Lorentz Transformations

Experts define Lorentz transformations as a one-parameter family of linear transformations. It is a linear transformation by nature and comprises of rotation of space. Moreover, Lorentz transformations facilitate the preservation of the space-time interval between any two events.

Introduction to Lorentz Transformations

Lorentz transformations refer to the relationship that exists between two different coordinate frames. Moreover, the movement of these frames takes place at a constant velocity and are relative to each other.

The name of this particular transformation is after a Dutch physicist named Hendrik Lorentz. Moreover, the derivation of this transformation is a lengthy process that takes place in a step by step manner.

How do we Measure Lorentz Transformations?

The relation of the Lorentz transformations is only with regard to change in the inertial frames. Moreover, this relationship usually exists in the context of special relativity. Furthermore, this transformation is a type of linear transformation in which mapping takes place between two modules that involve vector spaces.

In linear transformation, preservation of operations of scalar multiplications and additions takes place. Furthermore, this transformation is characterized by some instinctive features. An example of this is that an observer whose movement takes place at different velocity would be able to measure ordering of events, elapsed times, and different distances, but the condition over here is that the speed of light, in all inertial frames, should be the same.

There is a possibility that Lorentz transformations can also include rotation of space. Moreover, the Lorentz boost is a rotation that is free of this transformation. Due to Lorentz transformation, preservation happens of the space-time interval that is between any two events.

Formula of Lorentz Transformations

\(t{}’= \gamma \left ( t-\frac{vx}{c^{2}} \right )x{}’= \gamma \left ( x-vt \right )y{}’=yz{}’= z\)

Where,

• (t, x, y, z) and (t’, x’, y’, z’) are the coordinates of an event in two frames
• v is the velocity confined to x-direction
• c is the speed of light

Derivation of the Formula of Lorentz Transformations

From Galilean transformation below whose analysis took place for a beam of light, we can derive Lorentz transformations:

x′= a₁x + a₂ty′ = yz′ = zt′ = b₁x + b₂t

The origin of the primed frame x’ = 0, with speed v in unprimed frame S. For the beam of light, the location x = vt at time t in unprimed frame S.

∴ x′ = 0 = a₁x + a₂t → x = −a₂t/a₁ = vt Where,
a₂a₁ = −v

Rewriting the equation for the purpose of lorentz transformation derivation:

⇒ x′ = a₁x + a₂t = a₁(x + a₂t/a₁) = a₁(x − vt)

⇒ \(a_{1}^{2}\left ( x – vt^{2} \right ) + y{}’^{2} + z{}’^{2} – c^{2}\left ( b_{1}x + b_{2}t \right )^{2} =  x^{2} + y^{2} + z^{2} – c^{2}t^{2}\)

⇒ \(a_{1}^{2}x^{2} – 2a_{1}^{2}xvt + a_{1}^{2}v^{2}t^{2} – c^{2}b_{1}^{2}x^{2} -2c^{2}b_{1}b_{2}xt-c^{2}b_{2}^{2}t^{2}= x^{2} – c^{2}t^{2}\)

⇒ \(\left ( a_{1}^{2}\- c^{2}b_{2}^{1} \right )x^{2}= x^{2}ora_{1}^{2}\-c^{2}b_{1}^{2}=1\)

⇒ so, \(\left ( a_{1}^{2}v^{2} – c^{2}b_{2}^{2} \right )t^{2} = -c^{2}t^{2}orc^{2}b_{2}^{2} – a_{1}^{2}v^{2} = c^{2}\)

⇒ \(\left ( 2a_{1}^{2}v + 2b_{1}b_{2}c^{2} \right )xt = 0orb_{1}b_{_{2}}c^{2} = -a_{1}^{2}v\)

⇒ \(b_{2}^{1}c^{2} = a_{1}^{2} – 1\)

⇒ \(b_{2}^{2}c^{2} = c^{2} + a_{1}^{2}v^{2}\)

⇒ \(a_{1}^{2}c^{2} – a_{1}^{2}v^{2} = c^{2}\)

⇒ \(a_{1}^{2} = \frac{c^{2}}{c^{2}-v^{2}} = \frac{1}{1 – \frac{v^{2}}{c^{2}}}\)

⇒ \(b_{1}^{2}c^{2} = \frac{1 – \left ( 1 – \frac{v^{2}}{c^{2}} \right )}{1 – \frac{v^{2}}{c^{2}}} = \frac{\frac{v^{2}}{c^{2}}}{1 – \frac{v^{2}}{c^{2}}} . \frac{1}{1 – \frac{v^{2}}{c^{2}}}\)

⇒ \(b_{2}^{2} = \frac{1}{1 – \frac{v^{2}}{c^{2}}}\)

⇒ \(b_{2} = \frac{1}{\sqrt{1 – \frac{v^{2}}{c^{2}}}} ⇒ b22=11−v2c2\)

⇒ \(\gamma = \frac{1}{\sqrt{1 – \frac{v^{2}}{c^{2}}}}\)

We can also write it as:

a₁ = \(\gamma\)

⇒ a₂ = \(\gamma\)v

⇒ b₁ =

⇒ b₂ = -v\(\gamma\)/c²

Following are the final form of Lorentz transformations:

∴ x′= \(\gamma\)(x − vt)

⇒ y′ = y

⇒ z′ = z

⇒ t′ = \(\gamma\)(t – vx/c²)

FAQs For Lorentz Transformations

Question 1 What is Lorentz transformation definiton?

Answer 1: Lorentz transformations is the relationship between two different coordinate frames that move at a constant velocity and are relative to each other. Moreover, the movement of these frames happens in a manner such that they are relative to each other and the velocity is constant.

Question 2: How does mapping happen in Lorentz transformations?

Answer 2: The mapping in Lorentz transformations takes place between two modules that involve vector spaces.