March 20, 2008UT Relativity Seminar Spin-boost vs. Lorentz Transformations Application to area invariance of Black Hole horizons Sarp Akcay

March 20, 2008UT Relativity Seminar Foreword Area invariance of a black hole’s (BH) 2-dim. Apparent horizon (AH) under Lorentz trans. is well known. But hard to show explicitly. Usual derivation is based on spin-boost transformations. It turns out that spin-boost trans. do not always yield meaningful trans. on a BH spacetime. Meaning: not all spin-boosts are physical boosts. Spin-boost derivation is not a reliable way to show AH area invariance under Lorentz boosts.

March 20, 2008UT Relativity Seminar How we shall proceed Area invariance: The area of Black Hole’s (BH) apparent horizon (AH) is invariant under Lorentz transformations. Tetrad formalism: a basis of 4 lin. indep. vectors → a tetrad → null tetrad Special transformation: Null rotation of a null tetrad looks like a Lorentz boost, called a Spin-Boost transformation (type III rotation). AH geometry (metric) remains invariant under spin- boost transformations. (→ area invariance) Spin-boost is not really a Lorentz boost? (Poisson).

March 20, 2008UT Relativity Seminar Area Invariance of AH BH event horizon (EH) is a 3-dim. null surface in spacetime. (see fig.1) AH is a 2-dim. cross section of EH at a t = constant slice. Any t will do, S 2 topology. Area of AH is invariant under Lorentz transformations. Null directions do not contribute to the area. (see fig. 2)

March 20, 2008UT Relativity Seminar Area Invariance of AH contd. Explicitly shown in gr-qc: for Kerr BH w/ arbitrary boost. Area = ∫(det h AB ) 1/2 = 4π(r a 2 )+ ∫ sinφ dφ Same answer as unboosted Kerr BH.

March 20, 2008UT Relativity Seminar Usual derivation for Area Invariance Let ℓ α be a geodesic tangent to EH, λ be the affine parameter for ℓ α i.e. ℓ β ∂ β ℓ α = ∂ℓ α /∂λ Lorentz transformations change the parametrization of the null vector: 2-metric → where and e α A gets a null contribution under the transformation Therefore h AB remains invariant i.e. 2-metric is invariant.

March 20, 2008UT Relativity Seminar Questions? 1. Why did g α β remain invariant? (no bars?) Under a type III rotation (spin-boost), g α β remains invariant Given a null tetrad e µ (a) = (ℓ µ, n µ, m µ, m µ *) for a spacetime: e (a) 2 = 0 and ℓ∙n = -1, m∙m* = 1 (by choice) The metric is given by g α β = - ℓ α n β – n α ℓ β + m α m β * + m α *m β A type III rotation on the tetrad is as follows: ℓ µ → A -1 ℓ µ, n µ → An µ, m µ → e iθ m µ, m µ * → e -iθ m µ * Metric is invariant under type III rotation

March 20, 2008UT Relativity Seminar Questions 2. Why a type III rotation? It looks like a Lorentz boost Construct unit timelike T µ = (ℓ µ + n µ )/ √2 unit spacelike s µ = (ℓ µ - n µ )/ √2 Type III rotation transforms these Letting v/c ≡ β = (A 2 – 1)/ (A 2 + 1) we get Boost along s µ

March 20, 2008UT Relativity Seminar So Far Spin-boost transformation gives Lorentz boost along spacelike vector s µ. 2-metric h AB remains invariant under spin- boost transformation. But the 2-metric from slide 4 is not invariant yet gives invariant area. What is going on?

March 20, 2008UT Relativity Seminar Spin-Boost in Schwarzschild Spacetime Apply the spin-boost formalism to Schwarzschild (Sch.) spacetime 4-metric: Null tetrad: (ℓ µ, n µ, m µ, m µ *) ℓ µ is null, tangent to EH, geodesic with λ = r This gives Under spin-boost trans. → Boost along r

March 20, 2008UT Relativity Seminar Spin-Boost for Sch. Spacetime contd. Obvious choice for a tetrad in Sch. spacetime gives boost along radial direction r. Boost along radial direction makes no sense. Let us pick another tetrad Start by first picking the spacelike boost direction s µ. Use ADM formalism to pick timelike T µ. Construct the null tetrad from these.

March 20, 2008UT Relativity Seminar Spin-Boost for Sch. Spacetime contd. Boost along x-direction i.e. X µ = (0, 1, 0, 0) s µ must be unit spacelike, therefore in KS coord. (t, x, y, z) Unit timelike T µ is obtained from 3+1 ADM breakdown. ℓ µ is given by ℓ µ = (T µ + s µ )/ √2

March 20, 2008UT Relativity Seminar Spin-Boost for Sch. Spacetime contd. Put this ℓ µ into geodesic eqtn. in Sch. spacetime Not geodesic! Worse: this is not tangent to the Sch. event horizon located at r = 2M. Easy to see this in Sch. spherical coordinates (x = r sinθ cosφ) → can not be used

March 20, 2008UT Relativity Seminar Observations Under spin-boost transformation, null vectors tangent to EH yield radial boost directions. Boost direction must be rectilinear. Picking an a priori rectilinear direction results in null vectors that are not tangent to EH, thus can not be used to show the area invariance.

March 20, 2008UT Relativity Seminar Conclusions Although Spin-boost trans. look like regular Lorentz boosts, this is not always the case. Certain choice of null tetrads give radial boost directions. 2-metric h AB remains invariant under spin-boost but does NOT under Lorentz-boost. Spin-boost derivation is not the correct way to show area invariance of AHs. However, Area of AH still is invariant.