Lorentz Transformation Derivation

What is Lorentz Transformation?

Lorentz’s transformation in physics is defined as a one-parameter family of linear transformations. It is a linear transformation that 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.

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Lorentz Transformation Formula

Following are the mathematical form of Lorentz transformations:

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

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:

\(\begin{array}{l}\gamma =(\sqrt{1-\frac{v^{2}}{c^{2}}})^{-1}\end{array} \)

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

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

Lorentz Transformation Derivation

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

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

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.

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

Rewriting the equation:

\(\begin{array}{l}⇒{x}’=a_{1}x+a_{2}t=a_{1}(x+\frac{a_{2}}{a_{1}}t)=a_{1}(x-vt)\end{array} \)
\(\begin{array}{l}⇒a_{1}^{2}(x-vt^{2})+{y}’^{2}+{z}’^{2}-c^{2}(b_{1}x+b_{2}t)^{2}=x^{2}+y^{2}+z^{2}-c^{2}t^{2}\end{array} \)
\(\begin{array}{l}⇒a_{1}^{2}x^{2}-2a_{1}^{2}xvt+a_{1}^{2}v^{2}t^{2}-c^{2}b_{1}^{2}x^{2}-2c^{2}b_{1}b_{2}xt-c^{2}b_{2}^{2}t^{2}=x^{2}-c^{2}t^{2}\end{array} \)
\(\begin{array}{l}⇒(a_{1}^{2}-c^{2}b_{2}^{1})x^{2}=x^{2} or a_{1}^{2}-c^{2}b_{1}^{2}=1\end{array} \)
\(\begin{array}{l}⇒(a_{1}^{2}v^{2}-c^{2}b_{2}^{2})t^{2}=-c^{2}t^{2} or c^{2}b_{2}^{2}-a_{1}^{2}v^{2}=c^{2}\end{array} \)
\(\begin{array}{l}⇒(2a_{1}^{2}v+2b_{1}b_{2}c^{2})xt=0 or b_{1}b_{2}c^{2}=-a_{1}^{2}v\end{array} \)
\(\begin{array}{l}⇒b_{1}^{2}c^{2}=a_{1}^{2}-1\end{array} \)
\(\begin{array}{l}⇒b_{2}^{2}c^{2}=c^{2}+a_{1}^{2}v^{2}\end{array} \)
\(\begin{array}{l}⇒b_{1}^{2}b_{2}^{2}c^{4}=(a_{1}^{2}-1)(c^{2}+a_{1}^{2}v^{2})=a_{1}^{4}v^{2}\end{array} \)
\(\begin{array}{l}⇒a_{1}^{2}c^{2}-c^{2}+a^{4}v^{2}-a_{1}^{2}v^{2}=a_{1}^{4}v^{2}\end{array} \)
\(\begin{array}{l}⇒a_{1}^{2}c^{2}-a_{1}^{2}v^{2}=c^{2}\end{array} \)
\(\begin{array}{l}⇒a_{1}^{2}(c^{2}-v^{2})=c^{2}\end{array} \)
\(\begin{array}{l}⇒a_{1}^{2}=\frac{c^{2}}{c^{2}-v^{2}}=\frac{1}{1-\frac{v^{2}}{c^{2}}}\end{array} \)
\(\begin{array}{l}⇒a_{2}=-v\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\end{array} \)
\(\begin{array}{l}⇒b_{1}^{2}c^{2}=\frac{1}{1-\frac{v^{2}}{c^{2}}}-1\end{array} \)
\(\begin{array}{l}⇒b_{1}^{2}c^{2}=\frac{1-(1-\frac{v^{2}}{c^{2}})}{1-\frac{v^{2}}{c^{2}}}=\frac{\frac{v^{2}}{c^{2}}}{1-\frac{v^{2}}{c^{2}}}=\frac{v^{2}}{c^{2}}.\frac{1}{1-\frac{v^{2}}{c^{2}}}\end{array} \)
\(\begin{array}{l}⇒b_{1}^{2}=\frac{v^{2}}{c^{4}}.\frac{1}{1-\frac{v^{2}}{c^{2}}}\end{array} \)
\(\begin{array}{l}⇒b_{1}=-\frac{v}{c^{2}}.\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\end{array} \)
\(\begin{array}{l}⇒b_{2}^{2}c^{2}=c^{2}+v^{2}.\frac{1}{1-\frac{v^{2}}{c^{2}}}=\frac{c^{2}(1-\frac{v^{2}}{c^{2}})+v^{2}}{1-\frac{v^{2}}{c^{2}}}=\frac{c^{2}-v^{2}+v^{2}}{1-\frac{v^{2}}{c^{2}}}=\frac{c^{2}}{1-\frac{v^{2}}{c^{2}}}\end{array} \)
\(\begin{array}{l}⇒b_{2}^{2}=\frac{1}{1-\frac{v^{2}}{c^{2}}}\end{array} \)
\(\begin{array}{l}⇒b_{2}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} (\textup{ which is similar to }a_1)\end{array} \)

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

We can also write it as:

\(\begin{array}{l}a_{1}=\gamma\end{array} \)
\(\begin{array}{l}⇒ a_{2}=-\gamma v\end{array} \)
\(\begin{array}{l}⇒ b_{1}=-\frac{v}{c^{2}}\gamma\end{array} \)
\(\begin{array}{l}⇒ b_{2}=\gamma\end{array} \)

Following are the final form of Lorentz transformations:

\(\begin{array}{l}∴ {x}’=\gamma (x-vt)\end{array} \)
\(\begin{array}{l}⇒ {y}’=y\end{array} \)
\(\begin{array}{l}⇒ {z}’=z\end{array} \)
\(\begin{array}{l}⇒ {t}’=\gamma (t-\frac{v}{c^{2}}x)\end{array} \)

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’s 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:
    \(\begin{array}{l}T×\gamma\end{array} \)
  • It can be used for transforming one reference frame to another.
  • Lorentz transformation uses the speed of light for referring to frames as it is constant.

Frequently Asked Questions – FAQs

Q1

What do you mean by Lorentz Transformation?

Lorentz’s transformation in physics is defined as a one-parameter family of linear transformations. It is a linear transformation that includes rotation of space and preserving space-time intervals between two events.
Q2

After whom is the Lorentz Transformation named?

These transformations are named after the Dutch physicist Hendrik Lorentz.
Q3

What is the final form of Lorentz Transformation?

\(\begin{array}{l}{t}’=\gamma (t-\frac{v}{c^{2}}x)\end{array} \)
Q4

Among Lorentz and Galilean Transformation, which can be used for any random speed?

Unlike Galilean, Lorentz’s transformation is applicable for any speed.
Q5

Among Lorentz and Galilean Transformation, which perceives time to be independent and universal?

According to Galilean transformation, time is independent of the observer and universal.
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  1.  shrish kumar
    July 21, 2022 at 2:14 pm

    IT’S VERY USEFUL. THANK YOU BYJU’S

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