 #1
amathie
 2
 0
 Homework Statement:
 Take the line element ds^2= (a^2t^2−c^2)dt^2+ 2atdtdx+ dx^2+ dy^2+ dz^2, where a and c are constants. Calculate the components of the inverse metric. By identifying a suitable coordinate transformation, show that the line element can be reduced to the Minkowski line element.
 Relevant Equations:

Minkowski line element: ds^2 = dt^2 + dx^2 + dy^2 + dz^2
Matrix diagonalisation: S^1AS=D
The line element given corresponds to the metric:
$$g = \begin{bmatrix}a^2t^2c^2 & at & 0 & 0\\at & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$
Using the adjugate method: ##g^{1}=\frac{1}{g}\tilde{g}## where ##\tilde{g}## is the adjugate of ##g##. This gives me:
$$g^{1}=\begin{bmatrix}\frac{1}{c^2} & \frac{at}{c^2} & 0 & 0\\\frac{at}{c^2} & \frac{a^2t^2}{c^2}+1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$
The question now asks me to identify a coordinate transformation to reduce the line element given to the Minkowski line element.
I think what I need to do is find a matrix that operates on ##g## to give ##\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}##.
I've tried to find the eigenvalues of ##g## by the standard method (i.e. finding the characteristic equation and attempting to solve it), but as far as I can see it just makes an unfactorisable mess.
I get ##(a^2t^2c^2\lambda)(1\lambda)(1\lambda)^2=a^2t^2(1\lambda)^2##, which gives one degenerate eigenvalue of ##\lambda=1## with multiplicity...3? But I can't find the fourth.
I'm selfteaching this stuff so have no idea what I'm doing really. Any help very appreciated!
$$g = \begin{bmatrix}a^2t^2c^2 & at & 0 & 0\\at & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$
Using the adjugate method: ##g^{1}=\frac{1}{g}\tilde{g}## where ##\tilde{g}## is the adjugate of ##g##. This gives me:
$$g^{1}=\begin{bmatrix}\frac{1}{c^2} & \frac{at}{c^2} & 0 & 0\\\frac{at}{c^2} & \frac{a^2t^2}{c^2}+1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$
The question now asks me to identify a coordinate transformation to reduce the line element given to the Minkowski line element.
I think what I need to do is find a matrix that operates on ##g## to give ##\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}##.
I've tried to find the eigenvalues of ##g## by the standard method (i.e. finding the characteristic equation and attempting to solve it), but as far as I can see it just makes an unfactorisable mess.
I get ##(a^2t^2c^2\lambda)(1\lambda)(1\lambda)^2=a^2t^2(1\lambda)^2##, which gives one degenerate eigenvalue of ##\lambda=1## with multiplicity...3? But I can't find the fourth.
I'm selfteaching this stuff so have no idea what I'm doing really. Any help very appreciated!