Galilean invariance or Galilean relativity states that the laws of motion are the same in all
inertial frames of reference
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
.
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
first described this principle in 1632 in his ''
Dialogue Concerning the Two Chief World Systems
The ''Dialogue Concerning the Two Chief World Systems'' (''Dialogo sopra i due massimi sistemi del mondo'') is a 1632 Italian-language book by Galileo Galilei comparing the Copernican system with the traditional Ptolemaic system. It was tran ...
'' 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.
The young
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
"was engrossed in analyzing Galileo's principle of inertia (Galilean relativity)".
Formulation
Specifically, the term ''Galilean invariance'' today usually refers to this principle as applied to
Newtonian mechanics
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
, that is,
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
hold in all frames related to one another by a
Galilean transformation
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
. In other words, all frames related to one another by such a transformation are inertial (meaning, Newton's equation of motion is valid in these frames). In this context it is sometimes called ''Newtonian relativity''.
Among the axioms from Newton's theory are:
#There exists an ''
absolute space Absolute may refer to:
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* Absolute Entertainment, a video game publisher
* Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK
* Absolute Software Corporation, specializes in security and data risk manage ...
'', 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
Universal Time (UT or UT1) is a time standard based on Earth's rotation. While originally it was mean solar time at 0° longitude, precise measurements of the Sun are difficult. Therefore, UT1 is computed from a measure of the Earth's angle with ...
''.
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'' in ''S'', and ''r' '' = (''x' '', ''y' '', ''z' '') and time ''t' '' in ''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 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. 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
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
.
Some of the assumptions and properties of Newton's theory are:
#The existence of infinitely many inertial frames. Each frame is of infinite size (the entire universe may be covered by many linearly equivalent frames). Any two frames may be in relative uniform motion. (The relativistic nature of mechanics derived above shows that the absolute space assumption is not necessary.)
#The inertial frames may move in ''all'' possible relative forms of uniform motion.
#There is a universal, or absolute, notion of elapsed time.
#Two inertial frames are related by a
Galilean transformation
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
.
#In all inertial frames, Newton's laws, and gravity, hold.
In comparison, the corresponding statements from special relativity are as follows:
#The existence, as well, of infinitely many non-inertial frames, each of which referenced to (and physically determined by) a unique set of spacetime coordinates. Each frame may be of infinite size, but its definition is always determined locally by contextual physical conditions. Any two frames may be in relative non-uniform motion (as long as it is assumed that this condition of relative motion implies a relativistic dynamical effect – and later, mechanical effect in general relativity – between both frames).
#Rather than freely allowing all conditions of relative uniform motion between frames of reference, the relative velocity between two inertial frames becomes bounded above by the speed of light.
#Instead of universal elapsed time, each inertial frame possesses its own notion of elapsed time.
#The Galilean transformations are replaced by
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s.
#In all inertial frames, ''all'' laws of physics are the same.
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 10
7 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 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.
[Taylor and Wheeler']
''Exploring Black Holes - Introduction to General Relativity'', Chapter 2
2000, p. 2:6. 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.
Electromagnetism
There are two consistent Galilean transformations that may be used with electromagnetic fields in certain situations.
A transformation
is not consistent if
where
and
are velocities. A consistent transformation will produce the same results when transforming to a new velocity in one step or multiple steps. It is not possible to have a consistent Galilean transformation that transforms both the magnetic and electric fields.
There are useful consistent Galilean transformations that may be applied whenever either the magnetic field or the electric field is dominant.
Magnetic field system
Magnetic field systems are those systems in which the electric field in the initial frame of reference is insignificant, but the magnetic field is strong. When the magnetic field is dominant and the relative velocity,
, is low, then the following transformation may be useful:
where
is free current density,
is magnetization density. The electric field is transformed under this transformation when changing frames of reference, but the magnetic field and related quantities are unchanged.
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.
Electric field system
Electric field systems are those systems in which the magnetic field in the initial frame of reference is insignificant, but the electric field is strong. When the electric field is dominant and the relative velocity,
, is low, then the following transformation may be useful:
where
is free charge density,
is polarization density. 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
Work, kinetic energy, and momentum
Because the distance covered while applying a force to an object depends on the inertial frame of reference, so depends the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
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
In physical science, an isolated system is either of the following:
# a physical system so far removed from other systems that it does not interact with them.
# a thermodynamic system enclosed by rigid immovable walls through which neither m ...
also depends on the inertial frame of reference: it is the sum of the total kinetic energy in a
center-of-momentum frame
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system i ...
and the kinetic energy the total mass would have if it were concentrated in the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. Due to the
conservation of momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
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
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
of an object also depends on the inertial frame of reference, its change due to a change in velocity does not.
See also
*
Absolute space and time
Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.
Before Newton
A version of the concept of absolute space (in the sense of a prefe ...
*
Faster-than-light
Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with zero ...
*
Galilei-covariant tensor formulation
The Galilei-covariant tensor formulation is a method for treating non-relativistic physics using the extended Galilei group as the representation group of the theory. It is constructed in the light cone of a five dimensional manifold.
Takahashi et ...
(no relation to Galileo)
*
Superluminal motion
Notes and references
{{DEFAULTSORT:Galilean Invariance
Classical mechanics
Invariance
he:מערכת ייחוס#עקרון היחסות של גלילאו