Metric Engineering Investigations 1.8
L&L p. 228 explicitly say that the guv curved metric form is only for a non-inertial frame not an inertial frame. Since in 82 they are still operating in global special relativity they do not distinguish global from local noninertial frames at this stage in their pedagogical exposition:
"Thus, in a noninertial system of reference the square of the interval appears as a quadratic form of general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k (82.1)
i,k = 0,1,2,3
... Thus, when we use a noninertial system, the four-dimensional coordinate system x^0,x^1,x^2,x^3 is curvilinear. The quantities gik, determining all the geometric properties in each curvilinear system of coordinates, represent, we say, the space-time metric."
Many mathematicians have careers in relativity without paying attention to the measurement theory behind the formalism. Each choice of non-inertial coordinates is a possible physical configuration of detectors requiring non-gravity forces to sustain. For example, in the vacuum black hole Schwarzschild solution where, outside the event horizon where gtt = 0:
gtt = - grr^-1 = (1 - 2GM/c^2r)
2GM/c^2r < 1
Where A = 4pir^2 is area of sphere concentric with origin behind the event horizon
These are HOVERING REST LNIF observers that must fire their rockets toward the event horizon to maintain constant r. That is, they must fire rockets in order to stay still relative to the curvature field.