# Transformation question; first order shift of a scalar field

1. Jun 27, 2013

### llorgos

Hi to all! I have the following transformation

$\tau \to \tau' = f(\tau) = t - \xi(\tau).$

Also I have the action

$S = \frac{1}{2} \int d\tau ( e^{-1} \dot{X}^2 - m^2e)$

where $e = e(\tau)$. Then in the BBS String book it says that

$${X^{\mu}}' ({\tau}') = X^{\mu}(\tau)$$

and that the first order shift is

$$\delta X^{\mu} = {X^{\mu}}'(\tau) - X^{\mu}(\tau) = \xi(\tau)\dot{X}^{\mu}$$

Can someone explain why this is true? How can I realize it?

Then it say similarly for the $e(\tau)$ that

$$\delta e = e'(\tau) - e(\tau) = \frac{d}{d\tau} (\xi e)$$

again, I cannot see how this comes!