I'm reading Sean Carroll's book on general relativity, and I have a question about the definition of an inertial reference frame. In the first chapter that's dedicated to special relativity, the author describes a way of constructing a reference frame in the following manner:
"The spatial coordinates (x, y, z) comprise a standard Cartesian system, constructed for example by welding together rigid rods that meet at right angels. The rods must be moving freely, unaccelerated. The time coordinate is defined by a set of clocks, which are not moving with respect to spatial coordinates. The clocks are synchronized in the following sense. Imagine that we send a beam of light from point 1 in space to point 2, in a straight line at a constant velocity c, and then immediately back to 1 (at velocity -c). Then the time on the coordinate clock when the light beam reaches point 2, which we label $t_2$, should be halfway between the time on the coordinate clock when the beam left point 1 ($t_1$) and the time on the same clock when it returned ($t^{'}_{1}$): $$t_2=\frac{1}{2}(t^{'}_{1}+t_1)$$ The coordinate system thus constructed is an inertial frame".
First of all, it is not completely clear what does "the rods must be moving freely, unaccelerated" exactly mean. Unaccelerated compared to what?
Secondly, and this is my main question, is the ability to synchronize clocks is unique to inertial frames? If the frame is not inertial, in the sense that Newton's second law $\vec{F}=\frac{d\vec{p}}{dt}$ does not hold, is it still possible that for a set of clocks which are not moving with respect to the spatial coordinates of this frame, that the equation $t_2=\frac{1}{2}(t^{'}_{1}+t_1)$ will always hold for any 2 points in space and a beam of light traveling between them? Can the ability to synchronize clocks be used as a criteria for inertial frames?
