Here is a new, well-written explanation by R. J. Cook and M. S. Burns as to why the accepted physical attributes of GR are entirely frame-dependent. This concept is fundamental to understanding GR. Jon
A flat spacetime with a cosmological constant does violate special relativity. Once has to make adjustments to make it fit again. Also obviously a non flat spacetime with a cosmological constant does not converge to Newtonian gravity in the limit.
Hi MeJennifer, Einstein was right when he characterized the cosmological constant as an ugly add-on to GR. And it lacks a convincing physical embodiment. Too bad the cosmological constant currently is the most robust and reliable mathematical explanation for the concordance model. Jon
OK, so now that I understand better how GR physical attributes are frame-dependent, I still need help sorting out the following "paradox": Let's hypothesize a tiny, compact, slowly expanding, homogeneous and isotropic toy universe, without Lambda, which is above its critical density and so, according to the Friedmann calculations of observers at rest in its co-moving frame, has overall positive spatial curvature and is topologically manifested as a tiny 3-sphere. One of these observers (the "traveller") boards a fast spaceship and travels in an arbitrary direction, and does not pass near any significant masses. In due course, the traveller finds that she has returned exactly to the point of departure, because she has circumnavigated the 3-sphere. She shakes hands with her fellow observers. Meanwhile, another observer has adopted a "rigid" set of coordinates, which are not co-moving with the expansion. According to this observer, the same universe is spatially flat and space is not expanding per se. Therefore, this rigid-frame observer cannot see the traveller's flight continuing in a constant direction and returning to the departure point. Questions: 1. Does the handshake ever occur in the rigid-frame observer's view? 2. Does the rigid-frame observer see the traveller proceeding in a constant direction in perpetuity? 3. Does the rigid-frame observer see the traveller proceeding on a curved path which returns to its origin? If so, which specific direction does the path curve in, and why that direction only? Thanks! Jon
Co-ordinates are not real. Observations are. It doesn't matter what co-ordinates you choose(as long as those that you use are among the infinite set of co-ordinates valid for that spacetime, but not among the infinite set that are not), as long as you do that maths correctly you will get the same predictions for things that can be observed as someone using different co-ordinates. It is meaningless to ask what two observers in the same place who are 'using different co-ordinates' see. Clearly they see exactly the same thing!
Too late to edit, sorry for the double post...here is a the correct link to the paper in the OP, the original link was broken.
Hi Wallace, OK good point. So please bear with me while I try to frame my question in a way that elicits a substantive response. Surely there is some way to phrase this question which would illustrate that the traveller travels a course which returns to the departure point, but which ought not physically do that in another observer's different frame, because of the different interpretation of spatial curvature in the two frames. In order for my question to make sense, is it necessary for the rigid-frame observer to be at rest in the rigid frame, meaning that he is moving with respect to the co-moving frame? Alternatively, how could the 2nd observer, using the rigid-frame coordinates, "do the math correctly" in order to calculate that the traveller moved in a constant direction and yet returned to the departure point. Jon
Frames, co-ordinates etc are not real, so you cannot sensibly ask what you are doing above regardless of how you phrase it. The maths must accord with reality, not the other way around. This is a better question. I don't have the complete answer on hand ( I think it would be somewhat involved) but my best guess would be that making a Minkowski like rigid co-ordinate system would probably only have a finite range of validity in this spacetime, so you possibly couldn't answer this question at all in those co-ordinates. If someone did want to work this out, I think the conformal transformations for a open universe presented in this paper might be a good starting point. You'd have to derive the conformal function for a closed universe first and I think this might be where it goes pear shaped, since that function may blow up (or go complex or something like that) after a finite range from the origin. I'm just guessing really, but the point is that you can't answer your question unless you actually have the specific relationship between the FRW co-ordinates and the 'rigid' co-ordinates in this particular case. You need to have this first in order to be sure that it is even possible to construct sensibly such a co-ordinate system The paper in the OP does this only for an empty universe, so you can't just extrapolate the results blindly to a different physical situation. It's clear in fact that SR is not valid if there is any matter in the Universe. The paper I linked to above shows how to make co-ordinates that make the metric looks similar to the Minkowski metric of SR but they are still different from SR, due to the conformal function (that goes to unity as matter density goes to zero).
Hi Wallace, Thanks for fixing my broken link, and thanks for citing the Chodorowski paper. It looks like Chodorowski's conformal coordinates could be useful. But I'm a little hesitent to ascribe the physical meaning he does to them, given that he seems fixated on the proposition that space is not expanding. Others critisize his interpretations, as exemplified by this 7/07 paper by Lewis, Francis, Barnes & James. Chodorowski also seems to disagree with Davis & Lineweaver on this score. Jon
The message of the Lewis et al paper is not so much to criticise Chodorowski in particular, but more generally point out that any attempt to answer the question of whether space really expands (either in the affirmative of the negative) is futile since it is a description of one co-ordinate system only. The Abromowicz et al paper that Chodorowski is responding to makes the error of assuming that an observer who sends out a light beam that bounces off a distant galaxy and then returns can say something about the relative lengths or times of the forward and return journeys. Of course they cannot do this without violating relativity. All they can do is measure the total time of the journey. Effectively what they do is mistake co-ordinates with reality. It is vital that any discussion of relativity first establishes what is being measured by who and how they are doing it. Interestingly, the Lewis et al paper shows that the conformal Minkowski co-ordinates still describe superluminal recession of distant galaxies as long as the Universe is not empty, showing really that the gravitational effect of matter cannot be ignored. In the end, the important thing to remember is that it is matter, not some strange embodied 'space', that causes these effects.
Hi Wallace, I assume you agree that the cosmological constant (or other DE), if it exists, has an independent effect on test particles, distinctly separable from any influence from gravitating objects or particles. This can be described as an indirect case of empty space affecting test particle motions, since the cosmological constant is then an inherent feature of empty space. Jon
Sorry yes, when I said 'matter' I really meant 'energy', so the cosmological constant (under a vaccum energy interpretation) is included in that statement. The cosmological constant as a geometric effect (i.e. just how gravity works) has the same ambiguity as anything else on the question. Is it a property of 'space' that pushes things apart, or is it just a description of how bodies interact, i.e. their joint gravitational effect is not entirely attractive? You could play the same co-ordinate games with a cosmological constant only universe as you can with a Milne universe to try and support either interpretation.
Hi Chronos, I assume that your typically cryptic comment is aimed at my cosmological constant example specifically. I'm willing to go along with Wallace's response to that example, although I think it's a bit of a fine line. The cosmological constant seems to be an inherent attribute which is permanently attached to, and inseparable from, each tiny "unit" of space, so treating the energy content of space as something separate from space itself could not unreasonably be viewed as arbitrary. It's like treating the gravitational energy of matter as something separate from the matter itself. Nevertheless, I'll accept Wallace's answer, because energy is energy, regardless of what it is associated with. So here's a different question along the same lines: The peculiar velocity of a massless test particle will inherently decay over time as it travels through an expanding universe with no cosmological constant. Any gravitational sources are too distant to have any significant influence. In a comoving frame, the test particle's movement subjects it to an ongoing succession of Lorentz transformations. Is it wrong to describe this as an example of the expansion of space itself changing the motion of a test particle? Jon
No (but see caveat) it isn't wrong as long as what is meant by the term 'expansion of space' is understood. As long as you (where by you I mean anyone reading or writing such an explanation) understand that the entire concept of expanding space is a convenient shorthand, not a physical theory, then in the language of that shorthand it is the expansion of space that washes out peculiar velocities. Of course we could always relate this to the effect that matter has, and remove any notion of expanding space, by analyzing the problem more deeply. Caveat: The thought experiment you have setup is problematic since you haven't specified the matter content of the Universe, you have said that any massive bodies are 'too distant to have any effect'. In this case, if we want to use FRW solutions we would have to assume we have an empty Universe. In this case the perculiar velocities decay in co moving co-ordinate, but in Minkowksi co-ordinates remain constant since there is no matter present to decelerate them.
Hi Wallace. Your answer is very interesting, and I want to understand it better. Let's assume that the mass of the universe is distributed homogeneously as dust. Lambda is zero, and the matter is at critical density. Can you please explain how a deeper analysis of the effect of matter enables us to remove any notion of expanding space? Why does the fact that the distance between the dust particles is increasing cause the peculiar velocity of the test particle to decelerate? My understanding is that peculiar velocity decays at the rate of 1/a. This tells me that gravitational deceleration of the expansion causes peculiar velocity to decay more slowly (as a function of time) than if the expansion rate were decelerating less, or not at all. Jon
Indeed it does, and on the other hand in a Lambda dominated Universe, peculiar velocities decay very quickly. The most straightforward way to see why is to look at the maths, the details are presented in this paper. In terms of a handwaving explanation, consider first the empty universe and the tethered galaxy experiment. We set up a distant galaxy such that it has a peculiar velocity towards us that exactly balances the recession velocity away. We then let it go and see what happens. In the empty universe described in SR co-ordinates the particle simply has no motion with respect to our chosen origin. As per the Milne model, we can postulate massless co-moving particles that have initial velocities proportional to their distance from the origin, this defines the FRW like co-moving co-ordinates. What we see is as time goes by, particles that have slower and slower recession speeds pass our test particle. Since peculiar velocity is defined as the velocity relative to local objects in the Hubble flow the peculiar velocity decays, even though the origin and the test particle have no relative motion at any point. Now consider what happens when we add matter. It doesn't matter if the universe is flat, closed or open but we will restrict the analysis to an expanding but decelerating epoch (not a collapsing phase of a closed universe). In this case what we see is that the presence of matter will cause all velocities between all particles to decelerate. Relative to the empty universe then, the co-ordinate defining co-moving particles will start to move more slowly past our test particle, i.e. a co-moving particle midway between the origin and the tethered galaxy initially will take longer to move past the tethered galaxy in this decelerating universe than it did in the coasting universe. All of this means that the particles velocity relative to the local hubble flow remains greater for longer when there is matter in the universe. In the case of Lambda, the reverse occurs, the co-ordinate defining particles get pushed out more quickly, hence the peculiar velocity decays rapidly. This is just hand waving though, it's much clearer to go through the maths yourself, and see what terms would change in what way by the addition of matter.
Hi Wallace, First, thanks for the link to the Barnes & Francis paper. I've read several of their papers and this one is excellent, nicely explanatory, as were the others. I don't think their point by point dissection of terminology is at all pedantic; it is only through careful exposition of all of the terminology that the fog is lifted. Their conclusion doesn't surprise me: "We contend that the problem is not that expanding space has mislead us, but that describing the decay of v(pec) as joining the Hubble flow is a misnomer." I agree that the notion of "rejoining the Hubble flow" is rather unhelpful and non-intuitive. It is better just to say that peculiar velocity decays in fixed non-moving (proper distance) coordinates in an expanding universe, period. Converting to comoving coordinates only complicates and confuses this particular analysis. Peculiar velocity may or may not decay asymtoptically close to zero (in fixed coordinates) as time approaches infinity. I think the converse terminology about gravitationally bound objects "breaking away from the Hubble flow" also is a misnomer for exactly the same reason. So you agree with me that the only role of matter is to decelerate the Hubble recessional flow in an expanding universe with Lambda = 0. Adding matter to the universe seemingly does not affect the decay of peculiar velocities at all in fixed coordinates. Therefore, regardless of the matter density, matter cannot be the cause of the decay of peculiar velocity. Clearly then, "the expansion of space itself" is the sole cause of the decay of peculiar velocity. Which was my original point. Jon
Peculiar velocities don't exist in Minkowski like co-ordinates, so you cannot sensibly talk about 'the decay of peculiar velocities at all in fixed coordinates'. I've explained why peculiar velocities, as defined in co-moving co-ordinates (which is the only co-ordinates that are defined in) decay in general, and why the addition of matter changes the rate of decay. In an empty universe peculiar velocities still decay, so it is clear that there is no 'expansion of space' causing this decay, it is merely a property of a co-ordinate system. The addition of matter (or any other form of energy) changes the rate of decay, showing that it is the gravitational effect of that energy that causes any difference in the rate of decay. At no point is any co-ordinate independent 'expansion of space' required to explain this.