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!