gentsagree
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Hi, my first thread. The Hawking temperature for a BH can be derived through compactifying the timelike dimension and hence identifying the time coordinate of the euclidean metric such as a periodic coordinate τ with period β.
Now, very interestingly in the context of AdS/CFT correspondence, it is useful to derive said periodicity from the euclideanised AdS Schwarzschild metric, i.e.
ds^2=Vdτ^(2)+V^(-1)dr^(2)+r^(2)dΩ
where V=1-2M/(m^(2)r)+r^(2)/b^(2).
The result in AdS4 is given by β=4πb^(2)r+/(b^(2)+3r+^(2))
which is easily generalised to higher dimensions.
Anyone knows how to derive the above result from the metric?
I tried to linearise the metric element V, to change to Rindler coordinates and solving for the differential equations, but I only get a "close enough" answer. Perhaps I need a different linearisation of V.
Thanks
Now, very interestingly in the context of AdS/CFT correspondence, it is useful to derive said periodicity from the euclideanised AdS Schwarzschild metric, i.e.
ds^2=Vdτ^(2)+V^(-1)dr^(2)+r^(2)dΩ
where V=1-2M/(m^(2)r)+r^(2)/b^(2).
The result in AdS4 is given by β=4πb^(2)r+/(b^(2)+3r+^(2))
which is easily generalised to higher dimensions.
Anyone knows how to derive the above result from the metric?
I tried to linearise the metric element V, to change to Rindler coordinates and solving for the differential equations, but I only get a "close enough" answer. Perhaps I need a different linearisation of V.
Thanks