The Dirac Equation in a Gravitational Field

Transcription

1 8/28/09, 8:52 PM San Franiso, p. 1 of 7 sarfatti@pabell.net The Dira Equation in a Gravitational Field Jak Sarfatti Einstein s equivalene priniple implies that Newton s gravity fore has no loal objetive meaning. It is an inertial fore, i.e. a ontingent artifat of the ovariantly tensor aelerating (non-zero g s) Loal Non-Inertial Frame (LNIF i ) detetor. ii Indeed, Newton s gravity fore disappears in a loally oinident non-aelerating (zero g) Loal Inertial Frame (LIF iii ). The presene or absene of tensor spaetime urvature is ompletely irrelevant to this fat. In the ase of an etended test body, these remarks apply only to the Center of Mass (COM). Stresses aross separated parts of the test body aused by the loal objetive tensor urvature are a logially independent separate issue. Garbling this distintion has generated not-even-wrong ritiques of the equivalene priniple among philosophers of physis and even among some venerable onfused theoretial physiists. Non-standard terms oupling the spin-onnetion to the ommutator of the Dira matries and to the Lorentz group Lie algebra generators are onjetured. & e LNIF and 24 spin onnetion! LNIF = "! LNIF LNIF I LIF IJ LIF JI LIF The 16 tetrad e I ( LIF) oeffiient fields desribe the transformations between loally oinident LIFs and LNIFs. The tetrads desribe translational ovariant tensor aelerations on massive test partiles pushed off the zero g-fore time-like geodesis of the symmetri torsion-free Levi-Civita onnetion. The spin-onnetions desribe ordinary 3D spae rotations of the LNIF about its enter of mass as well as the spae-time rotations (Lorentz boosts). iv Negleting the eletro-weak-strong internal symmetry loal gauge fores at first, the transformation of partial derivatives proeeds as! = e I ( )! I L IJ (1.1) where! I & L IJ = "L JI are the 10 elements of the Lie algebra of the Poinare group that generate the 4 spae-time translations and the 6 spae-time rotations. The inverse is e I ( L JK ) =! I (1.2) ( )! " # JK Imagine an LIF with a small displaement 4-vetor! J from its origin. Small means relative to any loal tensor urvature radii fields R IJKL!1 that may or may not be there and then. Note that ""is the origin i.e. event at whih the small detetor is loated. Then

2 8/28/09, 8:52 PM San Franiso, p. 2 of 7 sarfatti@pabell.net L JK! " J # K $ " K # J! " [ J # K ] (1.3) The Lagrangian for a neutral Dira spinor seond quantized free field on a spaelike slie of 4D globally flat Minkowski spaetime in the absene of gravity is L( t) = d 3 L!!! L ( m + $ I % I )# = "# (1.4) The global dynamial ation in a finite flat spaetime region is S =! dtl( t) (1.5) The Feynman quantum amplitude for that partiular lassial field onfiguration history is symbolially A! e i S! (1.6) We annot do these global integrals in urved spaetime so easily beause of the pathdependent anholonomy analogous to history dependent irreversibility in non-equilibrium thermodynamis where we no longer have state funtions (e.g. eat Cartan 1-forms). Similarly, as in lassial mehanis of partiles when we no longer have a simple stati potential, but also have veloity-dependent non-entral fores. However, we an look at the loal terms.! ( ) = e I! I #$! I,! J % & (1.7) The seond spin-onnetion term on the RHS is non-standard and is only an empirial onjeture that is Popper falsifiable. It is also a new torsion field oupling if torsion fields are present. The Ansatz is then! I " I #! " I ' ( e ) = e I ( )! I + $ IJ %&! I,! J I ' J ' ( ( )" I ' + $ ' (1.8) )! I " I However, Einstein s loal frame differential spae-time interval ds 2 ( ) is invariant under LIF! LNIF ( )

3 8/28/09, 8:52 PM San Franiso, p. 3 of 7 sarfatti@pabell.net ds 2 ( ) = g! ( LNIF) ( )e ( )e! e!! ( ) ( LNIF) = e I ( )e ( I ) ( ) ( LIF) e I! ( ) # $! $ I = " IJ LIF e I ( )e J (1.9) where, again, is the loal oinidene of two detetors lose to eah other ompared to the loal radii of urvature if there is any. The e's are loal frame (o) vetors forming a basis in the sense of the algebra of vetor spaes. The important gravitational aeleration loal T4() gauge potential is A I ( ) where e I ( ) =! I I + A ( ) (1.10) For a simple eample of a non-relativisti uniformly slowly rotating non-inertial frame about the z ais with ignorable time dilation and tangential length ontration, i.e.,! '/! 1 et. z'! z ( LIF) = z ( LNIF)! z t '! t ( LIF) = t ( LNIF)! t et. ' = os"t + ysin"t y' = #sin"t + yos"t ' e! $' $ = os"t % A ' = os"t # 1 ' Lim A = 0 " %0 ' e t! 1 $' $t Lim A ' t = 0 " %0 et. ' = A t = " ( #sin"t + yos"t ) (1.11) ds 2 =! IJ ( LIF) d I d J = " 2 dt ' 2 + d' 2 + dy' 2 + dz' 2 (1.12) = g # ( LNIF) d d # Eah loally oinident frame arries its own non-intrinsi historially ontingent representation g! ( LNIF) ( ) & " IJ ( LIF) of the metri tensor field. Only the loally variable spaetime interval ds 2 (, + d)! ds 2 ( ) has objetive invariant physial meaning. g! ( LNIF) " #! + h! (1.13)

4 8/28/09, 8:52 PM San Franiso, p. 4 of 7 sarfatti@pabell.net h! ( rotatinglnif) = $ = 2 + y 2 = $ os#t y = $sin#t " # 2 $ 2 2 # y " # #$sin#t #$ os#t " (1.14) Einstein s loal equivalene priniple is mathematially epressed in terms of the nontensor Levi-Civita onnetion that desribes the ontingent inertial g-fores felt on massive strutures that are pushed off timelike v geodesis by non-gravity fores.! IJK ( LIF) = 1 ( 2 " g + " g # " g J IK K IJ I JK ) = 0! $% ( LNIF) = 1 ( 2 " g + " g # " g $ % % $ $% ) & 0 (1.15) The presene or absene of urvature is ompletely irrelevant to the validity of the equivalene priniple and some physiists are very onfused on this issue. The urvature is measured in a ompletely different way from the measurement of the onnetion field even though the urvature is the ovariant url of the onnetion field. One uses pairs of losely spaed freely falling test partiles eah on timelike geodesis to measure the omponents of the urvature tensor. All inertial g-fores are eliminated in the urvature measurement, whereas the onnetion field measurement is that of inertial g-fores! For eample, in the above slowly rotating frame! "# ( LNIF) = 1 ( 2 $ g + $ g % $ g " # # " "# ) i=!,y,z 00 & 1 ( 2 'i# 2$ 0 h 0# % $ # h 00 )! 00 = % ( 2 $ t)sin(t + $ ( y 2 = 1 2 ' 2$ 0 h 0 % $ h 00 * = % 2( 2 - +, 2. / *, /. (1.16)

5 8/28/09, 8:52 PM San Franiso, p. 5 of 7 sarfatti@pabell.net y Similarly,! 00 = " 2# 2 y 2. Therefore transforming this Cartesian representation to ylindrial oordinates gives 2! " 00 / 2 = #$ 2 " the radially inward entripetal aeleration inertial g-fore loally equivalent to Newton s gravity fore. Imagine a rigid rotating ylinder. The eletrial fores of the ylindrial wall provide the push off the timeline geodesi that you feel as weight, e.g. artifiial gravity on a rotating wheel spae station. e I ( )! I (! I,! ) J I ' e I ' J #$ % & ( )' I ' + " ' = e I ' I ( )e ( )! I ' I ' + e I ( )! I I ' J " ' #$! I,! J I ' % & e ( )' I ' #$! I,! J I ' J % & " ' (1.17) =! I ' I + e I ( )! I I ' J " ' #$! I,! J I ' % & e ( )' I ' #$! I,! J I ' J % & " '

6 8/28/09, 8:52 PM San Franiso, p. 6 of 7 sarfatti@pabell.net i LNIFs have indies,!,"... raised and lowered with the loally variable urvilinear metri tensor field g! ( ) et. ii What is taught as Newton s gravity fore in elementary physis is simply the universal ovariant aeleration of a stati LNIF detetor in Spherially Symmetri Stati urved spaetime. iii LIFs have indies I, J,K... that are raised and lowered with the SR onstant Minkowski metri tensor! IJ,,! I J,! IJ iv The Levi-Civita onnetion is mistakenly ompared to a loal gauge field by some physiists. This leads to the issue of the non-renormalizability of anonial top down quantum gravity, e.g. starting with lassial ADM formalism of lapse, shift and 3D geometry dynamial variables. The real gauge potentials are the tetrads and spin onnetions. The tetrads ome from loalizing the global 4-parameter translation subgroup T4 T4() of the globally rigid 10-parameter Poinare universal spaetime symmetry group of Einstein s 1905 Speial Relativity. If we stop there, we get preisely Einstein s 1915 General Relativity (GR) with urvature equivalent to dislination defets in a 4D Lorentzian world rystal lattie (Hagen Kleinert) without torsion gaps in the infinitesimal parallelograms where one parallel transports one edge by the other and ompares the relative rotation of a third non-planar vetor around the parallelogram losed loop. The angle of rotation (dislination) is the area of the parallelogram multiplied by the setional urvature in the limit that the area shrinks to zero. In that ase, the spin onnetions are dynamially redundant determined ompletely by the tetrads and their first partial derivatives in ompliated antisymmetri ombinations shown as Rovelli s eq in his book Quantum Gravity. If we loalize the full Poinare group we also get a new dynamial torsion gap disloation tensor field in addition to the urvature dislination tensor field. A dislination is a line defet in whih rotational symmetry is violated. In analogy with disloations in rystals, the term, disinlination, for liquid rystals first used by F. C. Frank and sine then has been modified to its urrent usage, dislination. It is a defet in the orientation of diretor whereas a disloation is a defet in positional order. We an ontinue, loalizing the dilatation group and the four onformal boosts to uniform hyperboli motion in SR to get a non-metriity tensor field. The Levi-Civita non-tensor onnetion field then gets additional tensor piees. The term tensor is always relative to a group of physial loal frame transformations. Unless otherwise stated, tensor always means loal LNIF LNIF among ovariantly aelerating non-inertial frames. Formal oordinate transformations in a fied loal frame are an equivalene lass and are gauged away. The LNIF is an equivalene lass of formal urvilinear transformations that are simply different desriptions of the same material detetor. v Timelike means inside the loal light one. Lightlike means on the loal light one. Spaelike means outside the loal light one. Eah light one has a future and past piee. Advaned Wheeler-Feynman eletromagneti spherial Huygens waves from the point origin of the light one propagate epanding fronts bakward in time to the past with

7 8/28/09, 8:52 PM San Franiso, p. 7 of 7 sarfatti@pabell.net positive energy along the past light one with phase! adv = kr + "t. Retarded spherial waves propagate epanding fronts forward in time with positive energy along the future light one with phase! ret = kr " #t. The spherial wave form in general is! spherial " Ae i# / r