Lorentz Transformation Derivation

What is Lorentz Transformation?

Lorentz transformation in physics is defined as a one-parameter family of linear transformations. It is a linear transformation which includes rotation of space and preserving space-time interval between any two events. These transformations are named after the Dutch physicist Hendrik Lorentz. The derivation of Lorentz Transformation is explained below in a step by step manner.

Lorentz Transformation Formula

Following are the mathematical form of Lorentz transformations:

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

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

Lorentz factor:

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

Frames of reference are divided into two groups inertial which includes relative motion with constant velocity and non-inertial which includes rotational motion with constant angular velocity and Lorentz transformations are used for inertial frames.

The Lorentz transformations are derived from Galilean transformation as it fails to explain why observers moving at different velocities measure different distance, a different order of events even after the same speed of light in all inertial reference frames.

Lorentz Transformation Derivation

From Galilean transformation below which was studied for a beam of light, we can derive Lorentz transformations:

\({x}’=a_{1}x+a_{2}t\) \({y}’=y\) \({z}’=z\) \({t}’=b_{1}x+b_{2}t\)

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

\(∴ {x}’=0=a_{1}x+a_{2}t\rightarrow x=-\frac{a_{2}}{a_{1}}t=vt\) Where,
\(\frac{a_{2}}{a_{1}}=-v\)

Rewriting the equation:

⇒ \({x}’=a_{1}x+a_{2}t=a_{1}(x+\frac{a_{2}}{a_{1}}t)=a_{1}(x-vt)\)

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

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

⇒ \((a_{1}^{2}-c^{2}b_{2}^{1})x^{2}=x^{2} or a_{1}^{2}-c^{2}b_{1}^{2}=1\)

⇒ \((a_{1}^{2}v^{2}-c^{2}b_{2}^{2})t^{2}=-c^{2}t^{2} or c^{2}b_{2}^{2}-a_{1}^{2}v^{2}=c^{2}\)

⇒ \((2a_{1}^{2}v+2b_{1}b_{2}c^{2})xt=0 or b_{1}b_{2}c^{2}=-a_{1}^{2}v\)

⇒ \(b_{1}^{2}c^{2}=a_{1}^{2}-1\)

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

⇒ \(b_{1}^{2}b_{2}^{2}c^{4}=(a_{1}^{2}-1)(c^{2}+a_{1}^{2}v^{2})=a_{1}^{4}v^{2}\)

⇒ \(a_{1}^{2}c^{2}-c^{2}+a^{4}v^{2}-a_{1}^{2}v^{2}=a_{1}^{4}v^{2}\)

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

⇒ \(a_{1}^{2}(c^{2}-v^{2})=c^{2}\)

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

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

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

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

⇒ \(b_{1}^{2}=\frac{v^{2}}{c^{4}}.\frac{1}{1-\frac{v^{2}}{c^{2}}}\)

⇒ \(b_{1}=-\frac{v}{c^{2}}.\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\)

⇒ \(b_{2}^{2}c^{2}=c^{2}+v^{2}.\frac{1}{1-\frac{v^{2}}{c^{2}}}=\frac{c^{2}(1-\frac{v^{2}}{c^{2}})+v^{2}}{1-\frac{v^{2}}{c^{2}}}=\frac{c^{2}-v^{2}+v^{2}}{1-\frac{v^{2}}{c^{2}}}=\frac{c^{2}}{1-\frac{v^{2}}{c^{2}}}\)

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

⇒ \(b_{2}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\) (which is similar to a1)

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

We can also write it as:

\(a_{1}=\gamma\)

⇒ \(a_{2}=-\gamma v\)

⇒ \(b_{1}=-\frac{v}{c^{2}}\gamma\)

⇒ \(b_{2}=\gamma\)

Following are the final form of Lorentz transformations:

\(∴ {x}’=\gamma (x-vt)\)

⇒ \({y}’=y\)

⇒ \({z}’=z\)

⇒ \({t}’=\gamma (t-\frac{v}{c^{2}}x)\)

You may also want to check out these topics given below!

Difference between Galilean and Lorentz transformation

Galilean transformation Lorentz transformation
Galilean transformation cannot be used for any random speed Lorentz transformation is applicable for any speed
According to Galilean transformation time is independent of the observer and universal According to Lorentz transformation time is relative

What is the significance of Lorentz transformation?

Following are the significance of Lorentz transformations:

  • It is a relativistic transformation which is applied when the relativistic effects like time dilation are noticeable. It is given as the product of time dilation T and Lorentz transformation γ is given as follows:
    T×\(\gamma\)
  • It can be used for transforming one reference frame to another.
  • Lorentz transformation uses speed of light to for referring frames as it is constant.

Leave a Comment

Your email address will not be published. Required fields are marked *