Galilean Transformation: UNIZOR.COM - Relativity 4 All - Galilean

Notes to a video lecture on http://www.unizor.com

Galilean Transformations

Here and in many other places of this course we will be dealing with two inertial reference frames moving relative to each other with constant speed.

Let's call one of these reference frames primary or α and its Cartesian axes will be called X,Y,Z.
The other system will be called secondary or β with axes x,y,z.

For simplicity, we assume that β reference frame at zero-time t=0 coincides with α.
We also assume that β frame is moving along the X-axis of α with constant speed V, preserving the parallelism of corresponding axes of these reference frames.

Time is considered absolute and the same for both reference frames.

Assume, there is a given stationary point P with coordinates {X_P,Y_P,Z_P} not changing with time in α frame.
Our task is to determine its coordinates {x_P(t),y_P(t),z_P(t)} in β frame that moves relative to α as described above.

It is geometrically obvious that coordinates Y_P and Z_P of this point in α will be equal to coordinates y_P(t) and z_P(t) of this point in β.
Coordinate x_P(t) of this point in β will be shifted in time from its coordinate X_P in α. This shift will be zero at zero-time t=0 and, as time goes by, it will shift more and more proportional to speed V of the movement of β relative to α.

More exactly, since the origin of β moves relative to the origin of α with speed V along X-axis, the X-coordinate of β's origin in α at time t is X_β(t)=V·t.
Therefore, since X-coordinate of point P is constant X_P, its x-coordinate in β frame is x_P(t)=X_P−V·t.

We can summarize the transformation of coordinates between primary and secondary frame as follows
x_P(t) = X_P−V·t
y_P(t) = Y_P
z_P(t) = Z_P

If point P is not a stationary point in α frame, but is moving, and its coordinates are functions of absolute time {X_P(t),Y_P(t),Z_P(t)}, the transformation to β frame would be analogous
x_P(t) = X_P(t)−V·t
y_P(t) = Y_P(t)
z_P(t) = Z_P(t)

Sometimes these transformations are complemented with t=T, implying that T is time in α frame, while t is time in β frame, to emphasize that the time is absolute and the same in both reference frames.

Since P is any fixed point in α frame, there is no need to index these formulae of transformation and they can be written as
t = T
x(t) = X(T)−V·T
y(t) = Y(T)
z(t) = Z(T)

These transformations of coordinates from one inertial reference frame to another, moving relative to the first as described above, are called
Galilean Transformations.

Let's consider the same relative movement of the same two reference frames α and β from the point of view of an observer at rest in the β frame.
From his perspective its own β frame is stationary and α frame is moving along x-axis with speed −V.

All the considerations above can now be applied to this situation and the transformation of coordinates from β to α will be similar to the above, but with a reversed speed of movement of α frame relative to β:
T = t
X(T) = x(t)+V·t
Y(T) = y(t)
Z(T) = z(t)

Linear transformation of coordinates can be expressed in matrix and vectors form.
We can express the transformation from α frame to β frame as

t

x

y

z

=

1	0	0	0
−V	1	0	0
0	0	1	0
0	0	0	1

·

T

X

Y

Z

The transformation from β frame to α frame is similar, but the speed is reversed

T

X

Y

Z

=

1	0	0	0
V	1	0	0
0	0	1	0
0	0	0	1

·

t

x

y

z

Just to check if two above transformations are inverse to each other, we can multiply the transformation matrices according to the rule of multiplication of matrices and will get an identity matrix 4x4 with all diagonal elements equal to 1 and all other elements equal to 0, as supposed to be with inverse matrices.

Incidentally, the determinant of both transformation matrices equals to 1.