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#2




Galaxies without dark matter halos?
on 25/10/03 9:28 pm, at
wrote: In article , Dag Oestvang wrote: Ted Bunn wrote: I claim that it the following procedure is a perfectly meaningful, consistent, and moreover extremely useful way to describe the expanding Universe on small scales: Well, in my opinion you are mistaken, see below. Rather than following Usenet convention and replying point by point (which would make for a long and unwieldy post that no one would want to slog through), I'll try to summarize where we disagree and explain my position reasonably succinctly. Please let me know if you think I've omitted an important facet of the discussion or misrepresented your position in any significant way. Everyone agrees on the following: A comoving observer who looks at a comoving object in an expanding FRW spacetime will observe that object to be redshifted. I would like to defend the following proposition: A. If the distance between observer and observed is much less than the scale of spacetime curvature, then it makes sense to describe that redshift as a Doppler shift. (Here "scale of spacetime curvature" means either the horizon distance or the radius of spatial curvature, whichever is smaller. I labeled this proposition A because I want to compare it later with another proposition, which I'll call B.) snip further clarification Now let me try to explain what I mean with an example. Suppose I stand on top of a tall tower and drop a baseball out. I track it with a radar gun as it falls to measure its speed. I claim the following: B. If the ball travels a distance that is small compared to the curvature scale of spacetime in my neighborhood, then it makes sense to interpret the observed redshift as a Doppler shift. By the way, spacetime is very weakly curved in the vicinity of Earth's surface, so the assumption happens to be fairly unconstraining in this case. The observed redshift will include a gravitational redshift as well as a Doppler shift, of course, but the latter will be very tiny under the assumed conditions; calling the observed redshift a Doppler shift is a kickass approximation. I can't speak for Dag Oestvang, but your comparison seems to present the following difficulties: 1) The geometry of spacetime in the immediate vicinity of the earth is not even approximately FRW. 2) You and the baseball are (a fortiori) not even approximately comoving. Crucially, the following more specific statements are true: 1') The geometry of spacetime in the immediate vicinity of the earth is (to a very good approximation) static. 2') The baseball is following a geodesic, but you aren't: you have a proper acceleration of 9.8 m s^2 due to the upward force exerted by the tower. The bulk of the baseball's redshift as measured by you is uncontroversially due to the nonzero relative velocity of the ball wrt you. You mention an additional 'gravitation redshift', but I maintain that this redshift isn't gravitational in the strict sense, but is actually _accelerational_. That is, it's the extra redshift due to acceleration that would also be measured by an accelerating observer in flat Minkowski spacetime. In short, gravity is completely irrelevant in example B. The whole thing could be done without significant alteration in SR. Note the irrelevance of tidal effects, which are the identifying mark of GR. By contrast, cosmological redshift is essentially gravitational; it cannot be reproduced in flat spacetime. The role of the relative motion of the galaxies is to produce framedragging, a GR effect, and it is the framedragging that is responsible for the 'expansion of space' and hence the cosmological redshift. The approximation in which the curvature is negligible is precisely the approximation in which the redshift is negligible. I suppose it is possible that the following idea might be made to work, although it doesn't feel right to me: If we assume that our FRW model is a spatially flat one, then the spatial slice at any given moment of cosmological time is Euclidean, and hence has an unambiguous notion of distance between us and the distant galaxy. We can then calculate the rate of increase of distance between us and the galaxy as a function of cosmological time (or our proper time, which hopefully isn't too different), and get a relative velocity, to which we could attribute the redshift. (Note that this description is coordinatefree.) However, conceptually this feels dubious to me. With a genuine doppler shift, the redshift is basically determined at the point that the light is emitted (assuming the observer doesn't accelerate in the meantime). With the gravitational redshift, the redshift depends on what happens to space during the time of flight of the light. That, as it were, is 'when' the wavelength gets 'stretched'. I also haven't done the calculations to check that this would come out numerically right; if it didn't, that would of course throw off the whole idea anyway. snip Tim [Mod. note: it does come out numerically right: d(proper distance)/d(cosmological time) is equal to the naively calculated recession speed in the limit z  0. Personally, I think this isn't a coincidence  mjh] 
#3




Galaxies without dark matter halos?
on 25/10/03 9:28 pm, at
wrote: In article , Dag Oestvang wrote: Ted Bunn wrote: I claim that it the following procedure is a perfectly meaningful, consistent, and moreover extremely useful way to describe the expanding Universe on small scales: Well, in my opinion you are mistaken, see below. Rather than following Usenet convention and replying point by point (which would make for a long and unwieldy post that no one would want to slog through), I'll try to summarize where we disagree and explain my position reasonably succinctly. Please let me know if you think I've omitted an important facet of the discussion or misrepresented your position in any significant way. Everyone agrees on the following: A comoving observer who looks at a comoving object in an expanding FRW spacetime will observe that object to be redshifted. I would like to defend the following proposition: A. If the distance between observer and observed is much less than the scale of spacetime curvature, then it makes sense to describe that redshift as a Doppler shift. (Here "scale of spacetime curvature" means either the horizon distance or the radius of spatial curvature, whichever is smaller. I labeled this proposition A because I want to compare it later with another proposition, which I'll call B.) snip further clarification Now let me try to explain what I mean with an example. Suppose I stand on top of a tall tower and drop a baseball out. I track it with a radar gun as it falls to measure its speed. I claim the following: B. If the ball travels a distance that is small compared to the curvature scale of spacetime in my neighborhood, then it makes sense to interpret the observed redshift as a Doppler shift. By the way, spacetime is very weakly curved in the vicinity of Earth's surface, so the assumption happens to be fairly unconstraining in this case. The observed redshift will include a gravitational redshift as well as a Doppler shift, of course, but the latter will be very tiny under the assumed conditions; calling the observed redshift a Doppler shift is a kickass approximation. I can't speak for Dag Oestvang, but your comparison seems to present the following difficulties: 1) The geometry of spacetime in the immediate vicinity of the earth is not even approximately FRW. 2) You and the baseball are (a fortiori) not even approximately comoving. Crucially, the following more specific statements are true: 1') The geometry of spacetime in the immediate vicinity of the earth is (to a very good approximation) static. 2') The baseball is following a geodesic, but you aren't: you have a proper acceleration of 9.8 m s^2 due to the upward force exerted by the tower. The bulk of the baseball's redshift as measured by you is uncontroversially due to the nonzero relative velocity of the ball wrt you. You mention an additional 'gravitation redshift', but I maintain that this redshift isn't gravitational in the strict sense, but is actually _accelerational_. That is, it's the extra redshift due to acceleration that would also be measured by an accelerating observer in flat Minkowski spacetime. In short, gravity is completely irrelevant in example B. The whole thing could be done without significant alteration in SR. Note the irrelevance of tidal effects, which are the identifying mark of GR. By contrast, cosmological redshift is essentially gravitational; it cannot be reproduced in flat spacetime. The role of the relative motion of the galaxies is to produce framedragging, a GR effect, and it is the framedragging that is responsible for the 'expansion of space' and hence the cosmological redshift. The approximation in which the curvature is negligible is precisely the approximation in which the redshift is negligible. I suppose it is possible that the following idea might be made to work, although it doesn't feel right to me: If we assume that our FRW model is a spatially flat one, then the spatial slice at any given moment of cosmological time is Euclidean, and hence has an unambiguous notion of distance between us and the distant galaxy. We can then calculate the rate of increase of distance between us and the galaxy as a function of cosmological time (or our proper time, which hopefully isn't too different), and get a relative velocity, to which we could attribute the redshift. (Note that this description is coordinatefree.) However, conceptually this feels dubious to me. With a genuine doppler shift, the redshift is basically determined at the point that the light is emitted (assuming the observer doesn't accelerate in the meantime). With the gravitational redshift, the redshift depends on what happens to space during the time of flight of the light. That, as it were, is 'when' the wavelength gets 'stretched'. I also haven't done the calculations to check that this would come out numerically right; if it didn't, that would of course throw off the whole idea anyway. snip Tim [Mod. note: it does come out numerically right: d(proper distance)/d(cosmological time) is equal to the naively calculated recession speed in the limit z  0. Personally, I think this isn't a coincidence  mjh] 
#4




Galaxies without dark matter halos?
In article ,
Tim S wrote: 1) The geometry of spacetime in the immediate vicinity of the earth is not even approximately FRW. That's certainly true, but it's not relevant to the point I was trying to make. My argument was based on the fact that the geometry of spacetime is approximately *flat* (i.e., Minkowski), not FRW. To be a bit more precise, I'm basing my argument on the utterly banal observation that everything is linear to first order. If you want to study phenomena on length scales much smaller than the curvature scale, then you can view spacetime as approximately Minkowski, with small perturbations of order epsilon = (length scale of observations) / (curvature scale of spacetime). When you observe a frequency shift near the Earth's surface, and you interpret it as a Doppler shift, that's precisely what you're doing: you're pretending that spacetime is flat and applying special relativity. You know that you're making errors because spacetime is not perfectly flat, but you also know that those errors are small (because epsilon is small under the circumstances). In short, As long as you're willing to ignore errors of order epsilon, you're allowed to pretend that spacetime is flat and interpret spectral shifts as Doppler shifts. I claim that interpreting the redshift of a nearby galaxy in an FRW Universe as a Doppler shift relies on exactly the same assumptions, and should be regarded as exactly as valid, as using a radar gun to measure the speed of a Pedro Martinez fastball. [OK, I confess I'm exaggerating a bit there. The quantity epsilon is much smaller in the latter case than in the former, so you're making less of an error. But the two cases are exactly the same in principle: the act of interpreting the observed spectral shift as a Doppler shift (as opposed to a gravitational shift) rests on neglecting corrections due to spacetime curvature. If those errors are small, and you're willing to ignore them, you can ignore them.] 1') The geometry of spacetime in the immediate vicinity of the earth is (to a very good approximation) static. Maybe I'm missing something, but I don't see the relevance of this observation. 2') The baseball is following a geodesic, but you aren't: you have a proper acceleration of 9.8 m s^2 due to the upward force exerted by the tower. Ditto. I'll change my example to eliminate this feature, if you like. Put yourself on a satellite in an elliptical orbit. Use a radar gun to measure the speed of another satellite in an elliptical orbit. Again, to a kickass approximation, the number that you get out can be interpreted as a Doppler shift (as long as conditions are such that epsilon is small, which is an easy condition to satisfy in these circumstances). Again, the reason that's true is that you can approximate spacetime as flat + perturbations of order epsilon to a kickass approximation. Again, the above statements apply equally well to a small neighborhood of an FRW spacetime. The bulk of the baseball's redshift as measured by you is uncontroversially due to the nonzero relative velocity of the ball wrt you. The reason this is "uncontroversial" is precisely because spacetime is flat to an excellent approximation. If that weren't true, then you couldn't even unambiguously defined "relative velocity." For instance, suppose you moved the experiment to a region just barely outside the Schwarzschild radius of a black hole, such that the quantity epsilon was not small. Then the question of whether an observed spectral shift was Doppler or gravitational would not be "uncontroversial": different people might analyze the situation using different coordinates (one using Schwarzschild coordinates, one using Kruskal coordinates, one using EddingtonFinkelstein coordinates). They'd all agree on the observations, of course, but they'd disagree on the interpretation. The reason this doesn't happen for experiments near the Earth's surface is that the quantity epsilon is small, which means that there's an obvious coordinate system in which to analyze things (the coordinate system that makes spacetime look flat). In short, gravity is completely irrelevant in example B. The whole thing could be done without significant alteration in SR. Note the irrelevance of tidal effects, which are the identifying mark of GR. I don't really understand why that's relevant, but if it's bothering you, then let's switch from the dropped baseball example to my new satellite example above. Let the two satellites by in very different orbits (so that their separation is of order the orbital radii). It's still much smaller than the curvature scale, so epsilon is still small, but now tidal effects are important. By contrast, cosmological redshift is essentially gravitational; it cannot be reproduced in flat spacetime. The role of the relative motion of the galaxies is to produce framedragging, a GR effect, and it is the framedragging that is responsible for the 'expansion of space' and hence the cosmological redshift. This sort of statement is precisely the reason I want to harp on this point. I think it's a very misleading way to think about the lowredshift FRW Universe. One way to understand why is to think of the Omega  0 limit of the FRW spacetime (i.e., the Milne model). When Omega = 0, spacetime becomes exactly Minkowski. It would surely be absurd to deny that the observed spectral shift of test particles in that spacetime was a Doppler shift. Yet as soon as Omega becomes nonzero (even if it's arbitrarily small), all of a sudden those observed redshifts are 100% due to "frame dragging" by the (aribtrarily close to massless) stuff flying through the Universe? I can't imagine seriously believing that. The approximation in which the curvature is negligible is precisely the approximation in which the redshift is negligible. That last statement is just factually false. I promise. Do the calculation. Figure out the coordinate system that best approximates a neighborhood of FRW spacetime as flat. Those coordinates will not be comoving coordinates. They'll be coordinates in which the galaxies are flying away from the origin at speeds given by Hubble's law. Ted  [Email me at , as opposed to .] 
#5




Galaxies without dark matter halos?
In article ,
Tim S wrote: 1) The geometry of spacetime in the immediate vicinity of the earth is not even approximately FRW. That's certainly true, but it's not relevant to the point I was trying to make. My argument was based on the fact that the geometry of spacetime is approximately *flat* (i.e., Minkowski), not FRW. To be a bit more precise, I'm basing my argument on the utterly banal observation that everything is linear to first order. If you want to study phenomena on length scales much smaller than the curvature scale, then you can view spacetime as approximately Minkowski, with small perturbations of order epsilon = (length scale of observations) / (curvature scale of spacetime). When you observe a frequency shift near the Earth's surface, and you interpret it as a Doppler shift, that's precisely what you're doing: you're pretending that spacetime is flat and applying special relativity. You know that you're making errors because spacetime is not perfectly flat, but you also know that those errors are small (because epsilon is small under the circumstances). In short, As long as you're willing to ignore errors of order epsilon, you're allowed to pretend that spacetime is flat and interpret spectral shifts as Doppler shifts. I claim that interpreting the redshift of a nearby galaxy in an FRW Universe as a Doppler shift relies on exactly the same assumptions, and should be regarded as exactly as valid, as using a radar gun to measure the speed of a Pedro Martinez fastball. [OK, I confess I'm exaggerating a bit there. The quantity epsilon is much smaller in the latter case than in the former, so you're making less of an error. But the two cases are exactly the same in principle: the act of interpreting the observed spectral shift as a Doppler shift (as opposed to a gravitational shift) rests on neglecting corrections due to spacetime curvature. If those errors are small, and you're willing to ignore them, you can ignore them.] 1') The geometry of spacetime in the immediate vicinity of the earth is (to a very good approximation) static. Maybe I'm missing something, but I don't see the relevance of this observation. 2') The baseball is following a geodesic, but you aren't: you have a proper acceleration of 9.8 m s^2 due to the upward force exerted by the tower. Ditto. I'll change my example to eliminate this feature, if you like. Put yourself on a satellite in an elliptical orbit. Use a radar gun to measure the speed of another satellite in an elliptical orbit. Again, to a kickass approximation, the number that you get out can be interpreted as a Doppler shift (as long as conditions are such that epsilon is small, which is an easy condition to satisfy in these circumstances). Again, the reason that's true is that you can approximate spacetime as flat + perturbations of order epsilon to a kickass approximation. Again, the above statements apply equally well to a small neighborhood of an FRW spacetime. The bulk of the baseball's redshift as measured by you is uncontroversially due to the nonzero relative velocity of the ball wrt you. The reason this is "uncontroversial" is precisely because spacetime is flat to an excellent approximation. If that weren't true, then you couldn't even unambiguously defined "relative velocity." For instance, suppose you moved the experiment to a region just barely outside the Schwarzschild radius of a black hole, such that the quantity epsilon was not small. Then the question of whether an observed spectral shift was Doppler or gravitational would not be "uncontroversial": different people might analyze the situation using different coordinates (one using Schwarzschild coordinates, one using Kruskal coordinates, one using EddingtonFinkelstein coordinates). They'd all agree on the observations, of course, but they'd disagree on the interpretation. The reason this doesn't happen for experiments near the Earth's surface is that the quantity epsilon is small, which means that there's an obvious coordinate system in which to analyze things (the coordinate system that makes spacetime look flat). In short, gravity is completely irrelevant in example B. The whole thing could be done without significant alteration in SR. Note the irrelevance of tidal effects, which are the identifying mark of GR. I don't really understand why that's relevant, but if it's bothering you, then let's switch from the dropped baseball example to my new satellite example above. Let the two satellites by in very different orbits (so that their separation is of order the orbital radii). It's still much smaller than the curvature scale, so epsilon is still small, but now tidal effects are important. By contrast, cosmological redshift is essentially gravitational; it cannot be reproduced in flat spacetime. The role of the relative motion of the galaxies is to produce framedragging, a GR effect, and it is the framedragging that is responsible for the 'expansion of space' and hence the cosmological redshift. This sort of statement is precisely the reason I want to harp on this point. I think it's a very misleading way to think about the lowredshift FRW Universe. One way to understand why is to think of the Omega  0 limit of the FRW spacetime (i.e., the Milne model). When Omega = 0, spacetime becomes exactly Minkowski. It would surely be absurd to deny that the observed spectral shift of test particles in that spacetime was a Doppler shift. Yet as soon as Omega becomes nonzero (even if it's arbitrarily small), all of a sudden those observed redshifts are 100% due to "frame dragging" by the (aribtrarily close to massless) stuff flying through the Universe? I can't imagine seriously believing that. The approximation in which the curvature is negligible is precisely the approximation in which the redshift is negligible. That last statement is just factually false. I promise. Do the calculation. Figure out the coordinate system that best approximates a neighborhood of FRW spacetime as flat. Those coordinates will not be comoving coordinates. They'll be coordinates in which the galaxies are flying away from the origin at speeds given by Hubble's law. Ted  [Email me at , as opposed to .] 
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