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I'm trying to understand if cosmological redshift is just a secondary form of doppler redshift, or something else entirely.

Suppose the two galaxies in the picture are receding from each other, but there is a tether tied between two planets, many, many light years apart such that their velocities with respect to each other are exactly zero (or, if that strains the imagination too much, each planet has huge engines and effectively keeps the velocity between them zero). The planets no longer orbit stars, and only keep the tether exactly taut. This relationship has been in place for longer than the light travel time between the two planets.

Tethered Planets

The doppler redshift ought to be $z = \frac {v} {c} = \frac {0} {c} = 0$ because of the nonexistent velocity difference. Is this correct?

Would there be any cosmological redshift between the two planets, provided they were distant enough apart that $ \frac {a_{now}} {a_{then}} $ is sufficiently distant from 1?

If so, how could this redshift be explained? Where is the energy 'going' from the photons, lost to the expansion of space?

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  • $\begingroup$ Also, +1 for your artwork! $\endgroup$ – pela Jan 20 '15 at 8:13
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    $\begingroup$ A green line drawn on top of an image of two galaxies in Paint.NET? Thanks :) $\endgroup$ – Ehryk Jan 20 '15 at 8:17
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The observed redshift of a distant source is given by the sum of its cosmological velocity and its velocity wrt. the local Hubble flow, i.e. its so-called peculiar velocity.

That means that if you were able to accelerate the two planets at redshifts $z_1$ and $z_2$ up to velocities such that their relative velocity vanishes, they would measure each others redshifts to zero, whether you did so by means of a huge engine or a magical rope. Attaching the planets to each other means detaching them from the Hubble flow, i.e. giving them an actual velocity in comoving coordinates.

If they're sufficiently far apart, this perculiar velocity would exceed the speed of light. Since this is not allowed, this means that it's simply not possible to build an engine so powerful, and that no matter how powerful your magic rope is, it will break. The distance at which the expansion velocity of the Universe becomes larger than the speed of light is roughly 14 billion lightyears, so no objects farther apart than this distance can be attached to each other (in practice the limit is somewhat smaller).

You end your question with a second question that also applies to photons from "normal", non-attached objects: "Where does the energy from redshifted photons go?" That is a great question, that I'm sure must have been asked and answered before in this forum. Essentially, the key to the answer is "Forget about energy conservation in general relativity".

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  • $\begingroup$ As far as "forget about energy conservation in general relativity", for the example above let's say I pick the reference frame in which both planets are stationary. Within that reference frame, would I not have energy conservation and thus non-redshifted photons? $\endgroup$ – Ehryk Jan 20 '15 at 8:15
  • $\begingroup$ And or the main point, why distinguish between peculiar and non-peculiar velocities, then, and have some distinction between 'doppler redshift' and 'cosmological redshift'? Why not just consider the vector sum of the two velocities as the doppler effect redshift? $\endgroup$ – Ehryk Jan 20 '15 at 8:19
  • $\begingroup$ I wouldn't say there's energy conservation in the reference frame in which the two planets are stationary, since this is still an expanding and hence generally relativistic frame; the two planets just happens to have a non-zero velocity in that frame toward each other, hence emitting light that is initially blueshifted and then gradually redshifts as it moves through space in such a way that it exactly reaches a total non-redshift when measured at the other planet. $\endgroup$ – pela Jan 20 '15 at 8:34
  • $\begingroup$ Why is it initially blueshifted? $\endgroup$ – Ehryk Jan 20 '15 at 8:36
  • $\begingroup$ As for the second comment, "Why distinguish?": You don't need to distinguish in order to calculate their velocities wrt. each other, but there is a physical difference: The cosmological redshift is caused by the expansion of space, which has a cosmological origin, whereas the peculiar velocity must be caused by some astrophysical process, e.g. a star being slung (is slung a word?) out from a binary system. $\endgroup$ – pela Jan 20 '15 at 8:38
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This is just a footnote to Pela's answer as he has covered the main points.

One way of understanding the redshift is that the energy in the light ray becomes spread out over a large region of space so the energy density, i.e. the energy per unit distance, is reduced as the spacetime it's passing through expands. In fact if we take the scale factor to be one at the present time, the redshift is simply:

$$ \frac{\lambda}{\lambda_0} = \frac{1}{a(t_0)} $$

where $\lambda$ is the observed wavelength, $\lambda_0$ is the original wavelength and $a(t_0)$ is the scale factor at the time the light was emitted (so $1/a(t_0)$ is the factor the universe has expanded since then).

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  • $\begingroup$ So according to this, then, the planets would see redshifted light, as the energy density of the space between the planets has diminished from the expansion of the space between them. So - the planets would see somewhat redshifted light from the other? $\endgroup$ – Ehryk Jan 20 '15 at 8:21
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    $\begingroup$ @Ehryk: the distance the light is spread over doesn't change because the planets are stationary wrt each other. So there is no change in the energy density and therefore no change in the wavelength. In effect the redshift due to expansion is balanced by the blue shift due to the planets peculiar velocity towards each other. $\endgroup$ – John Rennie Jan 20 '15 at 8:24
  • $\begingroup$ Why would they have 'peculiar velocity' toward each other, if they are at rest to each other? By the definition of peculiar velocity in the wiki article, would that not mean a peculiar velocity of $0\ m/s$? $\endgroup$ – Ehryk Jan 20 '15 at 8:28
  • $\begingroup$ But... the universe HAS expanded (even if the planets have not receded), and thus $a(t_0)$ would be != 1, and thus redshift, would it not? Let's say they're REALLY far apart, and the universe has doubled in size since the light was emitted from one planet. Would that not mean $\frac {\lambda} {\lambda_0} = \frac {1} {2} $ ? $\endgroup$ – Ehryk Jan 20 '15 at 8:33
  • $\begingroup$ I guess the fundamental part I'm asking is whether or not the cosmological redshift happens as the light travels through expanding space, or just due to observer velocities, peculiar or proper. $\endgroup$ – Ehryk Jan 20 '15 at 8:35

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