Why Lorentz Transformation Matters in Modern Physics
Introduction
In physics, we study the magnetic electric force charge to find out its true nature while observing and researching. We analyze the force exerted into space with a constantly varying flow of current.
The word electromagnetism can be elaborated as the summation of the electric and magnetic force acting simultaneously towards the space due to the motion of charged particles. The forces which are available around us can be studied through electromagnetism.
Lorentz transformation is put forward by the Dutch scientist Hendrik Antoon Lorentz. The frame of reference is any kind of that you are measuring something. For example, if you are standing on the floor and looking at some physical event such as a firecracker explosion or collision of two stones. that floor will become your frame of reference. If you are traveling inside the train and looking outside to a physical event, the train will be your frame of reference.
What is Lorentz Transformation?
Lorentz transformation can be elaborated as the linear transformation which consists of a rotation of space along with the constant distance between space and time. In physics, we can study Lorentz transformation which comes under linear transformation.
Lorentz transformations are a set of equations used in relativity physics to connect the space and time coordinates of two systems moving at the same speed. Lorentz transformations are used to describe extremely fast phenomena that approach the speed of light. They formally describe the relativity ideas of space and time not being absolute, length, time, and mass all being dependent on the observer's relative motion, and the speed of light in a vacuum being constant and independent of the observer's or source's motion. In 1904, a Dutch physicist named Hendrik Antoon Lorentz devised the equations.
Lorentz Transformation Derivation
We are using mathematics to elaborate and predict the events that happen in the world.
In general, an event indicates something that occurs at a given location in space and time. The Lorentz transformation transforms between two reference frames when one is moving with respect to the other.
The Lorentz transformation can be derived as the relationship between the coordinates of a particle in the two inertial frames on the basis of the special theory of relativity.
The Lorentz transformations are exclusively related in terms of change in inertial frames. Furthermore, this link is frequently found in special relativity. This transformation is also a type of linear transformation in which mapping occurs between two modules involving vector spaces.
The operations of scalar multiplications and additions are preserved throughout linear transformation. Furthermore, this transition has some innate characteristics. An observer moving at a different velocity, for example, could measure event ordering, elapsed periods, and various distances, but the constraint here is that the speed of light should be the same in all inertial frames.
Here,
S and S‘ = two inertial frames out of which S‘ is moving relative to S with v velocity along positive x-axis
At the beginning t = t‘ = 0
Origin O and O’ will coincide
Lorentz Transformation Equation Derivation
The wavefront of light emitted at t = 0 when reaches at P, the position and time observed by observers at O and O‘ are ( x, y, z , t ) and (t‘, x‘, y‘, z‘)respectively.
Time taken to reach O from P as observed in frame S is:
t = \[\frac{OP}{c}\] = \[\frac{\sqrt{x^{2} + y^{2} + z^{2}}}{c}\]
or, x\[^{2}\] + y\[^{2}\] + z\[^{2}\] = c\[^{2}\]t\[^{2}\]
or, x\[^{2}\] + y\[^{2}\] + z\[^{2}\] - c\[^{2}\]t\[^{2}\] = 0
The same equation can be obtained for t‘ time taken by light to reach from O‘ to P,
x\[^{‘2}\] + y\[^{‘2}\] + z\[^{‘2}\] - c\[^{2}\]t\[^{‘2}\] = 0
Since both equations represent the same spherical wavefront in S and S‘ frame, they can be equated as:
x\[^{2}\] + y\[^{2}\] + z\[^{2}\] - c\[^{2}\]t\[^{2}\] = x\[^{2}\] + y\[^{2}\] + z\[^{2}\] - c\[^{2}\]t\[^{2}\] ………….(1)
Since frame S and S‘ are moving relative to S along the x-axis, length in direction is perpendicular to direction of motion are unaffected.
i.e., y‘ = y and z‘ = z ……….(2)
From (1) and (2), we have
x\[^{2}\] - c\[^{2}\]t\[^{2}\] = x\[^{‘2}\] - c\[^{2}\]t\[^{‘2}\] (a)
In the frame S
x = v*t
or, x – v*t = 0
Whereas for frame S‘
x‘ = k (x – v*t) (3)
Since both are relative, we can assume s is moving relatively along with s‘ having a velocity v along the negative x – axis.
So, position of O at any instant of t‘ relative to observer is
x‘ = - v ∗ t‘
or, x‘ + v ∗ t‘ = 0
Whereas position of O relative to observer O in frame S is x = 0 , so x and x‘ must be related as
x‘ = k‘ (x‘ - v ∗ t‘) (4)
Where, K‘ is another constant
Substituting the value of x‘ from equation 3, we get;
x = [K‘(x - vt) + vt‘]
t‘ = \[(kt - [\frac{x}{v}])(1 - [\frac{1}{k‘}])\] (5)
Putting x’ from eq.3 and t’ from eq.5, in eq.(1) we get
\[x^{2} - c^{2}t^{2} = K^{2}(x - vt)^{2} - c^{2}k^2 (t - [\frac{x}{v}])(1 - [\frac{1}{k‘}])^{2}\]
By simplifying and equating coefficient of t² we get
K = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] (7)
Similarly, we get,
K‘ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]
Substituting the value of K and K’ in equation (3) and (5), we get,
x‘ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] (x - vt) and
t‘ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] (t - \[\frac{x}{v}\] x \[\frac{v^{2}}{c^{2}}\])
= = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] (t - \[\frac{vx}{c^{2}}\])
The Lorentz transformations are
x‘ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] (x - vt) ; y‘ = y ; z‘ = z
And t‘ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] (t - \[\frac{vx}{c^{2}}\])
What are the Differences Between Galilean and Lorentz Transformations?
FAQs on Step-by-Step Derivation of Lorentz Transformation
1. What are the Lorentz transformation equations?
The Lorentz transformation equations relate the spacetime coordinates of an event in one inertial frame (S), denoted as (x, y, z, t), to the coordinates of the same event in another inertial frame (S'), denoted as (x', y', z', t'), which is moving with a constant velocity 'v' along the x-axis relative to S. The equations are:
- x' = γ(x - vt)
- y' = y
- z' = z
- t' = γ(t - vx/c²)
Here, 'c' is the speed of light in a vacuum, and γ (gamma) is the Lorentz factor, defined as γ = 1 / √(1 - v²/c²).
2. What are the fundamental postulates used in the derivation of the Lorentz transformation?
The derivation of the Lorentz transformation is based on the two fundamental postulates of Albert Einstein's special theory of relativity:
- The Principle of Relativity: The laws of physics are the same (invariant) in all inertial frames of reference. This means there is no absolute motion or preferred inertial frame.
- The Principle of Constancy of the Speed of Light: The speed of light in a vacuum, 'c', is the same for all observers in inertial frames, regardless of the motion of the light source or the observer.
These postulates replace the assumptions of classical physics and lead directly to the relativistic nature of space and time.
3. How is the Lorentz transformation for the x-coordinate and time derived?
The derivation starts by assuming a linear relationship between the coordinates of two inertial frames, S and S'. Based on the postulates of special relativity, a light pulse emitted from the origin at t=0 must form a spherical wavefront in both frames. This leads to the invariance of the spacetime interval: x² + y² + z² - c²t² = x'² + y'² + z'² - c²t'². By assuming the transformation for the x-coordinate is of the form x' = k(x - vt) and solving for the time transformation t' while ensuring the constancy of 'c', we can determine the constant 'k' to be the Lorentz factor (γ). This process yields the final transformation equations for both position and time, linking them inseparably.
4. What is the significance of the Lorentz factor (γ) in these equations?
The Lorentz factor (γ) is a crucial element that quantifies the strength of relativistic effects. Its value depends on the relative velocity 'v'.
- When v is much smaller than c (v << c), γ is very close to 1, and the Lorentz transformations reduce to the classical Galilean transformations.
- As v approaches c, the value of γ increases and approaches infinity. This increase is directly responsible for relativistic phenomena such as time dilation (moving clocks run slower) and length contraction (moving objects appear shorter in the direction of motion).
Essentially, γ acts as a correction factor to classical mechanics that becomes significant only at speeds approaching the speed of light.
5. How do Lorentz transformations fundamentally differ from Galilean transformations?
The primary difference lies in their treatment of time and the speed of light. Galilean transformations, used in classical mechanics, assume time is absolute (t' = t) and that velocities add up linearly. In contrast, Lorentz transformations treat time as relative (t' depends on both t and x) and are built on the principle that the speed of light is constant for all observers. This leads to counter-intuitive but experimentally verified results like time dilation and length contraction, which are absent in the Galilean framework.
6. What are the inverse Lorentz transformation equations and when are they used?
The inverse Lorentz transformation equations are used to transform coordinates from the moving frame (S') back to the stationary frame (S). They are derived by simply replacing the velocity 'v' with '-v' in the original equations, which reflects the symmetry of relative motion. The inverse equations are:
- x = γ(x' + vt')
- y = y'
- z = z'
- t = γ(t' + vx'/c²)
These are essential for problems where measurements are made in the moving frame and need to be understood from the perspective of the stationary frame.
7. What happens to the Lorentz transformation if the relative velocity 'v' is zero?
If the relative velocity 'v' between the two inertial frames is zero, the frames are at rest with respect to each other. In this case, the Lorentz factor γ = 1 / √(1 - 0²/c²) = 1. Substituting v=0 and γ=1 into the Lorentz equations gives:
- x' = 1(x - 0*t) = x
- y' = y
- z' = z
- t' = 1(t - 0*x/c²) = t
This shows that the coordinates in both frames are identical, which is the expected result when there is no relative motion.
8. How does the Lorentz transformation for time lead to the concept of 'Relativity of Simultaneity'?
The relativity of simultaneity is a direct consequence of the time transformation equation: t' = γ(t - vx/c²). It shows that two events that are simultaneous in frame S (occurring at the same time, Δt = 0) but at different locations (Δx ≠ 0), will not be simultaneous in frame S'. For an observer in S', the time difference would be Δt' = -γ(vΔx/c²), which is non-zero. This means that two observers in relative motion can disagree on whether two events happened at the same time, shattering the classical idea of a universal, absolute 'now'.
