# Length contraction and angle change

I am new to special theory of relativity and it puzzles me at some points. For example, if we have a rod of length 1 meter, tilted at angle of 45 degrees ( x axis) and it moves at the speed of $v=0.95c$ in the direction of x axis, how come the angle at which it is tilted changes? I understand the x-length of the rod would change but why does the angle change?

## 2 Answers

The angle changes because the rod contracts only in the direction of motion. This means if we broke the rod into two components (like a vector)

$$\vec{L}=L_{x}\hat{x}+L_{y}\hat{y}$$

only the $L_{x}$ would "feel the length contraction" since its the only part in the direction of motion. So the new length vector would be

$$\vec{L}^{\prime}=L_{x}^{\prime}\hat{x}+L_{y}\hat{y}$$

Where the $L_{x}^{\prime}$ has the usually length contraction. Since the y remains the same and the x changes between the "moving" frame and the "rest" frame (and the triangle is still a right triangle) the triangle of the x and y components will have different angles. I left the algebra for you (and future users to enjoy)

• Thank you, your answer cleared the problem up. The textbook had a somewhat confusing and short solution. Thanks again ;) – user16688 Mar 2 '14 at 15:56

Because the angle depends on the length in the $x$ direction and the length in the $y$ direction, specifically the ratio between them. Therefore, if the $x$ direction is contracted while the $y$ direction remains unaffected, changing the ratio between the two, the angle will change too.