Length Contraction from Lorentz Transformations

Defining Length and Proper Length

Before we jump into the mathematics, we must be very precise about what we mean by "length." In classical physics, this is trivial. In relativity, it requires care.

• The length of an object in a given reference frame is the distance between its endpoints, determined by measuring the positions of both ends simultaneously in that frame.

The emphasis on simultaneity is the crucial point. As we know from the relativity of simultaneity, two events that are simultaneous in one frame are generally not simultaneous in another. This is the origin of length contraction.

Analogous to the concept of proper time, we also define:

• The proper length, $L_0$, of an object is its length as measured in the reference frame in which the object is at rest. The proper length is the longest possible measured length of an object.

Let's watch a short video clip that introduces the concept and sets up the problem.

Deriving Length Contraction from Lorentz Transformations

Now we have all the pieces to perform the derivation. We will follow a careful, step-by-step process.

1. Establish the Frames of Reference:

   • Let's have frame S' be the rest frame of an object, say a rod. The rod is aligned along the x'-axis. The length of the rod in this frame is its proper length, $L_0$. If its ends are at $x_1^{\prime }$ and $x_2^{\prime }$, then $L_0=x_2^{\prime }-x_1^{\prime }$.
   • Frame S is a stationary frame (e.g., a lab) with respect to which frame S' (and the rod) is moving with a constant velocity $v$ along the positive x-axis.

2. Define the Measurement in Frame S:

   • An observer in frame S wants to measure the length of the moving rod. To do this, they must record the positions of the two ends of the rod, $x_1$ and $x_2$, at the same instant in time.
   • This condition of simultaneity in the observer's frame is key: $t_1=t_2$, which means the time interval between the two measurements is $\Delta t=t_2-t_1=0$.
   • The length measured by the observer in frame S is then $L=x_2-x_1$.

3. Apply the Lorentz Transformation:
   We need to relate the coordinates in S to those in S'. We use the Lorentz transformation for the spatial coordinate $x$:
   $x^{\prime }=\gamma (x-vt)$
   Let's apply this to our two measurement events (measuring end 1 and measuring end 2):
   $x_1^{\prime }=\gamma (x_1-v{t}_1)$
   $x_2^{\prime }=\gamma (x_2-v{t}_2)$

4. Calculate the Length:
   To find the proper length $L_0$, we subtract the first equation from the second:
   $x_2^{\prime }-x_1^{\prime }=\gamma ((x_2-x_1)-v(t_2-t_1))$
   Now we can substitute our defined terms: $L_0=x_2^{\prime }-x_1^{\prime }$, $L=x_2-x_1$, and the simultaneity condition $\Delta t=t_2-t_1=0$.
   $L_0=\gamma (L-v(0))$
   $L_0=\gamma L$

5. The Result:
   Finally, we rearrange the equation to solve for $L$, the length measured in the frame where the rod is moving:
   $L=\frac{L_0}{\gamma }$
   Since $\gamma =1/\sqrt{1-v^2/c^2}$ is always greater than or equal to 1, this shows that $L\le L_0$. The length of a moving object is measured to be shorter than its proper length. This effect is length contraction.

A Common Pitfall

The condition of simultaneity ($\Delta t=0$) is what makes the derivation work. It's easy to make a mistake by confusing which frame the simultaneous measurements are made in. Let's watch a video that first demonstrates a common but incorrect derivation before showing the correct one. This is an excellent way to solidify your understanding.

For a concise, written version of the correct derivation, these notes from a Yale course are excellent.

The Muon Experiment, Revisited

In the last lesson, we explained how muons created in the upper atmosphere can reach sea level. From our perspective on Earth, the muon's internal clock runs slow due to time dilation, allowing it to survive a journey that would be impossible according to classical physics.

But what does the situation look like from the muon's perspective? In its own rest frame, its lifetime is just its proper lifetime, ${\tau }_0\approx 2.2\mu s$. There is no time dilation for the muon. So how does it reach the ground?

The answer is length contraction. From the muon's frame of reference, it is stationary, and the Earth's atmosphere is rushing up towards it at high speed (say, $v=0.995c$, giving $\gamma \approx 10$). The distance the muon needs to travel is the thickness of the atmosphere. Let's say this distance in the Earth's rest frame is $L_0=6.6$ km.

From the muon's point of view, this distance is contracted:
$L=\frac{L_0}{\gamma }\approx \frac{6.6\text{ km}}{10}=0.66\text{ km}=660\text{ m}$
Can the muon travel 660 meters in its 2.2 µs lifetime? Let's check:
$d_{\text{travel}}=v\times {\tau }_0\approx (0.995\times 3\times 10^8\text{ m/s})\times (2.2\times 10^{-6}\text{ s})\approx 657\text{ m}$
Yes, it can! The contracted distance is short enough for the muon to traverse before it decays.

This is a beautiful example of the consistency of relativity. Whether you analyze the situation from the Earth's frame (and see time dilation) or the muon's frame (and see length contraction), you arrive at the same physical conclusion: the muon reaches the ground.

Conclusion

In this lesson, we have built directly upon our knowledge of the Lorentz transformations to derive length contraction.

Key Takeaways:

• Proper Length ($L_0$) is the length of an object in its own rest frame.
• To measure the length of a moving object, an observer must locate its endpoints simultaneously in their own frame.
• Length contraction is derived from the Lorentz transformations, yielding the formula $L=L_0/\gamma$.
• An object is measured to be shorter in the direction of its motion when viewed from a frame moving relative to it. Dimensions perpendicular to the motion are unaffected.
• The phenomenon is symmetric: observers in each frame measure the other's objects to be contracted.
• The muon experiment can be explained equally well by time dilation (in the Earth's frame) or length contraction (in the muon's frame), demonstrating the self-consistency of relativity.

We have now formally derived the two most famous consequences of the Lorentz transformations. While we can always use algebra, physicists often rely on a graphical tool to visualize and solve problems in relativity. In our next lesson, we will introduce this tool: spacetime diagrams, also known as Minkowski diagrams. They provide a powerful, intuitive way to understand the geometric nature of spacetime and the effects we've just studied.