Introduction To Special Relativity - Reference Frames and Galilean Relativity: A Classical Prelude

Reference Frames and Galilean Relativity: A Classical Prelude

A reference frame is simply a selection of what constitutes a stationary object. Once the velocity of a certain object is arbitrarily defined to be zero, the velocity of everything else in the universe can be measured relative to that object.

One oft-used example is the difference in measurements of objects on a train as made by an observer on the train compared to those made by one standing on a nearby platform as it passes.

Consider the seats on the train car in which the passenger observer is sitting.

The distances between these objects and the passenger observer do not change. Therefore, this observer measures all of the seats to be at rest, since he is stationary from his own perspective.

The observer standing on the platform would see exactly the same objects but interpret them very differently. The distances between herself and the seats on the train car are changing, and so she concludes that they are moving forward, as is the whole train. Thus for one observer the seats are at rest, while for the other the seats are moving, and both are correct, since they are using different definitions of "at rest" and "moving". Each observer has a distinct "frame of reference" in which velocities are measured, the rest frame of the platform and the rest frame of the train – or simply the platform frame and the train frame.

Why can't we select one of these frames to be the "correct" one? Or more generally, why is there not a frame we can select to be the basis for all measurements, an "absolutely stationary" frame?

Aristotle thought that all objects tended to cease moving and came to rest if there were no forces acting on them. He imagined the Earth lying at the centre of the universe (the geocentric model), unmoving as other objects moved about it. In this worldview, one could select the surface of the Earth as the absolute frame. However, as the geocentric model was challenged and finally fell in the 1500s, it was realised that the Earth was not stationary at all, but both rotating on its axes as well as orbiting the Sun. In this case the Earth is clearly not the absolute frame. But perhaps there is some other frame one could select, perhaps the Sun's?

Galileo challenged this idea and argued that the concept of an absolute frame, and thus absolute velocity, was unreal; all motion was relative. Galileo gave the common-sense "formula" for adding velocities: if

1. particle P is moving at velocity v with respect to reference frame A and
2. reference frame A is moving at velocity u with respect to reference frame B, then
3. the velocity of P with respect to B is given by v + u.

In modern terms, we expand the application of this concept from velocity to all physical measurements – according to what we now call the Galilean transformation, there is no absolute frame of reference. An observer on the train has no measurement that distinguishes whether the train is moving forward at a constant speed, or the platform is moving backwards at that same speed. The only meaningful statement is that the train and platform are moving relative to each other, and any observer can choose to define what constitutes a speed equal to zero. When considering trains moving by platforms it is generally convenient to select the frame of reference of the platform, but such a selection would not be convenient when considering planetary motion and is not intrinsically more valid.

One can use this formula to explore whether or not any possible measurement would remain the same in different reference frames. For instance, if the passenger on the train threw a ball forward, he would measure one velocity for the ball, and the observer on the platform another. After applying the formula above, though, both would agree that the velocity of the ball is the same once corrected for a different choice of what speed is considered zero. This means that motion is "invariant". Laws of classical mechanics, like Newton's second law of motion, all obey this principle because they have the same form after applying the transformation. As Newton's law involves the derivative of velocity, any constant velocity added in a Galilean transformation to a different reference frame contributes nothing (the derivative of a constant is zero).

This means that the Galilean transformation and the addition of velocities only apply to frames that are moving at a constant velocity. Since objects tend to retain their current velocity due to a property we call inertia, frames that refer to objects with constant speed are known as inertial reference frames. The Galilean transformation, then, does not apply to accelerations, only velocities, and classical mechanics is not invariant under acceleration. This mirrors the real world, where acceleration is easily distinguishable from smooth motion in any number of ways. For example, if an observer on a train saw a ball roll backward off a table, he would be able to infer that the train was accelerating forward, since the ball remains at rest unless acted upon by an external force. Therefore, the only explanation is that the train has moved underneath the ball, resulting in an apparent motion of the ball. Addition of a time-varying velocity, corresponding to an accelerated reference frame, changed the formula (see pseudo-force).

Both the Aristotelian and Galilean views of motion contain an important assumption. Motion is defined as the change of position over time, but both of these quantities, position and time, are not defined within the system. It is assumed, explicitly in the Greek worldview, that space and time lie outside physical existence and are absolute even if the objects within them are measured relative to each other. The Galilean transformations can only be applied because both observers are assumed to be able to measure the same time and space, regardless of their frames' relative motions. So in spite of there being no absolute motion, it is assumed there is some, perhaps unknowable, absolute space and time.