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Spin-boost vs. Lorentz Transformations Application to area invariance of Black Hole horizons

Spin-boost vs. Lorentz Transformations Application to area invariance of Black Hole horizons. Sarp Akcay. 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.

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Spin-boost vs. Lorentz Transformations Application to area invariance of Black Hole horizons

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  1. Spin-boost vs. Lorentz TransformationsApplication to area invariance of Black Hole horizons Sarp Akcay UT Relativity Seminar

  2. 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. UT Relativity Seminar

  3. 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). UT Relativity Seminar

  4. 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, S2 topology. • Area of AH is invariant under Lorentz transformations. • Null directions do not contribute to the area. (see fig. 2) UT Relativity Seminar

  5. Area Invariance of AH contd. • Explicitly shown in gr-qc:0708.0276 for Kerr BH w/ arbitrary boost. • Area = ∫(det hAB)1/2 = 4π(r+2 + a2)+ ∫ sinφ dφ • Same answer as unboosted Kerr BH. UT Relativity Seminar

  6. Usual derivation for Area Invariance • Let ℓα be a geodesic tangent to EH, λbe the affine parameter for ℓα i.e. ℓβ∂βℓα = ∂ℓα/∂λ • Lorentz transformations change the parametrization of the null vector: • 2-metric → where and • eαAgets a null contribution under the transformation • Therefore hAB remains invariant i.e. 2-metric is invariant. UT Relativity Seminar

  7. Questions? • Why did gαβ remain invariant? (no bars?) • Under a type III rotation (spin-boost), gαβ remains invariant • Given a null tetrad eµ(a) = (ℓµ, nµ, mµ, mµ*) for a spacetime: e(a)2 = 0 and ℓ∙n = -1, m∙m* = 1 (by choice) • The metric is given by gαβ = - ℓαnβ – nα ℓβ + mαmβ* + mα*mβ • A type III rotation on the tetrad is as follows: ℓµ→ A-1ℓµ , nµ → Anµ, mµ → eiθmµ, mµ* → e-iθmµ* Metric is invariant under type III rotation UT Relativity Seminar

  8. Questions • Why a type III rotation? • It looks like a Lorentz boost • Construct unit timelike Tµ = (ℓµ + nµ)/ √2 unit spacelike sµ = (ℓµ - nµ)/ √2 • Type III rotation transforms these • Letting v/c ≡β = (A2 – 1)/ (A2 + 1) we get Boost along sµ UT Relativity Seminar

  9. So Far • Spin-boost transformation gives Lorentz boost along spacelike vector sµ. • 2-metric hAB remains invariant under spin-boost transformation. • But the 2-metric from slide 4 is not invariant yet gives invariant area. • What is going on? UT Relativity Seminar

  10. Spin-Boost in Schwarzschild Spacetime • Apply the spin-boost formalism to Schwarzschild (Sch.) spacetime • 4-metric: • Null tetrad: (ℓµ, nµ, mµ, mµ*) ℓµ is null, tangent to EH, geodesic with λ = r • This gives • Under spin-boost trans. → Boost along r UT Relativity Seminar

  11. 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 sµ. • Use ADM formalism to pick timelike Tµ. • Construct the null tetrad from these. UT Relativity Seminar

  12. Spin-Boost for Sch. Spacetime contd. • Boost along x-direction i.e. Xµ = (0, 1, 0, 0) sµ must be unit spacelike, therefore in KS coord. (t, x, y, z) • Unit timelike Tµ is obtained from 3+1 ADM breakdown. • ℓµ is given by ℓµ = (Tµ + sµ)/ √2 UT Relativity Seminar

  13. Spin-Boost for Sch. Spacetime contd. • Put this ℓµ 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 UT Relativity Seminar

  14. 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. UT Relativity Seminar

  15. 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 hAB 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. UT Relativity Seminar

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