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

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

Galilean Invariance

We stated previously that all laws of Physics are supposed to be expressed in the same form in all inertial reference frames (Principle of Relativity or Galilean Invariance).
This statement we take as an axiom because it corresponds to our experience and agrees with physical experiments.

We have also suggested the Galilean transformations of coordinates as a tool to express the known laws of Physics in different inertial reference frames to verify the invariance of these laws.

Let's check some laws of Mechanics comparing them in two inertial reference frames - α{T,X,Y,Z} and β{t,x,y,z} moving relative to α with constant speed v along the X-axis of α, maintaining parallelism of corresponding axes.
Assume, at zero-time T=t=0 both reference frames coincide and, therefore, any point in space at zero-time has the same coordinates in both reference frames.
We will check if some familiar physical laws look the same in these two inertial systems and are invariant relative to Galilean transformation of coordinates.


Newton's First Law

The Newton's First Law of Motion (Law of Inertia) states that every object in absence of force acting on it will remain at rest or in uniform motion in a straight line.
Assume, an object in α{T,X,Y,Z} frame is moving with a constant speed
V={V_X,V_Y,V_Z}
along a straight line.
Its position then can be described as linear functions of time
X(T) = X₀ + V_X·T
Y(T) = Y₀ + V_Y·T
Z(T) = Z₀ + V_Z·T
where {X₀,Y₀,Z₀} are coordinates of an initial position of an object at zero-time (t=T=0) in both reference frames and {V_X,V_Y,V_Z} are constant components of an object's velocity vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes.

Let's apply the transformation to β{t,x,y,z} frame moving along the X-axis of α frame with constant speed v.
t = T
x(t) = X(T) − v·T =
= X₀ + (V_X − v)·T =
= X₀ + (V_X − v)·t
y(t) = Y(T) = Y₀ + V_Y·T =
= Y₀ + V_Y·t
z(t) = Z(T) = Z₀ + V_Z·T =
= Z₀ + V_Z·t

Thus, the motion in β frame described by these linear functions of time t
x(t) = X₀ + (V_X − v)·t
y(t) = Y₀ + V_Y·t
z(t) = Z₀ + V_Z·t
is indeed a uniform motion along a straight line.
The Newton's First Law is preserved by a transformation from one inertial reference frame to another.


Newton's Second Law

The Newton's Second Law of Motion establishes the relationship between the force F, mass m and acceleration a of an object
F = m·a
where F and a are vectors, while m is a positive constant.
That means that acceleration vector is directed along the same direction as a vector of force.

This vector equation in α{T,X,Y,Z} reference frame can be rewritten in coordinate form
F_X = m·a_X
F_Y = m·a_Y
F_Z = m·a_Z
where {F_X,F_Y,F_Z} are constant components of a force vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes and
{a_X,a_Y,a_Z} are constant components of an acceleration vector, also the same in both systems for the same reason.

As we know, an acceleration is the second derivative of a position (a function of time) by time.
Therefore, in α frame the above equations can be written as
F_X = m·X"(T)
F_Y = m·Y"(T)
F_Z = m·Z"(T)

Since coordinates in β
{t,x,y,z} frame are related to coordinates in α frame as
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
and v is a constant speed of β{t,x,y,z} reference frame relative to α{T,X,Y,Z},
the first derivative by time from coordinates is
x'(t) = X'(T) − v
y'(t) = Y'(T)
z'(t) = Z'(T)
and the second derivative is
x"(t) = X"(T)
y"(t) = Y"(T)
z"(t) = Z"(T)

As we see, the acceleration of an object is the same in both inertial frames.

As mentioned before, it's important to notice that the components of the force and acceleration vectors are, correspondingly, the same in both reference frames since the axes of coordinates are correspondingly parallel.

Therefore,
F_x = F_X = m·X"(T) = m·x"(t)
F_y = F_Y = m·Y"(T) = m·y"(t)
F_z = F_Z = m·Z"(T) = m·z"(t)

That determines the equations of motion in β frame.
F_x = m·x"(t) = m·a_x
F_y = m·y"(t) = m·a_y
F_z = m·z"(t) = m·a_z

As we see, equations of motions in both reference frames are identical, which confirms the identical form of the Newton's Second Law if we switch from one inertial frame to another.


Velocity Addition Law

Let's examine the velocity of a moving object in two different inertial reference frames, α{T,X,Y,Z} and β{t,x,y,z}, assuming β frame is moving along X-axis of the frame α with constant speed v along X-axis.

The coordinates of this object in these reference frames are related by a familiar relations
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)

Consider an object moving in α frame with a constant vector velocity V^α(T).
The coordinates of a velocity vector are first derivatives of coordinates of the position. So, if our object moves in the α{T,X,Y,Z} frame with velocity vector
V^α(T) =
= {V_X^α(T),V_Y^α(T),V_Z^α(T)}

The components of this vector are, correspondingly
V_X^α(T) = X'(T)
V_Y^α(T) = Y'(T)
V_Z^α(T) = Z'(T)

Viewed from the β{t,x,y,z} reference frame, its velocity is
V^β(t) = {V_x^β(t),V_y^β(t),V_z^β(t)}
and the components of this vector are
V_x^β(t) = x'(t)
V_y^β(t) = y'(t)
V_z^β(t) = z'(t)

The equations of coordinate transformation
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
applied to components of velocity vector produce
V_x^β(t) = x'(t) = X'(T) − v
V_y^β(t) = y'(t) = Y'(T)
V_z^β(t) = z'(t) = Z'(T)

In vector form the movement of the β reference frame along X-axis of α with speed v means moving along a vector v^α(v,0,0)

Therefore, we can write in vector form the relationship between expressions of a movement of an object in two reference frames as
V^β(t) = V^α(T) − v^α
where V^β(t) is the velocity vector in β reference frame,
V^α(T) is the velocity vector in α reference frame,
v^α is a vector describing the motion of β reference frame along X-axis of α.

It's easy to generalize this relationship to a case when reference frame β moves along any constant vector v^α(v_X,v_Y,v_Z) within frame α, not necessarily along its X-axis.

The equations of coordinate transformation in this more general case are
t = T
x(t) = X(T) − v_X·T
y(t) = Y(T) − v_Y·T
z(t) = Z(T) − v_Z·T
applied to components of velocity vector produce
V_x^β(t) = x'(t) = X'(T) − v_X
V_y^β(t) = y'(t) = Y'(T) − v_Y
V_z^β(t) = z'(t) = Z'(T) − v_Z

In vector form this relationship between expressions of a movement of an object in two reference frames looks exactly as above:
V^β(t) = V^α(T) − v^α

The above is the velocity addition law of Galilean transformation.