is reached, starting from rest, by performing the direct product of two operations: 1) bringing the rest frame into a Lorentz frame with velocity v and vanishing ...
LETTERE AL NUOVO CII~ENTO
VOL. 34, N. 4
Remarks on the Maxlmal-Acceleration
22 Maggio 1982
Hypothesis.
E . i~. CAIANIELLO
Scuola di Per]ezionamento i n Seienze Cibernetiehe e I'isiche dell' Universit~ Salerno, I t a l i a
S. D ~ FILIPPO I s t i t u t o di .Fisica dell' Universit~ . Salerno, I t a l i a
G. MARco Is~iiuto di ,Fisica Teorica dell' Universith - N a p o l i I s t i t u t o N a z i o n a l e F i s i c a N u c l e a t e - Sezione di N a p o l i , I t a l i a
G. VILASI I s t i t u t o di ~ i s i c a dell' Universitd . Salerno, I t a l i a
(rieevuto il 12 Marzo 1982)
I n a p r e v i o u s n o t e (1) (referred to h e n c e f o r t h as I) it was shown t h a t t h e a d d i t i o n of some assumptions to a g e o m e t r i c a l m o d e l of QM (3) leads n a t u r a l l y , at a (~p r e - q u a n t u m , quasi-classical level ~>,to t h e n o t i o n of ~ m a x i m a l acceleration ~)for (~particles ~>. A ~(particle ~>, in this c o n t e x t , is an e x t e n d e d s t r u c t u r e (3,4), w h i c h we m a y visualize, if we h a v e to, as a (( rigid cloud ,) whose global p h y s i c a l properties are a t t a c h e d to a (~p o i n t ~>, after t h e fashion of K a l u z a and Klein, Synge (5) and m a n y others. W e propose to show h e r e some f u r t h e r consequences of t h e assumptions m a d e in I. T h e c o n t e n t of I is in f a c t i n d e p e n d e n t of t h e v a l i d i t y of t h e original m o d e l a n d should be an object of criticism p e r se. W e p u t t h e r e for a p a r t i c l e
(1)
(1) E. (~) E. (8) E. (4) E. curved (~) J.
112
1
ds~ = d ~ - - ~ dx~ +
1
2
R. CAIANIELLO: Left. Nuovo C~mento, 32, 65 (1981). R. GAIANIELLO: Nuovo Gimento B, 59, 350 (1980). R. CAIANIELLO and G. VXLASI: Lett. Nuovo Ctmento, 30, 469 (1981). R. CAIANIELLO, S. DE FILIPPO and G. VILASI: Extended Dirac particles and their spectra in phase space, Dreprint Salerno University (1981). L. SYNGE: Proc. R. Soe. London Set. A, 319, 307 (1970).
113
R]~MARK$ ON TH]~ M A X I M A L - A C C E L E R A T I O N H Y P O T H E S I S
with
(2)
h = 2"/,0.
T h e ensuing u p p e r b o u n d for acceleration (*) (3)
d~
a =
KAy=
(1--v~'/c2)~A [1 - - (v~/c 3)
sin S (v, a)] 89
with
(4)
/~ ~ c 3
c~
l.*c3
moh
m o2~
mo
is c o m p u t e d b y t a k i n g
(5)
E : cC~r ~+p~= ~.
T h e m e a n i n g of A is o b v i o u s : if # ~ m o (rest mass) and 2 ~ linear extension or ((radius ~>of t h e particle, (3) just says t h a t any v e l o c i t y v on a circular orbit cannot excced c(a = v~/r < c2/~). Use of (5) for d e r i v i n g (3) implies a r a t h e r drastic l i m i t a t i o n on t h e general m o d e l (s), i.e. a clear-cut separation b e t w e e n t h e x, t space and t h e i n t e r n a l space of t h e e x t e n d e d p a r t i c l e (described h e r e b y p, E). One supposes t h u s t h a t t h e state v, a is reached, s t a r t i n g from rest, b y p e r f o r m i n g t h e direct p r o d u c t of two operations: 1) bringing t h e rest f r a m e into a L o r e n t z f r a m e w i t h v e l o c i t y v and vanishing accelerat i o n ; 2) bringing, w i t h c o n s t a n t v, t h e acceleration to t h e v a l u e a. This assumption is t h e simplest one can make, all t h e others l e a d to complications such as found in t h e D i r a c - E l i z i e r classical electron t h e o r y (nonrigid clouds?); we m a i n t a i n it here. I t is t h e n consistent to take, a f t e r n o t i n g t h a t (1) implies
(dq (6)
\ dr/
(7)
=(
1
=
O
c~/k
ds/-
'
-
v3/+v'i
- - a o/A~3"
Suppose, now, v
