Schwarzschild Metric - Rindler coordinates

[aCHCa]

Hello, well I just read a paper by Atish Dabholkar and Ashoke Sen, titled "Quantum Black Holes", pp. 4-5 as shown below

and I tried to find $d\xi^{2}\frac{2GM}{\xi}=d\rho^{2}$ like this

which is different from the eq. in the paper.

So, could somebody please help me to find my mistake in the calculation? Thank you

 

[Bill_K]

ρ² = 8GM ξ
ρ = √8GM √ξ
dρ = √8GM ½dξ/√ξ = √2GM dξ/√ξ
dρ² = 2GM dξ²/ξ

 

[aCHCa]

Ah thank you so much :D