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Lorentz transformation is the process of transforming a coordinate frame to another to check the relevance of experimental measurement values in four-dimension (4D) or in space. It is a linear transformation we usually find in the study of electromagnetics. Developed by the Dutch physicist Hendrik Lorentz, the Lorentz transformation was later used by Einstein to prove the wrong assumption of the Galilean transformation in classical physics. Hence, the Lorentz transformation is an integral part of Einstein’s special relativity theory.
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Key Takeaways: Lorentz transformation, Lorentz force, Electromagnetics, Velocity, Hyperbolic rotation, Electric field, Magnetic field
What is Lorentz Transformation?
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Lorentz transformation is a family of linear transformations that consist of six parameters to observe an object from one inertial frame to another keeping compatibility of velocity with the former one in 4D. Unlike Galilean transformation, this one considers time as a relative factor. Just by adjusting the velocity with the previous frame, any amount of velocity is measurable with Lorentz transformation.
Lorentz Transformation
Sometimes the Lorentz transformation includes rotational motion as well. In the mathematical model "Minkowski space", the origin of the Lorentz transformation is considered fixed on the left, which is called hyperbolic rotation.
The rotation-less Lorentz transformation is called the Lorentz boost.
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What is Lorentz Force?
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Lorentz force is a force that acts on a point charge q when it moves with a velocity v from its position r at time t under a magnetic and electric field combination.
The expression of Lorentz force is given by,
F = q(E+ vB)
B → Magnetic field
E → Electric field
v → Velocity of the charge
q → The moving point charge
t → Time
r → Position of the charge before it moves.
Formula of Lorentz Transformation
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The formula of Lorentz transformation consists of six parameters. The formula is given by,
t’ = 11 – (vc)2. (t – vxc2)
Here,
t → Time in coordinate of unprimed frame
t’ → Time in coordinate of the primed or the second frame
11 – (vc)2 = → Lorentz factor
v → Speed of prime frame with respect to the unprimed frame along the x-axis
c → Speed of light
x → Axis of unprimed frame
x’ → 11-(vc)2.(x-vt) [with same notations]
Derivation of Lorentz Transformation
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Let's take two inertial frames S and S. The S’ frame moves with a relative velocity with respect to S along the x-axis. Therefore, the perpendicular axis y and z remain the same in the S’ frame.
So,
y = y’ and z = z’ ….(1)
Only the x-axis and the time(t) will transform into the S’ frame.
x = vt in the S frame and x’ = A(x-vt) in S’ frame. Since S’ is constant with S.
The light wavefront t = O emitted from a point (let's say O).
Light takes a time of t =SOc =x2+y2+z2c or (ct)2=x2+y2+z2
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)
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Significance of Lorentz Transformation
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The Lorentz transformation was developed to make predictions confirmed by experimental measurements. The Lorentz transformation is relatable to all the events of physics. A Lorentz transformation is also known as a hyperbolic rotation for having a left-fixed origin. In the formula of relative time in the Lorentz transformation, c represents the velocity of light, which is also the relativistic invariant. The velocity of light c is unaffected by the Lorentz boost and is the same regardless of the velocity connected to the Lorentz boost. That pretty much sums up why c is the same in all frames of reference.
Lorentz Transformation
The Lorentz transformation represents the fundamental truth about the Universe. Every event depends on an observer's reference frame. So, there is an effect on every possible event by the Lorentz transform. The Lorentz transformation is just another mathematical application in special relativity theory because it has a hyperbolic rotation.
Things to Remember
- The Lorentz transformation is a linear transformation that considers time as a relative factor.
- A Lorentz force acts on the moving charge under a combination of electric field and magnetic field.
- Lorentz factor is =11 – (vc)2
- The Y and Z axes remain the same in the new coordinate frame.
- In the Lorentz transformation, the origin is left fixed. So, it has a hyperbolic rotation.
- The Lorentz Boost is a rotationless Lorentz transformation.
- The derivation of the Lorentz transformation follows a calculation consisting of matrices.
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Sample Questions
Ques: Write down the significance of Lorentz transformation. [2 marks]
Ans: Lorentz transformation considers random velocities to measure different types of distances with it. There is no universality while considering time. Which makes the measurement more accurate, unlike Galilean transformation. Lorentz transformations also consider the speed of light in its relativity factor.
Ques: Derive the Lorentz transformation. [5 marks]
Ans: Let's take two inertial frames S and S. The S’ frame moves with a relative velocity with respect to S along the x-axis. Therefore, the perpendicular axis y and z remain the same in the S’ frame.
So,
y = y’ and z = z’ ….(1)
Only the x-axis and the time(t) will transform into the S’ frame.
x = vt in the S frame and x’ = A(x-vt) in S’ frame. Since S’ is constant with S.
The light wavefront t = O emitted from a point (let's say O).
Light takes a time of t =SOc =x2+y2+z2c or (ct)2=x2+y2+z2
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: Which electromagnetic wave is used to study the crystal structure of solids? Mention the frequency range. [2 marks]
Ans: X-ray is used to study crystal solid structures.
The frequency range is 1016-1020 Hz.
Ques: Name the six parameters of Lorentz transformation. [3 marks]
Ans:- The six parameters of Lorentz transformation depend on two inertial frames. These parameters are-
- Relative velocity (with respect to the previous frame)
- Speed of light
- Time coordinate axis of both the frame( t, t’)
- X-axis of both of the frame
- Y-axis of both frame
- Z-axis of both of the frame
Ques: Write down the difference between Galilean and Lorentz transformation. [3 marks]
Ans:
| Galilean transformation | Lorentz transformation |
|---|---|
| Galilean transformation is a part of classical mechanics. | Though Lorentz transformation is a part of electromagnetism, it contributes a vast role in special relativity theory in modern physics. |
| The Galilean transformation doesn’t consider various kinds of velocity of an object. | Lorentz transformations consider every random velocity keeping consistency with the previous frame. |
| Galilean transformation assumes that time is universal everywhere in any coordinate. | Lorentz transformation takes time as a relative factor that changes according to the observer’s coordinate. |
| Galilean transformations usually do not use the speed of light. | Lorentz transformation takes the speed of light as an integral factor to consider while measuring. |
Ques: Write down the formula of the Lorentz factor and Lorentz transformation formula of time. [3 marks]
Ans: The Lorentz factor is- ()=11-(vc)2
The Lorentz time transformation is t’=11-(vc)2. (t-vxc2)
t= time in coordinate of unprimed frame
t’=time in coordinate of the second frame
11-(vc)2 = = Lorentz factor
v = the speed of prime frame with respect to the unprimed frame along the x-axis
c = speed of light
x = axis of the unprimed frame
Ques: What is a Lorentz force? Mention the formula. [2 marks]
Ans: Lorentz force is a force that acts on a point charge q when it moves with a velocity v from its position r at time t under a magnetic and electric field combination.
The expression of Lorentz force is given by,
F = q(E+ vB)
Ques: Write any two uses of infrared rays and microwave rays. [2 marks]
Ans: Uses of infrared rays are:
- In TV remote and VCR.
- Used in grillers for cooking.
Uses of microwave rays are:
- To analyze the structure of an object at an atomic level.
- For a securely connected RADAR communication.
Ques. Depict the trajectory of a charged particle moving with velocity v as it enters a uniform magnetic field perpendicular to the direction of its motion. (CBSE 2012) [2 marks]
Ans. The force acting on the charged particle will be perpendicular to both v and S and therefore will describe a circular path.
Ques. Using the concept of force between two infinitely long parallel current-carrying conductors, define one ampere of current. (CBSE 2013) [2 marks]
Ans. One ampere of current is the value of steady current, which when maintained in each of the two very long, straight, parallel conductors of negligible cross-section; and placed one metre apart in vacuum, would produce on each of these conductors a force of equal to 2 × 10?? newtons per metre (Nm?¹) of length.
Ques. Why do the electrostatic field lines not form closed loops? (CBSE 2015) [2 marks]
Ans. Electric field lines do not form closed loops because the direction of an electric field is from positive to negative charge. So one can regard a line of force starting from a positive charge and ending on a negative charge. This indicates that electric field lines do not form closed loops.
Ques. (a) In what respect is a toroid different from a solenoid? Draw and compare the pattern of the magnetic field lines in the two cases.
(b) How is the magnetic field inside a given solenoid made strong? (CBSE 2011) [2 marks]
Ans.
(a) Solenoid consists of a long wire wound in the form of a helix where the neighbouring turns are closely spaced, whereas, the toroid is a hollow circular ring on which a large number of turns of a wire is closely wound.
(b) Magnetic field inside a given solenoid is made strong by putting a soft iron core inside it. It is strengthened by increasing the amount of current through it.
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