of of Institute of Theoretical Physics, Charles University in Prague April 28th, 2014 scholtz@troja.mff.cuni.cz 1 / 45
Outline of 1 2 3 4 5 2 / 45
Energy-momentum in special Lie algebra of the Killing vectors of the Minkowski spacetime of Killing 1-form in the Minkowski spacetime K a = T µ a x µ + M µν (x ν a x µ x µ a x ν ) 4+6 decomposition of the Lie algebra K = T R T translations; commutative ideal; constant vector fields R = K/T = so(1, 3) L R L = [(Λ, a)] = {(Λ, b) b T} fix the origin o: L R o Lo = o R o boost-rotations with respect to o 3 / 45
Energy-momentum in special Different definitions of the energy-momentum tensor of Canonical energy-momentum tensor θ µ ν = L M µ φ r ν φ r g µν L M translational invariance Belinfante-Rosenfeld tensor BR T ab = θ ab + c ( σ c[ab] + σ a[bc] + σ b[ac]) µ θ µ ν = 0 Hilbert energy-momentum tensor H T ab = 2 g δi M δg ab diffeomorphism invariance a T ab = 0 In GR, H T ab = BR T ab 4 / 45
Energy-momentum in special Quasi-local quantities of Conserved current for Killing vector K a j a = T ab K b a T ab =0 (a K b) =0 The 3-form ω = j is closed and exact dω = 0 H 3 (R 4 )=0 Quasilocal conserved quantity Q S [K] = V ab = S Σ ω = dv a j a = 0 K e T ef ɛ fabc 5 / 45
Energy-momentum in special Quasi-local quantities of Q S [K] defines linear maps P S : T R, J S : R o R by where P µ S = Σ Q S [K] = T µ P µ S + M µν J µν S θ µ e T ef ɛ fabc J µν S = Σ (x ν θ µ e x µ θ ν e ) T ef ɛ fabc and θ µ a = a x µ 6 / 45
Energy-momentum in special Quasi-local quantities of Correct behaviour under Poincaré transformations Under the Lorentz transformation x µ Λ µ ν x ν : P µ S Λµ ν P ν S J µν S Under translation x µ x µ + η µ : P µ S P µ S J µν S J µν S Quasi-local mass and Pauli-Lubanski spin Λµ α Λ ν β J αβ S + ην P µ S ηµ P ν S m 2 = g µν P µ S P ν S S µ = 1 2 ɛ µναβ P α S J αβ S 7 / 45
Energy-momentum in general Energy-momentum tensor of Action of gravity + matter: I[g, Φ] = I H [g] + I M [g, Φ] Variation of I M δi M δφ Variation of I H field equations δi M δg ab energy-momentum tensor δi H δg ab Einstein s equations 8 / 45 δi H δ? gravitational E-M tensor
Energy-momentum in general Diffeomorphism invariance of Principle of equivalence No geometric background to describe the dynamics of g ab Diffeomorphism invariance In general, no Killing vector But any diffeomorphism φ is only a gauge transformation Spacetimes (M, g ab ) and (M, φ g ab ) are equivalent Interpretation of T ab It is a source of gravity Not a priori related to energy-momentum 9 / 45
Quasilocal quantities Why not local? of By gauge freedom g ab φ g ab, the values of the fields at single point are meaningless Example: the connection Genuinely non-local object We can set Γ α µν = 0 locally, but not globally Connection connects the fibres of the tangent bundle over different points of the base manifold Any reasonable expression based on the connection must be non-local 10 / 45
Quasilocal quantities What does quasilocal mean? of Quasilocal quantities should be associated with: globally hyperbolic domains D M spacelike hypersurfaces Σ with the boundary closed, orientable two-surfaces (boundaries Σ) 11 / 45
Geometry of spacelike 2+2 decomposition of S closed spacelike two-surface V a (S) restriction of T M to S V a (S) = T S NS T S tangent bundle of S NS normal bundle of S Projections operators t a timelike unit normal v a spacelike unit normal Π a b : V b (S) T a S O a b : V b (S) N a S Π a b = δ a b t a t b + v a v b O a b = t a t b v a v b 12 / 45
Geometry of spacelike Connections on V a (S) of Intrinsic covariant derivative δ a X b = Π c a Π b d cx d Annihilates the fibre metric, δ c g ab = 0 Annihilates the projection, δ c Π a b = 0 Curvature S R a bcd Xb = (δ d δ c δ c δ d ) X a = 1 S R (q ac q bd q ad q bc ) 2 The Gauss-Bonnet theorem S R ds = 8π(1 g) S 13 / 45
Geometry of spacelike Connections on V a (S) of The Sen connection a X b = Π c a c X b Annihilates only the fibre metric, c g ab = 0 Torsion and curvature Q e ab = Πe c a Π c b ( a b b a ) φ = 2 Q e [ab] eφ ( c d d c ) X a = F a bef Xb 2 Q e [cd] ex a Spinor form of the torsion Q E af = 1 2 γr F aγ E R o A o B o C ō C Q ACC B = σ o A o B ι C ō C Q ACC B = ρ 14 / 45
Geometry of spacelike 2-surface spinors of 2-surface tensors = Tc..d a..bs, i.e. Πa e..π b f Πr c..π s d te..f r..s = t a..b c..d Spinor form of the normals t a and v a t a = 1 2 (o A ō A + ι A ῑ A ) ( v a = 1 2 o A ō A ι A ῑ A ) Complex metric on the spinor space γ A B = 2 tar v BR = 2 o (A ι B) γ A C γc B = δa B γ A A = 0 (S A, γ A B ) space of 2-surface spinors 15 / 45
Geometry of spacelike Decomposition of 2-surface spinors of Chirality ± π A B = ( 1 2 δ A B ± γ B) A For any ξ A, spinor ± π A B ξb is an eigenspinor of γ A B Usual spinor decomposition φ AB = φ (AB) + 1 2 ɛ AB φ X X In the presence of γ A B, the irreducible parts are φ (AB) + 1 2 γ AB γ XY φ XY 1 2 γ AB γ XY φ XY ɛ AB φ X X 16 / 45
Geometry of spacelike Decomposition of the Sen covariant derivative of Irreducible parts of a λ B The Sen-Witten operator A Rλ R T R A AB λ R = A (Aλ B) + 1 2 γ AB γ RS A Rλ S γ trace is reducible Left/right handed operators ± AA λ A = ± π B A AB λ A ± T R A AB λ R = ± π B R T A B AB λ R 17 / 45
Motivation of For a Killing vector K a Q S [K] = K e T ef? ɛ fabc = κ R abcd f cd Σ S Spinor decomposition of the Rieman tensor R abcd = Ψ ABCD ɛ A B ɛ C D + Ψ A B C D ɛ ABɛ CD + Φ ABC D ɛ A B ɛ CD + Φ CDA B ɛ ABɛ C D 2Λɛ C(A ɛ D)B ɛ A B ɛ C D 2Λɛ C (A ɛ D )B ɛ ABɛ CD Decomposition of anti-self-dual f cd f cd = ω CD ɛ C D 18 / 45
... of The two integrals are equal if the twistor equation holds AA ω BC = iɛ A(B K C)A A (A ω BC) = 0 Non-trivial solutions only in (conformally) flat The existence of the solution implies that K a is a Killing vector How to weaken the twistor equation? ω CD = α (C β D) Spinors α C and β C satisfy the twistor equation A (A α C) = 0 A (A β C) = 0 Still only the trivial solution 19 / 45
... of Twistor equation A (A α B) = 0 We are interested in the mass associated to S Hence, we take only projections tangential to S In GHP-formalism ðα 1 σα 0 = 0 ð α 0 σ α 1 = 0 In terms of the Sen connection T α = 0 + T α = 0 20 / 45
...solution of Atiyah-Singer index theorem Let T : E F be an elliptic operator and T : F E its adjoint. Then the analytical index of T defined by index T = dim ker T dim ker T is a topological invariant (and is equal to the topological index). Let T be the twistor operator ± T The index can be computed for round spheres, index ± T = 4 For general surface, index ± T = 4(1 g) The index does not change under homeomorphisms 21 / 45
The construction of Choose a surface S Solve the twistor equation ± T ω = 0 The solutions form 4 dimensional 2-twistor space T (S) Choose any two elements α A, β A of T (S) Set f ab = α (A β B) ɛ A B Associated quasilocal quantity is Q S [α, β] = 1 R abcd f cd 8πG S = 1 [ α 0 β 0 (Φ 01 Ψ 1 ) + 2α (0 β 1) (Φ 11 + Λ Ψ 2 ) 8πG S +α 1 β 1 (Φ 21 Ψ 3 ) ] ds 22 / 45
Energy-momentum and angular momentum in GR of In general spacetime no universal definition of energy-momentum of gravitational field Main reason: absence of the fixed background, e.g. Minkowski spacetime Asymptotically flat ADM mass 23 / 45
Asymptotically flat Penrose s definition of Idea: representing isolated sources Physical spacetime (M, g) Unphysical spacetime ( ˆM, ĝ) ĝ = Ω 2 g Conformal factor Ω Ω = 0 is the boundary M dω 0 on M Ω as coordinate: infinity is Ω 0 24 / 45
Asymptotically flat Example: Minkowski spacetime of ˆM = R S 3 i + I + M i 0 i I 25 / 45 Red line: r = constant (time-like geodesics) Blue line: t = constant (space-like geodesics) Black arrow: t r = constant (light-like or null geodesics)
The ADM mass Arnowitt-Deser-Misner of Object (e.g. star) moving along time-like geodesics I + time-like geodesic r = 0 r =const. 26 / 45
The ADM mass Arnowitt-Deser-Misner of At any instant of time t 1 we measure the mass along space-like hypersurface I + time-like geodesic space-like hypersurface r = 0 r =const. 27 / 45
The ADM mass Arnowitt-Deser-Misner of Object suddenly emits a burst of radiation I + time-like geodesic space-like hypersurface burst of radiation r = 0 r =const. 28 / 45
The ADM mass Arnowitt-Deser-Misner of The mass along space-like hypersurface remains the same because all geodesics cross both hypersurfaces I + time-like geodesic space-like hypersurface burst of radiation space-like hypersurface r = 0 r =const. 29 / 45
The Measured along asymptotically null surfaces of Object (e.g. star) moving along time-like geodesic I + time-like geodesic r = 0 r =const. 30 / 45
The Measured along asymptotically null surfaces of We choose the hypersurface to be asymptotically null I + time-like geodesic asymptotically null hyperusrface burst of radiation r = 0 r =const. 31 / 45
The Measured along asymptotically null surfaces of Emitted radiation does not cross later hypersurfaces I + time-like geodesic asymptotically null hyperusrface burst of radiation asymptotically null hyperusrface r = 0 r =const. 32 / 45
The Properties of the in vacuum of Strictly positive (zero only for flat spacetime) Non-increasing function of time Constant in stationary (with time-like Killing vector) Decreases whenever system emits radiation 33 / 45
Newman-Penrose expression for the of 34 / 45 Maxwell M B = 1 2 π ( Ψ 0 2 + σ 0 σ 0) ds S Ṁ B = 1 2 π ( σ0 σ 0 + φ 0 φ 2 0 ) 2 ds S M B = 1 ( 2 Ψ 0 2 + σ 0 σ 0 + 1 π 3 ( u φ 0 )) φ0 ds Ṁ B = 1 2 π S S ( σ 0 σ 0 + 2 φ 0 φ0 ) ds
Electro-scalar of The Lagrangian of coupled electro-scalar fields L = (D a φ)(d a φ) m 2 φ φ 1 4 F ab F ab The gauge-covariant derivative D f Tc..d a..b = f Tc..d a..b + i e A f Tc..d a..b A a connection on principal U(1) bundle charge of the field T a..b c..d Under the gauge transformation 35 / 45 φ e iχ(x) φ, A a A a + 1 e aχ
Electro-scalar Equations of EM and scalar field of equation ( Da D a + m 2) φ = 0 Maxwell s equations a F ab = J b with the current J b = i e ( φdb φ φd b φ) 36 / 45
Spinorial field equations of Equation for the 4-potential A A A BA = φ AB Equation for electromagnetic spinor A B φ AB = ie 2 ( φdb φd b φ) Equation for the scalar field D X A D BX φ = 1 2 m2 ɛ AB i e φ φ AB 37 / 45
Electro-scalar Gravitation field equations in spinor form of Ricci identities CD ξ A = Ψ ABCD ξ B 2Λ ɛ A(C ξ D) C D ξ A = Φ ABC D ξb Bianchi identities D A Ψ ABCD = B (A Φ BC)A B BB Φ ABA B = 3 AA Λ 38 / 45 Einstein s equations Φ ABA B = ( D (A(A φ ) ( D φ ) B )B) + φ AB φa B Λ = 1 [ 12 (Da φ) ( D a φ) + 2 m 2 φ φ ]
Electro-scalar Asymptotic solution of Analyticity on I implies m = 0 for the scalar field The spin coefficients ρ = Ω ( s s + φ 0 φ0 ) Ω 3 + O ( Ω 3), σ = s Ω 2 + O ( Ω 3) α = a Ω + (ð s + a s) Ω 2 + O ( Ω 3), β = a Ω a s Ω 2 + O ( Ω 3) π = (ð s) Ω 2 + O ( Ω 3) ( s λ = s Ω + 2 ðð s ) Ω 2 + O ( Ω 3) µ = 1 (ð 2 Ω 2 s + s s + Ψ 0 2 + 1 ) 6 u(φ 0 φ0 ) Ω 2 + O ( Ω 3) ( γ = að s a ðs 1 2 Ψ0 2 + 1 ) 6 u(φ 0 φ0 ) Ω 2 + O ( Ω 3) 39 / 45
Electro-scalar Asymptotic solution of The Ricci spinor and the Weyl spinor Φ 11 = 1 4 ( u φ 0 ) φ0 Ω 3 + O ( Ω 4) Λ = 1 12 ( u φ 0 ) φ0 Ω 3 + O ( Ω 4) Ψ 1 = Ψ 0 1 Ω 4 + Ψ 1 1 Ω 5 + O ( Ω 6) Ψ 2 = Ψ 0 2 Ω 3 + O ( Ω 4) Ψ 3 = ð s Ω 2 + O ( Ω 3) Ψ 4 = s Ω + ðð s Ω 2 + O ( Ω 3) 40 / 45
Electro-scalar Asymptotic solution of Example Ψ 1 1 = 3 φ 0 φ 0 0 1 ðψ 0 0 sð(φ 0 φ0 ) + 1 ( φ 0 ð 2 φ 1 + φ 0 ðφ 1) ( φ 1 ð φ 0 + φ 1 ðφ 0) + φ 0 ( φ0 3 e 2 A 0 0 A 0 1 ðs ) + 3 2 i e [ A 0 0 φ 0 ( ð φ 0 φ 1) A 0 φ 1 0 ( ðφ 0 φ 1)] + 1 2 s ð(φ 0 φ0 ) 41 / 45
Solution of asymptotic twistor equation of Solution of the 2-surface twistor equation A (A ω B) = 0 Asymptotic expansion of the twistor components ω 0 = ω 0 0 + ω 0 1 Ω + O ( Ω 2) ω 1 = ω 1 0 + ω 1 1 Ω + O ( Ω 2) Asymptotic solution ðω 1 0 = 0 ðω 1 0 = ω 0 0 ω 1 1 = 0 Nester-Witten 2-form u AB = 1 ( ) ω 2 (A C B) ω C ω C C (A ω B) F ab = u AB ɛ A B + ū A B ɛ AB 42 / 45
Construction of the of We define the integral I(Ω) = where S F ab l a n b ds F ab l a n b = ρ ω 0 ω 0 +µ ω 1 ω 1 +R ( π ω 1 ω 0 + ω 1 δω 0 ω 0 δω 1 ) The limit on I I = lim Ω 0 I(Ω) The asymptotic solution implies I(Ω) = O ( Ω 2) 43 / 45
Construction of the of Taking the limit Ω 0 yields ( I = R Ψ 0 2 + s s + 1 ) 6 u (φ0 φ0 ) ω0 1 ω 1 0 dŝ, The Bondi momentum covector is defined by ω 1 has the spin weight 1/2 I = P a t a, t a = ω A ω A ω0 1 = a 1 Y 1 2 2, 1 2 + b 1 Y 1 2 2, 1 2 ω 1 0 ω1 0 has the spin weight zero 44 / 45 Time-direction corresponds to 0 Y 00 We pick ω 1 0 ω1 0 = 1
for electro-scalar of The M B = ( Ψ 0 2 + s s + 1 6 ) u (φ0 φ0 ) dŝ The -loss [ Ṁ B = R ṡ s + φ 0 φ 2 0 0 2 + φ φ0 ( + i e A 0 2 φ 0 φ0 φ ) ] 0 φ0 + e 2 A 0 2 A 0 2 φ 0 φ0 dŝ In the gauge-invariant form [ṡ Ṁ B = s + φ 0 φ 2 0 2 + (D u φ 0 ) (D φ0 u ) ] dŝ 45 / 45
of Thank you for your attention! 46 / 45