A.5.2 Postulates of special relativity (HL only)

The Two Postulates of Special Relativity

In 1905, Albert Einstein introduced the theory of special relativity, which fundamentally changed our understanding of space and time.

The theory is based on two postulates:

Definition

The first postulate of special relativity

The First Postulate: The Principle of Relativity

The laws of physics are the same in all inertial reference frames.

Note

An inertial reference frame is one in which an observer is either at rest or moving with a constant velocity (not accelerating). (defined in A.5.1)

This postulate means that no inertial observer can perform an experiment to determine whether they are at rest or in uniform motion.

Example

Imagine you are in a train moving at a constant velocity.
If you drop a ball, it falls straight down, just as it would if the train were stationary.
This is because the laws of physics (such as gravity) are the same in both scenarios.

Definition

The second postulate of special relativity

The Second Postulate: The Constancy of the Speed of Light

The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or the observer.

This postulate is revolutionary because it contradicts our everyday experiences with relative motion.

Example

If a car is moving at 60 km/h and a ball is thrown forward at 20 km/h, an observer on the ground would measure the ball’s speed as 80 km/h.
However, light behaves differently.
Even if a spaceship is moving at 99% of the speed of light and emits a beam of light, an observer on the spaceship and an observer at rest will both measure the light’s speed as $3.00 \times 10^{8} m/s$ .

Note

The constancy of the speed of light was experimentally confirmed by the Michelson-Morley experiment, which failed to detect any variation in the speed of light due to Earth’s motion through space.

Implications of the Postulates

The two postulates of special relativity lead to profound changes in our understanding of space and time.

1. Time Dilation

Definition

Time dilation

Time dilation is the phenomenon where time passes more slowly for an observer in motion relative to a stationary observer.

This effect is described by the equation:

$Δ t = γ Δ t_{0}$

where:

$Δ t$ is the time interval measured by the stationary observer.
$Δ t_{0}$ is the proper time interval (measured in the moving observer’s frame).
$γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ is the Lorentz factor, where $v$ is the relative velocity and $c$ is the speed of light.

Definition

Proper time

Proper time ( $Δ t_{0}$ ) is the time interval between two events measured in the frame where the events occur at the same location.

Example question

Applying time dilation

Imagine a spaceship traveling at 80% of the speed of light ( $0.80 c$ ). If 10 years pass on the spaceship (proper time), how much time passes for an observer on Earth?

Solution

Calculate the Lorentz factor: $γ = \frac{1}{\sqrt{1 - (0.80)^{2}}} = 1.67$
Use the time dilation formula: $Δ t = γ Δ t_{0} = 1.67 \times 10 years = 16.7 years$

For the Earth observer, 16.7 years have passed while only 10 years have passed on the spaceship.

2. Length Contraction

Definition

Length contraction

Length contraction is the phenomenon where an object in motion appears shorter along the direction of motion to a stationary observer.

This effect is described by the equation:

$L = \frac{L_{0}}{γ}$

where:

$L$ is the contracted length measured by the stationary observer.
$L_{0}$ is the proper length (measured in the object’s rest frame).
$γ$ is the Lorentz factor.

Definition

Proper length

Proper length ( $L_{0}$ ) is the length of an object measured in the frame where the object is at rest.

Example question

Applying length contraction

A spaceship has a proper length of 100 meters. If it travels at 90% of the speed of light ( $0.90 c$ ), what is its length as measured by an observer at rest?

Solution

Calculate the Lorentz factor: $γ = \frac{1}{\sqrt{1 - (0.90)^{2}}} = 2.29$
Use the length contraction formula: $L = \frac{L_{0}}{γ} = \frac{100 m}{2.29} = 43.7 m$

The spaceship appears only 43.7 meters long to the stationary observer.

Why Do These Effects Matter?

These relativistic effects become significant at speeds close to the speed of light.
At everyday speeds, the Lorentz factor ( $γ$ ) is approximately 1, so time dilation and length contraction are negligible.
However, at relativistic speeds, these effects have profound implications for physics and technology.

Example

Muon Decay Experiments:

Muons are unstable particles with a short lifetime.
When muons are created in the upper atmosphere and travel towards Earth at near-light speeds, their lifetime (as measured by Earth observers) is extended due to time dilation.
This allows them to reach the surface before decaying, providing experimental evidence for relativity.

The Ultimate Speed Limit

One of the most revolutionary implications of special relativity is that the speed of light is the ultimate speed limit.
No object with mass can reach or exceed this speed.
As an object approaches the speed of light, its energy and momentum increase dramatically, requiring infinite energy to reach $c$ .

Note

This limitation has profound implications for our understanding of the universe, including the behavior of particles in accelerators and the nature of black holes.

Reflection

Theory of Knowledge

How do the postulates of special relativity challenge our perception of reality?
What does this tell us about the role of observation in defining physical laws?

The postulates of special relativity challenge our intuitive understanding of space and time, revealing that they are interconnected and relative.

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A.5.2 Postulates of special relativity (HL only)

The Two Postulates of Special Relativity

Implications of the Postulates

1. Time Dilation

Applying time dilation

2. Length Contraction

Applying length contraction

Why Do These Effects Matter?

The Ultimate Speed Limit

Reflection

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Introduction to Special Relativity