Suppose that we have found an inertial frame of reference. Let us set up a Cartesian coordinate system in this frame. The motion of a point object can now be specified by giving its position vector, , with respect to the origin of the coordinate system, as a function of time, . Consider a second frame of reference moving with some constant velocity with respect to the first frame. Without loss of generality, we can suppose that the Cartesian axes in the second frame are parallel to the corresponding axes in the first frame, that , and, finally, that the origins of the two frames instantaneously coincide at . (See Figure 2.1.) Suppose that the position vector of our point object is in the second frame of reference. It is evident, from Figure 2.1, that at any given time, , the coordinates of the object in the two reference frames satisfy
By definition, the instantaneous velocity of the object in our first reference frame is given by , with an analogous expression for the velocity, , in the second frame. It follows, from differentiation of Equations (2.1)-(2.3) with respect to time, that the velocity components in the two frames satisfy
Finally, by definition, the instantaneous acceleration of the object in our first reference frame is given by , with an analogous expression for the acceleration, , in the second frame. It follows, from differentiation of Equations (2.4)-(2.6) with respect to time, that the acceleration components in the two frames satisfy
According to Equations (2.7) and (2.11), if an object moves in a straight line with a constant speed in our original inertial frame (i.e., if ) then it also moves in a (different) straight line with a (different) constant speed in the second frame of reference (i.e., ). Hence, we conclude that the second frame of reference is also an inertial frame.
A simple extension of the preceding argument allows us to conclude that there exists an infinite number of different inertial frames moving with constant velocities with respect to one another. Newton thought that one of these inertial frames was special and defined an absolute standard of rest: that is, a static object in this frame was in a state of absolute rest. However, Einstein showed that this is not the case. In fact, there is no absolute standard of rest, which implies that all motion is relative--hence, the name ``relativity'' for Einstein's theory. Consequently, one inertial frame is just as good as another as far as Newtonian mechanics is concerned.
But what happens if the second frame of reference accelerates with respect to the first? In this case, it is not hard to see that Equation (2.11) generalizes to
A simple extension of the preceding argument allows us to conclude that any frame of reference that accelerates with respect to a given inertial frame is not itself an inertial frame.
For most practical purposes, when studying the motions of objects close to the Earth's surface, a reference frame that is fixed with respect to this surface is approximately inertial. However, if the trajectory of a projectile within such a frame is measured to high precision then it will be found to deviate slightly from the predictions of Newtonian mechanics. (See Chapter 6.) This deviation is due to the fact that the Earth is rotating about a north-south axis, and its surface is therefore accelerating towards this axis. When studying the orbits of objects around the Earth, a reference frame whose origin is the center of the Earth (or, to be more exact, the center of mass of the Earth-Moon system), and whose coordinate axes are fixed with respect to distant stars, is approximately inertial. However, if such orbits are measured to extremely high precision then they will again be found to deviate very slightly from the predictions of Newtonian mechanics. In this case, the deviation is due to the Earth's orbital rotation about the Sun. When studying the orbits of the planets in the solar system, a reference frame whose origin is the center of the Sun (or, to be more exact, the center of mass of the solar system), and whose coordinate axes are fixed with respect to distant stars, is approximately inertial. In this case, any deviations of the orbits from the predictions of Newtonian mechanics due to the orbital rotation of the Sun about the galactic center are far too small to be measurable. It should be noted that it is impossible to identify an absolute inertial frame--the best approximation to such a frame would be one in which the cosmic microwave background appears to be (approximately) isotropic. However, for a given dynamical problem, it is always possible to identify an approximate inertial frame. Furthermore, any deviations of such a frame from a true inertial frame can be incorporated into the framework of Newtonian mechanics via the introduction of so-called fictitious forces. (See Chapter 6.)