Special Relativity problem: Lorentz transformation and Reference Frames

[James Chase Geary]

Homework Statement

S and S' are in standard configuration with v=αc (0<α<1). If a rod at rest in S' makes an angle of 45^o with Ox in S and 30^o with O'x in S', then find α.

Homework Equations

We are in the world of Lorentz transformations so we have
t'=(t-vx/c²)/(1-v²/c²)^1/2
x'=(x-vt)/(1-v²/c²)^1/2

Angles are involved. For a rotation of frames through angle θ we have
x'=xcosθ + Tsinθ
T'=-xsinθ+Tcosθ
where T=ict and T'=ict'

The Attempt at a Solution

I am unclear how to apply the angles here. It is definitely not 45-30=15 is the angle to be used as the rotation angle. Going that route I always seem to get something like α=.268i.
Working through the formulas, I get to
α=√(1-(1/cos²θ))
θ=15 gives the answer above; the back of the book gives √(2/3) as the answer. Either I'm getting a wrong relation for α or else the angle I'm using is wrong.

Note this is from D'Inverno, Introducing Einstein's Relativity, Exercise 3.1

 

[PeroK]

Have you drawn a diagram?

PS I don't understand why you are looking at rotating axes, which I guess is what you are doing.

 

[James Chase Geary]

Yes, but I don't have the ability to add it on here right now. Two frames, S and S', in standard configuration (that is, with axes parallel and the motion of S' in the +x-direction, so we can use special Lorentz transformations), with the velocity v=αc in the +x-direction. The rod has one end at the origin in each frame; in the S-frame, its angle with the x-axis is 45 degrees, and in the S'-frame, it is 30 degrees.

The problem seems to be asking to find an expression for alpha in terms of the angle of rotation, then to substitute in the angle of rotation to get the value of alpha. I'm having difficulty finding out what the angle of rotation from S to S' is though. The book hasn't specifically discussed length contraction at this point, so I don't think I'm supposed to try using anything directly related to that for this problem.

 

[PeroK]

Yes, but I don't have the ability to add it on here right now. Two frames, S and S', in standard configuration (that is, with axes parallel and the motion of S' in the +x-direction, so we can use special Lorentz transformations), with the velocity v=αc in the +x-direction. The rod has one end at the origin in each frame; in the S-frame, its angle with the x-axis is 45 degrees, and in the S'-frame, it is 30 degrees.

The problem seems to be asking to find an expression for alpha in terms of the angle of rotation, then to substitute in the angle of rotation to get the value of alpha. I'm having difficulty finding out what the angle of rotation from S to S' is though. The book hasn't specifically discussed length contraction at this point, so I don't think I'm supposed to try using anything directly related to that for this problem.

If you really don't think you can use length contraction - although I doubt this very much - then you'll just have to use the full Lorentz Transformation. But, you are not going to get anywhere by rotating the axes of one frame.

 

[James Chase Geary]

Okay, I've got it figured out. The different angles come into play because the rod is only length-contracted in the x-direction. Because of this, the y-direction of the rigid rod in both S and S' are the same. So we can use trigonometry to get a relation between the lengths of the rods in both frames. Then, using the length contraction formula l=(1-v²/c²)^1/2*l₀, and substituting in v=αc, we can solve for α=(2/3)^1/2.

Phew!

 

[guijongustavo]

I made the answer in this video:

 

[kuruman]

I made the answer in this video $\dots$

Thank you for your detailed video solution. It complements the outline posted in #5 more than five years ago.