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:
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.
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
- Consider that 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.
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 $60 \text{ km h}^{-1}$ and a ball is thrown forward at $20 \text{ km h}^{-1}$, an observer on the ground would measure the ball’s speed as $80 \text{ km h}^{-1}$.
- 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 $3.00 \times 10^8 \, \text{m s}^{-1}$.
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.
Time Dilation
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:
$$\Delta t = \gamma \Delta t_0$$
where:
- $\Delta t$ is the time interval measured by the stationary observer.
- $\Delta t_0$ is the proper time interval (measured in the moving observer’s frame).
- $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor, where $v$ is the relative velocity and $c$ is the speed of light.
Proper time
Proper time ($\Delta t_0$) is the time interval between two events measured in the frame where the events occur at the same location.
Consider a spaceship traveling at 80% of the speed of light ($0.80c$). If 10 years pass on the spaceship (proper time), how much time passes for an observer on Earth?
Solution
- Calculate the Lorentz factor: $$\gamma = \frac{1}{\sqrt{1 - (0.80)^2}} = 1.67$$
- Use the time dilation formula: $$\Delta t = \gamma \Delta t_0 = 1.67 \times 10 \ \text{years} = 16.7 \ \text{years}$$
- For the Earth observer, 16.7 years have passed while only 10 years have passed on the spaceship.
