Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects

Abstract

We use the dynamics of a galaxy, set up initially at a constant proper distance from an observer, to derive and illustrate two counter-intuitive general relativistic results. Although the galaxy does gradually join the expansion of the universe (Hubble flow), it does not necessarily recede from us. In particular, in the currently favored cosmological model, which includes a cosmological constant, the galaxy recedes from the observer as it joins the Hubble flow, but in the previously favored cold dark matter model, the galaxy approaches, passes through the observer, and joins the Hubble flow on the opposite side of the sky. We show that this behavior is consistent with the general relativistic idea that space is expanding and is determined by the acceleration of the expansion of the universe -not a force or drag associated with the expansion itself. We also show that objects at a constant proper distance will have a nonzero redshift; receding galaxies can be blueshifted and approaching galaxies can be redshifted.

Tracing Hi Beyond the Local Universe

“…the motions of objects through space), V pec , as well as a proxy for redshift caused by the expansion of space, V cos . Beginning with the full relativistic expressions for these two velocities, for V pec , we have the special relativity expression (Einstein 1905;Davis, Lineweaver, & Webb 2003):…”

Comments on the tethered galaxy problem

“…This is not, in general, the same as v teth = −a ′ χ and it is this result which appears implicitly in the paper of Davis et al 1 . In Appendix C we recover the relationships v pec = f (v teth ) analytically for the cases of the Milne cosmology (Ω M = Ω Λ = 0) and the flat matter (Ω M = 1, Ω Λ = 0) and cosmological constant dominated (Ω M = 0, Ω Λ = 1) FRW models.…”

“…Following Davis et al 1 , we consider radial motion in an FRW universe. The metric (1) thus reduces to…”

“…In section II we review the 'tethered galaxy problem' as posed by Davis et al 1 . By removing the explicit redshift dependence from their calculations we recover their results as a combination of cosmic redshift and the special relativistic Doppler shift.…”

“…If we wish to study independently dynamical objects within such models we should like to quantify the effect of imposing such a cosmological background. We follow Davis et al 1 and consider the extreme case where a galaxy is endowed with sufficient peculiar velocity to escape the universal expansion, or Hubble flow, and remain at a fixed distance from a comoving observer. We study how light from such a galaxy would be redshifted and modify the previous calculations which suggest that a receding source could be significantly blueshifted.…”