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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?

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Welcome Andrey. Good Question. I hope it is attend to shortly. – Mew Dec 29 '12 at 15:13
For some more information on this method of synchronization, and its limitations (this coordinate setup is called "radar coordinates") see here. – twistor59 Dec 29 '12 at 17:34
Sean Carroll gives the appearence that the synchronization procedure is part of the construction. I think for an inertial frame it is sufficient to assert the rigidity of the rods and the non-acceleration. Then you want coordintes, for space and for time. The rods provide spatial coordinates, and you disseminate time to your clocks so that you're able to assign time coordinate consistently. I guess my point is: while you do need to disseminate time (synchronize the clocks), if the frame is non-accelerating it's an inertial frame anyway. – Cleonis Dec 29 '12 at 18:24
Take a look at my comment on accepted answer. – Evil Angel Jan 8 '13 at 5:28
up vote 4 down vote accepted

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.

Can the ability to synchronize clocks be used as a criteria for inertial frames?

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.

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This is wrong. Accelerometer would tell a freely falling reference frame that its accelerating when actually its not. A freely falling reference frame is an inertial reference frame in General Relativity. – Evil Angel Jan 8 '13 at 5:27
@Shekhar OK, let me state explicitly. With 'accelerometer' I mean the type of acceleration sensor that is in many of today's smartphones. When you throw such a smartphone up in the air then during its flight the smartphone's accelerometer will read zero. (Or very close to zero, if the accelerometer is sensitive enough to detect air drag.) So I mean detection of free fall motion. By contrast: a GPS based navigation device can evaluate successive positions and then compute acceleration with respect to some reference. Clearly I do not mean that type. Please correct your comment accordingly. – Cleonis Jan 9 '13 at 22:14
Which type of experiment you performed with by throwing your smartphone up in the air... Simply drop it and read the values. All accelerometers are same despite there can be sensitivity differences. – Evil Angel Jan 9 '13 at 22:34

The standard test to find out if you are in an inertial frame is to surround yourself with some non-interacting particles e.g. in a sphere. If the shape made up by the particles does not change with time then you are in a (locally at least) inertial frame. Curved space can be detected by either the volume and/or the shape made by the particles changing.

Bearing this in mind, in the example you give the rods are unaccelerated if they stay in the same position relative to each other i.e. the non-acceleration is with respect to each other and indeed everything else sharing their inertial frame.

Re your question about clocks, it isn't possible to synchronise clocks in different frames because they run at different rates. Well, I suppose it's possible to synchronise them once, but they won't stay in synch. In GR the clocks don't even need to be moving wrt each other. If I hover above a black hole and lower a clock on a rope to near the event horizon, that clock and my clock will run at different rates even though they aren't moving relative to each other.

In GR we don't really use the idea of inertial frames much. Instead we talk about spacetime being locally flat (Minkowski). In this sense it's possible to be in an inertial frame even when accelerating. The inside of the ISS is approximately an inertial frame even though it's accelerating towards the Earth at a fair clip.

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I didn't meant synchronizing clocks in different frames. Suppose I'm on spaceship accelerating at a constant acceleration relative to some inertial frame. A passenger could easily tell that the spaceships's reference frame is not inertial because Newton's second law doesn't hold, e.g for a pendulum like Vladimir proposed. But can we put a set of clocks at each point of the spaceship like Caroll described so that $t_2=\frac{1}{2}(t_1+t^{′}_{1})$ will always hold for light? – Andrey B Dec 29 '12 at 17:30
@ Andrey B - No, inconsistencies will build up, and they build up in more ways than one. As observed from a non-inertial frame the speed of light is not the same in all directions. Worse: in an accelerating spaceship the clocks in front are not in the same frame as the clocks in the rear. The effect is more pronounced with two spaceships moving co-linear, and both accelerating. Even when they both have the same proper acceleration their clocks cannot be synchronized consistently. Similar problem with a single spaceship for clocks in the front and the rear. – Cleonis Dec 29 '12 at 17:32
OK than, this answers my question. – Andrey B Dec 29 '12 at 17:58

First of all, it is not completely clear what does "the rods must be moving freely, unaccelerated" exactly mean. Unaccelerated compared to what?

Nice Question. I'd like to add: Moving freely with respect to what?
Well, its somewhat conventional. Newtonian mechanics defined an absolute inertial reference frame attached with heliocenter of Sun for that purpose. Einstein didn't defined such things, but borrowed one of conventionally defined inertial reference frames to define new ones unless new ones are available for defining purpose.
It means, the rods must be moving freely, unaccelerated with respect to another inertial reference frames. It may not look plausible, but an inertial reference frame can't be defined alone.

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, any 2 points in space and a beam of light traveling between them, that the equation $t_2=\frac{1}{2}(t^{'}_{1}+t_1)$ will always hold?

We are defining things here, not validating/verifying things. Clocks can be synchronized with the same rule in non-inertial reference frame as long as light reaches those points (in some of non-inertial cases, light can fail to reach another point due to change of local geometry of Spacetime).

Is the ability to synchronize clocks all over space is unique to inertial frames? Can the ability to synchronize clocks be used as a criteria for inertial frames?

No. Many of non-inertial cases can synchronize clocks with the same rule. As there is no absolute/universal clock, you can't really comment on those synchronizations (simply relativistic physics isn't that powerful for that comment).

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Acceleration is an absolute thing. Unaccelerated means that, say, a pendulum attached to these rods does not move at all at any time. Acceleration of rods would make it deviate depending on acceleration, like a passenger in a car. Clock synchronization can then be made in a "simple" way described by Sean.

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Inertial Frames of reference are fractals. You can imagine each Frame of reference as a box within a box within a box etc

You can zoom in or out. The observer in each inertial "box" see the behavior of matter according to the laws of "classical"Mechanics == That is clocks run normally, mass is constant as is length.

Example a car traveling at constant velocity on the surface of the earth (considered at rest to the car) are both in the same Inertial Frames of reference and conform to classical mechanical laws relative to each other.

It is only when you start to consider the observer in the car to the observer on the Earth what can be called a trans-inertial measurement that so called Relativistic effects are perceived.

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