Maximal tension, Born’s reciprocity, Discrete time and conformally flat Gaussian-like metric.

1. Maximal tension, Born’s reciprocity, Discrete time and conformally flat Gaussian-like metric. Lev.M.Tomilchik B.I.Stepanov Institute of Physics of NAS of Belarus, Minsk. Gomel, July 2009

2. Topics • Maximal Tension and Reciprocity; • Reciprocally-invariant generalization of the energy-momentum connection; • Reciprocally-invariant Hamiltonian one-particle dynamics; • Explicit expression for the classical time-dependent action; • Canonical quantization, semi classical approach; • Discrete time and quantized action; • The possible cosmological outcomes; • Connection between Born’s reciprocity and conformally-flat metric;

3. Maximal Tension Principle in GR Maximum Force is the reversal to Einstein’s gravitational constant. Gibbons (2002), Schiller (2003, 2005). The problem: MTP beyond the GR. Our proposition: to connect MTP with Born’s reciprocity.

4. Born’s reciprocityprinciple

5. M. Born’s Reciprocal Symmetry and Maximal Force 1

6. M. Born’s Reciprocal Symmetry and Maximal Force 2

7. Reciprocally-Invariant Quadratic Form in the QTPH Space 1

8. Case SB2=0 and hyperbolic motion

9. Reciprocally-Invariant Quadratic Form in the QTPH Space 2 The dimensionless variables

10. The self-reciprocal invariant (dimensionless values)

11. The maximum power

12. The maximum power (cont.)

13. The explicit form of action The conventional connection between the action S and Hamiltonian H is: Under supposition that integral of motion H0can be treated as a parameter, we can write the following:

14. The classical motion picture

15. The canonical quantization

16. The canonical quantization (cont.)

17. The action spectrum

18. The action spectrum (cont.)

19. Energy spectrum and discrete time

20. The possible cosmological outcome

21. Universe Expansion stages on energy scale

22. Born’s reciprocity and conformally-flat metrics We will show that in the Gaussian-like conformally-flat metric: the D’Alembert equation has the form of the M.Born’s equation; the solution of the geodesic equation describes the hyperbolic motion of the probe particle; there is a solution corresponding to the discrete spectrum;

23. The general covariant D’Alembert equation In conformally flat metric gμν= U2(x)ημν, ημν= diag{1, -1, -1, -1} gives ∂μ∂μφ + 2U-1(∂μU)(∂μφ) = 0 After substitution φ(x) = U-1(x)Φ(x) We obtain ∂μ∂μ Φ – (U-1 ∂μ∂μU) Φ = 0

24. In the case U(x) = exp(αx2) we have U-1 ∂μ ∂νU = 2αδνμ + 4α2xμxν In the case of pseudo Euclidian space with dimension D = Ns+1, were Ns- number of the space dimensions ημν= diag{1, -1, -1,… -1} Ns times In the Minkovsky space case: D = 4, Ns = 3. The equation for Φ(x) in general case (- ∂ξ2+ξ2 ±D)Φ(ξ) = 0, were ξ2 = xμ/l0, i.e. α = ± 1/2l02 Sign (±) corresponds to U2(ξ) = exp(±ξ2)

25. This equation coincide with the self-reciprocal M.Born’s equation in the general case of Ns space dimensions (- ∂ξ2+ξ2)Ψ(ξ) = λBΨ(ξ) For the case In the Minkovski space case: (- ∂ξ2+ξ2 ±4)Φ(ξ) = 0. For the Gaussian-like metric gμν= exp(±ξ2) ημνcorrespondingly.

26. The geodesic equation in the case of metric can be presented in the form Usingwe can write

27. In the case: Under condition , the geodesic line belongs to thehyperboloid. In this case: The geodesic equation under this condition transforms in The equations coincide (in the case of Minkovski space) with the SR equations for hyperbolic motion of the probe particle. Minkovski force ~

28. Multiplying by we have , ( corresponds to ) Using the identity We receive under condition This condition is satisfied for the upper sign (-), i.e. when c is thelimit for velocity

29. One interesting exact solution of M.Born’s equation (discrete spectrum) (- ∂ξ2+ξ2)Ψ(ξ) = λBΨ(ξ) In Cartesian coordinates - are the Hermitian polynomials Where ,and are the natural numbers in the case under consideration

30. Now we have the following conditions (2n0 + 1) - ( 2n + Ns) = ± (Ns + 1) The nonzero solutions exists when n0 = n - 1 in the case n0 = n + Ns in the case In the case I (II) states with n0 – n = -1 (n0 – n = Ns) we have infinite degeneracy. In the case of Minkovski space the condition I remain unchanged, condition II becomes the form n0 = n + 3

31. Einstein tensor for metric Energy-momentum tensor Minkovski force density : Energy density

32. Thank You for Your attention!