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
Topics








Maximal Tension and Reciprocity;
Reciprocally-invariant generalization of the energymomentum 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;
Maximal Tension Principle in GR
c
 dp 
MaximumForce :    F0 
.
4G
 dt lim
4
c
 dE 
MaximumPower : 
.
  cF0 
4G
 dt lim
5
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.
Born’s reciprocity principle
Reciprocity transformations : p  x, x   p.
RI quadratic form : S B2  p  p  x  x  B  inv.
RI equation :


2

 x x    B ;
  


x

x



SU(3,1) - invariance
M. Born’s Reciprocal Symmetry
and Maximal Force 1
RI Infinitesimal Interval
dS B 2

dp  

1
dp


2
 dx dx   2 dp dp   ds 1  2

0
  0 ds ds 
1
0 – universal constant with dimension
momentum/length or energy/time
(L. Tomilchik - 1974)
Choice:  0  c G (L. Tomilchik - 2003)
3
M. Born’s Reciprocal Symmetry
and Maximal Force 2

dp
If ds 2  c 2 d 2 ,
 f   Minkowski force
d
1
2

f  f 
than dS B  cd  2  , f  f   0.


F
0


In comoving reference frame
1
2
2

f 

2
f f    f , so dS B  cd  2  .
 F0 
One can see that F0  c 0 is the Maximum Force
(MF)  the upper limit of any force.
Reciprocally-Invariant Quadratic
Form in the QTPH Space 1
SB
2
1 
1 2
2
2
 x x  2 p p  x0  r  2  p0  p2   inv ,
κ0
κ0

μ
μ


x

x
,
r
;
p
  p0 , p ,
4-vectors
0
p and x are canonical variables
ημν  diag (1,1,1,1, ).
Poisson brackets (classical) are defined as
p μ , xν  ημν .
Reciprocity transformations : p   0 x, x   01 p.
The symmetry group — SU (1,3) .


Case SB2=0 and hyperbolic motion
Condition S B 2  p  p   02 x  x  0.
Twopossibilities : (A) p  p  x  x  0 (light cone)
(B) p02  p 2 = m 2 c 2 ; x02  x 2   r02 (hyperbolic motion).
In the case(B) we obtain the condition :
mc mG 1
2 2
2 2
m c   0 r0  0, hence r0 
 2  rg
0
c
2
( rg is the Schwartzchield radius)
F0
c2
c4
w(m)  

 is the maximal acceleration for the mass m.
r0 Gm m
Reciprocally-Invariant Quadratic
Form in the QTPH Space 2
The dimensionless variables
p
r
p
P   Q  , where qee   0 ,
pe
qe
pe qe  a  constant having dimension of action
1
2
1
 a 
 Planck constant, qe     pe   a 0  2 ,
 0 
 pe qe min  amin 

 c 
pextr    0   
  pP 

 G 

1
2
1

G


2
qextr   /  0    3   lP 
 c 

1
2
3
1
2
 Planck's parameters
The self-reciprocal invariant
(dimensionless values)
ct
S  H  P    Q  B  inv.,   .
qe
2
B
2
2
2
2
Thetime - dependentHamiltonian :
1
2 2
H ( P , Q, )  ( H   ) , where H  P  Q  B .
2
0
2
0
2
2
The canonical equations :
dQ H
dP
H
2
2

 P(H0  ) ;

 Q ( H 02   2 )
d P
dt
Q

1
2

1
2
Its easy to see, that H is the integral o f motion, and can be considered
0
as a constant, with respect to differentiation and integration by  .
The maximum power
The rate of change of the energy
dH
2
2
  ( H 0   )
d

1
2
dH
The functions H and
are real in the domain
d
H0    H0
dH
The average value of
:
d
 fin
H fin  H in
1
dH
dH
  fin   in  
d 
d
 fin   in
 in d
The maximum power (cont.)
Choosing  in  0,  fin  H 0 , we have H in  H 0 , H fin  0.
dH
Hence
 1 and in dimensional values :
d
5
dE c 2 pe dH
c

 c 2 0  cF0    the maximal power
d
qe d
G
The explicit form of action
The conventional connection between the action S and
Hamiltonian H is:
S ( H 0 )
H ( H 0 )  

Under supposition that integral of motion H0 can be treated as a
parameter, we can write the following:
1
dS ( H 0 )
 ( H 02   2 ) 2
d
Elementary integration gives :
2
  
H
2
0
S ( H 0 , )   ( H 0   ) 
arcsin 
  S0 ( H 0 )  const.
2
2
 H0 

1
2 2
The arbitary function S 0 ( H 0 ) and const
are to be defined from the initial conditions.
The classical motion picture
1. Initial conditions :
1
S0 ( H 0 )  (k  ) H 02 (k  0,1, 2,3...), const  0.
4
2. For the fixed value of H 0 the duration of the "particle's" motion.
qe
is restricted by the time interval t 
H0.
c
3. Change of energy : E  cpe H 0 .
E c 2 pe  dE 
4. Average rate of energy change :


 .
t
qe
 dt lim
5. Value of action S 

4
E t 
e
4
qe pe H 02 .
The canonical quantization
 k , Ql   i kl .

S ( H 0 , )  S N  N | S | N ,  is considered as a parameter.

1
2

SN    N | H | N   
2


2
0
2




1

2
 N | H 0 | N arcsin 
2


2
N
|
H

0 | N

1
2


1



2
2
  S0  N | H 0 | N 





The canonical quantization (cont.)

N | H 02 | N
For the definition of
we use the discrete spectrum of
 2
2
2
2 
Born's equation :  
  0          0 ,    B   0 ,   .
2
  0

Its solution : 
 ,     n
3
0
k 0
k
,
nk is the well - known oscillator eigen functions,
B    n0 , n    2n0  1   2n  3 , n  n1  n2  n3 .



n0 , n | H 02 | n0 , n  n0 , n | P 2  Q 2 | n0 , n   (n0 , n)  2n0  1
The action spectrum


1




S N   2 N  1 2 1 
1

2
   2 N  1 2

1




2
2

 





  S0  2 N  1 , N  n0  0,1, 2,...
  2 N  1 arcsin 
1


2
2
  2 N  1 
Linear dependence S  N  on N requires :
1



1)
1

2
2
N

1



2

  1; 2)S  2 N  1     2 N  1 ,
0


 is numerical parameter
The action spectrum (cont.)


Then (in dimentional units) : S N  pe qe      2 N  1 .
4

S N 1  S N  h is the condition for the choice of  and pe qe :
5
1)  
,
4
2) pe qe
 .
Finally :

1
S N  h  N   , N  0,1,2,...
2

Energy spectrum and discrete time
1
2
1
2
5


1
c


E N  EP  N   , EP  
 .
2

 G 
1
2
1
2
1

 G
tN  tP  N   , tP   5  .
2

 c 
The maximal power :
E E  N fin   E  N in  EN c 5  dE 


 

t
t
G
dt
t  N fin   t  N in 

lim
N
The possible cosmological
outcome
Early Universe : Radiation - Dominated Stage
(A) Standard picture : FLRW - model, scale factor R(t )
wavelength time dependence  (t )
R (t )
t
radiation energy dependence
1
2
 t fin 
E t ,

 .
fin
E
 tin 
(B) Discrete time picture :
 12
E in
E
fin


E
in
( N fin  2) 
  N fin 
N fin 1
1
2
1
2
1
2
t ,
Universe Expansion stages on
energy scale
Plankian magnitude
CUT (SU(5)) - breaking
SUL(2)  U(1) - breaking
Quark confinement
pp - annihilation
ee - annihilation
 separation
(final of RD stage)
Energy,
GeV
Time,
sec
1019
10–44
1015
102
100
10–3
10–4
10–9
10–36
10–10
10–6
100
102
1012
E in  EP
E fin
19
  N fin 
10
= 108 
15
10
1
2
1
1019
34 2
= 10 
102
19
10
= 1038 
0
10
19
10
= 10 44 
3
10
19
10
46
=
10


10 4
19
10
= 1056 
9
10
1
2
1
2
1
2
 t

 in

 t  tP 
fin
1
2
1
2
 10 
4
 44  = 10
 10 
36
1
 10 10  2
17
 44  = 10
 10 
1
2
 10 
19
 44  = 10
 10 
6
1
2
 10 
22
 44  = 10
 10 
0
1
1
2
1
2
 102  2
23
 44  = 10
 10 
1
2
 10 
28
 44  = 10
 10 
12
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;
The general covariant D’Alembert equation
1

g
  gg


  ( x)  0
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
= exp(αx2) we have
U-1 ∂μ ∂νU = 2αδνμ + 4α2xμxν
In the case U(x)
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)
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 B   N s  1
In the Minkovski space case:
(- ∂ξ2+ ξ2 ±4)Φ(ξ) = 0.
For the Gaussian-like metric gμν = exp(±ξ2) ημν correspondingly.
The geodesic equation


d 2 x
dx
dx

 
0
2
ds
ds ds
2
g

U
( x) can be presented in the form
in the case of metric 
d  2 dx   1 
2
U
 


U
0

2
ds 
ds  2U 1
 


ds

U
dx
dx
Using
 

2
 cUd
we can write
1 d U 2 dx  d 2 x 
c 2 
2




U
0

2
2
2
2U d d
d
2U
 
 x2 
In the case U ( x)  exp   : 
 2l 
0 

 
1 d x 2 dx  d 2 x  c 2  
 2

 2 x 0
2
l0 d d
d
l0
Under condition x 2  const, the geodesic line belongs to the
hyperboloid. In this case:
 
d x2
 dx
 2x
0
d
d
The geodesic equation under this condition transforms in
d 2 x c2
 2 x  0
2
d
l0
The equations coincide (in the case of Minkovski space) with the
SR equations for hyperbolic motion of the probe particle.
Minkovski force f  ~ x 
Multiplying by x  we have
2
2
d x c 2
x 2  2 x 0
d
l0
, (  corresponds to U ( x)  e

x2
l02
)
Using the identity
2
2
d x
 dx  1 d
2
x 2    
(
x
)
2
d
 d  2 d
2
We receive under condition x 2  const
2
2
 dx  c 2
    2 ( x  const )  0
 d  l0
This condition is satisfied for the upper sign (-), i.e. U ( x)  e
2
when  dx 
unparted 
2
2
2 




    с , x   l0 
 hyperboloi d
 d 
 of two sheets 
c is the
 upper 


 lower 
limit for velocity
x2
One interesting exact solution of M.Born’s equation (discrete
spectrum)
(- ∂ξ2+ ξ2)Ψ(ξ) = λBΨ(ξ)
In Cartesian coordinates
 ( )  e
 02

2
Ns
H n0 ( 0 ) e
k 1
H n ( )
 k2

2
H nk ( k )
- are the Hermitian polynomials
B  (2n0  1)  (2n  N s )
Ns
Where n   nk , n and nk are the natural numbers in the case
0
k 1
under consideration
B  ( N s  1)
Now we have the following conditions
(2n0 + 1) - ( 2n + Ns) = ± (Ns + 1)
The nonzero solutions exists when
(I)
n0 = n - 1 in the case U ( x)  exp  x 2 /( 2l02 ) 
(II)
n0 = n + Ns in the case U ( x)  exp x 2 /( 2l02 ) 
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
2
g

exp(

x
)
Einstein tensor G for metric

1
2
G   6    2 x x    x ,    2
l0
Energy-momentum tensor
C 4

T  
6    2 x x    x 2 
8G
2 4
3

С
Minkovski force density f    T: f  
x
2G
Energy density
3C 4  2C 4
T00  

3x02  r 2
4G
8G


Thank You for Your
attention!