How Does Rapidity Influence the Lorentz Transformation Matrix?

[Brian-san]

1. Homework Statement :
Consider a two dimensional Minkowski space (1 spatial, 1 time dimension). What is the condition on a transformation matrix [itex]\Lambda[/itex], such that the inner product is preserved? Solve this condition in terms of the rapidity.

2. Homework Equations :
Rapidity Relations:
[tex]\beta=tanh\theta, \gamma=cosh\theta[/tex]

Inner Product:
[tex]u^T \eta u[/tex]

3. The Attempt at a Solution :
From the definition of inner product, to preserve inner product when [itex]u'=\Lambda u[/itex], we must have [itex]\Lambda^T\eta\Lambda=\eta[/itex]

In matrix form:
[tex]\left[ \begin{array}{cc} \lambda_1 & \lambda_3 \\ \lambda_2 & \lambda_4 \end{array} \right]\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]\left[ \begin{array}{cc} \lambda_1 & \lambda_2 \\ \lambda_3 & \lambda_4 \end{array} \right]=\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right][/tex]

This gives three relations:
[tex]\lambda_1^2-\lambda_3^2=1, \lambda_2^2-\lambda_4^2=-1, \lambda_1\lambda_2=\lambda_3\lambda_4[/tex]

After substituting and solving the equations, letting [itex]\lambda_1=\lambda[/itex], I get the final form of the matrix as:
[tex]\Lambda=\left[ \begin{array}{cc} \lambda & \pm\sqrt{\lambda^2-1} \\ \pm\sqrt{\lambda^2-1} & \lambda \end{array} \right][/tex]

The two matrices are inverses of each other which can be shown easily. Since the Lorentz transformations are like rotations that mix space and time dimensions, I know the final result in terms of rapidity should be:
[tex]\Lambda=\left[ \begin{array}{cc} cosh\theta & \pm sinh\theta \\ \pm sinh\theta & cosh\theta \end{array} \right][/tex]

However, I'm not sure how to get the final step I need, by showing [itex]\lambda=cosh\theta[/itex]. All I can say for sure is based on how the transformations behave at v=0 (returns identity matrix), and v=c (rapidity is infinite), is that λ(0)=1 and the function is strictly increasing to infinity. Obviously hyperbolic cosine fits that description, but so do a lot of other functions. So, I'm not sure what specifically will let me get the function I need.

Thanks.

 

[Ben Niehoff]

What you have so far is correct. You have shown that you can write

[tex]\Lambda = \begin{pmatrix} \cosh \rho & \pm \sinh \rho \\ \pm \sinh \rho & \cosh \rho \end{pmatrix}[/tex]

for some quantity [itex]\rho[/itex]. What is left is to show that [itex]\rho[/itex] is, in fact, the rapidity. To accomplish that, you should take a particle at rest and then boost it to some velocity [itex]\beta = v/c[/itex]; then show how [itex]\rho[/itex] is related to [itex]\beta[/itex].

I.e., you should know independently what the velocity 4-vector (2-vector in this case) should look like after a boost from rest, so use that to find how to interpret the parameter [itex]\rho[/itex].