Tensor Taylor series method for vacuum effects

Transcription

1 Tensor Taylor series method for vacuum effects M. W. Evans, H. Eckardt Civil List, A.I.A.S. and UPITEC ( Numerical and graphical analysis 3.1 General formulas The detailed Taylor series of a function of vector arguments was given in Eq. (13). The isotropic averaging of f(r) can be performed on this abstract level, giving the nd order Taylor terms f () = δr δr ( ) d dx f + d dy f + d dz f, (55) the th order terms f () = 1 ( d ( dx f + d dy f + d d dz f + dy dz f + d dx dz f + d )) dx dy f (5) and the th order terms f () 3 ( d = 19 dx f + d dy f + d dz f (57) ( d + 15 dy dz f + d dy dz f + d dx dz f + d dx dy f + d dx dz f + d d ). +9 dx dy dz f In the following we consider some examples. 3. Vector potential of a magnetic dipole The vector potential of a magnetic dipole has been given by Eq. (). The vector components of A experience vacuum corrections of fourth order onward. The emyrone@aol.com mail@horst-eckardt.de ) dx dy f 1

2 X component in the first two non-vanishing orders is A X () = 35Y ( 3Z 5Y Z 3X Z + Y 5X Y + 3X ), 18(X + Y + Z ) 11 A X () = 3 9(X + Y + Z ) 15 (58) (59) 7X ( 8Z 75Y Z 15X Z 75Y Z + 3X Y Z 1X Z + 8Y 15X Y 1X Y + X ). The undistorted vector potential A has been graphed in Fig. 1 for constant values of δr. The averaged vacuum terms A X () and A X () are plotted in Figs. and 3. According to their degree of approxmiation, they have a high symmetry. There is a behaviour like a source field potential at the centre. When both contributions are added to the total potential of Fig. 1, the result of Fig. emerges. In the central part the corrections due to vacuum fluctuations dominate the structure, representing a multipole expansion of the vacuum terms. 3.3 Coulomb potential The isotropically averaged, non-vanishing fluctuations of the normalized Coulomb potential are and 1 U = () (X + Y + Z ) 1/ U () = 7 ( Z 3Y Z 3X Z + Y 3X Y + X ) U () = 3 9(X + Y + Z ) 15 18(Z + Y + X ) 9 (1) () 7X ( Z 15Y Z 15X Z 15Y Z + 18X Y Z 15X Z + Y 15X Y 15X Y + X ). These are formally similar to those of the vector potential components. Both contributions are graphed in Fig. 5 together with the Coulomb potential (in X direction). The fourth-order fluctuation is repulsive, while the sixth-order fluctuation is attractive like the original potential. All components together (Fig. ) give a correction to the potential which leads to a steeper descent near to the central charge. 3. Gravitational force The normalized gravitational force (identical to Coulomb force) r F = (3) (X + Y + Z ) 3/

3 leads to the non-vanishing Taylor series fluctuations (X component) F X () = 35X ( 3Z 3Y Z 5X Z + 3Y 5X Y + X ) 18(Z + Y + X ) 11 () and F X () = 3 9(X + Y + Z ) 15 (5) 7X ( 8Z 75Y Z 15X Z 75Y Z + 3X Y Z 1X Z + 8Y 15X Y 1X Y + X ). The fluctuations (Fig. 7) are similar in sign to those of the potential, but the force field (Fig. 8) has a pronounced saddle. Also in this case the vacuum fluctuations effect a broadening of the central region, like an effective central mass with extended radius, not a point mass. As stated in section, relativistic corrections to Newton s force law () can be expanded for small velocities v to a second-order correction in v /c: F 3 v mmg c r 3 r. () This term has to be equal to the Taylor expansion of F whose first two nonvanishing terms are given by Eqs. (, 5). Correctly, one has to use mmg ( ) 3/ r 3 r 1 v c = mmg r 3 r + F () + F () +... (7) The exponential term at the left hand side can be expressed by Newton s generalized binomial theorem: ( ) 3/ 1 v c = 1 + 7v 1 1c 1 + 3v 1 5c 1 + 3v 8 18c 8 + v 1c + 3v 8c 3v c (8) Using the terms of lowest order at the left hand side, we have mmg ( r 3 r 1 3 v c + 3 v 8 c + 1 v ) 1 c +... = F () + F () +... (9) The right hand side gives a dependence on δr δr so that in principle it is possible to compute the size of fluctuations δr from the orbit. Please note that the result depends on the orbital velocity v which is given by the orbital relation v (r). This relation is known if the orbital dynamics is known, for example from a Lagrange solution of (). Resolving Eq. (9) for δr gives a polynomial of degree in the case above. There is a real-valued solution for (δr) but the terms are such complicated that this method of determining δr cannot be handled even by computer algebra with reasonable effort. 3

4 Figure 1: Undistorted vector potential A of a magnetic dipole field. Figure : fluctuations of th order of the magnetic vector potential.

5 Figure 3: fluctuations of th order of the magnetic vector potential. Figure : Total magnetic vector potential with fluctuation terms. 5

6 Figure 5: Coulomb potential and th order and th order fluctuations. Figure : Total Coulomb potential with fluctuations.

7 Figure 7: Gravitational force and th order and th order fluctuations. Figure 8: Total gravitational force with fluctuations. 7