maximal tension, born’s reciprocity, discrete time and conformally flat gaussian-like metric
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Maximal tension, Born’s reciprocity, Discrete time and conformally flat Gaussian-like metric.
Lev.M.TomilchikB.I.Stepanov Institute of Physics of
NAS of Belarus, Minsk.
Gomel, July 2009
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;
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
0lim
5
0lim
.4
.4
dp cF
dt G
dE ccF
dt G
MaximumForce :
MaximumPower :
Born’s reciprocity principle
2
2
, .
.
;
B B
B
p x x p
S p p x x inv
x xx x
Reciprocity transformations :
RI quadratic form :
RI equation :
SU(3,1) - invariance
M. Born’s Reciprocal Symmetry and Maximal Force 1
RI Infinitesimal Interval
2 22 20 0
1 11B
dpdpdS dx dx dp dp ds
ds ds
0 – universal constant with dimension momentum/length or energy/time
(L. Tomilchik - 1974)
Choice: 30 c G (L. Tomilchik - 2003)
M. Born’s Reciprocal Symmetry and Maximal Force 2
2 2 2
1
2
20
12 2
22
0
0 0
, 0.
, .
B
B
dpds c d f
d
f fdS cd f f
F
f f dS cdF
F c
If Minkowski force
than
In comoving reference frame
so
One can see that is
the upper
,
ff
the Maximum Force
(MF) limit of any force.
Reciprocally-Invariant Quadratic Form in the QTPH Space 1
2 2 2 2 20 02 2
0 0
1 1BS x x p p x p inv
,
κ κr p
4-vectors ;, r0xxμ ,, p0pp μ p and x are canonical variables
.,,,, )1111( diagημν Poisson brackets (classical) are defined as
., μννμ ηxp
The symmetry group — (1,3)SU .
10 0, .p x x p Reciprocity transformations :
Case SB2=0 and hyperbolic motion
2 20
2 2 2 20 0 0
2 20 0 0 2
0
0.
0
; .
10,
2
B
2 2 2
2 2g
g
S p p x x
p p x x
p m c x r
mc mGm c r r r
c
r
Condition
Twopossibilities : (A) (lightcone)
(B) (hyperbolicmotion)
In thecase(B)weobtain thecondition :
hence
( is the Schwartz
p = x
2 40
0
( )Fc c
w m mr Gm m
is the maximal acceleration for the mass
chield radius)
.
Reciprocally-Invariant Quadratic Form in the QTPH Space 2
The dimensionless variables
12
12
12
12
121
2
0
00
3
0
0 3
, ,
,
/
e e
e e
e e min e emin
extr P
extr P
ee
pqp q
p q a
ap q a q p a
cp p
G
Gq l
c
where
constant having dimension of action
Planck constant,
Planck's parameter
p rP Q
s
The self-reciprocal invariant (dimensionless values)
2 2 2 2
12 2 2 2 220 0
1 12 2 2 22 20 0
0
., .
( , , ) ( ) , .
( ) ; ( )
2B B
e
B
H
ctS H inv
q
H H H
d H d HH H
d dt
Its easy to see, that is the integral o
Thetime - dependentHamiltonian :
The canonical equations :
where
P Q
P Q P Q
Q PP Q
P Q
.
f motion, and can be considered
as a constant, with respect to differentiation and integration by
The maximum power
12 2 20
0 0
1
( )
:
fin
in
fin infin in
fin in
dHH
ddH
Hd
H H
dH
d
H HdH dHd
d d
The rate of change of the energy
The functions and are real in the domain
The average value of
The maximum power (cont.)
0 0
2 52
0 0
0, , , 0.
1
in fin in fin
e
e
H H H H
dH
d
c pdE dH cc cF
d q d G
Choosing we have
Hence and in dimensional values :
the maximal power
The explicit form of actionThe conventional connection between the action S and
Hamiltonian H is:0
0
( )( )
S HH H
Under supposition that integral of motion H0 can be treated as a parameter, we can write the following:
12 20 20
( )( )
dS HH
d
0 0
1 22 2 02
0 0 0 00
( )
( , ) ( ) arcsin ( ) .2 2
S H const
HS H H S H const
H
Elementary integration gives :
The arbitary function and
are to be defined from the initial conditions.
The classical motion picture
20 0 0
0
0
14
( ) ( ( 0,1,2,3...), 0.
.e
S H k H k const
H
qt H
cE
1. Initial conditions :
2. For the fixed value of the duration of the "particle's" motion.
is restricted by the time interval
3. Change of energy :
)
0
2
lim
20
.
.
.4 4
e
e
e
ee e
cp H
c pE dE
t q dt
S E t q p H
4. Average rate of energy change :
5. Value of action
The canonical quantization
1
22 20
1
22 20 0 01
220
0
| |2
1| | arcsin | |
2| |
, .
( , ) | | ,
k l kl
N
N
Q i
S H
S N H
S N S
N
N H N S N H N
H N
N
N
is considered as a parameter.
The canonical quantization (cont.)
22 2 2
0 0 02
0
20
3
00
0
, , .
| |
, ,
, 2
B
kk
k
B
N H N
n
n
n n
For the definition of we use the discrete spectrum of
Born's equation :
Its solution :
is the well - known oscillator eigen functions,
2 2 2
0 0 0 0 0 0 0
0 1 2 3
, | | , , | | , (
1 2 3
, ) 2
, .
1
n n n n n
n n H n n n n n n n n
n
n
P Q
The action spectrum
12 2
1
21
2
0 01
2
0
2
1
2
2 1 12 2 1
12 1 arcsin 2 1 , 0,1, 2,...
2 2 1
2 1 2 11; ,2 1
NS NN
N S N N nN
S N N
S N N
N
Linear dependence on requires :
1) 2)
is numerical parameter
The action spectrum (cont.)
1
5,
4
2 1 .4
.
1 , 0,1,2,...2
N e e
N N e e
e e
N
S Np q
S S h p q
p q
S h N N
Then (in dimentional units) :
is the condition for the choice of and
1) 2)
Finally :
:
Energy spectrum and discrete time
115 22
1 1
2 2
5
5
lim
1, .
2
1, .
2
N P P
N P P
fin in N
Nfin in
cE E N E
G
Gt t N t
c
E N E N EE c dE
t t G dtt N t N
The maximal power :
The possible cosmological outcome
12
12
12
12
: ( ) ,
( ) ( )
, .in
fin
finin
R t t
t R t t
tEE t
E t
Early Universe : Radiation -Dominated Stage
(A) Standard picture FLRW -model, scale factor
wavelength time dependence
radiation energy dependence
(B
12
1
:
( 2)fin
in
fin finNfin
EN N
E
) Discrete time picture
Universe Expansion stages on energy scale
Energy,
GeV Time, sec
12
inP
finfin
E EN
E
12fin
inP
t
t t
Plankian magnitude 1019 10–44
CUT (SU(5)) - breaking 1015 10–36 812
19
15
10= 10
10
1236
444
10=10
10
SUL(2)U(1) - breaking 102 10–10 12
1934
2
10= 10
10
1210
1744
10=10
10
Quark confinement 100 10–6 12
1938
0
10= 10
10
126
1944
10=10
10
pp - annihilation 10–3 100
12
1944
3
10= 10
10
120
2244
10=10
10
ee - annihilation 10–4 102
12
1946
4
10= 10
10
122
2344
10=10
10
separation
(final of RD stage) 10–9 1012
12
1956
9
10= 10
10
1212
2844
10=10
10
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
In conformally flat metric
gμν = U2(x)ημν , ημν = diag{1, -1, -1, -1}gives
∂μ∂μφ + 2U-1(∂μU)(∂μφ) = 0After substitution
φ(x) = U-1(x)Φ(x) We obtain
∂μ∂μ Φ – (U-1 ∂μ∂μU) Φ = 0
0)(1
xggg
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)
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) ημν correspondingly.
1 sB N
The geodesic equation
in the case of metric can be presented in the form
Using we can write
02
2
ds
dx
ds
dx
ds
xd
02
1 22
2
U
Uds
dxU
ds
d
)(2 xUg
cUddxdxUds 2
1
022
1 22
2
2
22
2 U
U
c
d
xd
d
dx
d
Ud
U
In the case :
Under condition , the geodesic line belongs to the hyperboloid. 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 ~
0
2
2exp)(
l
xxU
0
120
2
2
22
20
xl
c
d
xd
d
dx
d
xd
l
constx 2
02
2
d
dxx
d
xd
020
2
2
2
x
l
c
d
xd
f x
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 the limit for velocity
x
0220
2
2
2
xl
c
d
xdx
20
2
)( l
x
exU
)(2
1 22
22
2
2
xd
d
d
dx
d
xdx
constx 2
0)( 220
22
constx
l
c
d
dx
2
)( xexU
dhyperboloi sheets twoof
unparted , 2
022
2
lxсd
dx
lower
upper
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
s
k
kN
kknn HeHe
1
20
2 )()()(
2
0
20
)( nH
)2()12( 0 sB Nnn
sN
kknn
10n kn
)1( sB N
Now we have the following conditions
(2n0 + 1) - ( 2n + Ns) = ± (Ns + 1)The nonzero solutions exists when
(I) n0 = n - 1 in the case
(II) 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
)2/(exp)( 20
2 lxxU
)2/(exp)( 20
2 lxxU
Einstein tensor for metric
Energy-momentum tensor
Minkovski force density :
Energy density
G )exp( 2xg
20
2 1 ,26
lxxxG
24
268
xxxG
CT
Tf
xG
Сf
2
3 42
220
424
00 384
3r x
G
C
G
CT
Thank You for Your attention!
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