | comment | added | Kyle Kanos | @user3229471: You should edit your comment into the post. | |
| Jan 25, 2015 at 17:22 | history | edited | Shadumu | CC BY-SA 3.0 |
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| Jan 25, 2015 at 17:16 | comment | added | Shadumu | what I mean is the fact that universe is expanding as we observed is only predicted by the FLRW metric in certain special coordinates. If I do a coordinate transformation, the space could be static. Does that mean we are in a special frame? | |
| Jan 25, 2015 at 17:14 | review | Close votes | |||
| Jan 25, 2015 at 23:17 | |||||
| Jan 25, 2015 at 16:39 | comment | added | Hypnosifl | What do you mean by "why we on Earth only confirm the standard coordinates used in FLRW metric through our observation"? Which observations are you talking about? In both special and general relativity, all coordinate systems predict the same things about local physical observations, like the proper time on an observer's clock at the moment they receive light from various distant events. | |
| Jan 25, 2015 at 16:37 | comment | added | John Rennie | I'm afraid it isn't clear to me what you mean. Can you rephrase your question in a more mathematical way to clarify exactly what you're asking? Are you comparing the different interpretations of the expansion given by comoving coordinates and "everyday" coordinates? | |
| Jan 25, 2015 at 15:30 | history | edited | Qmechanic♦ |
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| Jan 25, 2015 at 15:12 | history | asked | Shadumu | CC BY-SA 3.0 |