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logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation GR course 18 Tidal force Bin Chen School of Physics Peking University April 20, 2010 Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation GR course 18 Tidal force Bin Chen School of Physics Peking University April 20, 2010 Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Jj 1 Lie derivative 2 Geodesic deviation 3 Tidal force 4 Einstein equation Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Diffeomorphism It provides another way of comparing tensors at different points on a manifold, besides PT tensor between two points. Given a diffeomorphism φ : M →M and a tensor field Tµ1···µkν1···µl(x), we can form the difference between the value of the tensor at some point p and φ∗[Tµ1···µkν1···µl(φ(p))], its value at φ(p) pulled back to p. This may allow us to define another kind of derivative; However, we need a one-parameter family of diffeomorphism φt, which is defined as following: φt : R×M →M, s.t. φs ◦ φt = φs+t (1) with φ0 being the identity map; Given a point P ∈M , φt : P → R, i.e. φt(P ) defines a curve in M; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation vector fields One-parameter families of diffeomorphisms can be thought of as arising from vector fields (and vice-versa); φt(P ) defines a curve in M; Since the same thing will be true of every point on M , these curves fill the manifold (although there can be degeneracies where the diffeomorphisms have fixed points). Df: a vector field V µ(x) to be the set of tangent vectors to each of these curves at every point, evaluated at t = 0. An example on S2 is provided by the diffeomorphism φt(θ, φ) = (θ, φ+ t). Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Diagram for Diffeomorphism φ Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Integral curves Given a vector field V µ(x), we define the integral curves of the vector field to be those curves xµ(t) which solve dxµ dt = V µ . (2) Such a curve is called the orbit or trajectory of V µ; Solutions to (2) are guaranteed to exist as long as we don’t do anything silly like run into the edge of our manifold; The existence and uniqueness theorem ensures that a solution at least exists for some subset; Physical example: the “lines of magnetic flux” traced out by iron filings in the presence of a magnet are simply the integral curves of the magnetic field vector B. Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Lie derivatives How fast a tensor changes as we travel down the integral curves? For each t we can define this change as ∆tTµ1···µkν1···µl(p) = φt∗[T µ1···µk ν1···µl(φt(p))]−Tµ1···µkν1···µl(p) . Note that both terms on the right hand side are tensors at p. Define the Lie derivative of the tensor along the vector field Vˆ as £Vˆ T µ1···µk ν1···µl = limt→0 ( ∆tTµ1···µkν1···µl t ) . (3) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Diagram for Lie derivative T[ (p)]φt (p) p [T( (p))]φt tφ* T(p) x (t)μ φt M Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Properties of Lie derivatives £Vˆ : (k, l)-tensor → (k, l)-tensor; Independent of coordinate; Linear: £Vˆ (aTˆ + bSˆ) = a£Vˆ Tˆ + b£Vˆ Sˆ , (4) the Leibniz rule, £Vˆ (Tˆ ⊗ Sˆ) = (£Vˆ Tˆ )⊗ Sˆ + Tˆ ⊗ (£Vˆ Sˆ) , (5) where Sˆ and Tˆ are tensors and a and b are constants. it does not require specification of a connection (although it does require a vector field, of course). It reduces to the ordinary derivative on functions, £Vˆ f = Vˆ (f) = V µ∂µf . (6) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Lie derivatives on tensor It is always possible to introduce a coord. system s.t. the curve passing through P is given by x1 only; i.e. V µ ∼ δµ1 = (1, 0, · · · ) and Vˆ = V µ∂µ ∼ ∂1; In other words, x1 is the integral curve; In this special coord., Lie derivative reduces to ordinary one; The magic of this coordinate system is that a diffeomorphism by t amounts to a coordinate transformation from xµ to yµ = (x1 + t, x2, . . . , xn); the pullback matrix is simply (φt∗)µν = δνµ ; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Lie derivatives on tensor the components of the tensor pulled back from φt(p) to p are simply φt∗[Tµ1···µkν1···µl(φt(p))] = T µ1···µk ν1···µl(x 1 + t, x2, . . . , xn) . (7) In this coordinate system, then, the Lie derivative becomes £Vˆ T µ1···µk ν1···µl = ∂ ∂x1 Tµ1···µkν1···µl , (8) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Lie bracket For (1, 0)-vector, £Vˆ U µ = ∂Uµ ∂x1 . (9) It is clearly not covariant; Compared to the commutator [Vˆ , Uˆ ], which is a well-defined tensor [Vˆ , Uˆ ]µ = V ν∂νUµ − Uν∂νV µ = ∂U µ ∂x1 . (10) Since both are vectors, they must be equal in any coord. sys.: £Vˆ Uˆ = [Vˆ , Uˆ ]. (11) So the commutator is sometimes called the “Lie bracket.” Two relations £fXˆ Yˆ = f [Xˆ, Yˆ ]− Yˆ (f)Xˆ; (12) £Xˆ(fYˆ ) = f [Xˆ, Yˆ ] + Xˆ(f)Yˆ ; (13)Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Lie derivative of 1-form Consider the action on the scalar ωµU µ for an arbitrary vector field Uµ. The Lie derivative with respect to a vector field reduces to the action of the vector itself when applied to a scalar: £Vˆ (ωµU µ) = Vˆ (ωµUµ) = V ν∂ν(ωµUµ) = V ν(∂νωµ)Uµ + V νωµ(∂νUµ) . (14) the Leibniz rule on the original scalar: £Vˆ (ωµU µ) = (£Vˆ ωˆ)µU µ + ωµ(£Vˆ Uˆ) µ = (£Vˆ ωˆ)µU µ + ωµV ν∂νUµ − ωµUν∂νV µ . We see that £Vˆ ωµ = V ν∂νωµ + (∂µV ν)ων ; It is completely covariant, though not manifestly so. Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Lie derivative of general tensor For the Lie derivative of scalar, vector and 1-form, we can obtain the Lie derive of a general (k, l)-tensor iterally; It turns out to be £Vˆ T µ1µ2···µk ν1ν2···νl = V σ∂σT µ1µ2···µk ν1ν2···νl −(∂λV µ1)T λµ2···µkν1ν2···νl − (∂λV µ2)Tµ1λ···µkν1ν2···νl − · · · +(∂ν1V λ)Tµ1µ2···µkλν2···νl + (∂ν2V λ)Tµ1µ2···µkν1λ···νl + · · · . Its covariance is more transparent in the form £Vˆ T µ1µ2···µk ν1ν2···νl = V σ∇σTµ1µ2···µkν1ν2···νl −(∇λV µ1)T λµ2···µkν1ν2···νl − (∇λV µ2)Tµ1λ···µkν1ν2···νl − · · · +(∇ν1V λ)Tµ1µ2···µkλν2···νl + (∇ν2V λ)Tµ1µ2···µkν1λ···νl + · · · , where ∇µ represents any symmetric (torsion-free) covariant derivative. Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Example: metric tensor For the metric tensor, we have £Vˆ gµν = V σ∇σgµν + (∇µV λ)gλν + (∇νV λ)gµλ = ∇µVν +∇νVµ = 2∇(µVν) , (15) where ∇µ is the covariant derivative derived from gµν . Therefore, if ∇(µVν) = 0, then gµν is unchanged along the integral curves of Vˆ , and Vˆ is called the Killing vector; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Parallel lines? Euclidean geometry: initially parallel lines remain parallel forever (parallel postulate); This is not true anymore in curved space: on S2, all the lines will crossover; The notion of parallelism is ill-defined; Construct a one-parameter family of geodesics, γs(t): for each s ∈ R, γs is a geodesic parameterized by the affine parameter t; The collection of these curves defines a smooth two-dimensional surface (embedded in a manifold M of arbitrary dimensionality); The entire surface is the set of points xµ(s, t) ∈M ; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Congruence of geodesics The tangent vectors to the geodesics, Tµ = ∂xµ ∂t , (16) The “deviation vectors” Sµ = ∂xµ ∂s . (17) Sµ points from one geodesic towards the neighboring ones. Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Congruence of geodesics t s T S γ ( ) s tμ μ Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Geodesic deviation define the “relative velocity of geodesics,” V µ = (∇Tˆ Sˆ)µ = T ρ∇ρSµ , (18) the “relative acceleration of geodesics,” aµ = (∇Tˆ Vˆ )µ = T ρ∇ρV µ . (19) Since Sˆ and Tˆ are basis vectors adapted to a coordinate system, their commutator vanishes: [Sˆ, Tˆ ] = 0 . Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Geodesic deviation Eq. 1 Consider the case with vanishing torsion, then we have Sρ∇ρTµ = T ρ∇ρSµ ; (20) (Torsion free: ∇Xˆ Yˆ −∇Yˆ Xˆ − [Xˆ, Yˆ ] = 0. So we have ∇SˆTˆ = ∇Tˆ Sˆ) 2 The acceleration: aµ = T ρ∇ρ(T σ∇σSµ) = T ρ∇ρ(Sσ∇σTµ) = (T ρ∇ρSσ)(∇σTµ) + T ρSσ∇ρ∇σTµ = (Sρ∇ρT σ)(∇σTµ) + T ρSσ(∇σ∇ρTµ +RµνρσT ν) = (Sρ∇ρT σ)(∇σTµ) + Sσ∇σ(T ρ∇ρTµ) −(Sσ∇σT ρ)∇ρTµ +RµνρσT νT ρSσ = RµνρσT νT ρSσ . (21) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Geodesic deviation Eq. 1 geodesic deviation equation: aµ = D2 dt2 Sµ = RµνρσT νT ρSσ , (22) 2 The relative acceleration between two neighboring geodesics is proportional to the curvature; 3 Slightly formal derivation: 1 ∇Sˆ Tˆ = ∇Tˆ Sˆ; 2 ∇Tˆ∇Tˆ Sˆ = ∇Tˆ∇Sˆ Tˆ ; 3 From the definition of Riemann tensor: (∇Tˆ∇Sˆ −∇Sˆ∇Tˆ −∇[Tˆ ,Sˆ])Zˆ = Rˆ(Sˆ, Tˆ , Zˆ), (23) with Zˆ = Tˆ and geodesic equation ∇Tˆ Tˆ = 0, we get the geodesic deviation equation; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Remarks The vanishing of the commutator of Sˆ and Tˆ ⇔ the vector fields to be surface-filling; Note that Tˆ and Sˆ may not be orthogonal Tˆ · Sˆ 6= 0; But we may define a vector: ηˆ = Sˆ + Tˆ (Tˆ · Sˆ), (24) which is orthogonal to Tˆ : ηˆ · Tˆ = 0 since Tˆ · Tˆ = −1; It could be proved that the geodesic deviation equation of ηˆ is of the same form: D2 dt2 ηµ −RµνρσT νT ρησ = 0. (25) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Geodesic deviation equation in LIF For a free falling frame following a geodesic with velocity Tˆ ,eµ0 = T µ, {eˆ1, eˆ2, eˆ3} is a set of spacelike vectors orthogonal to Tˆ , with DDτ e µ m = 0. Df.: ηm = emµ η µ to be spatial frame components as η0 = e0µη µ = Tµηµ = 0. (26) Three-dim. eq.: D2 dτ2 ηi −RµνρσeiµT νT ρηjeσj = 0. (27) or D2 dτ2 ηi +Kijη j = 0 (28) with Kij = −RµνρσeiµT νT ρeσj . Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force in Newtonian theory 1 Consider two particles travel on curves C1 and C2; 2 Their worldlines are characterized by xi(t), x′i(t) = xi(t) + ηi(t) (29) where ηi(t) is a small connecting vector; 3 For the first particle, d2xi dt2 = −δij ∂Φ(x k) ∂xj ; (30) 4 For the second particle, d2x′i dt2 = d2(xi + ηi) dt2 = −δij ∂Φ(x k + ηk) ∂xj = −δij ( ∂Φ(xk) ∂xj + ∂ ∂xk ∂Φ(xk) ∂xj ηk + · · · ) ; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force in Newtonian theory 1 Newtonian deviation equation: η¨i = −δij ( ∂2Φ ∂xj∂xk ) ~x ηk; (31) 2 Df: Kij = ∂ i∂jΦ 3 s.t. the Newtonian deviation equation could be rewritten as η¨i +Kijη j = 0. (32) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Remarks 1 The tensor K measures differential acceleration and determines the force that tend to pull nearby particles apart or bring them closer; 2 They are called “tidal gravitational force” or “tidal acceleration tensor”; 3 From Laplace eq. of Newtonian gravity in vacuum: ∇2Φ = 0 (33) 4 We know that the tensor K in vacuum is traceless: Kii = 0. (34) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force outside a spherical mass 1 Newtonian potential: Φ = −M/r, (G = 1); 2 Tidal acceleration: aij = − ∂ 2Φ ∂xi∂xj = −(δij − 3ninj)M r3 (35) where ni = xi/r; 3 In the polar coord. (r, θ, φ), arr = 2Mr3 , aθθ = aφφ = −Mr3 ; 4 Obviously, the tensor K is traceless in this case; 5 An observer falling freely and radially might employ just such a basis where d2ηr dt2 = 2M r3 ηr, d2ηθ dt2 = −M r3 ηθ, d2ηφ dt2 = −M r3 ηφ, (36) 6 An object falling towards the central mass is stretched in the radial direction and compressed in the transverse direction by tidal grav. forces; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force on the Earth 1 The tidal force on the Earth is mainly caused by the Moon, (and also the Sun); 2 Assume that the moon is on z-axis (0,0,d) relative to the Earth (0,0,0), its Newton potential is Φm(z) = − GMm(x2 + y2 + (z − d)2)1/2 , (37) 3 The tidal acceleration:( ∂2Φ ∂xi∂xj ) ∣∣∣∣0 = GMmd3 diag(1, 1,−2) ; (38) 4 Consider an element of ocean of mass m located in (y,z)-plane at ~r = r(0,− sin θ, cos θ) with r << d, then the tidal force is ~Ftidal = GmMm d2 ( r d )(0,− sin θ, 2 cos θ); (39) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force on the Earth 1 Along z, θ = 0, the force is in the +z direction; 2 Along z, θ = pi, the force is in the −z direction; 3 Namely, the tidal force pull the ocean apart along z-direction; 4 By contrast, the tidal force push the ocean toward the center along the x- or y-axes; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force in GR 1 From EEP, when an astronomer stays in a freely falling spaceship, he does not feel any gravitational force; 2 Is the gravitational force canceled completely by the freely falling?Locally this is true; 3 For a drop of fluid, it is not exactly of a spherical shape, with a minor deformation due to the tidal force; 4 In GR, the deformation is due to the geodesic deviation; 5 In other words, the tidal force in GR could be understood as the geodesic deviation; 6 Compare the geodesic deviation with the tidal force equation, we find that ∂i∂jΦ = Kij = −RµνρσeiµT νT ρeσj ; (40) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Tidal force in GR 1 Introduce eµi = δ µ i , T σ = (1, 0, 0, 0), we find that Ri00i = 0, (41) 2 As a tensor equation, we have Rρ00ρ = 0, (42) which could be rewritten as Rρ00ρ = R ρ µνρT µT ν = −RµνTµT ν = 0; (43) 3 It’s a scalar, should vanish in all coord. system, Rµν = 0. (44) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Vacuum Einstein equations 1 In vacuum, the Poisson’s equation ∇2Φ = 0; 2 It leads to a tensor equation RµνT µT ν = 0; (45) 3 This should be true for any Tµ, so we have Rµν = 0; (46) 4 This is so-called vacuum Einstein equation; 10 independent eqs; Highly non-linear, hard to solve; 5 Two justifications: Newtonian limit; Schwarzschild solution; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Newtonian limit 1 Weak gravity: gµν = ηµν + �hµν ; 2 Slowly varying: ∂0hµν ∼ O(�); 3 The particle moves slowly; 4 Inverse of the metric tensor: gµν = ηµν − �hµν , where hµν = ηµσηνρhσρ; 5 Christoffel symbol: Γµσρ = 1 2 gµν(∂σgνρ + ∂ρgνσ − ∂νgσρ) = 1 2 ηµν�(∂σhνρ + ∂ρhνσ − ∂νhσρ) +O(�2) (47) 6 Since the particle move slowly, dx i dτ << dt dτ . 7 The geodesic equation reduces to d2xµ dτ2 + Γµ00 ( dt dτ )2 = 0 . (48) Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Newtonian limit: cont. 1 Γµ00 = −12�ηµλ∂λh00; 2 The geodesic equation is therefore d2xµ dτ2 = � 1 2 ηµλ∂λh00 ( dt dτ )2 . (49) 3 The µ = 0 component of this is just d2t dτ2 = 0 . (50) That is, dtdτ is constant. 4 For µ = i, we have d2xi dτ2 = � 1 2 ( dt dτ )2 ∂ih00 . (51) which could be cast into the form d 2xi dt2 = �12∂ih00. Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Newtonian limit: cont. 1 This is the law in Newtonian gravity, once we identify �h00 = −2Φ , (52) or in other words g00 = −(1 + 2Φ) . (53) 2 The curvature Rµ0ν0 = � 2 gµσ∂σ∂νh00 +O(�2) (54) 3 The (0,0)-component of Rµν = 0 is Rµ0µ0 = � 2 ∇2h00, (55) which allows us to obtain the Laplace equation ∇2Φ = 0, (56) from the vacuum Einstein equation; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Einstein equation Poisson equation: ∇2Φ = 4piGρ; (57) How to generalize it to a tensor equation? ρ→ Tµν , ∇2 → ∇µ∇µ, Φ→ gµν? (58) Since ∇µgσρ = 0, this does not work; Scalar theory: keep Φ. However this is not consistent with the experiment, 1/2 mismatch; Vector theory: it may lead to repulsive force; Rµν = κTµν? not consistent with the energy-momentum conservation ∇µTµν = 0; Bin Chen GR course 18 logo JJJjjj Lie derivative Geodesic deviation Tidal force Einstein equation Einstein equation However, as we know, the Einstein tensor satisfy ∇µGµν = 0, so we may guess that Gµν = Rµν − 12gµνR = κTµν ; (59) From the Newtonian limit, R00 = � 2 ∇2h00 = 12κT00, (60) we get κ = 8piG; (61) Einstein equation: Rµν − 12gµνR = 8piGTµν ; (62) Bin Chen GR course 18 Ìá¸Ù Lie derivative Geodesic deviation Tidal force Einstein equation