ATOMIC STRINGS, QUANTUM and FABRIC of SPACETIME

Atomic String (AString) is a soliton swing-like function by joining of which on a periodic lattice it is possible to build flat and curved continua and introduce the 'spacetime quantum' as an elementary block of the fabric of the universe. It leads to new models of spacetime as a combination of 'elementary strings' which may represent 'spacetime distortions/warps' studied by Stephen Hawking, or 'metriants' by A.Veinik, or 'ripples' of Higgs field permiating the space. AStrings can be used to create models of gravity and expain general relativity effects due to unique property of derivatives being expressed via AString itself. AStrings can also be used to construct models of different funamental blocks of matter, anti-matter (soliton-like atoms, breathers, cells, waves), cosmic strings and contribute to the theories of dark matter, chronal matter and multiverse where quanta can significantly differ from the ones in ordinary universe.

AStrings may find some applications in String theory, General and Special Relativity, Lattice physics, Quantum Gravity, Cosmology, Dark matter and Multiverse theories, Evolution of Biological Matter, as well as creating new type of computers, finite element methods, new regression analysis and artificial intelligence classification models.

Atomic String is introduced by the author, Professor Sergei Yu. Eremenko (https://au.linkedin.com/in/sergei-eremenko-3862079), in paper "Atomic Strings and Fabric of Spacetime" accepted for publication in Journal "Achievements of Modern Radioelectronics". Atomic String is a generalisation of well known Atomic Function (https://ru.wikipedia.org/w/index.php?oldid=82669103) discovered in 1967-1971 by my teacher Academician Rvachev V.L. (https://ru.wikipedia.org/w/index.php?oldid=83948367) and Professor Rvachev V.A. and advanced by many followers notably Professor Kravchenko V.F. (https://ru.wikipedia.org/w/index.php?oldid=84521570).

This is an Abstract of paper "Atomic Strings and Fabric of Spacetime"

Based on a generalization of well-known atomic function this paper introduces an atomic string (AString) as a swing-like function by joining of which on a periodic lattice it is possible to build one model for absolutely flat and curved spacetime and explain gravitational warping effects due to a unique property of all derivatives being expressed via AString itself. Physically AString may represent a soliton-like elementary string as a spacetime warp/distortion composing atomic quanta in multiple dimensions which can grow, shrink, group into atoms and compose smooth continua, solid and biological matter widespread in nature. This makes AString a good candidate for an elementary string as a fundamental block of matter and spacetime fabric. AString can also be used as a generating axiom for a new non-Archimedean atomic calculus to describe nonlinear metric within cosmic strings and build never overflowing atomic computer with the super fast calculation of derivatives. The AString along with closely related atomic function may find new areas of applications in lattice physics, string theory, relativity, quantum gravity, cosmic strings, multiverse, dark matter, solitons, dislocations, computing, evolution of solid and biological matter, finite element methods, new regression analysis and artificial intelligence classification models.

Script for Atomic Function (introduced in 1967 by V.L.Rvachev and V.A.Rvachev)



In [1]:

    
##########################################################
##This script introduces Atomic Function and Atomic String
##########################################################
import numpy as np
import pylab as pl
################### One Pulse of atomic function
def up1(x: float) -> float:
    #Atomic function table
    up_y = [0.5, 0.48, 0.460000017,0.440000421,0.420003478,0.400016184, 0.380053256, 0.360139056, 0.340308139, 0.320605107,
           0.301083436, 0.281802850, 0.262826445, 0.244218000, 0.226041554, 0.208361009, 0.191239338, 0.174736305, 
           0.158905389, 0.143991189, 0.129427260, 0.115840866, 0.103044024, 0.9110444278e-01, 0.798444445e-01, 0.694444445e-01, 
           0.598444445e-01, 0.510444877e-01, 0.430440239e-01, 0.358409663e-01, 0.294282603e-01, 0.237911889e-01, 0.189053889e-01, 
           0.147363055e-01, 0.112393379e-01, 0.836100883e-02, 0.604155412e-02, 0.421800000e-02, 0.282644445e-02, 0.180999032e-02, 
           0.108343562e-02, 0.605106267e-03, 0.308138660e-03, 0.139055523e-03, 0.532555251e-04, 0.161841328e-04, 0.347816874e-05, 
           0.420576116e-05, 0.167693347e-07, 0.354008603e-10, 0]
    up_x = np.arange(0.5, 1.01, 0.01)

    res = 0.
    if ((x>=0.5) and (x<=1)):
        for i in range(len(up_x) - 1):
            if (up_x[i] >= x) and (x < up_x[i+1]):
                N1 = 1 - (x - up_x[i])/0.01
                res = N1 * up_y[i] + (1 - N1) * up_y[i+1]
                return res
    return res

############### Atomic Function Pulse with width, shift and scale #############
def upulse(t: float, a = 1., b = 0., c = 1., d = 0.) -> float:
    x = (t - b)/a
    res = 0.
    if (x >= 0.5) and (x <= 1):
        res = up1(x)
    elif (x >= 0.0) and (x < 0.5):
        res = 1 - up1(1 - x)
    elif (x >= -1 and x <= -0.5):
        res = up1(-x)
    elif (x > -0.5) and (x < 0):
        res = 1 - up1(1 + x)
    res = d + res * c
    return res

############### Atomic Function Applied to list with width, shift and scale #############
def up(x: list, a = 1., b = 0., c = 1., d = 0.) -> list:
    res: list = []
    for i in range(len(x)):
        res.append(upulse(x[i], a, b, c, d))
    return res



In [2]:

    
x = np.arange(-2.0, 2.0, 0.01)
pl.title('Atomic Function')
pl.plot(x, up(x), label='Atomic Function')
pl.grid(True)
pl.show()

Atomic String as an Integral of Atomic Function (introduced in 2017 by Prof S.Eremenko)



In [3]:

    
############### Atomic String #############
def AString1(x: float) -> float:
    res = 1 * (upulse(x/2.0 - 0.5) - 0.5)
    return res

############### Atomic String Pulse with width, shift and scale #############
def AStringPulse(t: float, a = 1., b = 0., c = 1., d = 0.) -> float:
    x = (t - b)/a
    if (x < -1):
        res = -0.5
    elif (x > 1):
        res = 0.5
    else:
        res = AString1(x)
    res = d + res * c
    return res

###### Atomic String Applied to list with width, shift and scale #############
def AString(x: list, a = 1., b = 0., c = 1., d = 0.) -> list:
    res: list = []
    for i in range(len(x)):
        res.append(AStringPulse(x[i], a, b, c, d))
        #res[i] = AStringPulse(x[i], a, b, c)
    return res

###### Summation of two lists #############
def Sum(x1: list, x2: list) -> list:
    res: list = []
    for i in range(len(x1)):
       res.append(x1[i] + x2[i])
    return res



In [4]:

    
pl.title('Atomic String')
pl.plot(x, AString(x, 1.0, 0, 1, 0), label='Atomic String')
pl.grid(True)
pl.show()

Atomic String, Atomic Function and Atomic Function Derivative plotted together



In [5]:

    
#This Calculates Derivative
dx = x[1] - x[0]
dydx = np.gradient(up(x), dx)
pl.plot(x, up(x), label='Atomic Function')
pl.plot(x, AString(x, 1.0, 0, 1, 0), label='Atomic String')
pl.plot(x, dydx, label='A-Function Derivative')

pl.title('Atomic Function and String')
pl.legend(loc='best', numpoints=1)
pl.grid(True)
pl.show()

Properties of atomic function Up(x)

1) Remarkably, Atomic Function Derivative can be exressed via Atomic Function itself - up'(x)= 2up(2x+1)-2up(2x-1) meaning the shape of pulses for derivative function can be represented by shifted and stratched Atomic Function itself - remarkable property

2) The Atomic Function pulses superposition set at points -2, -1, 0, +1, +2... can exactly represent a Unity (number 1): 1 = ... up(x-3) + up(x-2) + up(x-1) + up(x-0) + up(x+1) + up(x+2) + up(x+3) + ...



In [6]:

    
pl.plot(x, up(x, 1, -1), linewidth=1, label='Atomic Function at x=-1')
pl.plot(x, up(x, 1, +0), linewidth=1, label='Atomic Function at x=0')
pl.plot(x, up(x, 1, -1), linewidth=1, label='Atomic Function at x=-1')
pl.plot(x, Sum(up(x, 1, -1), Sum(up(x), up(x, 1, 1))), linewidth=2, label='Atomic Function Compounding')
pl.title('Atomic Function Compounding represent 1')
pl.legend(loc='best', numpoints=1)
pl.grid(True)
pl.show()

3) Atomic Function (AF) is a 'finite' function (like spline) not equal to zero only on section |x|<=1

4) Atomic Function is a non-analytical function (can not be represented by Taylor's series), but with known Fourier Transformation allowing to exactly calculate AF in certain points, with tabular representation provided in script above.

Atomic String is a generalisation of an Atomic Function

1) Astring is an swing-like function - Integral of Atomic Function (AF) which can be expressed via AF itself: AString(x) = Integral(0,x)(Up(x)) = Up(x/2 - 1/2) - 1/2

2) Atomic Function can be represented via simple superposition of Atomic Strings: up(x) = AString(2x + 1) - AString(2x - 1)



In [7]:

    
#########   Presentation of Atomic Function via Atomic Strings ##########
x = np.arange(-2.0, 2.0, 0.01)

pl.plot(x, AString(x, 1, 0, 1, 0),  '--', linewidth=1,  label='AString(x)')
pl.plot(x, AString(x, 0.5, -0.5, +1, 0), '--', linewidth=2, label='+AString(2x+1)')
pl.plot(x, AString(x, 0.5, +0.5, -1, 0), '--', linewidth=2, label='-AString(2x-1)')
#pl.plot(x, up(x, 1.0, 0, 1, 0),     '--', linewidth=1, label='Atomic Function')
AS2 = Sum(AString(x, 0.5, -0.5, +1, 0), AString(x, 0.5, +0.5, -1, 0))
pl.plot(x, AS2, linewidth=3, label='Up(x) via Strings')
pl.title('Atomic Function as a Combination of AStrings')
pl.legend(loc='center left', numpoints=1)
pl.grid(True)
pl.show()

3) All derivatives of AString can be represented via AString itself: AString'(x) = AString(2x + 1) - AString(2x - 1)

4) Combination of Atomic Strings can exactly represent a straight line: x = AString(x) + Astring(x+1) + Astring(x+2)...



In [8]:

    
x = np.arange(-40.0, 40.0, 0.01)

#pl.plot(x, ABline (x, 1, 0),            label='ABLine 1*x')
pl.plot(x, AString(x, 10.0,-15, 10, -15),   '--',  linewidth=1, label='AString 1')
pl.plot(x, AString(x, 10.0, -5, 10, -5),    '--',  linewidth=1, label='AString 2')
pl.plot(x, AString(x, 10.0, +5, 10, +5),    '--',  linewidth=1, label='AString 3')
pl.plot(x, AString(x, 10.0,+15, 10, +15),   '--',  linewidth=1, label='AString 4')

AS2 = Sum(AString(x, 10.0, -15, 10, -15), AString(x, 10., -5, 10, -5))
AS3 = Sum(AS2, AString(x, 10, +5, 10, +5))
AS4 = Sum(AS3, AString(x, 10,+15, 10, +15))
pl.plot(x, AS4,  label='AStrings Joins', linewidth=2)

pl.title('Atomic Strings Combinations')
pl.legend(loc='best', numpoints=1)
pl.grid(True)
pl.show()

Representing of flat Spacetime Fabric by joining of Atomic Strings Quanta (Metriants)



In [9]:

    
x = np.arange(-30.0, 30.0, 0.01)
#pl.plot(x, ABline (x, 1, 0),            label='ABLine 1*x')
pl.plot(x, AString(x, 10.0,-15, 10, -15),   '--',  linewidth=1, label='AString Quantum 1')
pl.plot(x, AString(x, 10.0, -5, 10, -5),    '--',  linewidth=1, label='AString Quantum 2')
pl.plot(x, AString(x, 10.0, +5, 10, +5),    '--',  linewidth=1, label='AString Quantum 3')
pl.plot(x, AString(x, 10.0,+15, 10, +15),   '--',  linewidth=1, label='AString Quantum 4')

AS2 = Sum(AString(x, 10.0, -15, 10, -15), AString(x, 10., -5, 10, -5))
AS3 = Sum(AS2, AString(x, 10, +5, 10, +5))
AS4 = Sum(AS3, AString(x, 10,+15, 10, +15))
pl.plot(x, AS4,  label='Spacetime Dimension', linewidth=2)

pl.title('Representing Spacetime by joining of Atomic Strings')
pl.legend(loc='best', numpoints=1)
pl.grid(True)
pl.show()



In [10]:

    
x = np.arange(-50.0, 50.0, 0.1)
dx = x[1] - x[0]

CS6 = Sum(up(x, 5, -30, 5, 5), up(x, 15, 0, 15, 5))
CS6 = Sum(CS6, up(x, 10, +30, 10, 5))
pl.plot(x, CS6,  label='Spacetime Density')

IntC6 = np.cumsum(CS6)*dx/50
pl.plot(x, IntC6, label='Spacetime Shape')

DerC6 = np.gradient(CS6, dx)
pl.plot(x, DerC6, label='Spacetime Curvature')

LightTrajectory = -10 -IntC6/5
pl.plot(x, LightTrajectory, label='Light Trajectory')

pl.title('Fabric of Curved Spacetime')
pl.legend(loc='best', numpoints=1)
pl.grid(True)
pl.show()

Apart from standard Python code, this script and material is the intellectual property of Professor Sergei Yu. Eremenko (https://au.linkedin.com/in/sergei-eremenko-3862079). You may not reproduce, edit, translate, distribute, publish or host this document in any way without the permission of Professor Eremenko.