A5-2. Special Relativity: Postulates and Lorentz TransformationsHL ONLY

1. The Postulates of Special Relativity

In 1905, Albert Einstein revolutionized physics by recognizing that Galilean transformations fail for electromagnetism and light. He formulated Special Relativity based on two fundamental, unbreakable postulates.

  1. The Principle of Relativity: The laws of physics are identical and take the same mathematical form in all inertial frames of reference. (No inertial frame is "special" or "truly at rest").
  2. The Constancy of the Speed of Light: The speed of light in a vacuum ($c \approx 3 \times 10^8\text{ m/s}$) is exactly the same for all inertial observers, completely independent of the motion of the light source or the observer.

2. The Lorentz Factor ($\gamma$)

Because the speed of light is constant, time and space must warp to compensate when objects move incredibly fast. This relativistic scaling is dictated by the Lorentz Factor ($\gamma$).

$$\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}$$
  • At everyday speeds ($v \ll c$), the factor $\dfrac{v^2}{c^2}$ is practically zero, so $\gamma \approx 1$. Relativistic effects are unnoticeable.
  • As $v$ approaches $c$, $\gamma$ approaches infinity.
  • $\gamma$ is never less than 1.

3. Lorentz Transformations

To accurately translate coordinates between frames at relativistic speeds, we must replace the Galilean equations with the Lorentz Transformations.

For Frame $S'$ moving at velocity $v$ relative to Frame $S$:

Position ($x'$):
$$x' = \gamma (x - vt)$$
Time ($t'$):
$$t' = \gamma \left(t - \dfrac{vx}{c^2}\right)$$

Notice the time equation: Time ($t'$) is no longer absolute. It depends not only on the time in the other frame ($t$) but also on the spatial position ($x$).

4. Relativistic Velocity Addition

Because nothing can exceed the speed of light, we can no longer simply add velocities ($u = u' + v$) as Galileo did.

Velocity Addition Formula:

$$u' = \dfrac{u - v}{1 - \dfrac{uv}{c^2}}$$

Or, to find the velocity measured by the stationary observer:

$$u = \dfrac{u' + v}{1 + \dfrac{u'v}{c^2}}$$

Example 2: Adding Near-Light Speeds

Problem: A spaceship ($S'$) flies past Earth ($S$) at $v = 0.8c$. The spaceship fires a laser torpedo forward at $u' = 0.5c$ relative to the spaceship. What is the speed of the torpedo ($u$) as measured by observers on Earth?

Solution:

  • Using Galilean addition, it would be $0.8c + 0.5c = 1.3c$ (Impossible!).
  • Using Relativistic addition: $$u = \dfrac{0.5c + 0.8c}{1 + \dfrac{(0.5c)(0.8c)}{c^2}} = \dfrac{1.3c}{1 + 0.40} = \dfrac{1.3c}{1.40} \approx \mathbf{0.93c}$$
  • The torpedo remains safely below the speed of light.