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?
Best Answer
What Sean Carroll refers to is acceleration as indicated by an accelerometer that is right next to the rods, co-moving with the rods.
The readout of an accelerometer is a local measurement. That is important in this stipulation about the rigid rods. The demand is not about being unaccelerated with respect to some other object that may be at some distance, it's about strap-on accelerometers giving a readout of zero.
For the synchronisation procedure to work (to not run into inconsistencies), the speed of light must be the same in all directions. As we know, that is the case only for an observer in inertial motion.
Addressing your question from a more general perspective:
Thought experiments involving clocks being synchronized are pretty much always scenarios where the clocks are a great distance apart. Light is so fast, you want a good bit of distance. On the other hand, in the context of GR, when you take an inertial frame of reference as conceptual starting point, the scenario is that you are thinking really locally.
The frame that is co-moving with the International Space Station as it orbits the Earth is such a local inertial frame of reference. When you zoom out the next level is the inertial frame that is co-moving with the Earth's center of mass. And so you zoom out to ever larger perspectives.
In the GR concept of inertial frame of reference the inertial frames of each of those levels of perspective are in motion relative to each other. That is why as a starting point you start with a concept of a local inertial frame of reference.
Your question is not wrong, but I can't think of any useful thought experiment that features the combination that you ask about: the synchronization procedure, and the distinction between inertial and non-inertial frame.