# Q: Why does relativistic length contraction (Lorentz contraction) happen?

Physicist: This probably should have come before the last post.

Length contraction is a symptom of “tilted now planes”.  For someone moving past you events physically in front of them happen earlier than they should (according to you), and events physically behind them happen later (according to you).

Here’s the idea in a nutshell: you’re in the middle of a train when the front and back are hit by lightning.  People on the train will see the lightning at the front of the train a little earlier than someone on the tracks because they’re moving toward it, and will see the lightning at the back of the train a little later because they’re moving away from it.  At least, that’s the way someone on the tracks would explain it.

However, the laws of physics are blind to “uniform movement”.  That is, all physical laws are exactly the same whether you’re moving (at a constant speed) or not.  And that’s relativity.  So both points of view are equally correct.  That summary was a little fast because it’s covered in a lot more detail here: Q: According to relativity, two moving observers always see the other moving through time slower. Isn’t that a contradiction? Doesn’t one have to be faster?

Someone of the tracks (blue) sees lightning hit the front and back of a train, simultaneously, as it passes by. Someone on the train (red) sees the lightning at the front first. Both are right. So, in general (not just with lightning), when someone passes by events happening physically in front of them happen a little sooner than they should (according to you).

The traditional example is the barn-running pole-vaulter thought experiment.

A pole vaulter runs through a barn very, very fast with a pole that (when it’s standing still) is about as long as the barn.  From her point of view the barn, which is rushing past her, is contracted so that her pole (briefly) is sticking out of both ends of the barn.  The farmer, who leaves the doors to his barn open, because this happens all the time, sees the vaulter and her pole contracted so that (briefly) the entire pole is inside the barn.

They’re both right, and here’s why!  Consider the two events (A) the back of the pole entering the barn and (B) the front of the pole exiting the barn.

For the farmer event A happens first, then event B happens second.

For the farmer the back of the pole enters the barn before the front of the pole exits the barn. Obviously, the pole shrank.

The pole vaulter sees the same process, but sees the events in front of her happening sooner, and the events behind her happening later.  So, in this case, she sees event B first and event A second.

For the pole vaulter the front of the pole exits the barn first, and then the back of the pole enters the barn. Obviously, the barn shrank.

So, time dilation, length contraction, and the rearrangement of events are just three sides of the same weirdly shaped coin.

I should point out, that there’s a weakness in the language that makes it sound like relativistic effects aren’t real events; “from one point of view…”, “when one person looks at the other they see…”, etc.  Length contraction is a completely real effect.  At very high speeds objects really do contract in the direction of motion.  However, you have to be really trucking along before it becomes an issue.  What follows is answer gravy.

Answer gravy: The best way to describe how strong relativistic effects are is to use “$\gamma$” (“gamma”), and $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ where “c” is the speed of light and “v” is the relative speed of the object in question.  The amount that the mass is multiplied, the amount that time slows, and the amount that length contracts, are all $\gamma$.  Gamma is such a useful measure, that you’ll often hear physicists refer to things “moving with a gamma of ___”, instead of stating the actual speed.

That whole opening it to explain why relativistic effects are so unnoticable.  Below is a graph of speed on the x-axis (from 0 to c) and gamma on the y-axis.  Just to put things in perspective I’ve included some sample speeds, that people have experienced.

No one has ever experienced a relativistic effect that they could feel. We can measure the effects, but the effects are extremely small on the everyday scale. The horizontal line is "gamma = 1".

The Apollo 10 service and crew modules (the fastest manned vehicle ever) managed to get all the way up to 0.0037% of light speed, which is just unimaginably fast.  Commander Stafford, et al., experienced a $\gamma$ of approximately $\gamma = 1.00000000068$.  So from the perspective of everyone on the ground, the 11.03m long module shrank by approximately 7.5 nanometers.

More gravy!: Mathematically, the way a physicist might describe length contraction (more exactly) would be to use the “spacetime interval“.  If you have two events happening at the points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, at the times $t_1$ and $t_2$.  The spacetime interval, S, is defined as:

$S^2= c^2(t_2-t_1)^2-(x_1-x_2)^2 -(y_1-y_2)^2 -(z_1-z_2)^2$ $= c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2$

S is useful because, although relativistic effects change the x’s and t’s and whatnot, S remains constant for everybody.  Here goes!

When something is sitting still the way you measure it is to get out a ruler.  When something is moving past you the way to measure it’s length is to time how long it takes to get past a point.  If it’s moving at 10 kph and it takes 2 hours to pass, then it must be 20 km long.  So, we have two events.  When the front of the object passes the measuring point, and when the back of the object passes the measuring point.

The length of an object in the frame where something is sitting still, we’ll call “L”.  When something is moving it’s length is LM.

An object moving past an arbitrary point used for measurement. In the objects stationary frame the arrow is doing the moving, and in the arrows frame the object is doing the moving.

In the stationary frame:

$S^2 = c^2(\Delta t)^2 - (\Delta x)^2 = c^2 \left( \frac{L}{v} \right)^2 - L^2 = L^2 \left( \frac{c^2}{v^2}-1\right)$.  That is, the distance between is just the length, and the time between is how long it takes for the measuring point to traverse that distance.

Similarly, in the moving frame:

$S^2 = c^2(\Delta t^\prime)^2 - (\Delta x^\prime)^2 = c^2 \left( \frac{L_M}{v} \right)^2 - 0^2 = \frac{c^2}{v^2} L_M^2$.  That is, the events happen in the same place, but at different times given by how long it takes the object (now LM long) to pass.  The primes (the ” ‘ “) are to indicate that the position and time are different in the new frame.

But, the spacetime interval is always the same, even if everything else is different.  So:

$\begin{array}{ll}\frac{c^2}{v^2} L_M^2 = S^2 = L^2 \left( \frac{c^2}{v^2}-1\right)\\\Rightarrow L_M^2 = L^2 \left(1- \frac{v^2}{c^2}\right)\\\Rightarrow L_M = L \sqrt{1- \frac{v^2}{c^2}}\\\Rightarrow L_M = L\frac{1}{\gamma}\end{array}$

Gamma again!  Good times!

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### 10 Responses to Q: Why does relativistic length contraction (Lorentz contraction) happen?

1. oscar moya says:

is length contraction is real, and not only relativistic, does that mean that i am right now really contracting, and dilating, relative to many particles, starts etc moving at near c speeds from me?

2. The Physicist says:

Yup!

3. LITO says:

If you”re traveling just under the speed of light away from earth and send a radio message back, what would happen ?

4. The Physicist says:

It would be red shifted.