Time Dilation & Length Contraction (Symmetry)

• Symmetry of Observers:
  • A moving and stationary observer both perceive the other’s clock and rod differently.
  • For clocks:
    • A stationary observer will measure a moving clock to have a longer time interval between consecutive ticks (this is time dilation).
    • A moving observer will measure a stationary clock to have longer time intervals between ticks.
  • For rods:
    • A moving observer will measure a stationary rod to be shorter than its proper (rest) length (length contraction).
    • A stationary observer will measure a moving rod to be shorter than its rest length.

Time Dilation and Length Contraction Derivation:

Time Dilation Derivation

Concept: Time dilation describes how a moving clock is observed to tick more slowly compared to a stationary clock.

1. Consider a clock at rest in frame $K$ that ticks every $\mathrm{\Delta }t$ seconds.

2. In the stationary frame $K^{\prime }$, the clock moves with velocity $v$.

3. During one tick of the clock, light travels a distance $d$ to reflect off a mirror and return:

   • In the rest frame $K$, the light travels a total distance of $d+d=2d$.
   • In frame $K^{\prime }$, the time taken for the light to travel is $t^{\prime }=\frac{2d}{c}$.

4. The distances are related by the Lorentz transformation:

   $$d=vt$$

5. The time interval in the stationary frame $K$ is:

   $$d=c\mathrm{\Delta }t$$

6. Substituting these into the distance equation gives:

   $$2d=c\mathrm{\Delta }t=vt+vt$$

7. Solving for $t$ yields:

   $$t=\frac{2d}{c}=\frac{2v\mathrm{\Delta }t}{c}$$

8. Now, substituting back:

   $$\mathrm{\Delta }{t}^{\prime }=\frac{2d}{c}=\frac{2vt}{c}=\frac{\mathrm{\Delta }t}{\sqrt{1-\frac{v^2}{c^2}}}$$

9. Hence, the time dilation formula is:

   $$\mathrm{\Delta }{t}^{\prime }=\frac{\mathrm{\Delta }t}{\sqrt{1-\frac{v^2}{c^2}}}$$

Length Contraction Derivation

Concept: Length contraction states that the length of an object moving relative to an observer is measured to be shorter than its proper length.

1. Consider a rod of proper length $L_0$ at rest in frame $K$.

2. When the rod moves with velocity $v$ relative to an observer in frame $K^{\prime }$, its endpoints are measured simultaneously in $K^{\prime }$.

3. Let $x_1$ and $x_2$ be the positions of the endpoints of the rod in frame $K$:

   $$L_0=x_2-x_1$$

4. The Lorentz transformation gives the positions in the moving frame $K^{\prime }$:

   $$x_1^{\prime }=\gamma (x_1-v{t}_1)$$$$x_2^{\prime }=\gamma (x_2-v{t}_2)$$

5. For simultaneous measurements in $K^{\prime }$ ($t_1=t_2$), we have:

   $$L^{\prime }=x_2^{\prime }-x_1^{\prime }=\gamma (x_2-vt)-\gamma (x_1-vt)$$

6. This simplifies to:

   $$L^{\prime }=\gamma (x_2-x_1)=\gamma L_0$$

7. Since $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, the length contraction formula becomes:

   $$L^{\prime }=L_0\sqrt{1-\frac{v^2}{c^2}}$$

Derivation of Relative Velocity in the Lorentz Transformation

Let's consider a particle with position $ x(t) = u_x t $ moving with velocity $ u_x $ in the $ x $-direction, and apply the Lorentz transformation to calculate its velocity in the $ K' $ frame.

The Lorentz transformations are given by:

Now, differentiating both equations with respect to time:

For time:

Thus, the relative velocity in the $ K' $ frame becomes:

Similarly, for the $ y $- and $ z $-components of velocity:

Where:

• $u_x,u_y,u_z$ are the velocity components of the particle in the stationary frame $K$.
• $u_x^{\prime },u_y^{\prime },u_z^{\prime }$ are the velocity components in the moving frame $K^{\prime }$.
• $v$ is the relative velocity between frames.
• $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is the Lorentz factor.