Is General Relativity compatible with Mach's Principle? Einstein proposed what he called Mach's Principle. For the purposes of this question can we define Mach's Principle as being that e.g. if a bucket suspended by a rope contains water riding up on the sides, that phenomenon could be accounted for by (a) regarding the bucket and water as rotating, but could also be accounted for by (b) regarding the bucket and water being stationary (and the remainder of the masses in the universe rotating around it), and there is no detectable phenomenon which proves or could prove that (a) is correct and (b) incorrect, or vice versa. I.E. there is no such thing as "absolute" rotation - it is (like velocity) relative.
Einstein later proposed General Relativity. It is often said that GR is "not Machian" or "not very Machian".
My question is: Is GR incompatible with Mach's Principle (as defined above) - i.e. If GR is true does that make it logically impossible for Mach's Principle to also be true?
Or, to put it another way, when people say that GR is "not Machian" or "not very Machian" do they simply mean that GR does not require Mach's Principle to be true (but equally does not prove it to be untrue)?
 A: First off: as you mention: what is known as 'Mach's principle' was proposed by Einstein. (Also: Mach's austere philosophy of science is opposed to such a grand statement.) Einstein coined the name 'Mach's principle'. I want to emphasize that it was proposed by Einstein so I call it 'Einstein's Mach's principle.

Historians of physics describe the following:
Around 1915 Einstein was convinced that Einstein's Mach's principle is fundamental to GR.
Among the logical implications of the form of Einstein's Mach's principle that Einstein used was that in order for spacetime to exist there must be distribution of matter/energy in it.
A couple of years later the dutch astronomer Willem de Sitter found a solution to the Einstein Field Equations that describes a Universe with no matter in it.
This result was opposite to Einstein's expectation. Historians of physics describe that for quite a while Einstein tried hard to show that there was some mathematical error. Einstein worked to show that the de Sitter solution was actually invalid. But in the end Einstein had to admit that the de Sitter solution was in fact valid.
Historians of science describe that after that Einstein ceased to mention Einstein's Mach's Principle in articles about GR.
In 1954, in a letter written in reply to a specific question Einstein wrote: "Von dem Machschen Prinzip sollte man eigentlich überhaupt nicht mehr sprechen." (We shouldn't talk about Mach's principle anymore, really.)
(Written communication between Einstein and Felix Pirani)
Main source for this information:
Michel Jansen, 2008, Einstein's Quest for General Relativity, 1907-1920
Additional reading:
John Norton, Mach's principle before Einstein

My understanding is that there are multiple versions of "Mach's principle" in circulation. It seems to me that in order to begin to assess the question first the multiple versions of the principle would need to be categorized.


Einstein's version
I don't know which version the principle was in Einstein's thoughts, but it is interesting to see that Einstein's version was rendered untenable by the fact that de Sitter's solution is valid.
Decades later Johh Wheeler coined the phrase: "Curved spacetime is telling inertial mass how to move, inertial mass is telling spacetime how to curve." (Or words to that effect.) It appears to me that Einstein held to view where inertial mass is not only telling spacetime how to curve, but that in order for the Einstein Field to exist presence of inertial mass is necessary. (I use the expression 'Einstein Field' here as meaning: that which is described by the Einstein Field equations. The Einstein Field equations describe Einstein spacetime.)
It would appear that Einstein held a view (prior to being confronted by the de Sitter solution) that the existence of spacetime and inertial-mass-in-spacetime is fundamentally co-dependent. It would appear that this concept of co-dependency was sufficient to satisfy the version of Einstein's Mach's principle that Einstein used.
A: Mach's principle was one of the motivations for Einstein to develop General Relativity.  However when it was completed, it didn't incorporate Mach's principle and apparently later on Einstein doubted Mach's principle.
Your last sentence seems to summarise it well

GR does not require Mach's Principle to be true (but equally does not prove it to be untrue)

Section 1.3 of Mach's Priciple has the title 'Mach's principle and General Relativity' and is worth a read.
A: The question of Mach's principle already arises at the level of special relativity.
I recommend a very interesting book on the subject : "relativity Matters" J Rafelski, Springer.
I can't do better than quote this book (p 439):

This was the situation when about 150 years ago Ernst Mach returned to
the question raised by Newton. Mach proposed the ‘Universe at rest’
defined by the fixed stars, to be Newton’s absolute space. Adapting
these ideas to the post-relativity context of this book we would say
that we always measure acceleration against any know inertial frame,
and the fixed star frame is a suitable inertial reference frame.
Therefore choice of a preferred inertial frame such as that proposed
by Mach is not in conflict with special relativity. ….


The choice of an inertial frame made by Mach makes good sense for the merry-go-round
experiment in space, far from anything. By looking at starlight we can
determine who is rotating and who is not. Now the entire universe
dictates the answer to who is accelerated. However, this also prompts
us to reflect a question: “how does the water bucket know about the
universal inertial frame?” Especially in a book where we often check
if casual sequence of events is guaranteed, it seems doubtful that
this information is transferred from distant locations of fixed stars,
‘informing’ the water bucket to curve its surface. This information
about who or what is accelerated has to be available locally.  …..


One can see the situation as follow: the same quantum vacuum state
properties that assure quark confinement, and predominantly define the
inertial mass of matter, also define at a very microscopical scale the
inertial frame of reference.

