If space is curved, then the nature of the cosmological redshift and time dilation is the same effect as the gravitational redshift and time dilation in a gravitational field. This is confirmed by the fact that the ratio between redshift and time dilation is the same for both gravitational and cosmological redshifts.

Let us assume that the universe is infinite and:

> the universe is uniformly filled with matter with density ρ

> let's introduce two points in space : we are the point A. We are observing photons from the point B.

Let us consider a curved universe (

http://en.wikipedia.org/wiki/Ricci_curvature). In these coordinates the volume form expands in a Taylor series around the vicinity of any point p:

Thus, if the Ricci curvature Ric(ξ,ξ) is positive in the direction of a vector ξ, the conical region in M swept out by a tightly focused family of short geodesic segments emanating from p with initial velocity inside a small cone around ξ will have smaller volume than the corresponding conical region in Euclidean space, just as the surface of a small spherical wedge has lesser area than a corresponding circular sector. Similarly, if the Ricci curvature is negative in the direction of a given vector ξ, such a conical region in the manifold will instead have larger volume than it would in Euclidean space., such a conical region in the manifold will instead have larger volume than it would in Euclidean space[1].

Let's assume that the Ricci curvature is negative in all the directions. If the Ricci curvature is negative, the density integral of the current volume per volume unit, from the point of view A, will also be greater in B, just as the gravitational potential .

In other words, the gravitational potential (ψ) for the photon travelling from B (ψ 1) to A (ψ o) from the point of view A will decrease.

As a result, we get a gravitational redshift (in the linear approximation) Z = (Δ ψ) / c2 and the difference of potentials for B and A is proportional to the difference in volumes (ψ 1/ ψ 0 = V1/V0), moreover we also observe time dilation Δt1 = Δt0 (1 + (Δψ) / c2), observed at the point A for the photon from point B (which fully corresponds to the existing relation between the redshift Z and time dilation (Δt1 = (1 + Z) Δt0) in the spectra of galaxies))

Evidently, in the case of curved space there are no equal relative observation points (in any inertial reference systems, at least in contrast to SR), there are only absolute one, for each point in space.

From our point of observation (A), for a photon arriving from B, the gravitational potential at point B is greater (and vice versa).

And on the other hand, from our point of observation (A), for the photon radiated from A, the gravitational potential at point B is less, and the Ricci curvature is positive in relation to it.

And here we have a paradox, although one not greater than the difference of volumes from different points of observation.

And it concerns only the particle motion between points A and B

When a photon is radiated, its state at the given time is immediately beyond the event horizon for the source. Therefore it is more correct to consider a negative Ricci curvature (the state of photon source, the point A, is determined by events at the point A and by the outside information, not by the state of radiated photon at the given moment in time).

The above is also (must be) true for the gravitational potential (as the sum of gravitational waves and fields, where the contribution of gravitational waves may be greater).

source :

http://soundgraffiti.net/f/fp_100en.doc