Orthochronous transformations?

[myleo727]

How can one show that if det A = 1 and the 00th component of A > = 1 then A preserves the sign of the time component of time-like vectors? thanks!

 

[Daverz]

Hint: write out the terms of the time component of the transformed vector. You need to show this doesn't change sign.

A helpful relation is:

$$\Lambda^\mu_{\ \alpha}\Lambda^\nu_{\ \beta}\eta^{\alpha\beta}=\eta^{\mu\nu}$$

 

[Fredrik]

Use the definition of matrix multiplication on $(\Lambda x)_{00}$ (row 0, column 0 of the matrix $\Lambda x$). (Forget you ever even heard of tensors. This problem involves matrices and their components, nothing else). You need to translate the assumption that x is timelike to a relationship between its components, and use it.

I think you also need to know what I said about the velocity of a Lorentz transformation in this post. If you don't, you're going to have to prove algebraically that this velocity is <1 (using the condition in Daverz's post, which I prefer to write as $\Lambda^T\eta\Lambda=\eta$).

You also need to understand the Cauchy-Schwarz inequality for vectors in $\mathbb R^3$ with the standard inner product.

If you get stuck, show us where.