Register to reply 
Why expanding space ?by Peter Watkins
Tags: expanding space 
Share this thread: 
#73
Apr309, 08:06 AM

Sci Advisor
P: 1,909

Hi sylas,
I see that you're quite skilled in the art, so I'm looking forward to learning from you. in the future. 


#74
Apr309, 01:33 PM

P: 2,456

Just wanted to confirm: even in the particular case where space is flat, spacetime is not flat as it is expanding, right?



#75
Apr309, 02:16 PM

Sci Advisor
P: 1,909




#76
Apr809, 06:47 PM

P: 294

OK, I've read some more and thought some more about this.
I think we all agree that cosmological redshift includes no accumulation of SR time dilation, when considered in cosmological time coordinates. And I see no explanatory benefit in translating to global SR time coordinates in a hypothetical "empty" universe, as an alternative coordinate system, because isotropy and homogeneity require a distinctly hyperbolic (negative) spatial curvature in SR coordinates, which is inconsistent with actual observations. So I next want to explore Ich's assertion that cosmological redshift is nothing but an accumulation of classical Doppler shifts. Time dilation of the interval between two events (such as the beginning and end of an emitted light wave packet) is an inherent and commonly accepted outcome of applying the RW line equation. As Longair says, distant galaxies are observed at an earlier cosmic time when a(t) < 1 and so phenomena are observed to take longer in our frame of reference than they do in that of the source. I don't understand what physical action would cause an accumulation of incremental classical Doppler shifts to occur locally all along the light path, while also causing an accumulation of incremental elongations of the entire wave packet (photon stream) as it will eventually be observed in our observer frame of reference. The only purely kinematic cause I can see for such an elongation would be an ongoing acceleration of the wave packet (relative to our frame of reference). In that case, the leading edge of the wave packet would progressively "pull further ahead" of the trailing edge, because the leading edge experiences each successive temporal increment of acceleration before the trailing edge does. If such an ongoing acceleration is a real physical phenomenon, mustn't it be caused by the same cosmic gravitational spacetime curvature that causes gravitational blueshift (when the observer is considered to be at the center of the coordinate system)? I can't see any other kinematic explanation for ongoing incremental acceleration. However, an accumulation of gravitational blueshifts along the entire light path ought to reduce the total amount of cosmological redshift, as compared to a global classical Doppler shift calculation. But this is not what we observe. At high z's, the cosmological redshift is dramatically larger than the classical Doppler shift when calculated on a global basis. Thus gravitational blueshift seems to cut in the opposite direction it needs to. 


#77
Apr909, 08:57 AM

Sci Advisor
P: 1,909

For a measurement of redshift, both observers are connected by a unique path, the path that the light ray actually took. You can transport the wave vector along this path, and see that it got redshifted on arrival. Or, alternatively, you can compare the four velocities of the two observers by transporting the velocity vector of the emitter to the absorber. If you apply the SR doppler effect (including time dilatation) to this velocity, you get the same result. Both approaches always work. In the special case of a FRW spacetime, you can skip the procedures and get the result by simply comparing the scale factors at both events. The underlying symmetries make sure that it works. Cosmological coordinates reflect these symmetries, that's why they are so useful for this kind of calculation. But that does not mean that the other approaches, one of which including SR doppler and time dilatation, are no longer valid. You are still free to interpret the result as you like, and there is an exact mathematical framework for these different interpretations. But it has great explanatory power as a toy model. Not for predicting observations, but to make clear that cosmological coordinates are quite different from minkowski coordinates, even if one uses x=a*r as a spatial coordinate. No big deal, one should think, but I've seen that it's a common misconception among experts to neglect the difference and invent new physics to describe coordinate effects. I bet there are quite a few professionals who think that "cosmological proper distance" reduces to "(SR) proper distance" in an empty universe. We observe the wave packet in a succession of different reference frames. To get from one frame to the next includes a translation of the origin as well as a boost to the next velocity. That's effectively the acceleration you mention. I hope that clarifies your further points. 


#78
Apr909, 02:32 PM

P: 294

I think you are saying that SR time dilation can be part of the correct answer only if we transform from FRW coordinates to Milne or other nonFRW coordinates. I don't disagree with that limited conclusion, but I think in the particular context of the point I'm trying to make, it is unhelpful in nailing down the physical kinematic basis for cosmological redshift. First because as I said, an empty Milne SR universe depends upon distinctly hyperbolic spatial curvature which is inconsistent with actual observations. And second because no viable alternative global SR coordinate system exists (nor could it exist) which accurately accounts for the effects of cosmic gravitation on worldlines while preserving spatially flat global geometry, homogeneity and isotropy all at the same time. Therefore your statement  that inserting accumulated SR time dilation into the calculation does not change the cosmological redshift mathematical calculation one way or the other (presumably even if the accumulated SR time dilation is nonzero in any single selected coordinate system)  cannot be proven in a realistic model. Vague statements such as that "the underlying symmetries of FRW mathematics" ensure equivalence do not add clarity. And I think it's fair to say that you are the only author I've seen state that accumulated classical Doppler shift can be the sole basis for cosmological redshift. Obviously if the emitting location were set as the origin of the FRW coordinate system, and the gravitational sphere were drawn with it as the center, the wave packet would experience gravitational redshifting instead. But this arrangement seems to reflect what would be observed in the reference frame of the emitter rather than the receiver, which presumably is why it is not generally used. Moving ahead with the story, I want to further explore the kinematic action underlying cosmological redshift. Consider a scenario where a gun located at the emitting Galaxy "Ge" sequentially fires two massless test projectiles toward observing Galaxy "Go". Both projectiles have the same nonrelativistic muzzle velocity, which is far greater than Ge's escape velocity. Projectile 1 (P1) is launched at cosmological time t, and Projectile 2 (P2) at t + [tex]\Delta[/tex] t. Time t happens to be at z=3 in Go's reference frame. The scale factor increases by 4 during projectiles' journey, so the RW line equation says that P2 arrives at Go at an interval of 4[tex]\Delta[/tex] t after P1's arrival, in Go's reference frame. (Or at least the RW line equation would say that if the projectiles' velocities were relativistic.) Did cosmic gravitational acceleration cause the 4x increase in the arrival interval compared to the launch interval? It doesn't seem so. During the interval between the launch of P1 and P2, it is true that the sphere of cosmic massenergy centered on Go applies an acceleration to P1, increasing P1's velocity by the time P2 is launched. However, during the same interval the same cosmic gravitation applies an acceleration to Go, causing Go's recession velocity to decrease in approximately the same proportion as P1's velocity has increased. So when P2 is launched, its initial velocity toward Go should be approximately the same as P1's contemporaneous velocity. So this difference in launch times does not cause a significant increase in the distance between P1 and P2 at P2's launch time. Once both projectiles are launched, they both are subject to ongoing cosmic gravitational acceleration toward Go. However, since at each discrete moment during flight P2 is always further away from Go than P1 is, P2's position at that moment defines a gravitational sphere of slightly larger radius than the sphere affecting P1. (Both spheres have the same density). So if there is any gravitational effect on the inflight spacing between P1 and P2, it should be to decrease the distance between them because P2 experiences greater gravitational acceleration than P1. I can't see any kinematic mechanism for gravitational blueshift to be the cause of the time dilation of the arrival interval which is inherent in FRW cosmological redshift. P1 and P2 are not locally accelerated relatively away from each other. Of course I analogize P1 and P2 to the leading and trailing edge respectively of a wave packet. 


#79
Apr909, 04:20 PM

Sci Advisor
P: 1,909

Let's go back to the symmetry argument I mentioned earlier: In the standard FRW metric ds²=dt²a²dr², r does not appear explicitly. That means that at cosmological time t1 you can choose an arbitrary origin r1, start there a particle (say, a bullet), and it will be at r1+Dr at time t2. Consequently a particle started at the same time at arbitrary r2 under the same conditions will be at r2+Dr. Their comoving distance r2r1 will not change over time, therefore their "proper distance" a*r will increase with the scale factor. The underlying symmetry is the one concerning transformations r > r+dr. If you talk about particles started at the same pale but different times, this symmetry does not apply, except for light, where the speed is constant. Nonrelativistic particles startes under such conditions will simply stay at a constant proper distance. Relativistic particles will increase their distance only as length contraction (wrt the respective observers) gets smaller and smaller, and will eventually maintain constant distance also. Generally, the main contribution to the increasing distance in the symmetric ~a case is the relative velocity of the two starting points. If there is no such velocity difference, as in your scenario, the distance will not increase proportional to a. 


#80
Apr909, 05:32 PM

P: 294

I don't think a proof limited to z~0 is sufficient; even the authors who provide it don't claim that alone it is a complete proof. Edit: What specific underlying "symmetry" would account for an exact correspondence between the change in length contraction and the change in the scale factor? That correspondence implies to me that the universe isn't expanding at all, that the true scale factor (after correction for SRlike length distortion) is fixed for all time. This in turn seems to pose a fundamental circularity: if the scale factor does not expand with time (except to the extent that deceleration of recession velocities over time causes global length decontraction), then there wasn't a Hubble flow in the first place, and galaxies possessed no recession velocity with respect to each other; in which case the original justification for the occurrence of SRlike length contraction disappears! 


#81
Apr1009, 04:22 AM

P: 622

If by the "change in length contraction" you mean the change in Lorentz contraction (due to acceleration) and by "change in scale factor" the change (with time) in the separation of two objects moving with the Hubble flow (due to gravitation), then I think that you are asking about a gauge symmetry (in the original Weyl sense of a change of length scale). Here this gauge symmetry arises from a global uniformity of scale. In the case of SR this symmetry is uniaxial (along the axis of relative motion), in the case of a homogeneous FRW universe it is isotropic. The equivalence of these two symmetries is, I think, rooted in the Equivalence Principle of GR. 


#82
Apr1009, 02:43 PM

P: 294

I see a reason why such a "symmetrical" cosmic Lorentz contraction seems to be completely ruled out. If the Lorentz contraction occurred, it would require that the duration of the aging of a supernova in the supernova rest frame at the time of emission would be at a factor of 1 (compared to the duration of aging finally observed in a distant observer's rest frame), rather than the factor of 1 / (1 + z) which has been widely confirmed by observations of low and high z supernovae and is currently accepted as standard. Consider a supernova at z=3: In the supernova's rest frame at time of emission let's say the time between the first 2 spectra is 17 days, which is within the normal expected range. In the distant observer's frame that duration would initially be Lorentz contracted by 4x to 4.25 days, and then over the course of the wave packet's journey it would eventually "decontract" back to the original 17 day duration which the observer would finally measure. But in this example, actual observations have led us to expect a 4x dilation from the original dilation in the supernova frame, resulting in a 68 day duration measured by the observer. I think this exercise demonstrates that there is no place for ANY nonzero Lorentz contraction in lightpaths in the gravitational FRW model. So that idea for explaining a kinematic cause for FRW elapsed time dilation seems to be a dead end. 


#83
Apr1009, 04:06 PM

Sci Advisor
P: 1,909

Draw a spacetime diagram of the gedankenexperiment (empty model) in minkowski coordinates, and you have two paralle worldlines of the bullets. Their distance is measured by comoving observers at any point in the trajectory. You'll see that (for tardyons) it's the same as a ruler measured by observers with different relative velocities to it, and that therefore its length is maximal in the frame (for the observer) where it comes to rest. It does not expand indefinitely. In the empty model, the "change in length contraction" is not enough to give the result. It is important that there is an difference in velocity at the start, and that's exactly what F&R fail to account for. 


#84
Apr1009, 05:14 PM

P: 294

By the way, nonzero SR time dilation would be inconsistent with the Milne model too, except that the homogeneous, isotropic Milne model admits that it applies physically unrealistic hyperbolic global spatial curvature distortion for the express purpose of exactly negating the mathematical/geometric effect of nonzero SR time dilation between fundamental comoving observers. Of course I'm aware that unrealistic hyperbolic global spatial curvature is a standard theoretical analysis tool of GR and cosmology, which unfortunately can introduce confusion between what is physically real and what is mathematically possible. 


#85
Apr1109, 01:33 AM

P: 622

But remember that the eqivalence of acceleration and gravity is something raised to the status of a principle (the EP) because we don't understand why there is this equivalence; we like to conceal our ignorance in pompous ways. I'm suggesting that equivalence is due to an underlying gauge symmetry, namely the global uniformity of scale that seems to prevail in the universe we find ourselves in. But sadly I've not the least idea how or why this came about  so this is just regressing further into the unknown! 


#86
Apr1109, 02:26 PM

Sci Advisor
P: 8,515




#87
Apr1109, 02:28 PM

Sci Advisor
P: 1,909

If we can't get over this point, I fear that we'd better agree to disagree. I'd be happy to discuss this point with Wallace, if he likes to jump in. 


#88
Apr1209, 07:04 AM

P: 622




#89
Apr1209, 08:53 PM

P: 294

In any event, I'm talking about internal "rules" consistency within an individual coordinate system, as distinguished from the translation of coordinates between different systems. The homogeneous, isotropic FRW model by definition prohibits unsynchronized clocks as between fundamental comoving observers, so you must corrupt the metric if you try to insert it. Similarly, the homogeneous, isotropic Milne model (with hyperbolic spatial curvature) also prohibits unsynchronized clocks as between fundamental comoving observers. More trivially, even the spatially flat Minkowski metric does not support a homogeneous, isotropic matter distribution if it is expanding: instead, the matter field must be entirely at rest w/r/t itself, meaning zero recession velocity as between particles, which in turn means that zero SR time dilation is required as between fundamental "costatic" (opposite of "comoving") particles. (Hmm, I wonder if this pattern can be generalized, and homogeneity+isotropy is impossible in ALL coordinate systems that permit nonzero time dilation as between fundamental observers?) On the other hand, I think it's possible that SR time dilation and gravitational time dilation together could fit into the calculation of cosmological redshift. Since cosmic gravitational shift normally is interpreted by the observer as blueshift, it cuts in the opposite direction as SR time dilation. Yet for the same reason as for SR time dilation, the rules of the FRW metric rule out the possibility that nonzero gravitational time dilation could result (alone) as between fundamental comoving observers. So one is led to the thought that perhaps SR and gravitational time dilation exactly offset and negate each other mathematically in the FRW model. Each contributes an equal and opposite element of time dilation, such that when the two elements are combined, the net effect is zero. I'm skeptical that the math would work out so neatly, but I don't recall having seen any mathematical attempt to test this straightforward question. 


#90
Apr1609, 09:30 AM

Sci Advisor
P: 1,909

I'll give an example of what the paper claims: Consider an arbytrary function y=f(x). The claim is that, at each point, the arc length of the funtion between two nearby points can be approximated by rotating to a system where the two points lie parallel to x', and measure the difference dx'. To get the exact arc legth between two points at some distance, you repeat the procedure by applying it to infinitely many, infinitely small patches of the function. The parallels are: Via a (not really specified) coordinate transformation, you get a simple formula valid in the vicinity of an arbitrary point The formula is valid to first order only, second order contributions (such as curvature or relativistic corrections to the doppler effect) are neglected it gives nevertheless definitely the correct answer it is completely useless for all practical purposes, such as actually doing the calculation. The interesting point is the transformation. The authors specify it exactly, like I did here, by what it has to do. But they don't give its global mathematical form. In this example, you can get the difference dx' by applying dx'²=dx²+dy² in the global coordinate system. Nothing has changed in principle, the procedure is correct whether you define the transformation globally or not. But now it's useful, too. This last step should be done in the paper Old Smuggler referenced to. But it's not ok to pick one definition to establish synchronizity, and claim that procedures that give a different result are wrong. They aren't, they're simply different. "Homogeneous" means that after a certain proper time since the big bang, each comoving observer measures the same matter density in his/her vicinity. None is privileged. "Isotropic" means that thy universe looks the same to them in each direction. No direction is privileged. Both principles are, of course, also true in the minkowski coordinate representation, because they are defined independent of coordinates. It's just that FRW coordinates fully reflect that symmetry, while minkowski coordinates don't. But they have the advantage that space and time coordinates are defined the usual way, with velocities being velocities and such. So, by exploiting the symmetry, there is a simple redshift formula in FRW coordinates, namely anow/athen. But there is also a simple formula in minkowski coordinates, namely the SR doppler formula. 


Register to reply 
Related Discussions  
The universe is expanding & the universe is infinite  Astronomy & Astrophysics  21  
Difference between Identical , Equal , Equivalent  Calculus & Beyond Homework  9  
Green's functions for vector potential in curved SPACE  General Physics  4  
Expanding a function in GaussianHermites  Calculus & Beyond Homework  8  
Your weight on a Space Elevator ?  Astronomy & Astrophysics  8 