- #1

offscene

- 7

- 2

- Homework Statement
- Starting from the metric ##ds^2=-X^2dT^2+dX^2##, using a coordinate transformation of T and X to t and x, convert it to the form ds^2=-dt^2+dx^2 (standard flat 2D metric).

- Relevant Equations
- Not actually a homework problem ("confession" from example 7.3 in Hartle's book for Gravity and GR) and no other relevant equations I can think of besides the standard chain rule but thought that this was the most fitting place to ask.

I started by expanding ##dx## and ##dt## using chain rule:

$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$

$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$

and then expressing ##ds^2## as such:

$$ds^2 = \left(\left(\frac{dt}{dX}\right)^2+\left(\frac{dt}{dX}\right)^2\right)dX^2+\left(\left(\frac{dt}{dT}\right)^2+\left(\frac{dt}{dT}\right)^2\right)dT^2 + 2\left(\frac{dt}{dX}\frac{dt}{dT}+\frac{dx}{dT}\frac{dx}{dX}\right)$$

But after matching the coefficients to the original ##ds^2##, I am unable to solve the equations to come up with the right transformation and was wondering if anyone could point me in the right direction/show me.

$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$

$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$

and then expressing ##ds^2## as such:

$$ds^2 = \left(\left(\frac{dt}{dX}\right)^2+\left(\frac{dt}{dX}\right)^2\right)dX^2+\left(\left(\frac{dt}{dT}\right)^2+\left(\frac{dt}{dT}\right)^2\right)dT^2 + 2\left(\frac{dt}{dX}\frac{dt}{dT}+\frac{dx}{dT}\frac{dx}{dX}\right)$$

But after matching the coefficients to the original ##ds^2##, I am unable to solve the equations to come up with the right transformation and was wondering if anyone could point me in the right direction/show me.