Talk given to Matt Johnson's Cosmology group at York University. The focus was on applying the orthonormal frame formalism and dynamical systems theory to cosmological models.
This Lecture is based on Scientific Discoveries and Religious Scripture of Sikh religion " Sri Guru Granth Sahib". Surprisingly, Guru Nanak, founder of Sikh religion, was forerunner of Big Bang cosmology; his ideas on Creation of Space, Time and Universe find an echo in Big Bang Cosmological Models proposed 500 years after Guru Nanak's vision recorded in "Sri Guru Granth Sahib". Original quotes from Guru Nanak are recorded in Gurmukhi script/Fonts.
“The theoretical analysis result suggests that the impulsive electric field applied to the dielectric material may produce a sufficient artificial gravity to attain velocities comparable to chemical rockets.” ―Dr. Takaaki Musha
People: Al Bielek, Alcubierre, Andrew Bahnion, André Füzfa, Athanassios Nassikas, Biefeld, Bruce dePalma, Bud Rieken, Casimer, Cassenti, Dan Davidson, Daniel D. Home, David Cowlishaw, David Hammel, David Hooper, Dishington, Don Kelly, Douglas Torr, Dr. Albert Einstein, Dr. Berthart Heim, Dr. Charles F. Brush, Dr. Ed Witten, Dr. Edward Teller, Dr. Erwin Saxl, Dr. Eugene Podkletnov, Dr. Francis Nipher, Dr. Fredrick Alzofon, Dr. Ning Li, Dr. Richard Clark, Dr. Richard Clarke, Dr. Robert Dicke, Dr. Robert Forward, Dr. Schuman, Dr. Willaim Crookes, Dr. William Hooper., Ed Leedskelstein, Eric Laithwaite, Erich Halik, Floyd Sweet, Fran McCabe, Frost, Guido Fetta, Halvosky, Hans Nieper, Harold Aspden, Harold Puthoff, Harold Wilson, Hawasaka, Henry Wallace, Hoyassaka, James Clerk Maxwell, James Cox, James Hartman, James King Jr., James Woodward., Jefimenko, John Hutchingson, John Keely, John R. R. Searl, John Schnurrer, Kellogg, Le Sage, Leroy Cook, Lt. Col. Corso, Marcel Pages, Mark Tomion, Misner, Modanese, Moebuis (Faile), Montimen Delroy, Naudin, Neil Sorenzen, Nikola Tesla, Norman Dean, Oliver Heaverside, Otis T. Carr, P.M.S. Blackett, Paramahamsa Tewari, Patrick Bailey, Peter Kummel, Poliakov, Randall Mill, Richard Foster, Robert Collins, Robert Lazar, Sandy Kidd, Schnurer, Scpott Strachan, Shinicki Seike, T. Townsend Brown, Thorne, Tom Bearden, Travis Taylor, Viktor Schauberger, W. D. Clendonon, W. Peschaka, Wheeler, Wilbur Smith, William Littlejohn, William Rhodes, Yamashita Huaro
“Dr. Forward defines six different kinds of antigravity in his "Indistinguishable from Magic" book: Aside from the obvious "weightloss" in freefall as an astronaut in orbit; the first is to place a very heavy mass overhead to cancel the earth's gravity field. Secondly, we may use ultradense materials spaced very closely to the object we desire to be weightless. The last Newtonian idea, is to use various "guard" masses to reduce tidal forces on a test mass already in orbit on the space shuttle. From Einstein's theory, comes a fourth idea; that moving ultradense matter in a toroidal coil at high speed will generate a dipolar G-field. Fifth, the idea of dragging of inertial frames, which will pull objects along with them. Lastly, the sixth idea, is the possibility of negative matter (-m), just like negative charge, will reverse the direction of force; it moves towards you when you push on it! Negative, matter along with positive matter, will take off into space forming a space drive! Could the superconductor of Dr. Podkletnov be a form of negative matter?” ―James E. Cox
Alcubierre's drive allows to circumvent the speed of light limit. They are building a Negative Energy Generator to realize it.
This Lecture is based on Scientific Discoveries and Religious Scripture of Sikh religion " Sri Guru Granth Sahib". Surprisingly, Guru Nanak, founder of Sikh religion, was forerunner of Big Bang cosmology; his ideas on Creation of Space, Time and Universe find an echo in Big Bang Cosmological Models proposed 500 years after Guru Nanak's vision recorded in "Sri Guru Granth Sahib". Original quotes from Guru Nanak are recorded in Gurmukhi script/Fonts.
“The theoretical analysis result suggests that the impulsive electric field applied to the dielectric material may produce a sufficient artificial gravity to attain velocities comparable to chemical rockets.” ―Dr. Takaaki Musha
People: Al Bielek, Alcubierre, Andrew Bahnion, André Füzfa, Athanassios Nassikas, Biefeld, Bruce dePalma, Bud Rieken, Casimer, Cassenti, Dan Davidson, Daniel D. Home, David Cowlishaw, David Hammel, David Hooper, Dishington, Don Kelly, Douglas Torr, Dr. Albert Einstein, Dr. Berthart Heim, Dr. Charles F. Brush, Dr. Ed Witten, Dr. Edward Teller, Dr. Erwin Saxl, Dr. Eugene Podkletnov, Dr. Francis Nipher, Dr. Fredrick Alzofon, Dr. Ning Li, Dr. Richard Clark, Dr. Richard Clarke, Dr. Robert Dicke, Dr. Robert Forward, Dr. Schuman, Dr. Willaim Crookes, Dr. William Hooper., Ed Leedskelstein, Eric Laithwaite, Erich Halik, Floyd Sweet, Fran McCabe, Frost, Guido Fetta, Halvosky, Hans Nieper, Harold Aspden, Harold Puthoff, Harold Wilson, Hawasaka, Henry Wallace, Hoyassaka, James Clerk Maxwell, James Cox, James Hartman, James King Jr., James Woodward., Jefimenko, John Hutchingson, John Keely, John R. R. Searl, John Schnurrer, Kellogg, Le Sage, Leroy Cook, Lt. Col. Corso, Marcel Pages, Mark Tomion, Misner, Modanese, Moebuis (Faile), Montimen Delroy, Naudin, Neil Sorenzen, Nikola Tesla, Norman Dean, Oliver Heaverside, Otis T. Carr, P.M.S. Blackett, Paramahamsa Tewari, Patrick Bailey, Peter Kummel, Poliakov, Randall Mill, Richard Foster, Robert Collins, Robert Lazar, Sandy Kidd, Schnurer, Scpott Strachan, Shinicki Seike, T. Townsend Brown, Thorne, Tom Bearden, Travis Taylor, Viktor Schauberger, W. D. Clendonon, W. Peschaka, Wheeler, Wilbur Smith, William Littlejohn, William Rhodes, Yamashita Huaro
“Dr. Forward defines six different kinds of antigravity in his "Indistinguishable from Magic" book: Aside from the obvious "weightloss" in freefall as an astronaut in orbit; the first is to place a very heavy mass overhead to cancel the earth's gravity field. Secondly, we may use ultradense materials spaced very closely to the object we desire to be weightless. The last Newtonian idea, is to use various "guard" masses to reduce tidal forces on a test mass already in orbit on the space shuttle. From Einstein's theory, comes a fourth idea; that moving ultradense matter in a toroidal coil at high speed will generate a dipolar G-field. Fifth, the idea of dragging of inertial frames, which will pull objects along with them. Lastly, the sixth idea, is the possibility of negative matter (-m), just like negative charge, will reverse the direction of force; it moves towards you when you push on it! Negative, matter along with positive matter, will take off into space forming a space drive! Could the superconductor of Dr. Podkletnov be a form of negative matter?” ―James E. Cox
Alcubierre's drive allows to circumvent the speed of light limit. They are building a Negative Energy Generator to realize it.
This is a short presentation about Teleportation.
Most precise and accurate information about teleportation is shared here.
Let me know if you like it..
You can also tell me if there's something new to add in this hypothesis.
Space is not fundamental (although time might be). Talk at the 2010 Philosophy of Science Association Meeting, Montreal. By Sean Carroll, http://preposterousuniverse.com/
Slides from my presentation at the Joint CoEPP-CAASTRO Workshop (http://www.caastro.org/event/2013/coepp), 28 February 2013. Brief overview of the evidence for dark matter in the Universe, plus discussion of challenges, hints of possible signals, and some references for further reading.
The presentation time-slot was 30 minutes + 20 minutes discussion.
Vi på IVT Stuguns Bygg & Färg har marknadens bredaste sortiment. Hos oss hittar du bergvärmepumpar, jordvärmepumpar, sjövärmepumpar, grundvattenvärmepumpar, luft/vattenvärmepumpar, luft/luftvärmepumpar och solvärmelösningar för både villor och fastigheter. Dessutom är IVT den enda tillverkaren som klarar Svanens hårda kvalitets- och miljökrav. Samtliga värmepumpar installeras av kunniga och certifierade installatörer som är specialutbildade för att beräkna ditt hus energianvändning och ge dig lösningen med bäst besparing.
This is a short presentation about Teleportation.
Most precise and accurate information about teleportation is shared here.
Let me know if you like it..
You can also tell me if there's something new to add in this hypothesis.
Space is not fundamental (although time might be). Talk at the 2010 Philosophy of Science Association Meeting, Montreal. By Sean Carroll, http://preposterousuniverse.com/
Slides from my presentation at the Joint CoEPP-CAASTRO Workshop (http://www.caastro.org/event/2013/coepp), 28 February 2013. Brief overview of the evidence for dark matter in the Universe, plus discussion of challenges, hints of possible signals, and some references for further reading.
The presentation time-slot was 30 minutes + 20 minutes discussion.
Vi på IVT Stuguns Bygg & Färg har marknadens bredaste sortiment. Hos oss hittar du bergvärmepumpar, jordvärmepumpar, sjövärmepumpar, grundvattenvärmepumpar, luft/vattenvärmepumpar, luft/luftvärmepumpar och solvärmelösningar för både villor och fastigheter. Dessutom är IVT den enda tillverkaren som klarar Svanens hårda kvalitets- och miljökrav. Samtliga värmepumpar installeras av kunniga och certifierade installatörer som är specialutbildade för att beräkna ditt hus energianvändning och ge dig lösningen med bäst besparing.
The objective of this paper is to propose an approach to the unification of physics by attempting
to construct a physical worldview which can be used as the context for a unified physical theory.
The underlying principle is that we have to construct a clear description of the physical world
before we can build a unified physical theory.
The present state of physics is such that there are many theories which all differ in the descriptive
context in which they operate. The theories of general relativity, quantum theory, quantum
electrodynamics, string theory and the standard model of particle physics are based on differing
concepts of the nature of the physical world.
Dynamical Systems Methods in Early-Universe CosmologiesIkjyot Singh Kohli
Talk I gave at The Southern Ontario Numerical Analysis Day (SONAD): http://www.math.yorku.ca/sonad2014/ on General Relativity, Dynamical Systems, and Early-Universe Cosmologies.
I forgot... for a moment. He's in our lab in Sinaloa, too, and they brought his stuff.
Not only do we have the only real neuroscientists there, but also a gifted brain surgeon with a ton of info from a slightly different angle.
This paper presents the Physics Rotational Method of the simple gravity pendulum, and it also applies Physics Direct Method to represent these equations, in addition to the numerical solutions discusses. This research investigates the relationship between angular acceleration and angle to find out different numerical solution by using simulation to see their behavior which shows in last part of this article.
Reply to the Note of Jeremy Dunning-Davies Shafiq Khan
The open challenge was put forward by me and one professor of physics namely Jeremy Dunning-Davies accepted the challenge and wrote a note and that note was kept on General Science Journal & viXra. This is the published version of the reply to the note of Jeremy Dunning-Davies.
A Theological Approach to the Simplest Mathematical Explanation of the Dynami...AI Publications
Guided heuristically by the theism of ancient Egypt where religion underpinned cosmology, to explain the dynamics of the temporal universe (gravitation, rotation, and translation), this paper starts from the existence of individualities and the law of causality to build a cosmological argument, into a natural systematic theology (NST). According to this NST, God (the greatest possible being) includes the creator as one of his celestial manifestations. One deduces from the NST that, manifesting an individuality included in the indivisible and immutable God, the creator expresses the fullness of the Most High in an individual manner. This fullness (the Logos) is thus, like God, a constant power working in the creator. The existence of the temporal universe in God requires the existence of a principle of the mutability of God at the occurrence of creation ex nihilo; in accord to the law of causality this principle must greater than God, which is impossible. Moreover, the existence of creation outside of God implies an entity greater than God including him and creation, this is also impossible as God is the greatest possible being. Therefore, the temporal universe exists in the temporal consciousness of the creator as a mere appearance of, or perspective on, the celestial reality. Thus, the NST dictates the existence of an absolute space-time that includes the Euclidean space-time known in Newtonian physics and corresponds to the temporal consciousness of the creator. The NST proves that to create, the creator had first to leave the eternal plane for the temporal one. Then on, as a constant power, the Logos impels the creator to accelerate back toward the celestial-eternal level. This isotropic acceleration of the creator causes the absolute space-time to accelerate towards its nothingness, its non-existence in the celestial realm. Therefore, this background acceleration of the absolute space-time is the simplest, exhaustive, deterministic, and mathematical explanation of the dynamics of the temporal universe (gravitation, rotation, translation). This explanation offers the advantage of using elementary notions of algebra and analysis and the result applies to the astronomical level as well as to the subatomic one.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
4. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
5. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
One has to account for potential anisotropic/dissipative
effects which in fluid dynamics as characterized by
viscosity.
6. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
One has to account for potential anisotropic/dissipative
effects which in fluid dynamics as characterized by
viscosity.
For a review of the justification of including viscosity
terms in early-universe cosmological models see the
articles by Barrow (1988,1982) [3] [2] and the article by
Belinskii and Khalatnikov (1976) [4].
7. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Motivation
In this presentation, we are concerned with applying
dynamical systems theory to get information about the
early universe.
The early universe, that is, shortly after the big bang was
a hot and dense place, and it is not necessarily true that
the perfect fluid description used in the standard
cosmological models today would hold in such conditions.
One has to account for potential anisotropic/dissipative
effects which in fluid dynamics as characterized by
viscosity.
For a review of the justification of including viscosity
terms in early-universe cosmological models see the
articles by Barrow (1988,1982) [3] [2] and the article by
Belinskii and Khalatnikov (1976) [4].
As a result, we will keep our assumption of the spatial
homogeneity of the early universe, but now assume it is
8. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation I
In this section, we will derive the form of the required
energy-momentum tensor, namely, for that of a viscous fluid
without heat conduction. Recall that the energy-momentum
tensor for a perfect fluid takes the form
Tab
= (µ + p)ua
ub
− uc
ucgab
p. (1)
For the moment, letting µ + p = W, we obtain
Tab
= Wua
ub
− uc
ucgab
p, (2)
Denoting the viscous contributions by Vab, we seek a
modification of Eq. (2) such that
Tab = Wuaub − ucuc
gabp + Vab. (3)
9. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation II
To obtain the form of this additional tensor term, we note that
from classical fluid mechanics, the Euler equation is given as
(ρui ),t = −Πik,k, (4)
where Πik is the momentum flux tensor. Also, recall that for a
non-viscous fluid, one has the fundamental relationship
Πik = pδik + ρui uk. (5)
We simply add a term to Eq. (5) that represents the viscous
momentum flux, ˜Σik, to obtain
Πik = pδik + ρui uk − ˜Σik = −Sik + ρui uk. (6)
It is important to note that
Sik = −pδik + ˜Σik (7)
is the stress tensor, while, ˜Σik is the viscous stress tensor.
10. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation III
Note that, in what follows below, the viscous stress tensor,
˜Σik, is not to be confused with Σik, the Hubble-normalized
shear tensor.
The general form of the viscous stress tensor can be
formed by recalling that viscosity is formed when the fluid
particles move with respect to each other at different
velocities, so this stress tensor can only depend on spatial
components of the fluid velocity.
We assume that these gradients in the velocity are small,
so that the momentum tensor only depends on the first
derivatives of the velocity in some Taylor series expansion.
11. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation IV
Therefore, ˜Σik is some function of the ui,k. In addition,
when the fluid is in rotation, no internal motions of
particles can be occurring, so we consider linear
combinations of ui,k + uk,i , which clearly vanish for a fluid
in rotation with some angular velocity, Ωi . The most
general viscous tensor that can be formed is given by
˜Σik = η ui,k + uk,i −
2
3
δikul,l + ξδikul,l , (8)
where η and ξ are the coefficients of shear and
bulk/second viscosity, respectively [15] [13].
12. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation V
In Eq. (8), we note that δik ul,l is an expansion rate tensor,
and ui,k + uk,i − 2
3 δikul,l represents the shear rate
tensor. Since we would like to generalize this expression to
the general relativistic case, we replace the partial
derivatives above with covariant derivatives, and the
Kroenecker tensor with a more general metric tensor, that
is, δik → gik .
We thus have that
˜Σik = η ui;k + uk;i −
2
3
gikul;l + ξgikul;l . (9)
Denoting the shear rate tensor as σab, and the expansion
rate scalar as θ ≡ ua
;a, Eq. (9) becomes
Vab = −2ησab − ξθhab. (10)
13. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Energy-Momentum Tensor Derivation VI
Since we are interested in the Hubble-normalized approach,
we will make use of the definition θ ≡ 3H, where H is the
Hubble parameter. This means that Eq. (9) becomes
Vab = −2ησab − 3ξHhab. (11)
Substituting Eq. (11) into Eq. (3) we finally obtain the
required form of the energy-momentum tensor as
Tab = (µ + p) uaub − ucuc
gabp − 2ησab − 3ξHhab. (12)
For simplicity, we shall let πab = −2ησab denote the
anisotropic stress tensor, and commit to the metric
signature (−1, +1, +1, +1) such that Eq. (12) takes the
form
Tab = (µ + p) uaub + gabp − 3ξHhab + πab. (13)
14. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion I
We will now derive the dynamical equations. Most of this
derivation follows from Ellis [6] and Grøn and Hervik [9], but I
have tried to make thing slightly easier to understand. Let us
first recall some basic properties of non-relativistic fluid
mechanics. In fluid mechanics, we typically measure the fluid
acceleration through the material derivative:
Dv
dt
≡ v,t + (v · ∇)v, (14)
where v,t is the local derivative, and (v · ∇)v is known as the
convective derivative. Note that
Dva
dt
= va
,t + va
,bxb
,t = va
,t + vb
va
,b. (15)
15. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion II
So we can write the convective term as Mv, where M = vi
,j is a
matrix, that is, it is the gradient of the velocity.
We can now decompose vi
,j into symmetric and anti-symmetric
parts:
vi,j = θij + ωij , (16)
where
θi
j =
1
2
vi
,j + vj
i (17)
is the expansion tensor and
ωi
j =
1
2
vi
,j − vj
i (18)
is the vorticity tensor.
16. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion III
I can now decompose the expansion tensor into trace and
trace-free parts:
θi
j =
1
3
θδi
j + σi
j , (19)
where
θ = vi
,i , σi
j =
1
2
vi
,j + vj
,i −
1
3
δi
j vi
,i . (20)
Note that σij is the trace-free tensor, and is known as the shear
tensor. The shear tensor essentially measures deformations of
the fluid.
Combining all of these, I can essentially write:
vi,j =
1
3
θδij + σij + ωij. (21)
17. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion IV
Switching to the relativistic form we obviously have
aα = uα;mum
= ˙uα. (22)
We want to project the four-dimensional quantities onto the
spatial slices and see how the evolve, these are our “dynamical”
quantities. That is,
uα;β =
1
3
θhαβ + σαβ + ωαβ − ˙uαuβ. (23)
Note that this last term is a consequence of purely relativistic
effects. It measures how much the fluid deviates from being
geodesic. In our work, we always assume the fluid is geodesic,
so the four-acceleration vanishes.
18. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Basic Equations of Motion V
We can write the energy-momentum tensor of the previous
section in the more general form:
Tuv = µuuuv + phuv + πuv . (24)
Using the properties that
πu
u = 0, uu
πuv = 0, (25)
and the Bianchi identities Tv
u;v = 0 imply that
uu
Tv
u;v = 0
= uu
µ,v uuuv
+ µuu;v uv
+ µuuuv
;u + hv
u;v p + hv
up,v
+ πv
u;v
= − ˙µ − θµ − θp − σuv πuv
. (26)
21. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation I
The basic defining equation for General Relativity is the Ricci
identity. Let us start with:
−uu
uv
Raubv = uv
Rauvbuu
. (28)
I will write this as
−uu
uv
Raubv = ua;bv uv
+ ua;vuv
;b. (29)
If I now contract this equation, I obtain
− uu
uv
Ruv = ub
;bv uv
+ ua;v uv;a
. (30)
If I now substitute the Einstein field equations
Rab = Tab −
1
2
Tc
c gab. (31)
22. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation II
into the Ricci identity Eq. (30) and take the symmetric and
trace-free part, I obtain the shear-propagation equations:
ha
chb
d ˙σcd
= −2Hσab
− σa
c σbc
−
1
3
−2σ2
hab
+
1
2
πab
(32)
If I take the trace of Eq. (30), I get the Raychaudhuri equation:
˙θ +
1
3
θ2
+ σabσab
+
1
2
(µ + 3p) = 0. (33)
The assumption that ˙ua = 0 implies that ωuv = 0. In the 3 + 1
formalism, one can show that
ua;b = Kab, (34)
23. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation III
where Kab is the extrinsic curvature tensor. Gauss’ theorem
egregium (Page 171, [9]) states that
(n+1)
R =(n)
R + K2
− Kab
Kab + 2(−1)(n+1)
Rabua
ub
. (35)
Combining this with the Einstein field equations, we get:
Tab
uaub =
1
2
(3)
R − Kab
Kab + K2
. (36)
If we now use Eqs. (23) and (24), we get the Friedmann
equation:
1
3
θ2
=
1
2
σabσab
−
1
2
(3)
R + µ. (37)
This is a constraint equation on initial conditions as we will see
later.
24. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
The Raychaudhuri Equation IV
Note that, we still have a problem in that we really have not
simplified things much. Looking at the shear propagation
equation (32), for example, we have that
˙σab ≡ σab;mum
, (38)
which contains the covariant derivative, which contains the
Christoffel symbols, so we still need to work with the metric
tensor. This is where the grand theory of orthonormal frames
comes in!
25. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism I
The idea of orthonormal frames is based on the groundbreaking
1969 paper from Ellis and MacCallum [8], “A Class of
Homogeneous Cosmological Models”. First, what do we mean
when we say a space-time is spatially homogeneous? Let M be
a manifold with metric tensor g. The isometry group is defined
by
Isom(M = {φ : M → M|φ, isometry}, (39)
that is φ∗g = g, i.e., the metric is left unchanged after applying
an isometry. This isometry group is a Lie group, and the group
generators are Killing vectors. These Killing vectors span a
finite-dimensional space (the tangent space to the Lie group),
and we define the isotropy subgroup of a point p ∈ M by
fp(M) = {φ ∈ Isom(M)|φ(p) = p}. (40)
26. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism II
So a homogenous space is one that for each pair of points
a, b ∈ M, there exists a φ ∈ Isom(M) such that φ(p) = q.
For such a homogeneous space, there exists a set of Killing
vector fields ζi such that
[ζi , ζj ] = Dk
ij ζk. (41)
At a point p ∈ M, choose a basis set of vectors ei . It can be
shown that the frame ei span a Lie algebra:
[ei , ej ] = Ck
ij ek . (42)
Therefore, to construct a homogeneous space, one takes the
Ck
ij as the structure constants of the Lie algebra, and defines a
27. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism III
left-invariant frame by the above relation. For a dual basis, ωk
to ek , using the Cartan calculus, we have that
dwk
= −
1
2
Ck
ij ωi
∧ ωj
. (43)
Using these invariant one-forms, we can define the spatial
metric as
ds2
= gij ωi
⊗ ωj
, (44)
where the gij are constants. The amazing thing is that the
Killing vectors are ζi , and the basis vectors are ej , that is, we
can just read them off!
To proceed further, we also need to modify what we mean by a
connection on the manifold. First, consider a vector field u.
We have for any scalar function g that
u(g) = um
em(f ). (45)
28. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism IV
Now define
uv(g) = um
em(vn
en(g)), (46)
which is
uv(g) = um
em(vn
)en(g) + um
un
emen(g). (47)
This is clearly not a vector, since it has second-order partial
derivatives. However,
[u, v] = [um
em(vn
) − vm
em(un
)] en + um
un
[em, en] (48)
is indeed a vector. For an arbitrary basis,
[em, en] = cp
uv ep. (49)
These structure coefficients obviously vanish in a coordinate
basis, which is what most cosmologists usually work in.
29. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism V
We will also make use of a generalized connection ∇ that
associates ∇XY to any two vector fields X and Y. In a general
basis, we have that
∇v em = Γa
mnea, (50)
that is, the connection coefficients are defined as the
components of the directional derivative of the basis vectors.
For more information, see [9] and Abraham and Marsden’s
book [1]. For two vector fields, X = Xmem, and u = umem, we
have that
∇uX = (en(Xm
)un
+ Xa
Γm
anuv
) em. (51)
The commutator above therefore can now be written as:
[u, v] = ∇uv − ∇vu + (Γp
mn − Γp
nm + cp
mn) um
un
ep. (52)
30. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism VI
We defined the Torsion to be
ˆT(u ∧ v) ≡ ∇uv − ∇vu − [u, v] . (53)
We require spacetime in General Relativity to be torsion-free,
so we have
ca
mn = Γa
nm − Γa
mn. (54)
That is, all of the connection coefficients are now functions of
the structure constants!
In an orthonormal basis, gab = diag(−1, 1, 1, 1), so
Γamn =
1
2
gabcb
nm + gmbcb
an − gnbcb
ma . (55)
These are antisymmetric in the first two indices, so we can
write:
Γabt = −Γbat ≡ ϵabcΩc
, (56)
31. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism VII
where Ωc denotes the rotation of the spatial frame and is
defined by
Ωa
=
1
2
ϵabgd
ubeg · ˙ed . (57)
We can therefore write that
ca
tb = −θa
b + ϵa
bcΩc. (58)
The rest of the structure coefficients are all spatial in nature,
and we can write
ck
ij = ϵijl nlk
+ al δk
i δl
j − δk
j δl
i , (59)
where nlk and ai classify the Bianchi algebra, per the Behr
decomposition (See Stephani, Kramer, MacCallum,
Hoenselaers, and Herit [16]), Landau and Liftshitz [14], Grøn
and Hervik [9] for a further explanation:
33. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism IX
These classifications were due to Bianchi [5], but we follow the
conventions of Ellis, Maartens and MacCallum. [7].
One can find evolution equations for the nlk and the ai by
recognizing that the structure coefficients define a Lie algebra,
so that the Jacobi identity holds. That is, for the set of vectors
(u, ea, eb), we have that
0 = [u, [ea, eb]] + [ea, [eb, u]] + [eb, [u, ea]]. (60)
Go through some algebra, actually a lot of algebra (!), and find
that we get
u(ck
ab) + ck
td cd
ab + ck
ad cd
bt + ck
bd cd
ta = 0. (61)
34. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism X
Taking the trace of this equation, we get upon defining
u = ∂/∂t that
˙ai = −
1
3
θai − σij aj
− ϵijkaj
Ωk
, (62)
and the trace-free part yields
˙nab = −
1
3
θnab − 2nk
(aϵb)kl Ωl
+ 2nk(aσk
b). (63)
Returning to our propagation equations from before, we note
that for the shear propagation equation, we had
˙σab = um
∇mσab = u(σab) − Γm
anσmbun
− Γm
bnσamun
. (64)
35. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism XI
Using the orthonormal frame formalism, we can write this as
˙σab = u (σab) − 2σd
(aϵb)cd Ωc
. (65)
Setting a universal time gauge, u = ∂
∂t and using the
convention that H = 1
3θ, (where H is the Hubble parameter)
we have the Einstein field equations in an orthonormal frame:
˙H = −H2
−
2
3
σ2
− µ
1
6
+
1
2
w , (66)
˙σab = −3Hσab + 2ϵuv
(a σb)uΩv − Sab − 2ησab, (67)
µ = 3H2
− σ2
+
1
2
R, (68)
0 = 3σu
a au − ϵuv
a σb
unbv , (69)
36. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Orthonormal Frame Formalism XII
where Sab and R are the three-dimensional spatial curvature
and Ricci scalar and are defined as:
Sab = bab −
1
3
bu
uδab − 2ϵuv
(a nb)uav , (70)
R = −
1
2
bu
u − 6auau
, (71)
where bab = 2nu
a nub − (nu
u) nab. We have also denoted by Ωv
the angular velocity of the spatial frame.
˙nab = −Hnab + 2σu
(anb)u + 2ϵuv
(a nb)uΩv , (72)
˙aa = −Haaσb
a ab + ϵuv
a auΩv , (73)
0 = nb
a ab. (74)
38. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods I
With the propagation equations (66), (67), (75), and the
constraint equations (68) and (69) in hand, we can apply
techniques of dynamical systems theory to obtain a great deal
of information about the system’s future and past asymptotic
states. Let us introduce the following normalizations:
Σij =
σij
H
, Nij =
nij
H
, Ai =
ai
H
, Ri =
Ωi
H
, Sij =
(3)R⟨ij⟩
H2
,
(76)
Ω =
µ
3H2
, P =
p
3H2
, Qi =
qi
H2
, Πij =
πij
H2
, K = −
(3)R
6H2
.
(77)
The usefulness of this can be seen from the fact that, in
geometrized units, the Hubble parameter has units of inverse
length. Dividing each quantity above by an appropriate power
of H, makes the variables dimensionless.
39. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods II
We therefore obtain our dynamical system [11] [7] as:
Σ′
ij = −(2 − q)Σij + 2ϵkm
(i Σj)kRm − Sij + Πij
N′
ij = qNij + 2Σk
(i Nj)k + 2ϵkm
(i Nj)kRm
A′
i = qAi − Σj
i Aj + ϵkm
i AkRm
Ω′
= (2q − 1)Ω − 3P −
1
3
Σj
i Πi
j +
2
3
Ai Qi
Q′
i = 2(q − 1)Qi − Σj
i Qj − ϵkm
i RkQm + 3Aj
Πij + ϵkm
i Nj
kΠjm.(78)
These equations are subject to the constraints
Nj
i Aj = 0
Ω = 1 − Σ2
− K
Qi = 3Σk
i Ak − ϵkm
i Σj
kNjm. (79)
40. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods III
In the expansion-normalized approach, Σab denotes the
kinematic shear tensor, and describes the anisotropy in the
Hubble flow, Ai and Nij describe the spatial curvature, while
Ωi and Ri describe the relative orientation of the shear and
spatial curvature eigenframes and energy flux respectively.
Further the prime denotes differentiation with respect to a
dimensionless time variable τ such that
dt
dτ
=
1
H
. (80)
The remarkable thing about this method is as follows.The
system of equations (78) is a nonlinear, autonomous system of
ordinary differential equations, and can be written as
x′
= f(x), (81)
41. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods IV
where x = [Σij, Nij , Ai , Ω, Qi ] ∈ Rn, where the vector field f(x)
denotes the right-hand-side of the dynamical system.
Following [1], we first note that the vector field f(x) is clearly
at least C1 on M = Rn. We call a point m0 an equilibrium
point of f(x) if f(m0) = 0. Let (U, φ) be a chart on M with
φ(m0) = x0 ∈ Rn, and let x = [Σij , Nij , Ai , Ω, Qi ] denote
coordinates in Rn. Then, the linearization of f(x) at m0 in
these coordinates is given by
∂f(x)i
∂xj
x=x0
(82)
42. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods V
It is a remarkable fact of dynamical systems theory that if
the point m0 is hyperbolic, then there exists a
neighborhood N of m0 on which the flow of the system Ft
is topologically equivalent to the flow of the linearization
Eq. (82). This is the theorem of Hartman and Grobman
[17]. That is, in N, the orbits of the dynamical system can
be deformed continuously into the orbits of Eq. (82), and
the orbits are therefore topologically equivalent. We use
the following convention when discussing the stability
properties of the dynamical system. If all eigenvalues λi of
Eq. (82) satisfy Re(λi ) < 0(Re(λi ) > 0), m0 is local sink
(source) of the system. If the point m0 is neither a local
source or sink, we will call it a saddle point.
Given a linear DE x′ = Ax on Rn, we consider the
eigenvalues of A (complex in general, and not necessarily
43. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods VI
distinct) and the associated generalized eigenvectors. We
define three subspaces of Rn,
stable subspace, Es
= span(s1, . . . , sns ),
unstable subspace, Eu
= span(u1, . . . , unu ),
centre subspace, Ec
= span(c1, . . . , cnc ),
where s1, . . . , sns are the generalized eigenvectors whose
eigenvalues have negative real parts, u1, . . . , unu are those
whose eigenvalues have positive real parts, and c1, . . . , cnc
are those whose eigenvalues have zero real parts. It is
well-known then that
Es
⊕ Eu
⊕ Ec
= Rn
,
44. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods VII
and that
x ∈ Es
⇒ lim
t→+∞
eAt
x = 0,
x ∈ Eu
⇒ lim
t→−∞
eAt
x = 0.
These statements basically describe the asymptotic
behaviour of the system: All initial states in the stable
subspace are attracted to the equilibrium point 0, while all
initial states in the unstable subspace are repelled by 0. In
particular, if dim Es = n, all initial states are attracted to
0, which is referred to as a linear sink, while if
dim Eu = n, all initial states are repelled by 0, which is
referred to as a linear source.
45. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods VIII
We will consider a DE x′ = f(x) on Rn, where f is of class
C1. If a is an equilibrium point (f(a) = 0), the linear
approximation of f at a becomes
f(x) ≈ Df(a)(x − a),
where
Df(a) =
∂fi
∂xj x=a
(83)
is the derivative matrix of f. Thus, with the given DE
x′ = f(x), we associate the linear DE
u′
= Df(a)u, (84)
where u = x − a, called the linearization of the DE at the
equilibrium point a.
46. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods IX
Theorem
(Invariant Manifolds) Let x = 0 be an equilibrium point of the
DE x′ = f(x) on Rn and let Es , Eu, and Ec denote the stable,
unstable, and centre subspaces of the linearization at 0. Then
there exists
W s
tangent to Es
at 0,
W u
tangent to Eu
at 0,
W c
tangent to Ec
at 0.
Further, each equilibrium point of the dynamical system
represents a solution to the Einstein field equations!. This
was discussed at length by Jantzen and Rosquist [12] and
Wainwright and Hsu [18].
47. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods X
Therefore, we have transformed the Einstein field
equations from the pseduo-Riemannian manifold to
n-dimensional phase space. Solutions to the EFE are
equilibrium points of the dynamical system, and the
stability analysis performed by analyzing the eigenvalues of
Eq. (82) tells us what the future and past asymptotic
state of a particular Bianchi cosmology will be. It should
be noted that it is often that one gets eigenvalues with
more than one zero eigenvalue, or that, the Jacobian
matrix in Eq. (82) is undefined. In this case, we must use
methods from topological dynamical systems theory to
analyze the future and past asymptotic states.
For examples of our work, see:
48. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods XI
“Exploring Vacuum Energy in a Two-Fluid Bianchi Type I
Universe”, Ikjyot Singh Kohli and Michael C. Haslam -
arXiv:1402.1967 - Submitted to Phys. Rev. D - 2014
“On The Dynamics of a Closed Viscous Universe”, Ikjyot
Singh Kohli and Michael C. Haslam - arXiv:1311.0389 -
Phys. Rev. D 89, 043518 (2014)
“A Dynamical Systems Approach to a Bianchi Type I
Magnetohydrodynamic Model”, Ikjyot Singh Kohli and
Michael C. Haslam - arXiv:1304.8042 - Phys. Rev. D 88,
063518 (2013)
“The Future Asymptotic Behaviour of a Non-Tilted
Bianchi Type IV Viscous Model”, Ikjyot Singh Kohli and
Michael C. Haslam - arXiv:1207.6132 0 Phys. Rev. D 87,
063006 (2013)
49. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Dynamical Systems Methods XII
In these works, we found two new solutions to the Einstein field
equations, and also showed that the early universe quite
possibly had large magnetic fields. In our newest work, we have
discussed a possible mechanism for the existence of vacuum
energy. So, dynamical systems methods are very powerful in
cosmology!
As an example, to be possibly discussed for next time, consider
our parameter space depiction of the Bianchi Type I MHD
model:
52. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources I
[1] Ralph Abraham and Jerrold E. Marsden.
Foundations of Mechanics.
AMS Chelsea Publishing, second edition, 1978.
[2] John D Barrow.
Dissipation and unification.
Monthly Notices of the Royal Astronomical Society,
199:45–48, 1982.
[3] John D Barrow.
String-driven inflationary and deflationary cosmological
models.
Nuclear Physics B, 310:743–763, 1988.
53. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources II
[4] V.A. Belinskii and I.M. Khalatnikov.
Influence of viscosity on the character of cosmological
evolution.
Soviet Physics JETP, 42:205, 1976.
[5] L. Bianchi.
Sugli Spazi I A Tre Dimensioni Che Ammettono Un Gruppo
Continuo Di Movimenti.
Soc. Ital. Sci. Mem. di Mat., 1898.
[6] George F.R. Ellis.
Cargese Lectures in Physics, volume Six.
Gordon and Breach, first edition, 1973.
54. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources III
[7] George F.R. Ellis, Roy Maartens, and Malcolm A.H.
MacCallum.
Relativistic Cosmology.
Cambridge University Press, first edition, 2012.
[8] G.F.R. Ellis and M.A.H. MacCallum.
A class of homogeneous cosmological models.
Comm. Math. Phys, 12:108–141, 1969.
[9] Øyvind Grøn and Sigbjørn Hervik.
Einstein’s General Theory of Relativity: With Modern
Applications in Cosmology.
Springer, first edition, 2007.
55. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources IV
[10] S.W. Hawking.
Perturbations of an expanding universe.
Astrophysical Journal, 145:544, 1966.
[11] C.G. Hewitt, R. Bridson, and J. Wainwright.
The asymptotic regimes of tilted bianchi ii cosmologies.
General Relativity and Gravitation, 33:65–94, 2001.
[12] R.T. Jantzen and K Rosquist.
Exact power law metrics in cosmology.
Classical and Quantum Gravity, 3:281, 1986.
[13] Pijush K. Kundu and Ira M. Cohen.
Fluid Mechanics.
Academic Press, fourth edition, 2008.
56. Dynamical
Systems
Methods in
Early-Universe
Cosmologies
Ikjyot Singh
Kohli
Ph.D.
Candidate
York
University
Outline
Introduction
Description of
the Matter
Deriving the
Dynamical
Equations
The Theory of
Orthonormal
Frames
Dynamical
Systems
Techniques
Sources V
[14] L.D. Landau and E.M. Lifshitz.
Classical Theory of Fields.
Butterworth-Heinemann, fourth edition, 1980.
[15] L.D. Landau and E.M. Lifshitz.
Fluid Mechanics.
Butterworth-Heinman, second edition, 2011.
[16] Hans Stephani, Dietrich Kramer, Malcolm MacCallum,
Cornelius Hoenselaers, and Eduart Herlt.
Exact Solutions of Einstein’s Field Equations.
Cambridge University Press, second edition, 2009.
[17] J. Wainwright and G.F.R. Ellis.
Dynamical Systems in Cosmology.
Cambridge University Press, first edition, 1997.