Lorentz Transformations

Lorentz factor, often known as the Lorentz term, is a measurement that describes an object’s measurements of time, length, and other physical properties, which vary when it moves. The expression occurs in derivations of the Lorentz transformations and is found in a number of special relativity equations. It is named after the Dutch physicist Hendrik Lorentz, the term originates from its earlier use in Lorentzian electrodynamics. 

Lorentz Factor Definition

Lorentz factor is the factor that describes the dilated time of a moving clock evaluated in a stationary frame in the time dilation formula.

The Lorentz factor, which is typically represented by the Greek letter gamma (γ), is equal to:

γ = 1 / √(1-(v/c)²) 

where,
γ is Lorentz Factor
v is Relative velocity of two observers
c is the speed of light in a vacuum

Since the quantity (v/c) is often denoted by the β symbol, the equation above can be simplified as follows:

γ = 1 / √(1-(β)²) 

Inertial Frame of Reference

An inertial frame of reference is defined as, In relation to the imaginary inertial reference frame, the inertial frame is at rest or moves at a constant speed. A hypothetical inertial coordinate system is surrounded by an inertial frame that is either stationary or moving at a constant speed. A frame of reference that is inertial is one in which Newton’s law is valid. That means a body will remain at rest or continue to move uniformly if there is no outside force acting on it.

Basically, the reference frame is a component of the environment that is utilized to measure the motion of the moving item. If Newton’s law of motion holds true in reality, then any reference systems that rotate uniformly and precisely around a common system must also follow the law of inertia. It is known as an inertial frame whenever it is in a situation like this. A platform at a train station is an illustration of an inertial reference system. Since the platform is stationary, it satisfies the requirement of not accelerating. The platform’s objects satisfy the law of inertia.

Non-Inertial Frame of Reference 

A frame that is accelerated relative to the assumed inertial frame of reference is referred to as non-inertial. In these frames, Newton’s law will not hold true. To apply Newton’s law in a non-inertial frame of reference we need to apply a mysterious force known as pseudo force. A frame that is neither stationary nor moving at a constant speed is said to be non-inertial. One can either travel at a constant velocity on a circular road or at a constant speed in a non-inertial coordinate system while accelerating. The body in this frame experiences acceleration. 

Compared to the inertial coordinate system that is used by default, the non-inertial coordinate system is accelerated. When the frame is out of balance, accelerometers in such frames typically detect non-zero accelerometers. In the vehicle at a traffic signal, the car has stopped moving. As soon as the traffic light turns green, the car starts to go forward. The car is in a non-inertial frame of reference during this acceleration.

Lorentz Transformation

The Lorentz transformations are a one-parameter family of linear transformations from a frame in spacetime that is in a fixed position to a frame that is moving with constant speed. These transformations are named after a Dutch physicist, Hendrik Lorentz.

The formula for Lorentz transformation can be given as,

t’ = γ(t – (vx)/c²)

x’ = γ(x – vt)

y’ = y 

z’ = z

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

Since the Galilean transformation cannot explain why observers traveling at various speeds measure different distances and experience events in a different sequence even if light travels at the same speed in all inertial reference frames, the Lorentz transformations were developed from it.

From the Galilean transformation, we can derive Lorentz transformation as,

x’ = a₁x + a₂t

y’ = y 

z’ = z

t’ = b₁x + b₂t

With speed v in non-inertial frame S, the origin of the inertial frame is x’ = 0. Let x = vt represent the position in non-inertial frame S at time t for the light beam.

Therefore, x’ = 0 = a₁x + a₂t ⇒ x = -(a₂/a₁) t = vt

where, 

a₂/a₁ = -v

Now, the above equation can be written as,

x’ = a₁x + a₂t = a₁(x + (a₂/a₁)t) = a₁ (x – vt)

a₁²(x – vt₂) + y’² + z’² – c²(b₁x + b₂t)² = x² + y² + z² – c²t²

a₁²x² – 2a₁²xvt + a₁²v²t² – c²b₁²x² – 2c²b₁b₂xt – c²b₂²t² = x² – c²t²

(a₁² – c²b₂)x² = x²  OR  a₁² – c²b₁² = 1

(a₁²v² – c²b₂²)t² = -c²t²  OR  c²b₂² – a₁²v² = c²

(2a₁²v + 2b₁b₂c²)xt = 0  OR  b₁b₂c² = -a₁²v

b₁²c² = a₁² – 1

b₂²c² = c² + a₁²v²

b₁²b₂²c⁴ = (a₁² – 1) (c² + a₁²v²) = a₁⁴v²

a₁²c² – c² + a⁴v² – a₁²v² = a₁⁴v²

a₁²c² – a₁²v² = c²

a₁²(c² – v²) = c²

a₁² = c²/(c² – v²) = 1/(1 – v²/c²)

a₂ = -v(1 / √(1 – v²/c²))

b₁²c² = (1/(1 – v²/c²) – 1)

b₁²c² = (1-(1 – v²/c²))/(1 – v²/c²) = (v²/c²)/(1-(v²/c²)) = v²/c²(1/1-(v²/c²))

b₁² = v²/c⁴ (1/1-(v²/c²))

b₁ = -v/c² (1/√(1-(v²/c²)))

b₂²c² = (c² + v²(1/1-(v²/c²)) = c²(1 – v²/c²) + v² / 1-(v²/c²) = c²-v²+v²/1-(v²/c²) = c² / 1 – (v²/c²)

b₂² = 1/1-(v²/c²)

b₂ = 1/√1-(v²/c²)  (b₂ is close to a₁)

γ = 1 / √1 – (v²/c²)

the equation can also be written as,

a₁ = γ

a₂ = -γv

b₁ = -(v/c²)γ

b₂ = γ

The final form of Lorentz transformation is: 

• x’ = γ(x – vt)
• y’ = y
• z’ = z
• t’ = γ(t – (v/c²)x)

Time Dilation

Either a difference in gravitational potential between their locations or the relative velocities between the two frames of reference produce time dilation (gravitational time dilation taken from general relativity). “Time dilation” describes the velocity-related effect, when it cannot be determined.

Assume that in the reference frame, the time interval between the events is denoted by the symbol Δt0 and is known as proper time or one-position time. In another reference frame (i.e. the observers’ reference frame) the time interval between two events is denoted by the symbol Δt. the observer time will always be higher than the proper time. This is what we refer to as time dilation.

The time dilation formula can be written as,

Δt = Δt₀ / √(1-(v/c)²)

where,
Δt is Observer time or two-position time
Δt₀ is Proper time or one position time
v is Relative velocity of two observers
c is the speed of light in a vacuum

Properties of Lorentz Factor

Following are the properties of Lorentz Factor

• The value of the Lorentz Factor is always greater than 1. (γ > 1)
• The Lorentz factor is very close to 1, If the clock speed (v) is slow in comparison to the speed of light (c).
• If the clock speed, v, approaches the speed of light, c, the Lorentz factor increases significantly.

Also, Check

• Relative Motion
• Special Theory of Relativity

Solved Examples on Lorentz Factor

Problem 1: If the relative velocity between the two observers is 120 m/s, Determine the Lorentz factor. (Speed of light is 3 x 108 m/s).

Solution:

Given:

Relative Velocity (v) = 120 m/s

Speed of light (c) = 3 x 108 m/s

Therefore, Lorentz factor is given as,

γ = 1 / √(1-(v/c)²)

γ = 1 / √(1-(120/3 x 108)²)  

= 1 / √(1 – (14400 / 9 x 1016))

= 1 

Problem 2: If the relative velocity between the two observers is 300 m/s, Determine the Lorentz factor. (Speed of light is 2.99 x 108 m/s).

Solution:

Given:

Relative Velocity (v) = 300 m/s

Speed of light (c) = 2.99 x 108 m/s

Therefore, Lorentz factor is given as,

γ = 1 / √(1-(v/c)²)

γ = 1 / √(1 – (300/3 x 108)²)  

= 1 / √(1 – (90000 / 8.9401 x 1016))

= 1

Problem 3: The ratio of v to c is given as 26.7 x 10-8, Determine the Lorentz factor. (Speed of light is 2.99 x 108 m/s).

Solution:

Given:

The ratio of v to c (v/c) = β = 26.7 x 10-8

Therefore, Lorentz factor is given as,

γ = 1 / √(1-(v/c)²)

γ = 1 / √(1-(26.7 x 10^-8)²)  

= 1

Problem 4: If the time interval is 25 seconds and the observer velocity is 30,000 m/s, Find the relative time.

Solution:

Given,

Time interval (Δt₀) = 25 seconds

Observer velocity (v) = 30,000 m/s

Relative time Δt = Δt₀ / √(1 – v²/c²)

= 25 / √(1 – 30,000²/299,792,4582)

= 25 sec

Therefore, the relative time is 25 seconds.

Problem 5:  Find the relative time, If the time interval is 32 seconds and the observer velocity is 50,000 m/s.

Solution:

Given,

Time interval (Δt₀) = 32 seconds

Observer velocity (v) = 50,000 m/s

Relative time Δt = Δt₀ / √(1 – v²/c²)

= 32 / √(1 – 50,000²/299,792,4582)

= 32 sec

Therefore, the relative time is 32 seconds.

FAQs on Lorentz Factor

Question 1: What is the Lorentz force?

Answer:

The force exerted on a charged particle travelling along an electric field and magnetic field is known as the Lorentz force.

Question 2: Which devices use the Lorentz force? 

Answer: 

Lorentz force is used in magnetrons, cyclotrons and other circular route particle accelerators, mass spectrometers, velocity filters, and Lorentz force velocimetry.

Question 3: Explain the working of Lorentz transformation.

Answer: 

A Lorentz transformation is the relationship between two different coordinate frames that are travelling apart from one another at a constant speed.

Question 4: State a few effects of Lorentz’s transformation. 

Answer: 

The Lorentz transformation has several noticeable effects, but one of them is the requirement to give up simultaneity as a universal concept. Also relative is simultaneity.

Question 5: Give a few applications of Lorentz transformation.

Answer: 

Three accurate predictions of Lorentz transformations are as follows:

1. Length Contraction
2. Time Dilation
3. Relativity of Simultaneity

Question 6: What is time dilation? 

Answer: 

Assume, a clock that is watched over by two distinct people. One  observer is travelling at the speed of light, with one being at rest. Then, time dilation is the existence of a time discrepancy between the two clocks.