Relativity 3

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The Inertial or Galileian Frame of Reference

Outline the nature of inertial frames of reference.

Physicists, mathematicians, engineers and indeed most common people, use the Cartesian coordinate system as they find their way around the surface of the earth. This system requires three perpendicular x-, y- and z-axes (often north- east- and up-) with a common length scale (metres) along each axis. These systems extend uniformly throughout our three dimensional space. When bodies move through space, their position coordinates change with time. A particular position or displacement forms a vector from the origin of the chosen axis system. It is always possible to choose another Cartesian system to describe the same space, these may have different origins and different orientation of the axes. While we move about these systems we also experience the flow of time as a scalar. However we occasionally recall that we can still see back into the past as we look at distant stars.

As he speculated, Einstein was aware that we see all objects in the past because we must wait until the light from these objects reaches us before we see them. We usually forget this fact when we view our immediate environment and thus assume that we see objects and events simultaneously to their occurrence. Again here is a reason why we need to be patient and consider what Einstein taught although it often seems counter-intuitive. When considering the role of an observer in relativity, we mean something more than seeing. An observer is an intelligent and thoughtful person who has access to all forms of technology eg. clocks, rulers, telescopes video recorders etc.. An observation is made when an observer receives information and is able to scrutinise and analyse all aspects of that observation. Despite our advances in technology, there are still many things that we cannot yet do, or observe, for this reason Einstein pioneered the use of "thought experiments" so that logically possible, but technologically impossible, results could be anticipated and incorporated into our physical laws.

As moving bodies change their position co-ordinates in time we define the concepts of velocity, speed and acceleration. Although no acceleration can be constant, we find no need to give special names to the higher derivatives that describe changing acceleration etc.. In fact we can usually account quite well for our position and the position of objects around us by assuming that the accelerations (or at least their magnitudes) are constant. Some of the motion equations that should be familiar to you are:

As we use the relative displacement (s - s₀) and the relative velocity (v - v₀) we should also use relative time (t - t₀). Relative time is an important concept in simple but sequential models, however this is often obscured when we write t in place of t - t_0.

We also need an understanding of an inertial frame of reference, this was originally called a Galilean system of co-ordinates by Einstein. These are simply a system of co-ordinates in which the law, "A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion along a straight line." holds. A body in a state of rest or uniform motion along a straight line does not experience any unbalanced external forces. The converse, if a body feels no forces it must be at rest or in motion along a straight line, is not true. For instance you would feel no forces in free fall (we usually neglect air resistance) or, astronauts don't feel any forces while in orbit about the earth. In free fall you would observe your acceleration with reference to the earth to which you are falling. In the orbit you would observe that you are rotating about the earth and this rotation requires acceleration.

Einstein supposed that we could imagine we had an inertial reference frame that was far out in space removed from all significant gravitational fields. However for greater clarity he, and other teachers, drew on more practical examples such as trains, trams and more recently aeroplanes. These vehicles can move with uniform velocity (motion along a straight line), at the same time the local gravity can be felt but the vehicles are supported as they travel in straight lines. Within the vehicles the passengers feel just as if they were at rest on the earth's surface. This fact is a simple illustration of relativity, the familiar laws of science operate in a vehicle moving with constant velocity relative when close on the earth's surface. For instance you can safely pour a cup of coffee in an aeroplane without worrying about the sideways speed of say     250 m s^-1.

The bigger questions concerning what happens when the vehicles must turn on the earth's surface, the rotation of the earth or the orbital motion of the earth about the sun, etc. can be set aside on the assumption that for a short time the vehicle travels with a constant velocity. Somebody standing watching the vehicle pass is also at rest or moving with a relative velocity in the other direction to the vehicle. The vehicle is one inertial frame, the ground from which it observed is another, any other vehicle, also moving with a constant velocity, is yet another inertial frame of reference.

As Einstein pondered these matters, he became increasingly aware that his first Theory of Relativity rested on an assumption that motion along a straight line was possible. But what is a straight line? He also knew that he simply couldn't escape the universal gravitational field, no matter where he imagined his thought experiments. These considerations led to the second more General Theory of Relativity. For our purposes we will confine ourselves to initially considering just the first Special Theory of Relativity. We must be satisfied that we can imagine inertial frames where the reference axes (or rigid vehicle) does not accelerate, even if gravitational forces can be felt.

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