The Galilean transformation equations are a set of equations connecting the space-time coordinates of an event observed in two inertial frames, which are in relative motion. Consider two inertial frames S (unprimed) and S' (primed) with their corresponding axes parallel; the frame S' is moving along the common x-x' direction with velocity v relative to the frame S. Each frame has its own observer equipped with identical and compared measuring stick and clock. Assume that when the origin O of the frame S' passes over the origin O of frame S, both observers set their clocks at zero i.e., t = t' = 0.
The event to be observed is the motion of a particle. At certain moment, the S-observer registers the space-time coordinates of the particle as (x, y, z, t) and S'- observer as (x', y', z', t'). It is evident that the primed coordinates are related to unprimed coordinates through the relationship
x' = x – vt, y' = y, z' = z, t' = t ...(1.2.1) The last equation t' = t has been written on the basis of the assumption that time flows at the same rate in all inertial frames. This notion of time comes from our everyday experiences with slowly moving objects and is confirmed in analyzing the motion of such objects. Equations (1.2.1) are called Galilean transformation equations. Relative to S', the frame S is moving with velocity v in negative
direction of x-axis and therefore inverse transformation equations are obtained by interchanging the primed and unprimed coordinates and replacing v with –v. Thus
x = x' + vt', y = y', z = z', t = t' ...(1.2.2)
Fig. 1.2.1 Galilean transformation
Transformation of Length
Let us see how the length of an object transforms on transition from S to S'. Consider a rod placed in frame S along its x-axis. The length of rod is equal to the difference of its end coordinates: l = x2 – x1. In frame S', the length of rod is defined by the difference of its end coordinates measured simultaneously. Thus:
l' = x′2−x1′
Making use of Galilean transformation equations we have l' = (x2 – vt) – (x1 – vt) = x2 – x1 = l
Thus the distance between two points is invariant under Galilean transformation.
Transformation of Velocity
Differentiating the first equation of Galilean transformation, we have
′= −
′ =′ −
x x
dx dx dt dt v
u u v ...(1.2.3)
where ux and u'x are the x-components of velocity of the particle measured in frame S and S' respectively. Eqn. (1.2.3) is known as the classical law of velocity transformation. The inverse law is
ux = u' + v ...(1.2.4)
These equations show that velocity is not invariant; it has different values in different inertial frames depending on their relative velocities.
Transformation of Acceleration
Differentiating equation (1.2.3) with respect to time, we have
′ = ⇒ ′ =
x x
x x
du du
a a
dt dt ...(1.2.5)
where ax and a'x are the accelerations of the particle in S and S'. Thus we see that the acceleration is invariant with respect to Galilean transformation.
Transformation of the Fundamental Law of Dynamics (Newton’s Law)
The fundamental law of mechanics, which relates the force acting on a particle to its acceleration, is
ma = F ...(1.2.6)
In classical mechanics, the mass of a particle is assumed to be independent of velocity of the moving particle. The well known position dependent forces–gravitational, electrostatic and elastic forces and velocity dependent forces- friction and viscous forces are also invariant with respect to Galilean transformation because of the invariance of length, relative velocity and time. Hence the fundamental law of mechanics is also invariant under Galilean transformation. Thus
m a = F in frame S m' a' = F' in frame S'
The invariance of the basic laws of mechanics ensures that all mechanical phenomena proceed identically in all inertial frames of reference consequently no mechanical experiment performed wholly within an inertial frame can tell us whether the given frame is at rest or moving uniformly in a straight line. In other words all inertial frames are absolutely equivalent, and none of them can be preferred to others. This statement is called the classical (Galilean) principle of relativity.
The Galilean principle of relativity was successfully applied to the mechanical phenomena only because in Galileo’s time mechanics represented the whole physics. The classical notions of space, time and matter were regarded so fundamental that nobody ever felt necessity to raise any doubts about their truth. The Galilean principle of relativity did not worry physicists too much by the middle of the 19th century. By the middle of 19th century other branches of physics—electrodynamics, optics and thermodynamics—were developing and each of them required its own basic laws. A natural question arose: does the Galilean principle of relativity cover all physics as well? If the principle of relativity does not apply to other branches of physics then non-mechanical phenomena can be used to distinguish inertial frames thereby choosing a preferred frame. The basic laws of electrodynamics—Maxwell’s field equations—predicted that light was an electromagnetic phenomenon. The light propagates in vacuum with speed c = (m0e0) –ẵ = 3 ì 108 m/s. The wave nature of light compelled the then physicists to assume a medium for the propagation of light and hypothetical medium luminiferous ether was postulated to meet this requirement. Ether was regarded absolutely at rest and light was supposed to travel with speed c relative to the ether. If a certain frame is moving with velocity v relative to the ether; the speed of light in that frame, according to Galilean transformation, is c ± v; the plus sign when c and v are oppositely directed and minus sign when c and v have the same direction. Making us of this result that the light has different speed in different frames; the famous Michelson-Morley experiment was set up to detect the motion of the earth with respect to the ether.
When Galilean transformation equations were applied to the newly discovered laws of electrodynamics, the Maxwell’s equations, it was found that they change their shape on transition from one inertial frame to another. At first the validity of Maxwell’s equations was questioned and attempts were made to modify them in a way to make them consistent with the Galilean principle of
relativity. But such attempts predicted new phenomenon, which could not be verified experimentally.
It was then realized that Maxwell’s equations need no modifications.