Hi Armin. Here is my response which was cut short. I recall copying the text into the window, but then editing it. This is an updated version. OK. It does not like angle brackets, which are used in HTML tags.
1. - I am ignorant of many of the details of general relativity, including the one you point out. I think my statement ought to be evident after the succeeding argument about redshifts (and hence potential energy) requiring relativistic treatment which leads to the Machian escape velocity, regardless of those details. After all, that escape velocity leads to a different metric.
2.1 Can you derive it from the Einstein Field Equations (EFE)? - No.
2.2 ... is this because they require a modification? - Yes.
2.3 ... do you mean that its derivation from the EFE is faulty? - I do not contest the Schwarzschild solution of the EFE.
2.4 ... or that you need modified EFE? - Yes.
Either the EFE or the underlying differential geometry for gravity (or both) will need modification. For this latter consideration see the document tangent.pdf at the revisingnewton website in the References, which modifies the equation of the tangent to have the properties of a logarithmic differential instead of a linear differential. It seems to me that these are clues which might chip away at the problem of how to procede.
So, I have a relativistic gravitational potential energy function, yielding a Machian escape velocity, which has an associated Machian metric. It is almost like a textbook problem, where you are given a premise (Machian potential energy), and the answer (Machian metric), and are expected to provide the theory (something like general relativity, which I bypassed using G-P coordinates to get the metric).
3. Questioning Hubble recession seems like it might be a radical step for many cosmologists. - I have been working on tired light for quite a while. On posing a question to JV Narlikar about the possibility of publishing a paper on Hh, he responded, "Infinite patience is required". At this point I am just hoping that someone with sufficient background might see the logic in my arguments. I finished reading Bohm's Quantum Theory just before New Year, having Started last April. I read it like a novel - over 600 pages - mostly just scanning the math. What struck me was that, over and over again, he would take a classical formula, treat it using Planck's hypothesis, and get a quantum result. Radical as the viewpoint might be to a cosmologist, Planck's hypothesis is fundamental to quantum theory.
There are challenges with experimental accuracy. Eddington measured the deflection of starlight by the Sun to first order (1 part in 10^6). Today, measuring to second order (1 part in 10^12) is difficult, but seems achievable. This level of accuracy is required to discriminate between GR and other possible contenders, given that GR already passes first order tests.
Cheers,
Colin