Key Differences: Galilean vs. Lorentz Transformation in Physics
Galilean transformations can be represented as a set of equations in classical physics. They are also called Newtonian transformations because they appear and are valid within Newtonian physics. The Galilean transformation equation relates the coordinates of space and time of two systems that move together relatively at a constant velocity. Galilean transformations form a Galilean group that is inhomogeneous along with spatial rotations and translations, all in space and time within the constructs of Newtonian physics. Again, without the time and space coordinates, the group is termed as a homogenous Galilean group. The law of inertia is valid in the coordinate system proposed by Galileo.
What are Galilean Transformations?
To explain Galilean transformation, we can say that the Galilean transformation equation is an equation that is applicable in classical physics. Galilean equations and Galilean transformation of wave equation usually relate the position and time in two frames of reference. These two frames of reference are seen to move uniformly concerning each other.
Thus, the Galilean transformation definition can be stated as the method which is in transforming the coordinates of two reference frames that differ by a certain relative motion that is constant.
A group of motions that belong to Galilean relativity which act on the four dimensions of space and time and form the geometry of Galilean is called a Galilean group.
All these concepts of Galilean transformations were formulated by Gailea in this description of uniform motion. The topic of Galilean transformations that was formulated by him in his description of uniform motion was motivated by one of his descriptions. The description that motivated him was the motion of a ball rolling down a ramp.
Galilean Equations
The Galilean equations can be written as the culmination of rotation, translation, and uniform motion all of which belong to spacetime. Galilean transformation derivation can be represented as such:
To derive Galilean equations we assume that ‘x' represents a point in the three-dimensional Galilean system of coordinates. ‘t ’represents a point in one-dimensional time in the Galilean system of coordinates.
A uniform Galilean transformation velocity in the Galilean transformation derivation can be represented as ‘v’.
Thus, (x,t) → (x+tv,t) ; where v belongs to R3 (vector space).
A translation is given such that (x,t) →(x+a, t+s) where a belongs to R3 and s belongs to R.
A rotation is given by (x,t)→(Gx,t), where we can see that G: R3 →R3 is a transformation that is orthogonal in nature.
Galilean and Lorentz Transformation
Both the homogenous as well as non-homogenous Galilean equations of transformations are replaced by Lorentz equations. Galilean and Lorentz transformation can be said to be related to each other. Galilean transformation of the wave equation is nothing but an approximation of Lorentz transformations for the speeds that are much lower than the speed of light.
Between Galilean and Lorentz transformation, Lorentz transformation can be defined as the transformation which is required to understand the movement of waves that are electromagnetic in nature. To explain Galilean transformation, we can say that it is concerned with the movement of most objects around us and not only the tiny particles.
For the Galilean transformations, in the space domain, the only mixture of space and time is found that is represented as,
\[{x}' = (x-vt)\]; where v is the Galilean transformation equation velocity.
In Lorentz transformation, on the other hand, both ‘x’ and ‘t’ coordinates are mixed and represented as
\[{x}' = \gamma (x-vt) and {ct}'=(ct-\beta x)\]
Although, there are some apparent differences between these two transformations, Galilean and Lorentz transformations, yet at speeds much slower than light, these two transformations become equivalent. The differences become significant for bodies moving at speeds faster than light.
Did You Know?
According to the Galilean equations and Galilean transformation definition, the ideas of time, length, and mass are independent of the relative motion of the person observing all these properties.
According to the theory of relativity of Galileo Galilei, it is impossible by any mechanical means to state whether we are at rest or we are moving. If we consider two trains are moving in the same direction and at the same speed, the passenger sitting inside either of the trains will not notice the other train moving.
The coordinate system of Galileo is the one in which the law of inertia is valid.
Conclusion
It should always be remembered that the Galilean equations are applicable and physically valid in a Newtonian framework. They are definitely not applicable to the coordinate systems that are moving relative to each other at speeds that approach the speed of light. Galilean transformations formally express certain ideas of space and time and their absolute nature.
FAQs on Galilean Transformation Explained: Concepts & Applications
1. What is Galilean transformation and in what physical scenarios is it applicable?
In classical physics, a Galilean transformation is a set of equations used to transform the space and time coordinates of an event from one inertial frame of reference to another. It is applicable in scenarios where two reference frames are moving with a constant velocity relative to each other. Its validity is restricted to the realm of Newtonian physics, where the relative speeds involved are much lower than the speed of light.
2. What defines a Galilean or inertial frame of reference?
An inertial frame of reference, also known as a Galilean frame, is a coordinate system in which Newton's First Law of Motion is valid. This means that an object with no net force acting on it will either remain at rest or continue to move at a constant velocity. Any frame of reference moving with a constant velocity relative to another inertial frame is also an inertial frame.
3. Can you provide an example of how Galilean transformation is used in classical mechanics?
A classic example is calculating the velocity of an object in different reference frames. Imagine a person walking on a moving train. To find the person's velocity as seen by an observer standing on the ground, we use Galilean transformation. We add the person's velocity relative to the train to the train's velocity relative to the ground. This simple vector addition is a direct application of the Galilean transformation for velocities.
4. What are the standard Galilean transformation equations for position and velocity?
Let's consider two inertial frames, S and S', where S' moves with a constant velocity 'v' along the x-axis relative to S. The Galilean transformation equations are:
- Position Transformation:
x' = x - vt
y' = y
z' = z - Time Transformation:
t' = t (time is considered absolute) - Velocity Transformation:
u' = u - v (where u and u' are the velocities of an object in frames S and S', respectively)
5. What is the fundamental difference between Galilean and Lorentz transformations?
The fundamental difference lies in their treatment of time and the speed of light.
- Galilean Transformation assumes time is absolute (t' = t) for all observers and does not account for a universal speed limit. It is an excellent approximation for everyday, low-speed phenomena.
- Lorentz Transformation, a cornerstone of special relativity, treats time as relative and postulates that the speed of light in a vacuum ('c') is constant for all observers, regardless of their motion. It is accurate at all speeds, including those approaching the speed of light.
6. Why does the Galilean transformation fail when applied to electromagnetism?
The Galilean transformation fails because it contradicts one of the key predictions of Maxwell's equations of electromagnetism. Maxwell's equations predict that the speed of light ('c') is a universal constant. However, the Galilean velocity addition rule (u' = u - v) implies that an observer moving towards a light beam would measure its speed to be greater than 'c'. This discrepancy showed that the laws of electromagnetism were not invariant under Galilean transformation, leading to the development of special relativity and the Lorentz transformations.
7. In Galilean relativity, why is time considered absolute?
In the framework of classical mechanics, there was no experimental evidence or theoretical basis to suggest that the flow of time could be affected by motion. Time was conceived as a universal background parameter that ticked at the same rate for every observer in the universe, irrespective of their state of motion. This concept of absolute time, represented by the equation t' = t, is a foundational axiom of Galilean relativity that was later revised by Einstein's theory of special relativity.
8. What are the inverse Galilean transformation equations and what do they represent?
The inverse Galilean transformation equations are used to transform coordinates from a moving reference frame (S') back to a stationary reference frame (S). They essentially reverse the transformation. If S' moves with velocity 'v' relative to S, the inverse equations are:
- x = x' + vt
- y = y'
- z = z'
- t = t'
These equations are crucial for solving problems where we know the coordinates or velocity of an event in the moving frame and need to determine its properties as observed from the stationary frame.
