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In many books, a truth of gravitation field has been stated in the following way: "No matter what mass does matter have, all objects will follow exactly the same trajectory under the action done by gravitation field if they start with the same initial conditions."
But we know, if we state a truth of physics, we must say at first what reference frame we are using. Then what reference frame are we using here to state this truth? Inertial or non-inertial?
if we state a truth of physics, we must say at first what reference frame we are using.
That's not true. It's like saying that if you state a truth of Euclidean geometry, you must state what coordinate system you're using.
Truths of Euclidean geometry are independent of coordinates. No matter what coordinate system you pick, the statement is true in those coordinates. Otherwise it isn't a truth about geometry, it's just a property of particular coordinate systems.
Euclidean geometry was studied for millennia before the concept of coordinates was even invented. General relativity wasn't. But general relativity is fundamentally about geometry, and any truth about general relativity must also be independent of coordinates. That objects follow spacetime geodesics in the absence of nongravitational forces is a coordinate-independent geometric fact.
by definition because of the presence of a force, this is defined as a non-inertial reference frame.
the definition of an inertial frame is when Newton 1st law is held so everything stays the same speed.
so you don't need to stat the frame it can be implied but assumptions do need to be made, like the initial condition statement and it is the same gravitational field. you can see this statement is true from showing that the acceleration of any object in a field is $a=\frac{GM}{R^2}$ with M being the mass of the field we are in G being the gravitational constant and $R^2$ being the distance from one center to another. so we see that the object acceleration is not controlled by anything intrinsic to the object. this is a little rambly hope it helps.
