Inertial Frames of Reference

Consider a free body (call it A) i.e. a body not subjected to any kind of force or external disturbance. This body will execute some kind of motion in space about which we don’t know. A frame of reference attached to such a body is referred to as an inertial frame of reference. We can also call it as a body placed in a free space. A free space is one where the potential remains constant throughout the space.

We now make an assumption : free space is homogeneous. What this means is that the properties of free space at a point x1 are identical to properties at any other point x2 in space. This is the one of the most crucial assumptions in physics and is valid in special relativity also. In fact, the whole special relativity is based on this assumption (along with non-existence of signals of infinite velocity). The results obtained from the laws based on this assumption are found to be in agreement with the experimental observations. 

Now suppose that there is another free body in space, B say, (not subjected to any external influence and assume that these two bodies do not influence each other in any way), at a position different from that of A. Since B is placed in a free space which is homogeneous i.e. both A and B bodies are subjected to identical conditions, the body B must also behave in a manner identical to A. Thus, both the bodies will be at rest with respect to each other. Hence if we attach a frame of reference to B, this frame should also be called as an inertial frame of reference.

So far we have considered the case where both the bodies (and hence the frame of reference attached with them) were at rest with respect to each other. Now let us consider the case when one of the bodies moves with a constant velocity with respect to the other. Let us assume that the body B is moving with a constant velocity v with respect to A. Thus, if we stand on A and look at B, it will appear to be moving with a constant velocity v in a certain direction. Now if we go over to the second body B and look at the A we observe that the first body if moving with the same speed but in opposite direction. It is important to realise here that we cannot say which one is actually moving and which one is at rest. We cannot distinguish these frames on the basis of their mutual constant relative motion. Thus, all we know about them is that they are moving with a constant velocity relative to each other and hence both these frames of references must be treated equivalently. Therefore, we call both these frames of reference as inertial frames.

Using a similar argument we reach a very important conclusion that all other frames which are moving with constant velocity with respect to either A or B must also be indistinguishable from these two and must be called as inertial frames of reference. There are infinite number of such frames as there are infinite number of constant relative velocities possible. Since it is not possible to distinguish between these inertial frames in theory, the laws of physics should be the same in all of them. This is known as the principle of equivalence in physics. This is a very general concept and not limited to the realms of classical mechanics but also applies to relativity.

Let us now consider the case where a body is subjected to some external influence. If a body is disturbed by any external source in any way then its effect will be observed in the form of changes in its motion. The relative velocity with which the body may be moving (with respect to an inertial frame of reference) will no longer be a constant but will change depending upon the disturbance. The rate at which this velocity changes is known as acceleration

The frame of reference attached to such a disturbed body is called as a non-inertial frame of reference.