kerrnew.pdf - The Kerr Metric\u2014for a Rotating Black Hole In Boyer-Lindquist coordinates the Kerr metric may be written in the following form 2 dr 2mr 2

kerrnew.pdf - The Kerr Metricu2014for a Rotating Black...

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The Kerr Metric—for a Rotating Black HoleIn Boyer-Lindquist coordinates the Kerr metric may be written in the following form,ds2=Σ(dr2+2)+ (r2+a2) sin2θ dφ2-dt2+2mrΣ(asin2θdφ-dt)2,=Σ(dr2+2)+AΣsin2θ dφ2-22marΣsin2θ dφ dt-(1-2mrΣ)dt2,gμν(r, θ)drμdrν,(1)which describes the gravitational field exterior to a rotating black hole of massmand angularmomentum per unit mass of amounta,pointing along the positive ˆz-direction. I also note thatthe determinant of the metric satisfiesg=-Σ2sin2θ.It is reasonable to describe this system via a “locally, non-rotating,” (orthonormal) tetrad,i.e., a LNRF—also called a ZAMO, because this observer has zero angular momentum—of thefollowing form:ωr=Σdr ,ωθ=Σ dθ ,ωt=ΣAdt ,ωφ=AΣsinθ dφ-2marsinθΣAdt=AΣsinθ(-ω dt),ωˆμYˆμαdxαwithΣr2+ (acosθ)2,r2+a2-2mr ,ω2marA=-gφtgtt=-gφtgφφ,A(r2+a2)2-a2∆ sin2θ= (r2+a2)Σ+ 2ma2rsin2θ= ∆Σ+ 2mr(r2+a2).(2)On the other hand the associated, reciprocal basis of tangent vectors iser=Σr,eθ=1Σθ,e
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