Schwarzschild Metric - Rindler coordinates

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

1PF.jpg


2pf-1.jpg


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

3pf-1.jpg


which is different from the eq. in the paper.

So, could somebody please help me to find my mistake in the calculation? Thank you
 
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ρ2 = 8GM ξ
ρ = √8GM √ξ
dρ = √8GM ½dξ/√ξ = √2GM dξ/√ξ
dρ2 = 2GM dξ2/ξ
 
Ah thank you so much :D
 
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