# 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?

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.