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**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.