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Lorentz transformation derivation has a six-parameter family of linear transformations. It is a linear transformation found in electromagnetic waves which also comprises the rotation of space.
- This is developed to measure the preservation of the space-time interval between any two events.
- Lorentz transformation is the process of converting a coordinate frame to another to examine the importance of experimental measurement in 4D space.
- Dutch physicist Hendrik Lorentz developed it and later on, Einstein used this method to demonstrate the false assumptions of the Galilean transformation
- Therefore, the Lorentz transformation becomes an important part of Einstein's special relativity theory.
- The Lorentz transformation in Physics is relatable to each event.
Key Terms: Lorentz Transformation, Lorentz Factor, Galien Transformation, Electromagnetic Waves, Relativity
What is Lorentz Transformation derivation?
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Lorentz transformation is a family of linear transformations. It consists of six parameters. Lorentz transformation derivation helps to perceive an object from one inertial frame to another.
- In other words, the Lorentz transformation moves at a constant velocity v, and are relative to each other.
- Therefore it is a relationship between two different coordinate frames.
- The Lorentz transformation derivation can be simplified by using the Lorentz factor (γ).
- So instead of inputting all the factors into the equation, the Lorentz factor gives γ, which simply needs to be divided by or multiplied, depending on what one is trying to determine.
- The frames of reference is divided into two categories: Non-inertial frames & inertial frames.
Lorentz Transformation derivation
Any type of velocity is measurable with this method if the adjustment in the velocity in the previous frame is done. It also consists of rotational motion.
What is the significance of the Lorentz transformation derivation?
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The significance of lorentz transformation derivation are given below:
- It is a relativistic transformation, related only when relativistic effects are noticeable.
- All the events of physics are relatable to the Lorentz transformation.
- The other name for this method is ‘Hyperbolic Rotation.
- It is just another mathematical application in special relativity theory, as it has a hyperbolic rotation.
- It allows one to transform one reference frame to another.
- It can be simplified by using the lorentz factor.
- The speed of light is used in Lorentz transformation derivation.
Formula of Lorentz Transformation
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The Lorentz transformation derivation consists of six parameters. The given below is the formula:
t’ = 11 – (vc)2. (t – vxc2)
where,
t - Time (coordinate unprimed frame)
t’ - Time (coordinate primed frame)
11 – (vc)2 = → Lorentz factor
v - velocity
c - Speed of light
x - Axis of unprimed frame
x’ - 11-(vc)2.(x-vt) with same notations
Lorentz Transformation formula Derivation
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The Lorentz transformation derivation can be linked between the coordinate particles of the two frames. It is based on the theory of relativity. This method is only related to the modification of inertial frames.
- Take two inertial frames S and S’. Along the X axis the S frame moves also S’ frame with v.
- Therefore, in the S’ frame perpendicular axis-y and axis-z will remain unchanged.
So,
Y’ = y and z’ = z (1)
Only the t(time) and x axis will change into the frame S’
In the S frame x = vt
In the S’ frame x’ = A (x-vt)
Since S’ is constant with S.
t = O.
The time taken is t = SOc = x2 + y2 + z2 c
Similarly,
For the S’ frame, (ct')2=x'2 + y'2+z'2
With a similar wavefront represented by them, x2+y2+z2-(ct)2=x'2+y'2+z'2-(ct')2
From (1),
x2 – (ct)2 = x'2 – (ct')2
Or,
x2 – (ct)2 = (A(x-vt))2 – (c.A(t – xv )(1-1kk'))2
The coefficient of t2, A=11-(vc)2 = A’
∴ x’=11-(vc)2.(x-vt)
∴ t’=11-(vc)2.(t-vxc2)
What is the difference between Lorentz and Galilean Transformation?
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The difference between Lorentz and Gallien transformation is as follows:
| Gallien transformation | Lorentz transformation |
|---|---|
| Time is independent and universal to the observer | Time is relative |
| It is also known as Newtonian Transformation | It is also called Lorentz Boosts |
| It cannot be used at any random speed. | It can be used for any random speed. |
Things to Remember
- The Lorentz transformation moves at a constant velocity
- Lorentz transformation is a family of linear transformations.
- 11-(vc)2 = Lorentz factor
- It is a relativistic transformation, related only when relativistic effects are noticeable.
- Time is a relative factor in Lorentz transformation
- The Lorentz transformation derivation can be linked between the coordinate particles of the two frames.
- The difference between Lorentz and Gallien transformation Galien is time independent & Lorents is time relative.
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Sample Questions
Ques: Write all the parameters of Lorentz transformation? (3 marks)
Ans: The six parameters of Lorentz transformation are:
- Time coordinate axis of t, t’
- Relative velocity
- Speed of light
- X-axis of both of the frame
- Z-axis of both of the frame
- Y-axis of both frame
Ques: (a) How is a solenoid different from toroid? (CBSE 2011) (2 marks)
(b) Explain the reason behind strong solenoid of magnetic field?
Ans: a) Solenoid is a straight cylinder. On the contrary, toroid is a circular ring or hollow ring. It consists of a large amount of turns of wire firmly wound.
(b) The solenoid can be strengthen by two ways:
- By rising the amount of turns per unit.
- By increasing the strength of the current flowing the solenoid.
Ques: What are magnetic fields? (1 mark)
Ans: The magnetic field is the region around the magnet. In the magnetic field, the magnetism effect is felt.
Ques: Write the formula of Lorentz force and also explain the meaning. (2 marks)
Ans: The formula of lorentz transformation is F = q(E+ vB). Lorentz force is a mixture of the electric force and magnetic force, because of electromagnetic fields.
Ques: Write the derivation of Lorentz transformation? (5 marks)
Ans: For example, you take two inertial frames such as S and S’. Along the X axis the S frame moves also S’ frame with v. Therefore, in the frame S’ the perpendicular z axis and y axis remain the same.
So,
Y’ = y and z’ = z (1)
Only the t(time) and x axis will change into the frame S’
In the S frame x = vt
In the S’ frame x’ = A (x-vt)
Since S’ is constant with S.
t = O.
The time taken is t = SOc = x2 + y2 + z2 c
Similarly,
For the S’ frame, (ct')2=x'2 + y'2+z'2
With a similar wavefront represented by them,x2+y2+z2-(ct)2=x'2+y'2+z'2-(ct')2
From (1), we can conclude,
x2 – (ct)2 = x'2 – (ct')2
Or,
x2 – (ct)2 = (A(x-vt))2 – (c.A(t – xv )(1-1kk'))2
The coefficient of t2, A=11-(vc)2 = A’
∴ x’=11-(vc)2.(x-vt)
∴ t’=11-(vc)2.(t-vxc2)
Ques: What is the significance of lorentz transformation? (5 marks)
Ans: It is a relativistic transformation, related only when relativistic effects are noticeable.
- All the events of physics are relatable to the Lorentz transformation.
- The other name for this method is ‘Hyperbolic Rotation.
- It is just another mathematical application in special relativity theory, as it has a hyperbolic rotation.
- It allows one to transform one reference frame to another.
- It can be simplified by using the lorentz factor.
- The speed of light is used in Lorentz transformation derivation.
Ques: What is the Lorentz transformation? (2 marks)
Ans: Lorentz transformation is a family of linear transformations. It consists of six parameters. Lorentz transformation derivation, helps to perceive an object from one inertial frame to another. In other words, the Lorentz transformation moves at a constant velocity v, and are relative to each other. Therefore it is a relationship between two different coordinate frames.
Ques: Who has discovered the Lorentz transformation derivation? (1 mark)
Ans: The Dutch physicist Hendrik Lorentz has discovered the method of Lorentz transformation.
Ques: What is the formula of Lorentz transformation? (2 marks)
Ans: The formula of Lorentz transformation derivation is given by
Formula: t’ = 11 – (vc)2. (t – vxc2).
where,
t - Time (coordinate unprimed frame)
t’ - Time (coordinate primed frame)
11 – (vc)2 = → Lorentz factor
v - velocity
c - Speed of light
x - Axis of unprimed frame
x’ - 11-(vc)2.(x-vt) with same notations
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