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It is well established that the cosmological redshift causes both the wavelength and the separation of photons to increase in proportion with the increase in the cosmic scale factor a. The traditional explanation for the mechanism of the redshift is that ‘expanding space’ progressively forces apart both the wave crests of individual photons, as well as the separation between photons in a string, as a light beam crosses the expanding space between the emitter and the observer. See, e.g., textbooks by Misner, Peebles, Peacock, etc. This explanation has justifiably been criticized. Here is Alan Whiting’s take in his http://arxiv.org/abs/astro-ph/0404095v1" ‘The expansion of space: free particle motion and the cosmological redshift’, published in The Observatory:
“Misner, Thorne & Wheeler, p. 776 and Peebles (1993)12, p. 96-7, use the picture of a standing wave with expanding boundary conditions… It is not clear, for instance, why there should be a standing wave generated between comoving points in the universe, nor why it should maintain itself. More importantly, as pointed out by Cooperstock et al. (1988) (among others), electromagnetic radiation automatically tracking the universal expansion cannot be right. All our test equipment and comparisons are built of or use electromagnetic forces, and they should also expand with the universe; so any cosmological redshift would be undetectable in principle. At the very least, atoms in the Hubble flow would change their characteristic wavelengths with time (and perhaps with the state of the local gravitational field), leading to strange results indeed.”
The ‘tethered galaxy’ exercise definitively teaches that, if Lambda=0, ‘expanding space’ does not act like a force, and cannot cause an increase in the proper distance separation of wave crests, photons, or any other objects, if they didn’t already have a proper momentum away from each other in the initial conditions. It is clear that successive wave crests and photons do not have a positive radial proper velocity away from each other when they are emitted. If the observer is considered to be the coordinate origin, the slowing expansion rate must affect the emitter’s recession velocity at the time of each subsequent emission at least as much as it affects the approach rate of the previously emitted photons.
If ‘expanding space’ does not force wave crests and photons to separate, then what does?
Not the emitter’s recession velocity
The emitter’s velocity away from the observer does not contribute anything to the redshift. It may seem reasonable, as a number of authors claim, that the emitter’s recession velocity contributes an element of Doppler shift. But on closer examination, it turns out that the emitter’s Doppler shift is exactly offset by the gravitational acceleration of the observer toward the approaching photon over the course of the photon’s journey. This gravitational acceleration imparts a blueshift to the photon (in the observer’s frame) which exactly offsets the classical Doppler shift. In other words, on the photon's inbound trip, gravitational acceleration imparts the same proportional velocity increase to the photon as the gravitational deceleration it imparts to the cosmic Hubble rate.
There is no element of SR time dilation because the FRW metric does not include any time dilation between fundamental comovers, which allows them to share the same cosmological proper time. It is easy to see that the time part of the RW line element is linear:
[tex] ds^{2} = -dt^{2} + a^{2}(t) \left[ dx^{2} + \left\{sin^{2} \chi, \chi^{2}, sinh^{2} \chi \right\} \left( d\theta^{2} + sin^{2 \theta} d\phi ^{2} \right) \right] \\ \left\{ closed, flat, open \right\} [/tex]
The emitter’s progressive recession movement during the interval between the emission of each successive photon in a string causes the photons to initially be separated from each other by an additional distance factor (1 + v/c). This is a classical Doppler shift. If the photons were to arrive at the observer in this configuration, the observer would measure that the reception is time dilated accordingly. However, again this effect is canceled out en route. The Lambda=0 ‘tethered galaxy’ exercise demonstrates that if a string of two (or more) particles are launched from the same comoving coordinate at constant time intervals, cosmic gravitational tidal forces will cause the proper distance between the lead and tail particles to decrease in proportion to their proximity to the observer. If this effect is considered in isolation, each successive photon will arrive at the emitter after exactly the same time and distance interval at which they were originally omitted.
Traversing the Hubble velocity gradient causes the redshift
It is well established that the peculiar velocity of a non-relativistic particle decays in inverse proportion to the growing cosmic scale factor, or 1/a. A comoving observer interprets the decay in a non-relativistic particle’s peculiar velocity as a decrease in the particle’s momentum.
For a particle which is ‘outbound’ from the coordinate origin, this peculiar velocity decay reflects simply that the particle finds itself overtaking successive galaxies that have increasingly large Hubble velocities, since Hubble velocity is proportional to proper distance from the origin. For a particle which is ‘inbound’ toward the coordinate origin, the peculiar velocity decay is also 1/a, but the picture is less intuitive. An inbound particle observes the local Hubble velocities it passes through to be decreasing, all the way to exactly zero, as it approaches the origin. But since the particle’s peculiar velocity is negative (toward the origin), and the Hubble velocity is positive (away from the origin), the decreasing Hubble velocity in fact represents a decreasing peculiar velocity.
Of course the peculiar velocity of relativistic photons does not decay as the cosmic scale factor increases. Photons must retain a peculiar velocity of exactly c in every local frame they pass through. Tamara Davis comments on this behavior in her http://arxiv.org/abs/astro-ph/0402278v1" at the page numbered 51:
“It may seem strange that momentum decaying as 1/a means the peculiar velocities of massive objects decay until the objects are comoving, and yet the peculiar velocities of photons always stay at c. It seems that photons are getting some velocity boost that massive particles miss out on.”
The photons effectively are experiencing a local velocity boost: their velocity is boosted each time they pass from one infinitesimal local frame to the next which has a different Hubble velocity. But in this situation there is no external source of incremental energy (such as gravity) driving the velocity boost. Therefore, energy conservation requires the photon’s momentum, as experienced by a comoving observer, to decrease in the same proportion as its velocity increases. Thus in an expanding universe a photon’s comoving momentum decays at 1/a, the same momentum decay as a non-relativistic particle. The decay in the photon’s momentum at 1/a means that the photon’s wavelength is observed by a comover to be redshifted in exact proportion to the expansion of the universe since the photons were emitted.
Note that the photon’s comoving momentum does not decay from the perspective of the emitter’s frame.
It is logical to interpret the successive boosts of the photons’ velocity to also cause their increasing physical separation. The velocity boosts occur as a function of location (passing through the local frame of a galaxy with a particular local Hubble velocity) rather than as a function of time per se. Since the lead photon in the train arrives at each location before the tail photon, the lead photon experiences each velocity boost sooner than the tail photon does. Therefore, the lead photon’s average proper coordinate velocity is always relatively higher than the tail photon’s, despite (or rather, because of) the fact that the local peculiar velocity of each is always exactly c.
My analysis indicates that the cosmological redshift obeys the following simple equation:
[tex] \frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) = \int_{t_{e}}^{t_{o}} (1 + H_{t}dt) [/tex]
where the subscripts e and o respectively signify the time of emission and observation, H is in units of ly/y/ly, and t is in units of years. Note that in this equation dt is equal to dD, the change in the photon’s proper distance in light travel time. So this equation simply accumulates, in a linear multiplicative manner, the changes in the Hubble velocity (H * D) experienced by the photon over its journey. The formula for each frame-crossing is the same as for the classical Doppler shift. A rough calculation of this integration on my spreadsheet is within 10% of the correct result for z=1023, which suggests it is likely to be correct.
Note again that SR time dilation contributes nothing to this redshift, because the FRW metric has no place for time dilation between comovers. This is consistent with my rough spreadsheet calculations: when I include SR time dilation, the error increases to 40%.
In net, the cosmological redshift does not seem to occur as a discrete event at the observer due to the relative velocity between the emitter and receiver; instead it occurs progressively en route.
“Misner, Thorne & Wheeler, p. 776 and Peebles (1993)12, p. 96-7, use the picture of a standing wave with expanding boundary conditions… It is not clear, for instance, why there should be a standing wave generated between comoving points in the universe, nor why it should maintain itself. More importantly, as pointed out by Cooperstock et al. (1988) (among others), electromagnetic radiation automatically tracking the universal expansion cannot be right. All our test equipment and comparisons are built of or use electromagnetic forces, and they should also expand with the universe; so any cosmological redshift would be undetectable in principle. At the very least, atoms in the Hubble flow would change their characteristic wavelengths with time (and perhaps with the state of the local gravitational field), leading to strange results indeed.”
The ‘tethered galaxy’ exercise definitively teaches that, if Lambda=0, ‘expanding space’ does not act like a force, and cannot cause an increase in the proper distance separation of wave crests, photons, or any other objects, if they didn’t already have a proper momentum away from each other in the initial conditions. It is clear that successive wave crests and photons do not have a positive radial proper velocity away from each other when they are emitted. If the observer is considered to be the coordinate origin, the slowing expansion rate must affect the emitter’s recession velocity at the time of each subsequent emission at least as much as it affects the approach rate of the previously emitted photons.
If ‘expanding space’ does not force wave crests and photons to separate, then what does?
Not the emitter’s recession velocity
The emitter’s velocity away from the observer does not contribute anything to the redshift. It may seem reasonable, as a number of authors claim, that the emitter’s recession velocity contributes an element of Doppler shift. But on closer examination, it turns out that the emitter’s Doppler shift is exactly offset by the gravitational acceleration of the observer toward the approaching photon over the course of the photon’s journey. This gravitational acceleration imparts a blueshift to the photon (in the observer’s frame) which exactly offsets the classical Doppler shift. In other words, on the photon's inbound trip, gravitational acceleration imparts the same proportional velocity increase to the photon as the gravitational deceleration it imparts to the cosmic Hubble rate.
There is no element of SR time dilation because the FRW metric does not include any time dilation between fundamental comovers, which allows them to share the same cosmological proper time. It is easy to see that the time part of the RW line element is linear:
[tex] ds^{2} = -dt^{2} + a^{2}(t) \left[ dx^{2} + \left\{sin^{2} \chi, \chi^{2}, sinh^{2} \chi \right\} \left( d\theta^{2} + sin^{2 \theta} d\phi ^{2} \right) \right] \\ \left\{ closed, flat, open \right\} [/tex]
The emitter’s progressive recession movement during the interval between the emission of each successive photon in a string causes the photons to initially be separated from each other by an additional distance factor (1 + v/c). This is a classical Doppler shift. If the photons were to arrive at the observer in this configuration, the observer would measure that the reception is time dilated accordingly. However, again this effect is canceled out en route. The Lambda=0 ‘tethered galaxy’ exercise demonstrates that if a string of two (or more) particles are launched from the same comoving coordinate at constant time intervals, cosmic gravitational tidal forces will cause the proper distance between the lead and tail particles to decrease in proportion to their proximity to the observer. If this effect is considered in isolation, each successive photon will arrive at the emitter after exactly the same time and distance interval at which they were originally omitted.
Traversing the Hubble velocity gradient causes the redshift
It is well established that the peculiar velocity of a non-relativistic particle decays in inverse proportion to the growing cosmic scale factor, or 1/a. A comoving observer interprets the decay in a non-relativistic particle’s peculiar velocity as a decrease in the particle’s momentum.
For a particle which is ‘outbound’ from the coordinate origin, this peculiar velocity decay reflects simply that the particle finds itself overtaking successive galaxies that have increasingly large Hubble velocities, since Hubble velocity is proportional to proper distance from the origin. For a particle which is ‘inbound’ toward the coordinate origin, the peculiar velocity decay is also 1/a, but the picture is less intuitive. An inbound particle observes the local Hubble velocities it passes through to be decreasing, all the way to exactly zero, as it approaches the origin. But since the particle’s peculiar velocity is negative (toward the origin), and the Hubble velocity is positive (away from the origin), the decreasing Hubble velocity in fact represents a decreasing peculiar velocity.
Of course the peculiar velocity of relativistic photons does not decay as the cosmic scale factor increases. Photons must retain a peculiar velocity of exactly c in every local frame they pass through. Tamara Davis comments on this behavior in her http://arxiv.org/abs/astro-ph/0402278v1" at the page numbered 51:
“It may seem strange that momentum decaying as 1/a means the peculiar velocities of massive objects decay until the objects are comoving, and yet the peculiar velocities of photons always stay at c. It seems that photons are getting some velocity boost that massive particles miss out on.”
The photons effectively are experiencing a local velocity boost: their velocity is boosted each time they pass from one infinitesimal local frame to the next which has a different Hubble velocity. But in this situation there is no external source of incremental energy (such as gravity) driving the velocity boost. Therefore, energy conservation requires the photon’s momentum, as experienced by a comoving observer, to decrease in the same proportion as its velocity increases. Thus in an expanding universe a photon’s comoving momentum decays at 1/a, the same momentum decay as a non-relativistic particle. The decay in the photon’s momentum at 1/a means that the photon’s wavelength is observed by a comover to be redshifted in exact proportion to the expansion of the universe since the photons were emitted.
Note that the photon’s comoving momentum does not decay from the perspective of the emitter’s frame.
It is logical to interpret the successive boosts of the photons’ velocity to also cause their increasing physical separation. The velocity boosts occur as a function of location (passing through the local frame of a galaxy with a particular local Hubble velocity) rather than as a function of time per se. Since the lead photon in the train arrives at each location before the tail photon, the lead photon experiences each velocity boost sooner than the tail photon does. Therefore, the lead photon’s average proper coordinate velocity is always relatively higher than the tail photon’s, despite (or rather, because of) the fact that the local peculiar velocity of each is always exactly c.
My analysis indicates that the cosmological redshift obeys the following simple equation:
[tex] \frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) = \int_{t_{e}}^{t_{o}} (1 + H_{t}dt) [/tex]
where the subscripts e and o respectively signify the time of emission and observation, H is in units of ly/y/ly, and t is in units of years. Note that in this equation dt is equal to dD, the change in the photon’s proper distance in light travel time. So this equation simply accumulates, in a linear multiplicative manner, the changes in the Hubble velocity (H * D) experienced by the photon over its journey. The formula for each frame-crossing is the same as for the classical Doppler shift. A rough calculation of this integration on my spreadsheet is within 10% of the correct result for z=1023, which suggests it is likely to be correct.
Note again that SR time dilation contributes nothing to this redshift, because the FRW metric has no place for time dilation between comovers. This is consistent with my rough spreadsheet calculations: when I include SR time dilation, the error increases to 40%.
In net, the cosmological redshift does not seem to occur as a discrete event at the observer due to the relative velocity between the emitter and receiver; instead it occurs progressively en route.
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