Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer below the deck would not be able to tell whether the ship was moving or stationary.
Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws of motion hold in all frames related to one another by a Galilean transformation.
Among the axioms from Newton's theory are: Galilean relativity can be shown as follows.
By the second axiom above, one can synchronize the clock in the two frames and assume t = t' .
Suppose S' is in relative uniform motion to S with velocity v. Consider a point object whose position is given by functions r' (t) in S' and r(t) in S. We see that The velocity of the particle is given by the time derivative of the position: Another differentiation gives the acceleration in the two frames: It is this simple but crucial result that implies Galilean relativity.
Some of the assumptions and properties of Newton's theory are: In comparison, the corresponding statements from special relativity are as follows: 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.
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.
For example, an artificial satellite orbiting the Earth can be viewed as a cabin.
However, reasonably sensitive instruments could 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 (at a certain distance) be subjected to tidal forces strong enough to crush it in width and tear it apart in length.
[2] 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, the forces at that distance 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.
[2] There are two consistent Galilean transformations that may be used with electromagnetic fields in certain situations.
When the magnetic field is dominant and the relative velocity,
The electric field is transformed under this transformation when changing frames of reference, but the magnetic field and related quantities are unchanged.
[3]: 261 An example of this situation is a wire is moving in a magnetic field such as would occur in an ordinary generator or motor.
The transformed electric field in the moving frame of reference could induce current in the wire.
The magnetic field and free current density are transformed under this transformation when changing frames of reference, but the electric field and related quantities are unchanged[3]: 265 Because the distance covered while applying a force to an object depends on the inertial frame of reference, so depends the work done.
Due to Newton's law of reciprocal actions there is a reaction force; it does work depending on the inertial frame of reference in an opposite way.
The total work done is independent of the inertial frame of reference.
Correspondingly the kinetic energy of an object, and even the change in this energy due to a change in velocity, depends on the inertial frame of reference.
The total kinetic energy of an isolated system also depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center-of-momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.
Due to the conservation of momentum the latter does not change with time, so changes with time of the total kinetic energy do not depend on the inertial frame of reference.