Gravitational scattering of a quantum particle and the privileged coordinate system

Transcription

1 rxiv: v1 [physics.gen-ph] 5 Dec 2017 Grvittionl scttering of quntum prticle nd the privileged coordinte system A.I.Nishov July 20, 2018 Abstrct In grvittionl scttering the quntum prticle probes the Fourier-trnsforms of metric. I evlute the Fourier-trnsforms of Schwrzschild metrics in stndrd, hrmonic nd other coordinte systems in liner nd G 2 pproximtions. In generl different coordinte systems led to different scttering. This opens up the possibility to choose the privileged coordinte system which should led to scttering in greement with experiment. In generl reltivity ll permissible coordinte system re equivlent. Yet s V.Fock insisted there should be the privileged coordinte system nd the theory should give the unique solution, see 93 in [1]. We note here tht the theory of sources nturlly gives unique solution. As quntum scttering probes the Fourier trnsform of metric, the ide is to use it in choosing the privileged coordinte system. For simplicity reson I ssume tht the test prticle hs 0-spin nd interct only grvittionlly by one grviton exchnge. In cse of other spins similr results re expected. Then the scttering prticle my be hypotheticl drk mtter prticle, grviton or neutrino. The following nottion is used g = η +h (1) +h(2) +, h(n) Gn, h,l = h, h = h k k, η = dig( 1,1,1,1). x l The Ltin indexes run from 0 to 3, the Greek ones from 1 to 3. In ech coordinte system we denote coordintes by the sme x i. This is becuse we use it only s n integrtion vribles in obtining Fourier-trnsforms. We strt with the simplest cse of liner pproximtion to Schwrzschild metric nd ssume t first tht the point-le prticle is the source of grvittionl field. The stndrd coordinte system in liner nd G 2 -pproximtion hs the form h (1)st = 2GMx αx β r 3, h (2)st = (2GM)2 x α x β. (1) see S.Weinberg [2], Ch 8, 2 nd get the necessry pproximtion of unnumbered expression fter eq.(8.2.15) In liner pproximtion the hrmonic nd isotropic systems coinside h (1)hr = h (1)iso = 2MG δ r nd h (1) 00 = 2MG is the sme in ll coordinte systems. Now r h (1)st h (1)iso = MG(Λ (1) α,β +Λ(1) β,α ). Λ(1) α,β = (δ r x αx β r 3 ), Λ (1) α = x α r. (2)

2 So the difference in the l.h.s. is guge function. Now the scttering mplitude of two sclr prticles exchnging by one grviton is proportionl to T ij (p,p ) P ijkl q 2 Tkl (k,k ), T ij (p,p ) = 1 2 (pi p j +p j p i ) 1 2 ηij (pp +M 2 ), P ijkl = 1 2 (η η jl +η il η jk η ij η kl ), q = k k = p p, (3) seeforexmple 4.3inR.Feynmnndothers[3](noticethtin[3]η = dig(1, 1, 1, 1)). It is esy to check tht q i T ij (p,p ) = 0 nd similrly for T kl (k,k ). This is due to the conservtion lw for energy-momentum tensor.nd it holds for the prticle of ny spin. Next we ssume tht the prticle with momentum p,p 2 = M 2 is hevy one nd the prticle with momentum k,k 2 = m 2 is the probing one. Then T ij (p,p ) P ijkl = 1 M 2 q 2 2 q δ kl. 2 UptofctorwhichisnotinterestedusnowthisisFourier-trnsformofh (1)iso kl : d 3 xe i q x h (1)iso kl (x) = 4πδ kl 2MG/q 2. So the hevy prticle becomes the source of the Schwrzschild field in Hilbert (or Lorentz) guge: h kl,l = (h kl 1 2 ηkl h),l = 0. The coordinte system in this guge I consider s the privileged one [4]. In this liner pproximtion it coinsides with hrmonic system nd grees with Fock definition of privileged system. Now we evlute the Fourier-trnsforms of metrics in different coordinte system. We use the expression r n = q αq β q 2 A(q)+δ B(q). (4) Cses n = 3 nd n = 4 dmit the limit b = 0 where b is the rdius of the bll of mtter. So in these cses we put b = 0. The functions A(q) nd B(q) cn be obtined s fllows. We put α = β. Then d 3 xe i q x 1 = A(q)+3B(q). (5) rn 2 Next we ssume temporrily tht q α = (0,0,q) nd use x 3 = rt,t = cosθ. Then from (4) we hve d 3 i q x t2 xe = A(q)+B(q). (6) rn 2 To evlute the l.h.s. we twice differentite over the reltion 1 get dteit = 2 sin. Thus we dte it t 2 = 2[( )sin+ cos]. (7) 2 Using this eqution we find for the l.h.s. of (6) for n = 3 nd n = 4 d 3 xe i q xt2 r = 4π q 2, d 3 xe i q xt2 = 0. (8) r2 For the l.h.s. of (5) putting α = β nd n = 3 we obtin just s in Coulomb scttering d 3 xe i q x1 r = 4π = A(q)+3B(q). (9) q2

3 On the other hnd from (6) nd the first eqution in (8) for n = 3 we hve 4π = A(q)+B(q). (10) q2 Now from (9) nd (10) it follows tht A(q) = 8π/q 2,B(q) = 4π/q 2. Hence eqution (4) for n = 3 gives = 4π αq β r 3 q 2( 2q +δ q 2 ). (11) In similr mnner we obtin Putting here α = β we find Now from (9) nd (11) for Λ (1) α,β = π2 q ( q αq β q 2 +δ ). (12) d 3 xe i q x 1 r 2 = 2π2 q. (13) (defined in (2)) we get d 3 xe i q x Λ (1) α,β = 8πq αq β q 4, (14) similrly from (12) nd (13) we hve d 3 xe i q x Λ (2) α,β = 2π2q αq β q 3, Λ(2) α,β = (x α r 2),β = δ r 2 2x αx β. (15) Theequtions(2)nd(14)menthth (1)st ndh (1)hr (whichisequlh (1)iso )ledtothesme scttering due to the conservtion lw of energy-momentum tensor of the probing prticle: q α T (k,k ) = 0. We remind here tht q 0 = 0 when M >> k 0. Now in G 2 -pproximtion we hve h (2)hr h (2)hr 00 = h (2)iso ( = G 2 M 2 xα x β + δ r 2 00 = 2M2 G 2 r 2, h (2)hr ), h (2)iso = 3M2 G 2 2r 2, (15), h (2)iso = 1 2 M2 G 2 Λ (2) α,β. (15b) The first eqution in (15) cn be obtined from eqution (8.2.15) in [2], the second from Problem 4 in 100 in [8]. Now from (15) nd the lst eqution in (15b) it follows tht h (2)iso nd h (2)hr lso led to the sme scttering. In other words the differences in metrics by guge function does not ffect the scttering. On the other hnd the difference h (2)st h (2)hr = G 2 M 2 ( 3x αx β δ r 2 ) (16) it is not guge function nd the scttering is different in these coordinte systems in G 2 - terms. Indeed, in cse of hrmonic coordinte the contribution to scttering mplitude is proportionl to d 3 xexpi q x[t 00 (k,k )h (2)hr 00 +T (k,k )h (2)hr ] = G 2 M 2π2 q [5 2 k k k 13 2 m2 ]. (17)]

4 The sme result we get if insted of h (2)hr we use h (2)iso. In cse of stndrd system insted of (17) we hve d 3 xexpi q x[t (k,k )h (2)st = G 2 M 2π2 q [6k2 0 2 k k 6m 2 ], h (2)st 00 = 0. (18)] We see tht (17) nd (18) re different. Now we tke into ccount the finiteness of rdius b of the bll of mtter. Then for the privileged coordinte system nd for the stndrd system h (1)priv = { MG b r2 (3 )δ b 2, r < b, 2MGδ r, r > b, (19) { h (1)st x 2MG αx β, r < b, = b 3 2MG xαx β, r > b. r 3 Now for the Fourier-trnsforms using equtions (41)nd (42) below we obtin d 3 xe i q x h (1)priv = 2MG 4π 3 q 2{ 2(sin cos) cos}δ, (21) r>b d 3 xe i q x h (1)priv The sum of (21) nd (22) is d 3 xe i q x h (1)priv 0<r (20) = 2MG 4π q 2δ cos (22) = 2MG 4π q 2 3 2(sin cos)δ, (23) Similrly for the stndrd system d 3 xe i q x h (1)st = 2MG 4π αq β q 2{q q [(15 2 1)cos+ 2 r>b ( )sin]+δ [ 3 cos+( )sin]}, (24) d 3 xe i q x h (1)st = 2MG 4π αq β 3sin sin q 2{q (cos )+δ q2 }. (25) The sum of (24) nd (25) is d 3 xe i q x h (1)st = 2MG 4π αq β q 2{q q [15 2 cos+( )sin]+δ [ 3 cos+ 3 sin]}. (26) 2 3 0<r As terms contining q α q β do not contribute to scttering, it follows from (23) nd (26) tht in our pproximtion the privileged nd stndrd coordinte systems led to the sme scttering: dσ = M 2 G 2 ( k 2 0 m 2 /2 k 2 ) 2 dω sin 4 θ 2 F 2 (), F() = 3 2(sin cos), F() <<1 = (27)

5 So for << 1 we return to the scttering on point-le source [5]. For finite there re oscilltions in the ngulr distribution in (27) becuse = bq = b2 k sinθ/2. Finlly we consider the cse of finite rdius of the mtter bll in G 2 -pproximtion. Now the theory of sources suggests tht the privileged coordinte system is tht one in which the guge degrees of freedom re put to zero [4]. Assuming tht the 3-grviton vertex is given by generl reltivity, such system in G 2 -pproximtion ws obtined in [6]. It ws lso shown there tht the theory of sources leds to the ppernce in externl metric in G 2 -pproximtion of the term κm 2 G 2 b( x αx β δ r 5 3r3). (28) Here b is the rdius of the bll of mtter, κ depends on the ssumed form of the 3-grviton vertex nd (contrry to generl reltivity, where the exterior metric does not depends on interior rigion) on the energy-momentum tensor of the mtter. It is of order unity. Despite the fct tht (28) hve the form of guge function it is generted by source. Now we get the contribution to its Fourier-trnsform from the exterior region. In eqution (5) nd (6) we put n = 5. Using (7) we obtin d 3 xe i q xt2 r 3 = 4π + 1 cosx 3 x dx] dx[ 2sinx x 4 + 2cosx x 3 + sinx x ] = 4π[2 sinx 2 cosx 1 sinx 2 3 x 3 3 x 2 3 x = 4π[ 2 sin + 2 cos + 1 sin cosx dx]. (29) x Just s in eqution (8) nd (9) we find for B(q) nd A(q) in equtions (4) nd (5) for n = 5 nd from here B(q) = 4π[ 1 3 ( cos 3)sin cosx dx], (30) x A(q) = 4π[ sin + cos ], = qb, (31) 3 2 d 3 xe i q x ( x αx β δ r 5 3r 3) = A(q)(q αq β δ q 2 3 ). (32) It will be seen below tht thisterm cncontribute to scttering onlyin moreelbortepproximtion, which probbly include bsorption in the mtter bll or more then one grviton exchnge with grvittionl source. As shown below in the considered pproximtion nd h (2)hr led to the sme scttering. According to the theory of sources (ssuming tht the nonliner sources re given by generl reltivity) we hve, see equtions (18), (25) nd (19) in [6] = ( MG b )2 [ ( r 2 19b ) δ 2 28b 9 x α x β + 2 r 2 x α x β ], r < b 5 b 2 7 b 4 (33) = (MG) 2 [ 5δ 7x αx β ( r 2 35 b xα x β δ ) ], r > b, r 5 3r 3 (33b) 00 = h (2)hr 00 = h (2)iso 00 = ( MG b )2 [ r 2 2b b 4], r < b (34)

6 For hrmonic h (2)hr 00 = h (2)hr 00 = h (2)iso 00 = 2 (MG)2 r 2, r > b,. (34b) nd isotropic h (2)iso we hve, see see equtions (6) nd (13) in [7] h (2)hr = M2 G 2 b 2 {δ ( 9 5 3r2 2b r4 3x αx β 2 4b 4)+ b 2 h (2)hr 2r2 x α x β b 4 }, r < b, (35) = M 2 G 2 ( x αx β + δ ), r > b, (36) r2 h (2)iso = M2 G 2 ( 15 b 2 4 3r2 b + 3r4 ), r < b, 2 b4 (37) h (2)iso = 3 M 2 G 2 δ 2 r 2, r > b. (38) From (35) nd (33) we hve h (2)hr = M2 G 2 { 4δ b (δ r 2 b +2x αx β ) 4 2 b 2 7 (δ x α x β b 4 4r2 }, r < b (39) b 4 In the r.h.s. of (39) stnd guge functions. Similrly from (36) nd (33b) we hve h (2)hr = M 2 G 2 { 4 δ r 2 +8x αx β b(x αx β r δ )},r > b. (40) r3 As expected the right hnd sides of (39) nd (40) re the guge function. This is becuse the nonliner sources re the sme. Now we give building blocks for clculting the Fourier-trnsforms. For contributions from r < b: d 3 xe i q x 1 b = 4π 2 q (sin cos ), = qb, (41) 2 d 3 xe i q xr2 b = 4π 4 q {( )sin+( )cos}, (42), d 3 xe i q xr4 b = 4π 6 q {( ! 2 4 6)sin+( ! 3 5)cos}, (43), = 4π b 4 q {q αq β q [( )sin+( )cos]+ d 3 xe q xr2 x α x β b 6 δ [( )sin cos]}, (44), 3 = 4π q {q αq β q [( )sin ( )cos]+δ 3 5 [( )sin+( )cos]}, (45) 3 5

7 For contributions from r > b we hve: d 3 xe i q x = 4π q cos, 2 r 3 = 4π q {q αq β q [ 3sin + 3cos δ [ sin 2 2 cos r 5 Putting α = β in (46) nd (47) we get = 4π q 2{q αq β q 2 ( 3 sin+cos)+δ δ [( cos 33)sin d 3 xe i q x 1 r 2 = 4π q 1 2 u du]+ sin }, (45) du]}, (46) u = 4π{ q αq β q [ sin + cos 2 3 ]+ 2 d 3 xe i q x 1 r 3 = 4π ( sin + du]}, (47) u du, u (48) ) u du. (49) Now it is esy to verify tht the sum of contributions from r < b nd r > b to Fouriertrnsforms of differences in (39) nd (40) do not contin terms proportionl to δ. Similr sttement is true for h (2)hr h (2)iso. This mens tht if two h (2) differ only by guge function then they led to the sme scttering. So, it is shown tht with term (28) leds to the sme scttering s h (2)hr. References 1. V.A.Fock, The theory of spce-time nd grvittion, Pergmon, New York, S.Weinberg, Grvittion nd Cosmology,John Wesley, New York, R.Feynmn, F.Moringo, W.Wgner, Feynmn Lectures on Grvittion. Addison-Wesley. 4. A.I.Nishov, Grvity from the viewpoint of theory of sources, rxiv: v1[physics.genph] 16 My N.V. Mitskevich, JETP, 34, 1656 (1958) (In Russin). 6. A.I.Nishov, Grvity ccording to theory of sources, rxiv: v.1, (physics genph) 7 Dec A.I.Nishov, On the simplified tree grphs in grvity. rxiv: (physics genph).(2011). 8. L.Lndu nd E.Lifshitz, The clssicl theory of fields, Oxford,Pergmon Press,1983.