Gravitational effects are present even when the FRW metric is time independent, as is well known from the Schwarzschild and Kerr space–times. Static FRW metrics therefore do not simplify the redshift by eliminating one or more of the contributors. This is a crucial point because the cosmological redshift is therefore not just a kinematic effect (as in the Milne Universe) it is generally a combination of Doppler and gravitational effects (as one finds in de Sitter and Lanczos). It is important to stress as we proceed through this exercise that although the space–time curvature is constant in the cases we consider here, it is generally non-zero. We will consider each of these special cases in turn, including the Minkowski space–time, the Milne Universe, de Sitter space, the Lanczos Universe, and anti-de Sitter space. It is not difficult to show that there are exactly six FRW metrics with constant space–time curvature in each of these cases, a transformation of coordinates permits us to write these solutions in static form ( Florides 1980). The most complicated portion of this procedure is the search for an appropriate coordinate transformation that renders the FRW metric static. Steps two and three are rather standard in relativity (see, e.g. Finally, we obtain the apparent time dilation, which differs from its counterpart at the emitter’s location because the motion of the source alters the relative arrival times of the photon’s wave crests. Secondly, we use this transformed metric to calculate the time dilation at the emitter’s location relative to the proper time in a local free-falling frame. It goes without saying that equation (2) is not adequate for our purposes because the metric coefficients g μν generally depend on time t, through the expansion factor a( t). First, we find a set of coordinates permitting us to write the metric in stationary form. Our procedure for finding the cosmological redshift as a lapse function involves three essential steps. A different picture emerges when we derive z directly as a ‘lapse function’ due to Doppler and gravitational effects. We will therefore show for the static FRW metrics that the interpretation of z as a stretching of space is coordinate dependent. For these static FRW metrics, we will prove that the cosmological redshift can be calculated – with equal validity – either from the ‘usual’ expression ( equation 4) involving the expansion factor a( t), or from the well-known effects of kinematic and gravitational time dilation, using a transformed set of coordinates ( cT, η, θ, φ), for which the metric coefficients g μν are independent of time T. The complete treatment, including also those FRW metrics whose curvature changes with time, will be discussed elsewhere. In this paper, we will seek a partial answer to this question by considering a subset of FRW metrics – those that have a constant space–time curvature and can therefore be written in static form. But is cosmological redshift really due to ‘stretching,’ and therefore a different type of wavelength extension beyond those expected from Doppler and gravitational effects? Or is this different formulation – and therefore its interpretation – merely due to our choice of coordinates? In other words, is it possible to use another set of coordinates to cast the cosmological redshift into a form more like the ‘traditional’ lapse function used in other applications of general relativity? This is the principal question we wish to explore in this paper.īut finding a resolution to this important issue is quite difficult, as others have already discovered (see, e.g. It is this formulation, in particular, that seems to suggest that z is due to the aforementioned stretching of space, because it does not look like any of the other forms of redshift we have encountered before. 4 in terms of the expansion factor a( t), where t o and t e represent, respectively, the cosmic time at which the radiation is observed and that at which it was emitted.
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