First, an inertial object is one which is not subject to active forces. Literally, inertial means not active, from Latin, the language in which Newton wrote the Principia. Newton considered that an active force required contact. Ideally, we would
define inertial as meaning that there are no contact interactions with other matter (including interation with light, or photons). This is not strictly possible, but we can get arbitrarily close to it:
- An inertial object is one such that, in any reference frame, the effect on
its motion due to contact interactions with other matter is negligible.
We can now restate Newton's first law as a local law:
- N1*: An inertial body will locally remain at rest or in uniform motion
with respect to other local inertial matter.
Newton's first law was needed in Newtonian mechanics to determine "absolute space", but relativity replaces this with the idea of a local inertial reference frame, based on N1*:
- An inertial reference frame is one in which inertial bodies remain at rest or in uniform motion.
It is implicit in this definition that inertial reference frames are local. That is to
say, an inertial reference frame describes only a finite region of spacetime in
which deviation from rest or uniform motion is not measurable and can be
neglected. The size of this region depends upon the accuracy of measurement,
but it is worth noting that a second in time corresponds to a light-second in distance.
In terms of normal timescales, local refers to quite short intervals of time.
Einstein used a different definition "the laws of physics in their simplest form" but Newton's first law is more concrete, and it is sufficient to define inertial reference frames.
The Earth can be regarded as an inertial body, and, when you remove rotation, it defines an inertial reference frame, but this only works for a short amount of time. For a full description of spacetime you have to paste together inertial reference frames. The result uses the same mathematics, differential geometry, as for curved surfaces which can be seen as near flat in small enough regions. This is how we form "curved" spacetime but it must be understood that this is a mathematical definition of curvature - it does not mean that spacetime is actually curved in the same sense as a curved surface.