[varsigma (OP)]

I feel I should point out Mach's principle is quite loosely defined.
There have been attempts to make it more definitive, but the basic idea is that the rest of the universe is the real background against which everything locally rotates.

Today this has to explain gravitational waves and their finite speed; whether or not Einstein's SR restricts all massless waves with energy to the limit, c.
So I have to wonder what my friend is trying to do really. Maybe I should try to track him down.

[Colin2B]

I feel I should point out Mach's principle is quite loosely defined.
There have been attempts to make it more definitive, but the basic idea is that the rest of the universe is the real background against which everything locally rotates.

Today this has to explain gravitational waves and their finite speed; whether or not Einstein's SR restricts all massless waves with energy to the limit, c.
So I have to wonder what my friend is trying to do really. Maybe I should try to track him down.

Welcome to the forum @varsigma
I usually send out a welcome to newcomers, but apologies I’ve been been up to my ears lately with not a lot of spare time.
I think your post has suffered from a lack of focus and a question in the title (as @Kryptid requested) would have helped.
If I understand correctly you are really asking about Mach’s principle, which as you say was loosely defined. Another interpretation is that there is no absolute rotational frame and rotation is meaningless unless measured against some reference. It often gets linked to the idea that the mass of the outer regions of space is dragging our local frame.
As far as I am aware there was an attempt to develop Mach’s ideas into a gravitational theory. Your friend should check out C. Brans and R. H. Dicke, “Mach’s Principle and a Relativistic Theory of Gravitation,” Physical Review 124 (1961) 925.
They made testable predictions which were tested by the propagation of radio signals between earth and the Cassini-Huygens space probe back in 2003 and were found to be false, hence the Mach principle they were based on is considered false.
It would be worth your friend following that up because if he could find a flaw in the tests, or in the interpretation of Brans, Dicke, it would reopen the discussion on an alternative to GR.

[alancalverd]

But the speed of light and the speed of gravitational waves are fixed by the special theory;

No. The Special Theory  describes and predicts other phenomena on the assumption that c is fixed and limiting.

All scientific laws and theories are attempts to describe and predict what happens, they don't fix, affect or control nature.

[varsigma (OP)]

No. The Special Theory  describes and predicts other phenomena on the assumption that c is fixed and limiting.

What other phenomena does it predict, in fact? If those phenomena are predicted to have energy (and what kind of phenomenon wouldn't?) what does SR say about a limit?

All scientific laws and theories are attempts to describe and predict what happens, they don't fix, affect or control nature.

The speed of light is fixed in the theory though. Do you mean it can't be assumed to always be fixed?

[alancalverd]

There's a world of difference between "fixed in the theory", i.e. an axiom on which the theory is based, and "fixed by the theory", i.e. an experimental result predicted by the theory or a man-made fact (like a building that doesn't fall over when the wind blows).

Significant phenomena predicted by SR are incorporated in nuclear power generation and GPS navigation. 

[alancalverd]

If Isaac Newton could do it, why not you?

I've watched a gorilla deduce and verify Newton's theory of gravitation, which made me wonder  why the Catholic Church defended Aristotle's nonsense to the extent of excommunicating or even killing unbelievers.  Conclusion: gorillas are more intelligent than churchgoers. So "why not you?" may be answered by "religious education".

[varsigma (OP)]

Significant phenomena predicted by SR are incorporated in nuclear power generation and GPS navigation. 

It also holds a significant place in electronics. It's a commonly held assumption that you can't get anything to propagate faster than c, even a nanometre, or a picometre. These days, it matters quite a lot how far light can travel in a nanosecond. Ask Intel how important it is to chip design and fabrication (yep, even making chips depends on this assumption we're discussing).

[varsigma (OP)]

I've watched a gorilla deduce and verify Newton's theory of gravitation, which made me wonder  why the Catholic Church defended Aristotle's nonsense to the extent of excommunicating or even killing unbelievers.

Ok. Who was this gorilla married to? You know about that "the real brains behind the performing monkey" thing?

[Colin2B]

It also holds a significant place in electronics. It's a commonly held assumption that you can't get anything to propagate faster than c, even a nanometre, or a picometre.

Relativity also plays an essential part in why we get a force between two current carrying  wires. Which of course is where Einstein started from.