Galilean Relativity

Whilst, in the world of Physics, the term Relativity is usually taken to refer to Einstein’s theories of Special and General Relativity, the concept can be more generally applied to the study of how the laws of physics vary or remain the same to different observers, particularly to observers travelling with different velocities. Galilean Relativity, formulated by Galileo Galilei, is the theory that was most widespread before Einstein, and formed part of the basis of Newtonian Mechanics.

Inertial Frames

Like Einstein, Galileo postulated that the laws of physics remain the same for observers in all inertial (non-accelerating) frames of reference. That is to say that for two bodies moving at different (but constant) velocities, it is impossible to make an absolute determination as to whether one is moving and the other stationary. All that can be determined is their relative velocity. The thought experiment that Galileo used was to consider a passenger in the hold of the ship on a calm see, who cannot look outside. There is no experiment that he can perform within the hold that will allow him to determine whether the ship is moving or not. He formalized the idea as follows:

Any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments.

Galilean transformations

For a inertial frame of reference, , moving past another frame at speed , with the and aligned, the transformations are:

In relativity, these transformations are modified to produce the Lorentz transformations.

Relative Velocity

Where Galilean Relativity primarily differs from Special Relativity is in the calculation of relative velocity. If as observed by an observer in an inertial frame of reference two bodies have velocities of and , then their relative velocity, , is

(Note that if the two bodies are travelling towards each other then one of the velocities will be negative, thus if and are taken simply magnitudes, then the equation more familiar in school is produced if the bodies are travelling in opposite directions. If they are travelling in the same direction, then it becomes .)

Under special relativity, the relative velocity is calculated using the Lorentz Transformation, producing:

where is the speed of light in a vacuum.

(This is expressed in a simplified non-vector form, assuming that the two velocities are in a single dimension. For vectors, the calculation needs to be performed as a dot product.)

Note that at low velocities (where is small) then is close to zero and so the equation gives the same result as the Galilean formulation. Since is such a large number ( being ), the Galilean transformation is sufficiently accurate for the everyday situations which humans encounter, and thus the transformation is sometimes regarded as intuitive. Even for the speeds involved in modern space exploration, Lorentzian adjustments are small. It is only with extremely lightweight bodies (i.e. subatomic particles) that high enough speeds can be achieved, and thus devices such as particle accelerators need to take account of the differences.

Thus it is debatable whether the Galilean transformation is actually ‘wrong’ since it is still of practical use in many situations – special relativity is a refinement under extreme conditions. Similarly, Newtonian Gravitation is perfectly adequate in many situations – general relativity is a broadly equivalent refinement. It is often considered that Newtonian mechanics is an approximation to relativity at low speeds and small gravitational fields.

Moral Relativism

Many liberals (and others) see an analogy between Einsteinian Relativity and Moral Relativism, arguing that if there is no absolute frame of reference for velocity, then there is no absolute standard for moral behaviour. However, despite the fact that Galilean Relativity maintains exactly the same tenet, the association with Galilean Relativity is not made. This can only be put down to an ignorance of the history of science.