@UmeshKonduru I would ask in the Problem Solving Strategies room. That room is for help with JEE questions and a lot of the students who chat there are about to go to an IIT or are already at an IIT.
"Although the search for diffeomorphism invariant observables lasts for almost a century, there still seems to be confusion in the subject and claims ranging from denying existence of any observables (except usually for the ADM charges which are defined only in asymptotically flat cases) to claiming that the problem has been solved entirely appear in the literature."
In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable
$$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + B^a\partial_\mu A^{a\,\mu}+\partial_\mu\bar\eta^a(D^\mu\eta)^a,$$
and in the superfield formalis...
If we have two intertial system K and K', and K' moves with a relativistic speed in relation to K, basically the case in which we need to use the lorenz trasformations, when we observer an event, we can have one of the 3 following cases: the event happens in K, or it can happen in K' or it can happen in another reference frame that is none of the above . Is that correct?
but what I am trying to say is that, you can observe a bouncing ball inside the rocket. for the guy inside of it, the movement observed is different then the guy in the planet
but if the bouncing ball event ,takes place in the planet , the guy that is there observes a different type of event then the guy in the ship
But look at this definition, or distinction between frame and coordinate system
We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer".
In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers
I associate a reference frame with an entity (lack of better words) that has physical boundaries while the coordinate system an abstract mathematical tool to asign a set of numbers to an observed event
One more thing, why do we use the metric tensor when we want to calculate the square of the absolute value of a 4 vector? Can't we straight up do a scalar product of it with itself ?