The given formula for relatA = tB / sqrt (1- vBA2 / c2 )
tA = Time differential as measured by observer A
tB = Time differential as measured by observer B
vBA2 = Square of the difference in velocity between observer A and observer B
Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics -- the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C) if the ratio of the lengths of the two strings were a whole number. Pythagoras discovered this by looking and listening. Today that information is more precisely encoded into mathematics, namely the wave equation for a string with a tension T and a mass per unit length . If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t, then the equation of motion is the one-dimensional wave equation
When solving the equations of motion, we need to know the "boundary conditions" of the string. Let's suppose that the string is fixed at each end and has an unstretched length L. The general solution to this equation can be written as a sum of "normal modes", here labeled by the integer n, such that
According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. But then we have a problem, because a string oscillates in space and time, and as it oscillates, it sweeps out a two-dimensional surface in spacetime that we call a world sheet (compared with the world line of a particle).
In the nonrelativistic string, there was a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a two-dimensional spacetime of its own, where the division between space and time depends upon the observer.
The classical equation can be written as
These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet
The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as
For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and left-movers, and this difference will be important later in the supersymmetric heterotic string theory.
This is classical string. When we add quantum mechanics by making the string momentum and position obey quantum commutation relations, the oscillator mode coefficients have the commutation relations
So this is where the elementary particle arise in string theory. Particles in a string theory are like the harmonic notes played on a string with a fixed tension
In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts. This happens because of the minus sign in the space-time metric, which implies that
In 26 space-time dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore. bosonic string quantum mechanics is only consistent if the dimension of spacetime is 26.
By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in space time, with mass that depends on the spin according to the formula
The main alternative theory of the origin of the structures of the universe are the cosmic strings or super heavy strings which are predicted too form in the early universe by the Grand Unified Theory (GUT) in inflationary “ Big Bang model. Loops of cosmic strings were the seed of the galaxies. They were super heavy strings, formed at phase transition or condensation that took place when the universe was cooled after GUTS in the very early universe. Kibble had suggested that GUTS strings played an important role in the evolution of the Universe and the strings provided the inhomogenity leading to the formation of galaxies. In the very early universe Strings were predicted to be formed at symmetry breaking phase transition by those in grand unified theories (GUTS) in which vacuum had the appropriate topology. Cosmic Strings were the configuration of the matter fields, which owe their topology of the space of degenerate vacuum, produced in the phase transition, in the early universe. Let us ignore the internal structure of the strings and treat them as one-dimensional object with tension. In the resting frame of the strings, the mass per unit length μ to the tension. The equality of the line of the density and the tension caused the typical velocity associated with large vibration on the strings to be close to speed of light. The strings cannot end but can either close on themselves or can be extended to infinity. The closed strings are loops.
Whenever two long strings cross each other, they exchange ends, or `intercommute' (case (a) in the figure below). We had already encountered this apparently strange fact when we discussed the strings in the context of nematic liquid crystals. In particular, a long string can intercommute with itself, in which case a loop will be produced (this is case (b) below).
As with any object in tension, strings would also accelerate so as to try to become straight. Damping of the string motion was due to their non gravitational interaction with other matter, those become negligible as soon as the strings were formed. Strings that extended outside the horizon were conformably stretched by the cosmic expansion. Thus at a given epoch, these strings were straight on their length & scale, but were smaller then the horizon size, but was quiet convoluted on large scale lager then this. The typical velocity was associated with the straightening of a string and was close to the speed of light and the velocity field of the string extending outside the horizon was relativistic and approximately constant over scale much smaller than a horizon size. Once a loop entered the horizon it no longer expanded but rather started oscillate with a period comparable to light travel time across it. This motion was damped by gravitational radiation causing the size & period of the loop to decrease approximately linearly with time The Fractional decrease in size, period,& mass of the strings in one oscillation was given by equation* Gμwhere G is Gravitational constant. A string will decrease to zero size in a finite amount of time loosing its energy by Gravitational Radiation. The distribution of strings in our universe was not quite so well understood
These pictures show
1) a full three-dimensional simulation of the intercommoning of two cosmic strings... The reconnection and `exchange of partners' when two strings intersect. In this three-dimensional simulation, the strings approach each other at half the speed of light. Notice the radiation of energy and the production of a small interaction loop in the aftermath of the collision
[ Picture By Rupak Bhattacharya].
The scattering of two vortices is highly non-trivial; the two vortices approach and form a donut from which the emerge at right-angles have `exchanged halves'
Both long cosmic strings and small loops will emit radiation. In most cosmological scenarios this will be gravitational radiation, but electromagnetic radiation or axions can also be emitted in some cases (for some specific phase transitions). Here is a single, oscillating piece of string
4] Radiation fields from the oscillating shown above. A transverse cross-section of the fields has been made at the point of maximum amplitude. Notice the four lobes of the radiation (a quadrupole pattern) which is characteristic of all cosmic string radiation
5] The effect of radiation is much more dramatic for loops, since they lose all their energy this way, and eventually disappear. Here you can see what happens in the case of two interlocked loops. This configuration is unlikely to happen in a cosmological setting, but it is nevertheless quite enlightening. Notice the succession of compicated dynamic processes before the loop finally disappears
. After formation, an initially high density string network begins to chop itself up by producing small loops. These loops oscillate rapidly (relativistically) and decay away into gravitational waves. The net result is that the strings become more and more dilute with time as the universe expands. From an enormous density at formation, mathematical modelling suggests that today there would only be about 10 long strings stretching across the observed universe, together with about a thousand small loops!
In fact the network dynamics is such that the string density will eventually stabilize at an exactly constant level relative to the rest of the radiation and matter energy density in the universe. Thus the string evolution is described as `scaling' or scale-invariant, that is, the properties of the network look the same at any particular time t if they are scaled (or multiplied) by the change in the time. This is schematically represented below:
At GUT’s the Strong, Weak and Electromagnetic forces behaved as if, they had equal strength, much as line defects found in the crystal. They formed as a net work across the space& time. The GUT”s predicts that strings were formed at a temperature of about 1015 to ~1016 Gev. at a Cosmological time of about 10̃35 Second. The Cosmic Strings were formed at the mass scale of GUTs Symmetry breaking (Mx-̃ 2x1015 Gev) was typified by a mass per unit length μG/c2̃̃ ~ 2x10 6 in dimensionless unit.[ G= Gravitational Constant, C= speed of Light, which is corresponding to μ= ~ 2.6X10 21, Kgm-1~ 4x107 MOPC-1 where MO= Mass of Sun . Or in other words the strings were formed with a mass per unit length of about 1020 kg-1. They have a mass per unit length μ=ε/G [where ε= ~( <φ>/ mp)2 is the dimensionless amplitude of their Gravitational potential, mp is the Plank Mass and the Vaccum Expectation value of Higgs field is φ.] Because of their enormous tension ε/G , the net work of the Strings were formed in the phase transition. In this Theory the Strings contributed only a small fraction of mass of the Universe. The Galaxies were formed by Accreating of ordinary matter about the Strings. The Strings were stretched by subsequent expansion of the Universe on waves, on a given scale and began to oscillate then. The strings underwent Oscillation in which the Transverse intertia acted as weight and the restoring forces were provided by longitudinal tension of the strings. As a result of oscillation in such that the scale entered the particles horizon and whenever the strings crossed itself and exchanged particle partners and produced closed Oscillating loops of the Strings with long life(Peebles. P.G. Z- large scale Structure of the
There remain always a fundamental asymmetry in the distribution of the constituents of the universe. That is, there appears to be an excess of normal matter over antimatter in the most current and compelling models of the universe (cold dark matter [CDM]). The origin of this asymmetry remains yet unexplained before us as do the nature of both dark matter and dark energy. Dark matter and dark energy are required by the latest CDM models that have recently been shown to be very much in accord with the findings of the cosmic background surveys . However, most intriguingly, this fundamental cosmic asymmetry appears to manifest itself by way of other asymmetries observed in other more complex systems of universe. For example, there has been a much discussed thesis that the left-right symmetry encountered in simple as well as complex multicellular organisms, including human laterality and cerebral symmetry, are a consequence of symmetry at the molecular level . This, in turn, is thought to arise from asymmetry at the level of elementary particles. However, although connecting links between molecular—and subatomic—chirality and macroscopic handedness and asymmetry are not established, the implications of this asymmetry for biologic processes and evolution are profound. We today now know that proteins in life forms consist (almost) exclusively of L--amino acids, whereas nucleic acids contain only the D-isomers of ribose or deoxyribose. Although there exists considerable controversy concerning the questions of when and how this homochirality arose in world, but it appears to be the fundamental, but incompletely tested, assumption that life as we know it could not have arisen without it. Much less attention seems to have been paid in recent years to the reasons for homochirality and its connection to the origin of life. Older studies have held that the structure-destabilizing effects of ‘‘chiral defects’’ (i.e., the incorporation of D-amino acids or L-nucleotides into their respective polymers would render them incapable or unable to participate in ‘‘biology’’). However, although newer studies confirm some destabilization, they also indicate that there is more ability to accommodate unnatural enantiomers than was previously appreciated. These findings provide new insights into the constraints imposed on life’s origin with respect to chiral purity. We should note, however, that this is a subject that has attracted considerable interest and has been reviewed in the past . Indeed, one can even use one’s nose and establish that stereo isomers can smell different
Or on a more tragic note, the story of thalidomide where the R isomer is a teratogen while the S isomer is a tranquilizer. The primary amino acid sequence determines the structure and function of a protein. The two most common structural motifs are the -helix and ß-sheet. Although -helices are now more abundant in proteins than ß-sheets, it is thought that the ß-sheet occurred earlier during chemical evolution . Generally, L-amino acids form a right-handed helix; a right-handed helix exhibits optical rotation of its own. Similarly, ß-sheets are not flat but, if made of L-amino acids, exhibit a right-handed twist when viewed along their strands. This right-handedness of turn arises from energetic constraints in the bonding of L-amino acids; a chain consisting of D-amino acids would produce sheets with a left-handed twist . Indeed, the circular dichroism (CD) spectrum produced by the all-D enantiomer of the full-length ß-amyloid peptide (42 amino acid residues) was a mirror image of the spectrum obtained with the natural all-L enantiomer, indicating that the two enantiomers had opposite optical rotation . Furthermore, there are indications that such mirror image conformation translates into functional stereospecificity. When the D- and the L-enantiomer of the complete enzyme HIV-1 protease were chemically synthesized, they were found to have identical covalent structure and CD spectra of equal, but opposite, optical rotation . These data suggest that the folded forms of the D- and L-protease enzymes are mirror images when viewed in three dimension
. Most notably, the enantiomers exhibited reciprocal chiral specificity, the L-enzyme cleaving only the L-substrate and the D-enzyme showing activity only for the D-substrate. Although protein macromolecules are carriers of function, DNA macromolecules are the transgenerational informational carriers of most contemporary organisms . RNA plays the role of an intermediary between DNA and proteins in eukaryotes and can take on both informational as well as functional roles. As in proteins, the monomeric units of DNA and RNA are homochiral, each of the nucleotides containing either D-ribose or D-deoxyribose. Also like proteins, nucleic acids are able of taking on a variety of secondary structures, most famous among them the double helix. Both RNA and DNA are matrices for the assembly of a complementary replica, and homochirality has been postulated to be an absolute necessity for complementarity . Molecular modeling was interpreted as indicating that incorporation of a single T of the opposite chirality in double-stranded poly(A)/poly(T) would prevent base coupling, thereby destroying the template property and the ability to act as an information carrier . This does not, however, appear to be entirely correct. An NMR study with a dodecadeoxynucleotide containing a single nucleotide with a L-deoxyribose (the G4 residue) formed a stable base pair with the natural C-9 residue within a right-handed B-form conformation . Similarly, although substitution of a D-nucleotide with an L-nucleotide somewhat destabilized a short DNA duplex, the D-isomer could nonetheless be accommodated via changes in some of the backbone torsion angles around the phosphates and the glycosidic bond . Others have confirmed that cooperative binding between mixed L/D-oligodeoxynucleotides and single-stranded DNA and RNA is possible despite the destabilizing effect of L-substitutions . The magnitude of this destabilization was found to depend on the position and type of the nucleotide . In addition, there appears to be a limit to the number of substitutions that a sequence can tolerate .
Professor and Head of Pathology now convener In-charge of DCP &DLT course of WBUHS School of Tropical Medicine 108 CRAvenue Kol-73 , and EX Professor&HOD Ophthalmic pathology RIO, CR Avenue kol-73 and also of WBUHS and EX Professor of Pathology, Ex In charge of Histopathology unit, in charge of Blood Banks VCCTC, in charge of Cytogenetics ,Human Tech DNA & Gene cloning laboratoriesat Institute of Post Graduate medical Education & Research,244a AJC Bose Road, Kolkata-20.India, Member of Board of Studies Undergraduate, Post Graduate& post Doctoral Courses in Pathology of West Bengal University of HealthSciences(WBUHS), India