What is the question that Mach tried to address in his principle?
Mach's principle isn't as clear as people suggest, but IMHO what it tries to address is inertia. Resistance to change-in-motion.
I mean, we know how to detect the inertial and non-inertial frames (by Newton’s law).
I "root for relativity", but I have to say this: an inertial frame isn't some actual thing that has an objective existence. The universe exists, you exist, the Earth exists. But an inertial reference frame is little more than a steady state of motion, and a non-inertial frame is little more than a changing state of motion.
Once this is understood we also see that due to the acceleration of a non-inertial frame pseudo forces appear.
Yes, fire your boosters and you're pressed back into your seat. Because of your inertia, and because of your changing state of motion. But in truth your seat is pushing into your back.
Since there is no privileged inertial frame the acceleration of a non-inertial frame is quite unique i.e., one and the same with respect to every inertial frame. Right?
There is a privileged frame of sorts, which is the CMB rest frame, see this question. It isn't an absolute frame in the strict sense, but you can use it to gauge your motion with respect to the universe, and the universe is as absolute as it gets.
So what extra does the Mach’s principle try to answer? I’m a little confused.
Like CuriousOne said, you're right to be confused. Because Mach's principle is contradicted by E=mc². Inertia doesn't depend on distant rotating stars, it depends on local physics here and now. A photon has energy E=hf and momentum p=hf/c. These are two measures of resistance to change-in-motion for a wave travelling linearly through space at c. You divide by c to go from one to the other. Then remember the wave nature of matter: when you trap that wave in a mirror-box, it increases the inertia of the system. Because mass is a measure of energy-content, like Einstein said, and you divide by c again to say how much mass there is. But all it really is, is resistance to change-in-motion for a wave going round and round at c. Open the box, and it's a radiating body that loses mass. That radiation "conveys inertia between the emitting and absorbing bodies". Catch it in another mirror-box, and you increase the mass of that system. Having said all that, check out this article where Mark Hadley says large-scale rotation could be the cause of CP violation. It isn't quite Mach's principle as we normally understand it, but it relates to what's in the Wikipedia article, and IMHO is very interesting.
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.
Best Answer
Mach's principle has influenced Einstein but the final formulation of general relativity as of 1916 clearly invalidates Mach's conjecture. According to Mach's principle, motion - including accelerating and rotating one - may only be defined relatively to other objects. That would imply that there can't exist any gravitational waves.
However, general relativity predicts and experiments confirm that gravitational waves do exist: the relevant observations were awarded by the 1993 physics Nobel prize, too. The waves are vibrations of the space itself. It means that the metric tensor remembers the information about the geometry - and curvature at each point, even in the empty space, something that Mach's principle specifically wanted to prohibit.
Moreover, the perceptions and other effects of acceleration were supposed to be determined by comparisons with distant objects. This simple fact itself violates locality that has become important already in special relativity, and was simply inherited by general relativity.
If you care about history, the new cold relationship is mutual: much like general relativity rejected Mach's principle, Mach rejected general relativity - and already special relativity, in fact. ;-) If you care about sociology, there's been a poll among physicists active in relativity, and a vast majority of them would also say that Mach's principle is invalidated by general relativity.
Some people sometimes say that some effects predicted by general relativity, such as frame-dragging, are "Machian" in character. I think it is very misleading because it tries to make the listeners think that Mach's principle may be made compatible with the observations. It's very questionable what Mach's principle would predict about frame-dragging because Mach's principle has never become any viable candidate for a physical theory. But the idea that frame-dragging is Machian is more ideology and hype than a valid observation. Despite the vagueness about such very detailed effects, Mach's principle has said enough for us to be sure that it's incorrect in all of its forms.
Well, there's a lot of discussion on the Internet about long-dead ideas in physics - and maybe mostly about them. However, the Internet has nothing to do with the current state of physics.