Landau-Lifshitz Formulation Of GR - Department Of Physics
Landau-Lifshitz Formulation of GRPost-Newtonian and post-Minkowskian theory start with the Landau-LifshitzformulationDefine the “gothic” metric density g pgg Then Einstein’s equations can be written in the form@µ H µ H µ tLL16 G ( g) T tLL4c g gµ g g µ @g · @gAntisymmetry of Hαµβν implies the conservation equation@h( g) T t LLi 0 () r T 0
Landau-Lifshitz Formulation of GRThe Landau-Lifshitz pseudotensor(g)t LL(c4: @ g @µ g16 Gµ @ gg gµ @ g @ gµ 12g g8µg ggµ@µ gµ1 g g2µ @ ggµ @ g @ gµ g2g g µg @ gµ @ g @ g)g g @ g @µ g µ
Landau-Lifshitz Formulation of GRConservation equation allows the formulation of global conservation laws:E dE dtZI3( g) T 00 t00dxLL2( g)t0jdSjLLSimilar conservation laws for linear momentum, angular momentum, andmotion of a center of mass, withZ13P ( g) T j0 tj0dxLLcZ1 jklj3J ( g)xk T l0 tl0dxLLcZ1j3X ( g)xj T 00 t00dxLLEj
The “relaxed” Einstein equationsh Define potentialsg Impose a coordinate condition (gauge): Harmonic or deDonder gauge@ h 0Matter tellsspacetimehow to curveSpacetimetells matterhow to move h g x( ) 016 G 4c1 @2@2@2@2 2 2222c @t@x@y@z ( g) T [m, g] t [h] tLLH [h]( g)t H@ c4: @µ h @ h16 G 0Still equivalent to the exact Einstein equationsµhµ @µ h
The “relaxed” Einstein equations h 16 G 4cSolve for h as afunctional of mattervariables@ 0Solve for evolution ofmatter variables togive h(t,x)
Iterating the “Relaxed” Einstein EquationsAssume that hαβ is “small”, and iterate the relaxed equation:16 G (hN )4cZ 00 (h)(t xx /c,x) 3 04GN hN 1 4d x0c x x hN 1Start with h0 0 and truncate at a desired NYields an expansion in powers of G, called a post-Minkowskian expansionFind the motion of matter using@ (hN ) 0
Solving the “Relaxed” Einstein Equations 4 µ )N : r0 R ,W : r0 RR wavelength s/v N W Zµ(tC x xx0 /c, x0 ) 3 0d x0x
Solving the “Relaxed” Einstein Equations:Far zoneNear zone integral:NFor x x’, Taylor expand x-x’ µ(t x x1x0 /c, y) X ( 1) 0L µ(t x @Lx0 ! 0r/c, y).r Z1 X( 1)1@Lµ( , x0 )x0L d3 x0N (t, x) !r M 0A multipole expansion tR/cIntegrals depend on R
Solving the “Relaxed” Einstein Equations:Far zoneFar zone integral:WSince contributions to µ in the far zone come from retarded fields, theyhave the generic formµ f ( 0 , 0 ,0)/r0nChange variables from (r’, θ’, φ’)to (u’, θ’, φ’), where u’ cτ’ ct’-r’u0 r0 ct xx0
Solving the “Relaxed” Einstein Equations:Far zoneFar zone integral:W1 4 ZWudu01IS(u0 )f (u0 /c, 0 , 0 )r0 (u0 , 0 , 0 )n 2 ctd 0u0 n 0 · xIntegral also depends on RBut N Wis independent of R
Gravity as a source of gravityand gravitational “tails”
Solving the “Relaxed” Einstein Equations:Near zoneNear zone integral:NFor x x’, Taylor expand about tµ(t x 1 X(1)@00µ(t,x) xx /c) !c @t 0 Z1 X( 1)@0µ(t,x) xN (t, x) !c @tM 0 A post-Newtonian expansionin powers of 1/c Instantaneous potentials Must also calculate the far-zoneintegralWx0 x0 1d3 x0
Post-Newtonian approximation: Near zoneNewtonianplus expand h00 in the near zone:Illustration:corrections up toNo 0.5 PN term:2.5 PN orderconservation of Mwithin τ00h00N 4Gc4 Z M3 xZ00x0 d3 x0 21 @2c2 @t2ZM 00 x1 PNcorrectiond2X/dt2x0 d 3 x 04 Z1 @1@ 00 x x0 2 d3 x0 00 x x0 3 d3 x033446c @t M24c @t MZi1 @5000 4 3 06 xx dx O(c)120c5 @t5 M2 PN termPure function oftime – a coordinateeffect2.5 PNtermGmv2 2 rc2c
Near zone physics; Motion ofextended fluid bodiesMatter variables: rescaled mass density : pg(u0 /c)proper pressure : pinternal energy per unit mass : fourvelocity of fluid element :u u0 (1, v/c) @ r ( u ) 0 () r( v) 0@tSlow-motion assumption v/c 1:T 0j /T 00 v/c ,h0j /h00 v/c ,T jk /T 00 (v/c)2hjk /h00 (v/c)2
Post-Newtonian approximation: Near zoneRecall the action for a geodesicS mcmcmc2Z2d 1Z 2rZ121 Gmv2 2 2rccg 12dr drdtdt dtjUc2g00εε222vg0jcvc2ε2εi jvvc2ε2gij 1/2dtWe need to calculateg00to O( 2 )g0jto O( 3/2 )gijto O( )Two iterations of the relaxedequations required
Post-Newtonian approximation: Near zoneConversion between h and g( g) 11 2h h21 µ h hµ O(G3 ) ,211µh h µ hhh 22 g h 1 2 1 µ hh hµ O(G3 ) ,84To 1PN order:g00g0jgjk1 00 1 kk 3 00 1 h hh228 h0j O(c 5 ) , 1 00 jk 1 h O(c 4 ) ,22 O(c6),
Post-Newtonian limit of general relativityg00g0jgjk 221 1 2U 4 @tt Xcc245 U O(c),jc3 2 jk 1 2 U O(c 4 ) ,cU (t, x) : G(t, x) : GX(t, x) : GjU (t, x) : GZ 03 0dx ,0 x x Z 0 xZZ 03 022vU2 O(c6),U 0 0 3p0 / 0 3 0d x ,0 x x x0 d3 x0 , 0 v 0j 3 0d x x x0
Post-Newtonian HydrodynamicsFromr T 0Post-Newtonian equation of hydrodynamics dvjdt @ j p @ j U 11 2p 2v U @j p v j @t pc2 1 h 2 2 (v4U )@j U v j 3@t U 4v k @k Uci 4@t Uj 4v k @k Uj O(c4@j Uk @j) 1 @tt X2
N-body equations of motionMain assumptions:§ Bodies small compared to typical separation (R r)§ “isolated” -- no mass flow§ ignore contributions that scale as Rn§ assume bodies are reflection symmetricZ d 3 xAZ1position : rA (t) xd3 xmA AZ1velocity : vA (t) vd3 x mA AZ1acceleration : aA (t) ad3 x mA Amass :CmA BxrA (t)drAdtdvAdtAx̄x rA (t) x̄
Landau-Lifshitz Formulation of GR Conservation equation allows the formulation of global conservation laws: E Z (g) T 00 t00 LL d3x dE dt I (g)t0j LL d 2S j Similar conservation laws for linear momentum, angular momentum, and motion of a center of mass, with P j 1 c Z (g) T j0
The Landau-Lifshitz-Gilbert equation is the fundamental evolution equation for spin fields in the continuum theory of ferromagnetism,first proposed by Landau and Lifshitz [1]in 1935. In the simplest case,where the energy of spin interactions is modeled by E(u) 1 2 Ω u 2 for magnetic moment u: Ω Rn S2,the Landau-Lifshitz-Gilbert
Lev Landau 11. Landau-Lifshitz Equation M M H M @ M M H dt d s G G G G G G u 0 u u 0 DJP JP Landau-Lifshitz equation H Larmor Term Damping Term animations courtesy of Helmut Schultheiss Evgeny Lifshitz
Landau-forbidden phase transitions between Landau-allowed phases Naive expectation: Breakdown of LGW paradigm at QCP when one of the two proximate phases has non-Landau order. Very interesting that LGW can also break down at critical point between two Landau-allowed phases. LRO, no LRE LRO', no LRE Eg: AF - VBS in spin-1/2 square lattice
Landau Levels within Landau Levels So Fractional Quantized Hall States are understood as filled Landau Levels of particles in a fictitious magnetic field, within a partially-full Landau Level of the true magnetic field. The fictitious magnetic field is an emergent phenomenon, an effective interaction that emerges from
Landau Forte Academy QEMS is a member of the family of Academies operated by Landau Forte Charitable Trust. This is an educational charitable trust founded by Martin . Landau and the Forte family over twenty years ago. The Trust expects each of the Academies to provide a Landau . Forte style of education which is distinctive, high quality and
Solutions to the Landau-Lifshitz system with nonhomogenous Neumann boundary conditions arising fro
Landau, L. D., Lifshitz, E. M., 1959, Course of Theoretical Physics, Vol. 6 : Fluid Mechanics Pergamon Press, pp. 319, 325, 332. F. P. Pijpers: On the propagation of sound waves in a stellar wind traversed by periodic strong shocks 9 this sh
Certification Standard Animal Nutrition – V5 for January 2020 P a g e 7 81 Daily ration: Average total quantity of feedingstuffs, calculated on a moisture content of 12 %, required daily by an animal of a given species, age category and yield, to satisfy all its needs (Regulation 1831/2003).
