Unraveling Time Dilation and Length Contraction:
Author: Ganesh Swaminathan
Date: Sunday, 16th April 2023
Place: Shri Ram Spandhana, Bangalore
Introduction:
Time Dilation:
Time dilation occurs when time appears to run slower for an object in motion relative to an observer at rest. The difference in the rate at which time passes depends on the relative speed between the two objects. The Lorentz factor, denoted by γ (gamma), plays a crucial role in time dilation:
γ = 1 / √(1 - v²/c²) where:
v is the relative velocity between the two observers
c is the speed of light
The time dilation equation is given by:
Δt₀ is the time interval experienced by the stationary observer (the "proper" time)
As the relative velocity (v) increases, γ also increases, causing time to dilate or slow down for the moving observer. This phenomenon has been experimentally confirmed through various experiments.
Length Contraction:
Length contraction is another relativistic effect that arises due to the constancy of the speed of light. An object in motion appears shorter in the direction of its motion when observed by a stationary observer. The length contraction equation, which also involves the Lorentz factor, is given by:
L = L₀ / γ. where:
L is the contracted length of the object as measured by the stationary observer
L₀ is the proper length of the object as measured by an observer at rest relative to the object (in the object's rest frame)As the relative velocity (v) increases, the factor γ increases, resulting in a decrease in L, which in turn leads to length contraction.
Time Dilation Explanation with Example:.jpeg)
A light clock consists of two mirrors facing each other, with a light beam bouncing between them. The time it takes for the light beam to travel from one mirror to the other and back is considered one "tick" of the clock.
Now, let's consider two scenarios:
Stationary light clock:
Imagine a stationary light clock, with an observer (Observer A) also at rest relative to the clock. For Observer A, the light beam travels vertically between the mirrors, covering a distance of 2L (L being the distance between the mirrors). If 'c' is the speed of light, the time it takes for the light beam to complete one tick (proper time, Δt₀) can be calculated as: Δt₀ = (2L) / c
Moving light clock:
Now, let's assume the light clock is inside a spaceship moving horizontally at a constant velocity (v) relative to another observer (Observer B) outside the spaceship.
For Observer B, the light beam inside the moving spaceship appears to travel diagonally between the mirrors, as it moves up and down while the spaceship travels horizontally. The light beam's diagonal path forms a right triangle, with the distance between the mirrors (L) as one side and the horizontal distance traveled by the spaceship during one tick as the other side. The diagonal path (hypotenuse) has a length of 2D.
Since the spaceship is moving at a constant velocity (v), the horizontal distance it covers during one tick can be calculated as:
Horizontal distance = v * Δt where Δt is the dilated time interval as measured by Observer B.
Now, using the Pythagorean theorem, we can relate the diagonal distance (2D) to the vertical distance (2L) and the horizontal distance:
(2D)² = (2L)² + (v * Δt)²
The time it takes for the light beam to travel the diagonal path (dilated time, Δt) can be calculated as:
Δt = (2D) / c
By combining the equations, we can relate the proper time (Δt₀) and the dilated time (Δt):
Δt = Δt₀ / √(1 - v²/c²)
The term √(1 - v²/c²) is the Lorentz factor (γ).
As the relative velocity (v) between the spaceship and Observer B increases, the Lorentz factor (γ) also increases, causing the dilated time (Δt) to be longer than the proper time (Δt₀). This demonstrates time dilation, where the moving clock (light clock in the spaceship) appears to run slower as observed by the stationary observer (Observer B).
The light clock example with two mirrors demonstrates the concept of time dilation in Special Relativity. When the clock is in motion relative to an observer, the time it takes for the light beam to complete one tick appears longer for the stationary observer compared to an observer at rest with the clock. This effect arises due to the constancy of the speed of light and the relative motion between the observers and the moving clock.
Length Contraction Example
The fascinating concepts of time dilation and length contraction, born out of Einstein's theory of Special Relativity, have reshaped our understanding of space and time. By appreciating the deep connection between these effects, the Lorentz factor, and the constancy of the speed of light, we can better comprehend the astonishing nature of our universe. As we continue to explore the cosmos, these relativistic effects will play a crucial role in shaping our future discoveries and technological advancements.
Time Dilation:
Time dilation occurs when time appears to run slower for an object in motion relative to an observer at rest. The difference in the rate at which time passes depends on the relative speed between the two objects. The Lorentz factor, denoted by γ (gamma), plays a crucial role in time dilation:
γ = 1 / √(1 - v²/c²) where:
c is the speed of light
The time dilation equation is given by:
Δt = Δt₀ * γ where:
Δt is the time interval experienced by the moving observer (the "dilated" time)Δt₀ is the time interval experienced by the stationary observer (the "proper" time)
As the relative velocity (v) increases, γ also increases, causing time to dilate or slow down for the moving observer. This phenomenon has been experimentally confirmed through various experiments.
Length Contraction:
Length contraction is another relativistic effect that arises due to the constancy of the speed of light. An object in motion appears shorter in the direction of its motion when observed by a stationary observer. The length contraction equation, which also involves the Lorentz factor, is given by:
L = L₀ / γ. where:
L is the contracted length of the object as measured by the stationary observer
L₀ is the proper length of the object as measured by an observer at rest relative to the object (in the object's rest frame)As the relative velocity (v) increases, the factor γ increases, resulting in a decrease in L, which in turn leads to length contraction.
Time Dilation Explanation with Example:
.jpeg)
Now, let's consider two scenarios:
Stationary light clock:
Imagine a stationary light clock, with an observer (Observer A) also at rest relative to the clock. For Observer A, the light beam travels vertically between the mirrors, covering a distance of 2L (L being the distance between the mirrors). If 'c' is the speed of light, the time it takes for the light beam to complete one tick (proper time, Δt₀) can be calculated as: Δt₀ = (2L) / c
Moving light clock:
Now, let's assume the light clock is inside a spaceship moving horizontally at a constant velocity (v) relative to another observer (Observer B) outside the spaceship.
For Observer B, the light beam inside the moving spaceship appears to travel diagonally between the mirrors, as it moves up and down while the spaceship travels horizontally. The light beam's diagonal path forms a right triangle, with the distance between the mirrors (L) as one side and the horizontal distance traveled by the spaceship during one tick as the other side. The diagonal path (hypotenuse) has a length of 2D.
Since the spaceship is moving at a constant velocity (v), the horizontal distance it covers during one tick can be calculated as:
Horizontal distance = v * Δt where Δt is the dilated time interval as measured by Observer B.
Now, using the Pythagorean theorem, we can relate the diagonal distance (2D) to the vertical distance (2L) and the horizontal distance:
(2D)² = (2L)² + (v * Δt)²
The time it takes for the light beam to travel the diagonal path (dilated time, Δt) can be calculated as:
Δt = (2D) / c
By combining the equations, we can relate the proper time (Δt₀) and the dilated time (Δt):
Δt = Δt₀ / √(1 - v²/c²)
The term √(1 - v²/c²) is the Lorentz factor (γ).
As the relative velocity (v) between the spaceship and Observer B increases, the Lorentz factor (γ) also increases, causing the dilated time (Δt) to be longer than the proper time (Δt₀). This demonstrates time dilation, where the moving clock (light clock in the spaceship) appears to run slower as observed by the stationary observer (Observer B).
The light clock example with two mirrors demonstrates the concept of time dilation in Special Relativity. When the clock is in motion relative to an observer, the time it takes for the light beam to complete one tick appears longer for the stationary observer compared to an observer at rest with the clock. This effect arises due to the constancy of the speed of light and the relative motion between the observers and the moving clock.
Length Contraction Example
Imagine a train that is slightly longer than a tunnel when both are at rest. Now, let the train move at a very high speed (close to the speed of light) towards the tunnel.
From the perspective of an observer standing near the tunnel (stationary observer), the length of the moving train appears contracted due to its high speed. As a result, the train now seems short enough to fit entirely inside the tunnel. This demonstrates length contraction, where the length of the train (L) as observed by the stationary observer appears shorter than its proper length (L₀) when it was at rest.
On the other hand, from the perspective of an observer inside the train (at rest relative to the train), the train has its original length (L₀). However, due to the relative motion between the train and the tunnel, the observer on the train perceives the tunnel as contracted. Consequently, the tunnel appears too short to accommodate the train.
In this example, the moving train and the tunnel illustrate the concept of length contraction in Special Relativity. The length of an object in the direction of motion appears shorter to a stationary observer due to the relative motion between the object and the observer
Formula on a wall in Leiden, Netherlands. Lorentz was chair of theoretical physics at the University of Leiden 1877-1910
Wheels which travel at 9/10 the speed of light. The speed of the top of a wheel is 0.994 c while the speed of the bottom is always zero. This is why the top is contracted relative to the bottom.
From the perspective of an observer standing near the tunnel (stationary observer), the length of the moving train appears contracted due to its high speed. As a result, the train now seems short enough to fit entirely inside the tunnel. This demonstrates length contraction, where the length of the train (L) as observed by the stationary observer appears shorter than its proper length (L₀) when it was at rest.
On the other hand, from the perspective of an observer inside the train (at rest relative to the train), the train has its original length (L₀). However, due to the relative motion between the train and the tunnel, the observer on the train perceives the tunnel as contracted. Consequently, the tunnel appears too short to accommodate the train.
In this example, the moving train and the tunnel illustrate the concept of length contraction in Special Relativity. The length of an object in the direction of motion appears shorter to a stationary observer due to the relative motion between the object and the observer
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| By Thierry Dugnolle - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=90983525 |
Wheels which travel at 9/10 the speed of light. The speed of the top of a wheel is 0.994 c while the speed of the bottom is always zero. This is why the top is contracted relative to the bottom.
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