What is Lorentz Transformation?
Lorentz transformation refers to the relationship between two coordinate frames that move at a constant speed and are relative to one another. It is named after a Dutch physicist, Hendrik Lorentz.
We can divide reference frames into two categories:
Frames of inertial motion - motion with a constant velocity
Non-inertial Frames - Rotational motion with constant angular velocity and acceleration in curved paths
Lorentz Transformation in Inertial Frame
A Lorentz transformation can only be used in the context of inertial frames, so it is usually a special relativity transformation. During the linear transformation, a mapping occurs between 2 modules that include vector spaces. The multiplication and addition operations on scalars are preserved when using a linear transformation. As a result of this transformation, the observer who is moving at different speeds will be able to measure different elapsed times, different distances, and order of events, but it is important to follow the condition that the speed of light should be equivalent across all frames of reference.
Lorentz Boost
It is also possible to apply the Lorentz transform to rotate space. A rotation free of this transformation is called Lorentz boost. This transformation preserves the space-time interval between two events.
The Statement of the Principle
The transformation equations of Hendrik Lorentz relate two different coordinate systems in an inertial reference frame. There are two laws behind Lorentz transformations:
Relativity Principle
Light's constant speed
Simplest Derivation of Lorentz Transformation
We will start by scaling Galilean transformations by Lorentz factor (γ) which is-
γ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]
γ = \[\frac{1}{\sqrt{1 - β^{2}}}\]
Galilean transformations of Newtonian transformations: -
t’=t
z’=z
y’=y
x’=x- vt
Here, x’, y’ , z’ and ct’ are the new coordinates. We need to transform from x to x’ and ct to ct’.
This implies, x’ = γ(x - βct)
And, ct’ = γ(ct - βx)
Extending it to 4 dimensions,
y’=y
z’=z
Another form of writing the equations, is to substitute β = \[\frac{v}{c}\]
γ = \[\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\] = \[\frac{1}{\sqrt{1 - β^{2}}}\]
x’ = γ(x - ct\[\frac{v}{c}\])
x’ = γ(x - vt)
ct’ = γ(ct - βx)
ct’ = γ(ct - \[\frac{v}{c}\]x)
Dividing by c ,
\[\frac{ct’}{c}\] = γ(\[\frac{ct}{c}\] - \[\frac{vx}{c^{2}}\])
t’ = γ(t - \[\frac{vx}{c^{2}}\])
When , v << c , Then \[\frac{vx}{c^{2}}\] ≈ 0
and when γ is equal to 1,
t’ = γ(t - \[\frac{vx}{c^{2}}\]) becomes t’ ≈ t
x’ = γ(x - vt) becomes x’ = x - vt
Equation of the Lorentz Transformation
Lorentz transformations transform one frame of spacetime coordinates into another frame that moves at a constant speed relative to the other. The four axes of spacetime coordinate systems are x, ct, y, and z.
x’ = γ(x - βct)
ct’ = γ(ct - βx)
Extending it to 4 dimensions,
y’=y
z’=z
Space-Time
The concept of Lorentz transformation requires us to first understand spacetime and its coordinate system.
As opposed to three-dimensional coordinate systems having x, y, and z axes, space-time coordinates specify both space and time (four-dimensional coordinate system). The coordinates of each point in four-dimensional spacetime consist of three spatial and one temporal characteristic.
Need of a Spacetime Coordinate System
Earlier, time was viewed as an absolute quantity. Since space is not an absolute quantity, observers would disagree about the distance (thus, the observers would not agree about the speed of the light) even though they agree on the time it takes for the light to travel.
Consequently, time is no longer considered an absolute quantity due to the Theory of Relativity.
As a result, the distance between events can now be calculated as a function of time.
d = (1/2)c
Where,
d-distance of the event
t-time was taken by a pulse to reach the event and reflect back
c-speed of light
The theory of relativity has changed our understanding of space and time as separate and independent components. Therefore, space and time had to be combined into one continuum.
World-Line
The path that an object follows as it moves through a spacetime diagram is called its world line. Spacetime diagrams are important because world lines may not correspond to paths that objects traverse in space. For example, when a car moves with uniform acceleration, the graph in a velocity-time graph is no longer a straight line. In your reference frame, a world line is a stationary straight line whose x coordinate is always equal to zero.
Fun Facts about Lorentz Transformation
The world line of the speed of light is the only such path that does not change when followed by a series of contraction and expansion.
The world line of the speed of light is always at an angle of 45° to the spacetime coordinate system.
FAQs on Lorentz Transformation
1. Why was there a Need to Develop Lorentz Transformation Equations?
When objects start approaching the speed of light Newtonian physics starts to collapse. For example, if we think of a hypothetical situation where the sun was to cease to exist at this very moment, according to Newton's law of gravitation, the earth would feel its impact instantaneously. However, we know it is not possible. Therefore, the effect should be felt only after 8 minutes.
In Galilean Transformations, the speed of light is perceived differently depending on the reference frame. We know that nothing ever travels faster than the speed of light. Thus, there was a contradiction that formulated the need to develop a different set of transformational equations. Also, earlier transformations included rotation, scaling, reflection, and shear, which were not sufficient to incorporate spacetime coordinate systems.
2. What are the Various Applications of the Lorentz Transformation Equation?
A. Lorentz transformation gives an intuitive insight into: –
Time dilation- The faster you move through spacetime, the more the relative spacetime graph contracts, which are similar to the theory that the slower you move through time, the faster you move through space.
Relativity of mass- Mass of an object tends to approach infinity when its speed approaches the speed of light.
If you look at motion in the form of snapshots at various intervals of time, according to Lorentz transformation, the constant time snapshots from your perspective may not necessarily be constant time snapshots from someone else’s perspective. This visualization is in correlation with our belief that time is not absolute.
3. What are the Physical Implications?
Invariance of light speed is one of the critical requirements for the Lorentz Transformation, and this is also taken into account in the derivation of this equation.
In terms of Lorentz transformations, there are 3 unintuitive but correct predictions:
Contraction of length.
Dilation of time.
The relativity of simultaneity.
4. The Lorentz Transformation is used for What?
Lorentz transformation describes the relationship between the space and time coordinates of two continuous-motion systems interacting with each other.
5. What are the Features of the Lorentz Transformation?
Lorentz transformations are linear transformations. The rotation of space may be included in the transformation; a rotation-free transformation is known as a Lorentz boost. Lorentz transformations preserve the spacetime interval between any two events in Minkowski space, the mathematical model of spacetime in special relativity.
6. Why are Lorentz Transformations Linear?
In the same way as the Galilean transformation, the Lorentz transformation is linear since the relative velocity is constant. Otherwise, inertial forces might appear. They are sometimes referred to as Galilean and inertial frames, respectively.
7. What are the Applications of the Lorentz Transformation?
The worldline of the speed of light was kept constant by the relative contraction and expansion of the spacetime coordinate system.
When an object in motion is observed from a different coordinate system, the Lorentz transformation is integral to calculating its various attributes.
Various Lorentz transformations provide an insight into how bodies move, as well as topics such as length contraction, time dilation, and relative mass.
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