Restricted and full 3-body problem in effective field theories of gravity

Transcription

1 Restricted and full 3-body problem in effective field theories of gravity Emmanuele Battista* Giampiero Esposito* *Istituto Nazionale di Fisica Nucleare, Sezione Napoli Università Federico II, Napoli

2 Introduction: Restricted three-body problem (1) Examples of hybrid schemes in Physics: Non-relativistic quantum particle in curved space-time. Quantum Field Theory in curved space-time G αβ = < T αβ >. Quantization of General Relativity with the application of effective field theory point of view higher-derivative corrections of the Hilbert-Einstein Lagrangian. Restricted three-body problem: a quantum perspective.

3 Introduction: Restricted three-body problem (2) The restricted three-body problem: Body A with mass α. Body B with mass β<α. A and B move under Newtonian potential considered without correction. Planetoid P with mass m such that m<<α, m<<β. Center of mass C. P is subjected to the quantum corrected Newtonian attraction of A and B. Where:

4 Quantum corrected Lagrangian (1) The quantum corrected Lagrangian describing the motion of P assumes the form: Where:

5 Choice of constants: Quantum corrected Lagrangian (2) + choice choice We define U as T 0 V = GU, so that N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Phys. Rev. D 67, (2003). J. F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994).

6 Equilibrium conditions Derivative of U: with y=0 equilibrium points on the line joining A to B λ =0 equilibrium points not lying on the line joining A to B

7 Equilibrium points on the line joining A to B (1) Divide the line y=0 into three regions R 1 : x (-,-a) R 2 : x (-a,b) R 3 : x (b,+ ) In each region U(x,0) has one equilibrium point:

8 Equilibrium points on the line joining A to B (2) In Newtonian theory U,xx y=0 >0. In the quantum case we have + choice In Newtonian theory -a-l < n 1 < -a. In our model we have choice

9 Equilibrium points on the line joining A to B (3) Classically, N 2 lies between C and B. Thus + choice In Newtonian theory <0. In quantum case condition that can be violated as r 0 and with choice

10 Equilibrium points on the line joining A to B (4) We also note that at N 2 we have Classically U(n 2 ) > U(n 3 ) > U(n 1 ). + choice + choice

11 Equilibrium points not lying on the line joining A to B (1) From the condition λ =0 we get the equation which is no longer solved by r=s. This equation can be expressed in the form

12 Equilibrium points not lying on the line joining A to B (2) The equation of fifth degree can be put in the form where if we study the first derivative of f(w), we obtain the following result

13 Equilibrium points not lying on the line joining A to B (3) which therefore, apart from w=0, vanishes at which are real roots provided that w 1 and w 2 are both negative

14 Equilibrium points not lying on the line joining A to B (4) the second derivative of f(w) is given by w=0 is a flex point. The sign of f (w) is governed by g(w). The graphs of f (w) and f (w) are given by

15 Equilibrium points not lying on the line joining A to B (5) f (w) f (w)

16 Equilibrium points not lying on the line joining A to B (6) There are two equilibrium points not lying on the line joining A to B, which we write in the form N 4 ( x(l),y + (l) ) and N 5 ( x(l),y - (l) ), where In Newtonian theory at the points N 4 and N 5 the planetoid is at the same distance from A and B. Our quantum corrected model predicts tiny displacement from this case.

17 Equilibrium points not lying on the line joining A to B (7) For the Sun-Moon-Earth system the quantum corrected planetoid coordinates are while the Newtonian values are while for Jupiter-Ganimede-Adrastea we have

18 Stability of equilibrium points Using first-order stability criterion, for quantum case we have found that Points N 1, N 2 and N 3 remain points of unstable equilibrium provided we adopt the + choice. Points N 4 and N 5 remain points of first-order stable equilibrium if we use the + choice.

19 Introduction: Full 3-Body problem Bodies A 1, A 2 and A 3 with masses m 1, m 2 and m 3 u=(x,y,z) and v = (ξ,η,ζ) H center of mass of A 1 and A 2 Equation of motion : where (α 1 =m 1 /(m 1 +m 2 ), α 2 =1-m 1 ):

20 A choice of quantum corrected potential the potential U(r 1, r 2, r 3 ) assumes the form Therefore we have

21 Hamiltonian equation of motion (1) Hamiltonian equations of motion read as where:

22 Hamiltonian equation of motion (2) At this stage we can exploit Poincaré theorem: if our Hamiltonian equations, which depend on a parameter ρ=l P, possess for ρ=0 a periodic solution whose characteristic exponents are all nonvanishing, they have again a periodic solution for small values of ρ. In our case for ρ =0 we revert to three-body problem in post- Newtonian mechanics. A. Chenciner and R. Montgomery, Ann. Math. 152, 881 (2000). G. Huang and X. Wu, Phys. Rev. D 89, (2014).

23 Variational equations (1) Assume a periodic solution of our Hamiltonian equations has been found Consider small disturbances of these periodic solutions Variational equations

24 Variational equations (2) We try to integrate variational equations by setting the constant α is called characteristic exponent. If, when ρ=0, the characteristic exponents are vanishing, then for small but nonvanishing values of ρ we have the expansions

25 Variational equations (3) If the Hamiltonian has the asymptotic expansion by virtue of the previous expansions variational equations give,

26 In our case we have Variational equations (4) Therefore, on writing we have found, for all i=1,,6 the general solutions of variational equations

27 Variational equations (5) where while for higher-order terms we find the inhomogeneous equations

28 Variational equations (6) What if the characteristic exponent does not vanish at ρ=0? In that case the asymptotic expansion should be generalized by adding α 0 Variational equations become

29 The periodic solutions Variational equations (7) can be taken to be solutions of equations when ρ = 0. Therefore the matrix M 0 ik should be evaluated along the solutions of the coupled equations

30 Variational equations (8) The desired periodic solutions can be written in the form

31 Conclusions We have derived a recursive scheme for the analysis of variational equations, although we have not solved them explicitly. The evaluation of periodic solutions of the full three-body problem within the post-newtonian regime is still in its infancy: only results for the circular restricted three-body problem are available so far. The years to come will hopefully tell us whether the scheme described may have observational consequences in orbital motion physics and in the experimental search for quantum gravity effects. G. Huang and X. Wu, Phys. Rev. D 89, (2014).