Suppose you’re passing me as we both drive in the same direction along a freeway. My car’s speedometer reads 55 mph while your speedometer shows 60 mph. Is 60 mph your “true” speed? That is certainly your speed relative to someone standing beside the road, but your speed relative to me is only 5 mph. Your speed is 120 mph relative to a driver approaching from the other direction at 60 mph.

An object does not have a “true” speed or velocity. The very definition of velocity, $v=\text{Δ}x/\text{Δ}t$, assumes the existence of a coordinate system in which, during some time interval $\text{Δ}t$, the displacement $\text{Δ}x$ is measured. The best we can manage is to specify an object’s velocity relative to, or with respect to, the coordinate system in which it is measured.

Let’s define a reference frame to be a coordinate system in which experimenters equipped with meter sticks, stopwatches, and any other needed equipment make position and time measurements on moving objects. Three ideas are implicit in our definition of a reference frame:

• A reference frame extends infinitely far in all directions.

• The experimenters are at rest in the reference frame.

• The number of experimenters and the quality of their equipment are sufficient to measure positions and velocities to any level of accuracy needed.

The first two ideas are especially important. It is often convenient to say “the laboratory reference frame” or “the reference frame of the rocket.” These are shorthand expressions for “a reference frame, infinite in all directions, in which the laboratory (or the rocket) and a set of experimenters happen to be at rest.”

FIGURE 36.1 shows two reference frames called S and $\text{S}\prime$. The coordinate axes in S are x, y, z and those in $\text{S}\prime$ are $x\prime$, $y\prime$, $z\prime$. Reference frame $\text{S}\prime$ moves with velocity v relative to S or, equivalently, S moves with velocity $-v$ relative to $\text{S}\prime$. There’s no implication that either reference frame is “at rest.” Notice that the zero of time, when experimenters start their stopwatches, is the instant that the origins of S and $\text{S}\prime$ coincide.

FIGURE 36.1 The standard reference frames S and S$\prime$.

We will restrict our attention to inertial reference frames, implying that the relative velocity v is constant. You should recall from Chapter 5 that an inertial reference frame is a reference frame in which Newton’s first law, the law of inertia, is valid. In particular, an inertial reference frame is one in which an isolated particle, one on which there are no forces, either remains at rest or moves in a straight line at constant speed.

Any reference frame moving at constant velocity with respect to an inertial reference frame is itself an inertial reference frame. Conversely, a reference frame accelerating with respect to an inertial reference frame is not an inertial reference frame. Our restriction to reference frames moving with respect to each other at constant velocity—with no acceleration—is the “special” part of special relativity.