Relativity of Galilei

[size=18]Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his "Dialogue Concerning the Two Chief World Systems" using the example of a ship traveling at constant speed, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. Today one can make the same observations while travelling in an aeroplane with constant velocity. The fact that the earth on which we stand orbits around the sun at approximately 30 km/s offers a somewhat more dramatic example.

Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all inertial frames. In this context it is sometimes called Newtonian relativity.

A comparison can be made between Newtonian relativity and special relativity.

Some of the assumptions and properties of Newton's theory are:

The existence of infinitely many inertial frames. Each frame is of infinite size (covers the entire universe). Any two frames are in relative uniform motion. (The relativistic nature of mechanics derived above shows that the absolute space assumption is not necessary.)
The inertial frames move in all possible relative uniform motion.
There is a universal, or absolute, time.
Two inertial frames are related by a Galilean transformation.
In all inertial frames, Newton's laws, and gravity, hold.
In comparison, the corresponding statements from special relativity are:

Same as the Newtonian assumption.
Rather than allowing all relative uniform motion, the relative velocity between two inertial frames is bounded above by the speed of light.
Instead of universal time, each inertial frame has its own time.
The Galilean transformations are replaced by Lorentz transformations.
In all inertial frames, all laws of physics are the same (this leads to the invariance of the speed of light).
Notice both theories assume the existence of inertial frames. In practice, the size of the frames in which they remain valid differ greatly, depending on gravitational tidal forces.

In the appropriate context, a local Newtonian inertial frame, where Newton's theory remains a good model, extends to, roughly, 107 light years.

In special relativity, one considers Einstein's cabins, cabins that fall freely in a gravitational field. According to Einstein's thought experiment, a man in such a cabin experiences (to a good approximation) no gravity and therefore the cabin is an approximate inertial frame. However, one has to assume that the size of the cabin is sufficiently small so that the gravitational field is approximately parallel in its interior. This can greatly reduce the sizes of such approximate frames, in comparison to Newtonian frames. For example, an artificial satellite orbiting around earth can be viewed as a cabin. However, reasonably sensitive instruments would detect "microgravity" in such a situation because the "lines of force" of the earth's gravitational field converge.

In general, the convergence of gravitational fields in the universe dictates the scale at which one might consider such (local) inertial frames. For example, a spaceship falling into a black hole or neutron star would be subjected to tidal forces so strong that it would be crushed. In comparison, however, such forces might only be uncomfortable for the astronauts inside (compressing their joints, making it difficult to extend their limbs in any direction perpendicular to the gravity field of the star). Reducing the scale further, it might have almost no effects at all on a mouse. This illustrates the idea that all freely falling frames are locally inertial (acceleration and gravity-free) if the scale is chosen correctly.

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