Simple problem driving me nuts :-)

[Fredrik]

I'm embarrassed to ask, but I think this will save me some time...

I'm trying to use the condition \Lambda^T\eta\Lambda=\eta to show that \Lambda_{0i}=-\Lambda_{i0}, where i=1,2,3. This is the algebraic version of the physically obvious condition that if the velocity associated with a homogeneous Lorentz transformation is \vec{v}, then the velocity associated with its inverse is -\vec{v}. This should be easy, but I don't see it.

(X_{0i} is row 0, column i, of the matrix X. I'm putting all the indices downstairs because I feel that's less confusing when calculations include transposes of matrices).

 

[Fredrik]

I gave it another shot, and it turned out that the "obvious" condition isn't true. If my math is correct this time, the general form of \Lambda is

\Lambda=\gamma\begin{pmatrix}1 & v^T R \\ -v & -R\end{pmatrix}

where v is a 3x1 matrix that we can think of as "their" velocity in "our" frame, and R is a 3x3 matrix that isn't orthogonal in general, but is orthogonal in the limit v \rightarrow 0 (so its components must depend on v). The velocity that corresponds to \Lambda^{-1} the same way that v corresponds to \Lambda is -R^Tv. So it's only =-v when R=0, i.e. when we have a pure boost.

 

[DrGreg]

I gave it another shot, and it turned out that the "obvious" condition isn't true. If my math is correct this time, the general form of \Lambda is

\Lambda=\gamma\begin{pmatrix}1 & v^T R \\ -v & -R\end{pmatrix}

where v is a 3x1 matrix that we can think of as "their" velocity in "our" frame, and R is a 3x3 matrix that isn't orthogonal in general, but is orthogonal in the limit v \rightarrow 0 (so its components must depend on v). The velocity that corresponds to \Lambda^{-1} the same way that v corresponds to \Lambda is -R^Tv. So it's only =-v when R=0, i.e. when we have a pure boost.

If we ignore the possibility of a spatial rotation and consider boosts only, a boost in an arbitrary direction ought to be

{\Lambda^a}_b=\begin{pmatrix}\gamma & -\gamma \textbf{v}^T \\ -\gamma \textbf{v} & \textbf{I} + (\gamma - 1)\frac{\textbf{v} \textbf{v}^T}{\textbf{v}^T \textbf{v}}\end{pmatrix}

(assuming c = 1 or equivalently x⁰ = ct).

However, when you say you are "putting all the indices downstairs" I'm not sure whether you mean you are ignoring the index position or whether you really are calculating \Lambda_{ab} which differs from both \Lambda^{ab} and {\Lambda^a}_b.

 

[Fredrik]

If we ignore the possibility of a spatial rotation and consider boosts only, a boost in an arbitrary direction ought to be

{\Lambda^a}_b=\begin{pmatrix}\gamma & -\gamma \textbf{v}^T \\ -\gamma \textbf{v} & \textbf{I} + (\gamma - 1)\frac{\textbf{v} \textbf{v}^T}{\textbf{v}^T \textbf{v}}\end{pmatrix}

(assuming c = 1 or equivalently x⁰ = ct).

However, when you say you are "putting all the indices downstairs" I'm not sure whether you mean you are ignoring the index position or whether you really are calculating \Lambda_{ab} which differs from both \Lambda^{ab} and {\Lambda^a}_b.

Thank you. That makes my R equal to

-\frac{1}{\gamma}(I+(\gamma-1)\frac{vv^T}{v^Tv})

I should get -\gamma v^T when I calculate \gamma v^T R, and it turns out I do:

\gamma v^T R=-v^T(I+(\gamma-1)\frac{vv^T}{v^Tv})=-v^T-(\gamma-1)v^T=-\gamma v^T

And, yes, when I wrote \Lambda_{i0} in the OP I was referring to components of the tensor which in the abstract index notation would appear as \Lambda^a{}_b. I find that notation slightly easier to use in the type of calculations I had to do to get this result.

Hm, do you know if the general form of \Lambda (i.e. not a pure boost) is the same except that the identity matrix in your expression is replaced with an orthogonal matrix that's independent of v? Maybe it's more complicated than that.

 

[DrGreg]

Hm, do you know if the general form of \Lambda (i.e. not a pure boost) is the same except that the identity matrix in your expression is replaced with an orthogonal matrix that's independent of v? Maybe it's more complicated than that.

I'm no expert in this, but my geometrical intuition says it ought to be either Q \Lambda or \Lambda Q, where \Lambda is as above and Q is a space-only rotation, i.e. in the form

Q = \begin{pmatrix}1 & \textbf{0}^T \\ \textbf{0} & \textbf{Q}_0 \end{pmatrix}

where Q₀ is a 3 x 3 orthogonal matrix (\textbf{Q}_0^T \textbf{Q}_0 = \textbf{I}).


Perhaps someone with more experience in this area could confirm that?