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nutgeb
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Apparently my recent thread on the cosmological redshift assumed more general insight into this subject than is the case. So the purpose of this thread is to help dispel some popular misconceptions about the cosmological redshift and hopefully spur a vigorous discussion. This thread is NOT about comparing the 'expanding space' and 'kinematic' paradigms, nor about comparing Newtonian gravity to GR.
The starting point of course is that the cosmological redshift is proportional to the change in the cosmic scale factor a(t) between the time of emission and the time of reception:
[tex] \frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) [/tex]
Keep in mind that in addition to increasing the wavelength of photons by this factor, the cosmological redshift also increases the proper distance separation between photons by the same factor.
Misconception #1
The first widely disseminated misconception is that 'expanding space' -- which really means an 'expanding hypersphere' of geometry -- acts like a stretching rubber sheet -- as a force -- causing photons and photon wave crests to physically separate. This idea assumes the 'expanding hypersphere' paradigm for the expansion of the universe, applying the FRW metric.
However, it is well established in peer-reviewed analyses in recent years that if Lambda=0, an 'expanding hypersphere' metric does not act as a force or rubber sheet. An expanding hypersphere metric is incapable of motivating a physical separation in proper distance between photons, wave crests, or any other objects. The expanding hypersphere will cause coasting particles to continue separating only if they began with a proper velocity away from each other in the initial conditions. The hypersphere's action is like a flywheel's momentum (think 1st derivative of distance) rather than like an accelerator (2nd derivative of distance). This analysis is most intuitively confirmed by the https://www.physicsforums.com/showthread.php?t=315550&highlight=misconceptions", about which much has been much written.
Consider the scenario where the 'observer' is at the coordinate origin, and non-relativistic particles are fired from the distant 'emitter' at constant time intervals dt. Over a short period of time, the Hubble rate Ht will remain essentially constant. In a simplified scenario without gravitational effects, at the instant after each particle is fired, its proper velocity will be zero relative to the previously fired particle, dv=0. With this as the starting condition, the 'expanding hypersphere' will never have any effect on the proper distance between the particles, dD = Htdt = constant.
Next make the scenario more realistic by adding gravitational effects. From the observer's perspective, there is a sphere of cosmic matter centered at the observer with its radius at the proper distance of any given 'incoming' particle. Birkhoff's Theorem allows us to disregard all matter outside of that sphere. Since the cosmic matter density is deemed to be homogeneous, a larger sphere will exert a relatively greater gravitational acceleration at its surface than will a smaller sphere. Considering a string of equally spaced particles moving radially toward the observer, at all times the lead particles will be subject to less gravitational acceleration than the tail particles. Therefore the velocity of the tail particles will increase relative to the lead particles over time, and the particle string will compress longitudinally. So, quite contrary to what the 'rubber sheet' metaphor would suggest, the proper distance separation between particles actually will decrease rather than increase, despite the effect of the 'expanding hypersphere' on the background geometry.
You may say but wait, the lead particle experienced a period of gravitational acceleration before the second particle was fired, so doesn't that give it a head start? As it turns out, the answer is no, because the Hubble recession rate does not remain constant. The same sphere of cosmic matter causes the Hubble velocity (H * D) near the emitter to decelerate at essentially the same rate as the lead photon accelerates. The net effect is that at the time it is fired, the second particle has the same proper velocity relative to the observer as the first particle does. (Actually, after firing, the second particle has a small net proper velocity toward the first particle, since the gravitational sphere acting on the first particle was slightly smaller than the sphere acting on the emitter's Hubble velocity. This just further compresses the photon string.)
You may also wonder why the sphere of cosmic matter centered on the emitter doesn't cause the traveling particles to experience gravitational acceleration away from the observer. In a sense this does occur, but at the same time, the emitter experiences even greater acceleration toward the observer than the particles do (because the matter sphere is larger). These effects offset each other, and the particles' acceleration toward the emitter has no net effect whatsoever on their acceleration or velocity toward the observer. This may seem confusing, but it's helpful to remember that, assuming perfect homogeneity, the cosmic gravity causes everything to accelerate toward everything else, and never causes anything to accelerate away from anything else.
The entire discussion above has been about non-relativistic particles. Exactly the same effect applies to relativistic photons, with the same consequences. In addition, the gravitational acceleration causes a gravitational blueshift in the photons' wavelength.
However, photons have one attribute that non-relativistic particles do not: photons always must travel through every local frame at exactly c. This unique attribute causes photons to experience coordinate velocity acceleration toward the observer over their worldlines, as they ascend the Hubble velocity gradient. This coordinate acceleration results in a loss of comoving peculiar momentum, and additional redshifting, as viewed in the observer's frame. You can read about that topic in my thread about the mechanics of the cosmological redshift.
Misconception #2
Another widely disseminated misconception is that the cosmological redshift can be calculated by integrating (multiplying together) the SR redshifts that occur over a very large number of infinitesimal local frames along the photon's worldline. Presumably, although this usually isn't rendered explicit, each local SR redshift is calculated using the change in recession velocity (relative to the observer) at each local frame-crossing.
Anyone with a cosmic calculator and a spreadsheet can convince themselves that this math does not generate an SR redshift anywhere close to the expansion of the scale factor. In my own relatively crude spreadsheet with 70 integration intervals from z=127, the calculated SR redshift is over 400,000 times larger than the correct answer, 128. This should not be surprising, because the recession velocity is well above > c for the majority of the worldline, and the geometric mean of the recession velocity is > c. As velocity approaches c, the SR redshift becomes infinite.
There is another reason to discard this approach. As mentioned in my other post, the time portion of the FRW metric is obviously linear, so no time dilation at all can occur between fundamental comovers regardless of distance. They all share a common cosmological proper time. Since SR time dilation is an inherent component of the SR redshift, it is impossible for the SR redshift formula to apply between two local frames if no time dilation is permitted between those frames.
. . . . . . . . . .
Hopefully these two misconceptions can be dropped from our cosmology dialog, but let me know if you agree or disagree!
The starting point of course is that the cosmological redshift is proportional to the change in the cosmic scale factor a(t) between the time of emission and the time of reception:
[tex] \frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) [/tex]
Keep in mind that in addition to increasing the wavelength of photons by this factor, the cosmological redshift also increases the proper distance separation between photons by the same factor.
Misconception #1
The first widely disseminated misconception is that 'expanding space' -- which really means an 'expanding hypersphere' of geometry -- acts like a stretching rubber sheet -- as a force -- causing photons and photon wave crests to physically separate. This idea assumes the 'expanding hypersphere' paradigm for the expansion of the universe, applying the FRW metric.
However, it is well established in peer-reviewed analyses in recent years that if Lambda=0, an 'expanding hypersphere' metric does not act as a force or rubber sheet. An expanding hypersphere metric is incapable of motivating a physical separation in proper distance between photons, wave crests, or any other objects. The expanding hypersphere will cause coasting particles to continue separating only if they began with a proper velocity away from each other in the initial conditions. The hypersphere's action is like a flywheel's momentum (think 1st derivative of distance) rather than like an accelerator (2nd derivative of distance). This analysis is most intuitively confirmed by the https://www.physicsforums.com/showthread.php?t=315550&highlight=misconceptions", about which much has been much written.
Consider the scenario where the 'observer' is at the coordinate origin, and non-relativistic particles are fired from the distant 'emitter' at constant time intervals dt. Over a short period of time, the Hubble rate Ht will remain essentially constant. In a simplified scenario without gravitational effects, at the instant after each particle is fired, its proper velocity will be zero relative to the previously fired particle, dv=0. With this as the starting condition, the 'expanding hypersphere' will never have any effect on the proper distance between the particles, dD = Htdt = constant.
Next make the scenario more realistic by adding gravitational effects. From the observer's perspective, there is a sphere of cosmic matter centered at the observer with its radius at the proper distance of any given 'incoming' particle. Birkhoff's Theorem allows us to disregard all matter outside of that sphere. Since the cosmic matter density is deemed to be homogeneous, a larger sphere will exert a relatively greater gravitational acceleration at its surface than will a smaller sphere. Considering a string of equally spaced particles moving radially toward the observer, at all times the lead particles will be subject to less gravitational acceleration than the tail particles. Therefore the velocity of the tail particles will increase relative to the lead particles over time, and the particle string will compress longitudinally. So, quite contrary to what the 'rubber sheet' metaphor would suggest, the proper distance separation between particles actually will decrease rather than increase, despite the effect of the 'expanding hypersphere' on the background geometry.
You may say but wait, the lead particle experienced a period of gravitational acceleration before the second particle was fired, so doesn't that give it a head start? As it turns out, the answer is no, because the Hubble recession rate does not remain constant. The same sphere of cosmic matter causes the Hubble velocity (H * D) near the emitter to decelerate at essentially the same rate as the lead photon accelerates. The net effect is that at the time it is fired, the second particle has the same proper velocity relative to the observer as the first particle does. (Actually, after firing, the second particle has a small net proper velocity toward the first particle, since the gravitational sphere acting on the first particle was slightly smaller than the sphere acting on the emitter's Hubble velocity. This just further compresses the photon string.)
You may also wonder why the sphere of cosmic matter centered on the emitter doesn't cause the traveling particles to experience gravitational acceleration away from the observer. In a sense this does occur, but at the same time, the emitter experiences even greater acceleration toward the observer than the particles do (because the matter sphere is larger). These effects offset each other, and the particles' acceleration toward the emitter has no net effect whatsoever on their acceleration or velocity toward the observer. This may seem confusing, but it's helpful to remember that, assuming perfect homogeneity, the cosmic gravity causes everything to accelerate toward everything else, and never causes anything to accelerate away from anything else.
The entire discussion above has been about non-relativistic particles. Exactly the same effect applies to relativistic photons, with the same consequences. In addition, the gravitational acceleration causes a gravitational blueshift in the photons' wavelength.
However, photons have one attribute that non-relativistic particles do not: photons always must travel through every local frame at exactly c. This unique attribute causes photons to experience coordinate velocity acceleration toward the observer over their worldlines, as they ascend the Hubble velocity gradient. This coordinate acceleration results in a loss of comoving peculiar momentum, and additional redshifting, as viewed in the observer's frame. You can read about that topic in my thread about the mechanics of the cosmological redshift.
Misconception #2
Another widely disseminated misconception is that the cosmological redshift can be calculated by integrating (multiplying together) the SR redshifts that occur over a very large number of infinitesimal local frames along the photon's worldline. Presumably, although this usually isn't rendered explicit, each local SR redshift is calculated using the change in recession velocity (relative to the observer) at each local frame-crossing.
Anyone with a cosmic calculator and a spreadsheet can convince themselves that this math does not generate an SR redshift anywhere close to the expansion of the scale factor. In my own relatively crude spreadsheet with 70 integration intervals from z=127, the calculated SR redshift is over 400,000 times larger than the correct answer, 128. This should not be surprising, because the recession velocity is well above > c for the majority of the worldline, and the geometric mean of the recession velocity is > c. As velocity approaches c, the SR redshift becomes infinite.
There is another reason to discard this approach. As mentioned in my other post, the time portion of the FRW metric is obviously linear, so no time dilation at all can occur between fundamental comovers regardless of distance. They all share a common cosmological proper time. Since SR time dilation is an inherent component of the SR redshift, it is impossible for the SR redshift formula to apply between two local frames if no time dilation is permitted between those frames.
. . . . . . . . . .
Hopefully these two misconceptions can be dropped from our cosmology dialog, but let me know if you agree or disagree!
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