Galilean invariance

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.

Formulation

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.

Among the axioms from Newton's theory are:

#There exists an "absolute space", in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
#All inertial frames share a "universal time".

Galilean relativity can be shown as follows. Consider two inertial frames "S" and "S' ". A physical event in "S" will have position coordinates "r" = ("x", "y", "z") and time "t"; similarly for "S' ". 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 "r" = "r"("t") in "S". We see that

:$r\text{'}(t)\; =\; r(t)\; -\; v\; t,$

The velocity of the particle is given by the time derivative of the position:

:$u\text{'}(t)\; =\; frac\{d\}\{d\; t\}\; r\text{'}(t)\; =\; frac\{d\}\{d\; t\}\; r(t)\; -\; v\; =\; u(t)\; -\; v$

Another differentiation gives the acceleration in the two frames:

:$a\text{'}(t)\; =\; frac\{d\}\{d\; t\}\; u\text{'}(t)\; =\; frac\{d\}\{d\; t\}\; u(t)\; -\; 0\; =\; a(t)$

It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows Newton's laws of mechanics, if valid in one frame, must hold for all frames. But it is assumed to hold in absolute space, therefore Galilean relativity holds.

Newton's theory versus special 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. [Taylor and Wheeler's [http://www.eftaylor.com/pub/chapter2.pdf "Exploring Black Holes - Introduction to General Relativity, Chapter 2"] , p. 2-6.]

Electromagnetism

Maxwell's equations governing electromagnetism possess a different symmetry, Lorentz invariance, under which lengths and times "are" affected by a change in velocity, which is then described mathematically by a Lorentz transformation.

Albert Einstein's central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised such that Lorentz invariance replaces Galilean invariance. At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very different.

Higher geometry

One can introduce "line coordinates" ("k,s") in a Galilean plane::$m\; :\; y\; =\; k\; x\; +\; s$:$m1\; :\; y\; =\; k1\; x\; +\; s$ .The Galilean angle, according to Isaak Yaglom is “…the fundamental invariant of a pair of lines ("k,s"), ("k1,s1") under the [line] transformation …, the [Galilean] angle [between lines "m" and "m1" is] :δ("m" , "m1" ) = "k1" – "k" .”So in particular, if "m" is horizontal, then "k" = 0, and the Galilean angle to line"m1" : δ( "m" , "m1" ) is the slope "k1" .

In this sophisticated approach to the Galilean plane (taken by Isaak Yaglom and V.V. Kisil), there is a study of parabolas called cycles which generalize the concept of a circle to the peculiarities of Galilean geometry through the Galilean angle.

Work, kinetic energy, momentum

The work done by a force acting on an object depends on the inertial frame of reference, because the distance covered while applying the force does. Due to 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 of the kinetic 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.

By contrast, while the momentum of an object also depends on the inertial frame of reference, its change due to a change in velocity does not.

ee also

*Absolute time and space

Notes

References

*cite book |last=Yaglom |first=Isaak M. |authorlink=Isaak Yaglom |coauthors= |others=Abe Shenitzer (trans.) |title=A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity |year=1979 |publisher=Springer-Verlag |location=New York |id=ISBN 0387903321 (translated from the Russian)