I think that there may be an alternate approach than the one I described earlier for calculating the redshift which is easier than considering the null geodesic congruences. But it still involves considering the massive object to be stationary. As I've mentioned before, the reason for doing this is that we can then use the well-known Schwarzschild metric to describe the metric of space-time. This is part of the abstraction process I mentioned, whereby we regard the metric as a complete specification of the physics, which we then use to calculate what individual observers "see" by means of frame-fields.
The basic idea is that we can model a photon as a particle with some energy-momentum 4-vector.
Using this approach, though, requires one to have enough knowledge of tensors to deal with covariant and contravariant components.
I'll sketch it out below - as far as relativistic mass goes, I'll say that I don't use it in the calculations, and that I've never seen a GR textbook use it either.
Using this approach, we can derive the redshift for a moving observer as follows:
1) We sepcify the energy E of the photon in some "frame-field" of the source. A frame field is just a local (not global) frame with a locally Minkowskian metric, i.e. a Minkowskian metric just like that of special relativity.
We can then write the energy in this frame as E = \sqrt{|E^0 E_0|}, where E^0 and E_0 are the covariant and contravariant components of the energy in Schwarzschild coordinates.
MTW refers to E as E_{local}.
2) In the Schwarzschild geometry, we know that E_0 is a conserved quantity of motion, also called the "energy at infinity". We can further simplify the problem by assuming that all motion is radial, so that the angular momentum of the photon is zero. In this case, specifying the energy of the photon and that it is outgoing is enough to specify the complete energy-momentum 4-vector.
3) Since the Schwarzschild metric is diagonal
<br />
E_{xmit} = \sqrt{|E^0 E_0|} = \sqrt{|g^{00} E_0 E_0|} = \sqrt{|g^{00}|} E_0 = \frac{E_0}{\sqrt{1-\frac{2 r_s}{r_{xmit}}}}<br />
where r_{xmit} is the Schwarzschild coordinate of the transmitter, and r_s is the Schwarzschild radius of the large mass.
4) We can compute the energy-momentum 4-vector for a stationary (in Schwarzschild coordinates) receiver at some height r_{rcv} by
<br />
E_{rcv} = \sqrt{|g^{00}|} E_0<br />
because E_0 is a conserved quantity for the Schwarzschild geometry. This gives the well-known result
<br />
E_{rcv} = E_{xmit} \sqrt{\frac{1 - \frac{2 r_s}{r_{xmit}}}{1- \frac{2 r_s}{r_{rcv}}}}<br />
Because frequency and energy are proportional, to get the energy for the moving reciever, we can apply the relativistic doppler shift, because our reciever is in a locally Minkowskian metric.
Thus we take the above forumla, and multiply it by the relativistic doppler shift factor
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/reldop2.html
of \sqrt{\frac{1+ \beta}{1 - \beta}}
Here \beta is the velocity measured between a "stationary" observer at constant height r_{rcv} and our hypothetical moving reciever in the frame-field of the receiver (or equivalently the frame-field of the moving observer).
We can perform this Lorentz boost because near the reciever, we can ignore the curvature of space-time, just as we can ignore the curvature of the Earth when dealing with nearby objects. Thus we can use SR techniques such as a Lorentz boost to convert from the frame-field of the reciever moving relative to the large mass to the frame field stationary relative to the large mass, and vica-versa, and we can also use the standard SR doppler shift formula. To make this work, it is important that we measure the velocity locally as well (i.e. either the moving observer or the stationary observer measures the velocity of the other while they are located at the same point, or very close to one another).
