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6^th March 2003

Gravity, brane leaks and stringy networks.

 

Abstract

Gravitational attraction is described and the work of Isaac Newton (1642-1727) is taken as the basis. The Newtonian view was developed in Europe and elsewhere for a couple of hundred years or so until measurements of the orbit of the planet Mercury indicated that the model was incomplete. Revolutionary work in the 20^th Century progressed in two thematic branches of physical research – quantum theory and relativity theory and towards the end of the century, internet objects were instantiated[1] such that research connections increased inordinately. As the understanding of relativity theory, quantum mechanics and object-orientation spread, more and more discoveries were found to be parts of an underlying mathematical construct known as k-theory[2]. This branch of mathematics describes the geometry and topology of manifolds: the description is huge and the resources available to its study are spread very thinly around the planet: less than one tenth of one percent of the global, human population works towards its comprehension, development and dissemination. More recently, some workers have revealed a level of super-symmetric analysis, named M-theory, which both contains and dwarfs k-theory and includes the current view of gravitational action. M-theory makes forecasts for others to test.

 

 

The Newtonian view

 

In the 17th Century Isaac Newton reported that he felt as though he had learned a small amount concerning the workings of the physical universe and some of his thoughts are in the Principia of 1687.

 

 

Towards the end of his life, he reported that his knowledge amounted to nothing more than the ripples created when a pebble splashes in a pond in comparison with the vast ocean beyond: his understanding seemed very, very small.

 

The description is that, in the absence of other forces, object pairs attract each other with a force dependent upon the magnitude of the masses involved and the distance of their separation. The gravitational force is proportional to the product of the masses and inversely proportional to the separation of their centres squared. The relationship is known as Newton’s universal law of gravitation and it may be written as:

 

F_g = Gm₁m₂/r²

 

Where F_g is the gravitational force, m₁ and m₂ are the masses and r is the distance between their centres. The constant of proportionality, G, is determined by experiment to be close to 6.671 x 10^-11 Newton m² kg^-2. Since G is a small number when expressed in these SI-units[3], it could be guessed here that the force it represents is weak[4]. For a given object pair, the gravitational force diminishes as the square of the object separation and this is illustrated below: when the separation increases ten-fold (from 1 to 10 units) the force diminishes by a factor of a hundred (from 1 to 0.01 units).

 

 

To first order approximation, it is observed that the Earth’s distended mass[5] beneath an observer’s feet acts as though the force originates at its centre, around six thousand kilometres below. Some geologists use the observed differences from this description to gauge the beat of the planet beneath.

 

The gravitational attraction of planet Earth overwhelms that of lighter, negatively buoyant, electrically neutral, magnetically depolarised, separate, unrestrained, chemically inert, biologically dead and magic-free objects near to its surface. The rate that the Moon falls towards the Earth is related to the rate of fall of an apple, whether the apple is made of plutonium, water, sponge or “apple”. The Moon does not hit the Earth because it travels through space with enough sideways speed to miss. Through Newton’s second law of motion[6] and the principle of strong equivalence[7], it is possible to predict the energies of objects in such gravitational fields to high degrees of accuracy.

 

For a body of material near to the surface of the Earth the inertial and gravitational forces are of equal magnitude[8], F_i = F_g. We may substitute the inertial mass of the object in the universal law and with Newton’s second law of motion the result is an expression for the rate at which near-surface objects fall to the ground that is independent of their own mass:

 

dv/dt = g = G m_e/r_e². ‘g’ is calculated by substitution of the SI-unit values for ‘G’ and the mass and radius of the Earth: g ≈  9.8 m s^-2.

 

The ideas of Newtonian mechanics were developed by later mathematicians, tested by classes of students, practiced by teams of engineers and checked by conferences of physicists and all seemed fine for almost two hundred years.

 

 

Albert Einstein and Max Planck: Relativity and Quantum theory

 

Eventually, towards the end of the 19th Century, observations showed that the point at which the planet Mercury is closest to the sun on its orbit[9] describes a circular path around Sol as the orbit count increases. The perihelion of Mercury precesses at a rate of 42.9 ± 0.1 seconds of arc per century[10] or approximately one three hundredth of one per cent of the orbit length per orbit. This motion is neither predicted nor predictable from the Newtonian view and many were perplexed.

 

Calculations based on the postulated speed limit[11] in Einstein’s Relativity theory[12] closed the gap to these astronomical deviations. To late-21^st Century limits of observation, the paths of planetary orbits are described accurately by this theorem and its predictions continue to be tested with ever increasing precision. Tested where? Cavendish-like, desktop experiments: rocket science: solar-system dynamics: gravitational red shift: gravitational lensing.

 

Another consequence of Einstein’s analysis is that an energy change, ΔE = E₂-E₁, may be expressed as a change of mass through the relation ΔE = Δmc². This equivalence suggests correctly that energy and mass are manifestations of the same underlying property of the universe.

 

The limits of GR4 are reached in the description of objects known as black holes. Black holes contain sufficient mass, M, within a radius, r, such that their Newtonian escape velocity, V_escape = (2GM/r)^1/2 = c and no material or energetic thing can escape. The location of this event horizon is at a distance from the centre of the object known as the Schwarzschild radius[13]. A black hole with the mass of Sol has a Schwarzschild radius (r_s) of approximately[14] 3 km, for the mass of the Earth r_s is approximately 9 mm and for the material in, say, a kilogramme of apples r_s is around 10^-27 m.

 

Contemporary with the development of relativity theory a second revolutionary idea was becoming formalised. The German physicist Max Planck became aware that energy is not continuous but exists in discrete packets or quanta: E = ћω, where ω (=2πf) is the angular frequency of the radiation. Since ћ has a value around 10^-34 joule-seconds then it is clear that quantum theory describes object behaviour at very small scales. Other consequences of quantum mechanics include that observation has intrinsic uncertainty, all objects display wave-like behaviour and that packets of wave energy behave like material objects. The uncertainty principle is encapsulated in relationships between hermitian[15] object property pairs e.g. position and momentum, Δx Δp ≥ ћ, energy and time, ΔE Δt ≥ ћ and so on. The de Broglie wavelength, λ = h/(m_ov) and electron diffraction[16] demonstrate the wave-like behaviour of matter and when a one-in-a-million wave breaches one on foot, the material nature of wave energy is manifest.

 

The application of ΔE = Δmc² by weapons researchers in the 1940-1950s led Albert Einstein to report that if only he had known, he would have become a watchmaker. The description of these weapons also require an understanding of the quantum theory developed by Neils Bohr, Erwin Schrödinger and others yet, the “spooky” action at a distance inherent in these Copenhagen statements of quantum mechanics still clouded Einstein’s view.

 

Newton’s expression to Thomas Hooke that the reason he could see so far was that he stood on the shoulders of giants[17] (some say as a derogatory reference to Hooke’s physical stature) is used here in another sense: when stood on the shoulders of the early 20^th century physics ‘giants’ the view is spectacular and it will blow away the unwary.

 

 

Symmetry

 

Mirrors reflect. The light in a plane-mirror reflection was scattered by the object.

 

The description of a force in quantum field theory, QFT, is the description of a process of exchange: objects exchange other objects to maintain the force between them. An analogy is when two people play catch with a ball: from a great distance the ball cannot be discerned and the pair appear to be linked in a closely related dance whereas close-up it is obvious that the pair is connected by the action of throwing and catching the ball.

 

Four forces are observed frequently in nature: electromagnetism, strong nuclear, weak nuclear and gravity. In combination, these produce action that results in what Homo sapiens sense. Currently, there is debate over the existence of a fifth force (‘quintessence’) that is repulsive at large separations and, therefore, resolves the paradox that the rate of expansion of the universe is increasing.

 

In the 1960s the electromagnetic and weak nuclear forces were shown to be degenerate and this unification by Stephen Weinberg, Sheldon Glashow, Abdus Salaam, Richard Feynman et al. led to the prediction and discovery of the omega-minus, Ω^- baryon. Other baryons include the proton, the neutron, the lambda, Λ and their anti-particles.

 

The analysis includes definitions of spin and parity, the existence of pi mesons and symmetry breaking in QFT.

 

Within QFT the agents of forces are defined as bosons: the photon, the gluon and the W and Z particles. These objects are located at points in space-time: initially their action was centred. This, so-called standard model is charted at http://particleadventure.org/particleadventure/frameless/chart.html

 

The largest discrepancy between GR4 and the ideas of k-theory appears in H.Y. Cui, 2002[18], where the prediction is made that planetary perihelion precessions are about 8.3% lower in value than those predicted by GR4. The discrepancy is related to the symmetrical nature of the tensor geometry used to describe electromagnetic and gravitational forces and it is a measurable prediction.

 

String theory was first discovered in the1960s when Gabriele Veneziano was searching for the amplitude of action in space-time as the object separation shrank, A(s, t), and found that: A(s, t) = g_s²Γ(a(s))Γ(a(t))/Γ(a(s)-a(t)) where, the Dirac Γ-matrices are constructed through QFT. A Dirac Γ–matrix is an object to describe point behaviour – the discovery was that the smallest region of space-time from which an action originates has extent: it is not a zero-dimensional point but a one-dimensional object known colloquially as a string: the constant of action between object pairs is always non-zero, g_s<>0.

 

 

Strings and membranes in other dimensions

 

The minimum quantities for the properties length, time, energy, electric charge, spin, strangeness, charm, flavor and color were found to result from the relationships of quantum mechanics combined with relativity theory: the properties from spin to flavor arise directly from relativistic QFT.

 

   length scale[19], L_p = (G ћ /c³)^1/2 ~ 10^-35 metres.

   Planck time, T_p = (G ћ /c⁵)^1/2 ~ 10^-42 seconds.

   Planck mass, M_p = (ћc/G)^1/2 ~ 10^-9 kilograms or energy, E_p ~ 10²⁸ eV.

 

Scale: one eats apples; trees grow apples; this planet has apples. The word “apples” describes a different quantity in each clause but the absolute number of apples is in the context or scale of the clause: the quantum unit is one apple.

 

Dimension: to describe the location of an object near to the surface of our planet it is sufficient to specify four numbers or coordinates: latitude, θ°, longitude, φ°, distance from the centre of the planet, r, and the time of the observation[20]. This provides a description of four-dimensional space-time in a frame of reference with its origin at the centre of the earth at a given time: the here and now. In the mathematics of relativistic quantum mechanics other dimensions arise such as an object’s electrical charge or spin. The dimension of mass arises in the Higgs description and this brings the dimension count to ten. M-theory adds one more space-like dimension that is compactified[21] to regions less than the Planck length. The collection that the dimensions describe is known as a brane.

 

Sometimes the number of dimensions is expressed as the letter p and the term p-brane arises often in mathematical study. In the natural units of string theory ћ = c = 1 and their SI unit (or equivalent) values are substituted whenever a specific calculation is required. In the vocabulary of string theory, a centre is treated as an extended region in the quantum foam of the 10-dimensional, Planck-space-time with open or reconnected ends (loops).

 

The holographic principle: the theory may be written down in one less dimension than the theory occupies. For example, it is possible to describe holographic imagery by tracing rays on a piece of paper[22] [23] [24]. It is a requirement that a successful theory is “writable” in a common language.

 

Current theory[25] predicts the existence of spin 2 gravitons and their mass-energy[26] ~1 TeV/c² that is well below M_p. These energies will be attained in particle anti-particle colliding machines[27] by the year 2005.

 

String theory makes the claim that it is possible to fit more than 10³⁴ pieces of open string end to end in each metre of the space-like dimensions of the local Dirichlet[28] p-brane. In comparison, the number of people on earth is around 6x10⁹ and the number of stars in the Milky Way is approximately10¹¹.

 

In experimental particle physics, some of the individual properties of quantum objects[29] are tracked throughout their existence whilst hermitian property pairs are determined only after measurement is made.

 

What happens to hermitian properties after the instantiation of a pair of quobjects?

 

When measurements are made on a hermitian property of quobject₁ the corresponding “other” property of quobject₂ is determined precisely and immediately. The effect is known as quantum entanglement and research into its possible uses is underway at many places.

 

An analogous problem has a solution provided by string theory:

(In the example given, it is best if the selections[30] are made doubly blind[31].)

 

Place a blank, circular card in an envelope and seal the envelope.

Similarly, seal a marked, polygon of card of similar weight and size in another envelope. Deliver the envelopes to separate people.

 

Each envelope contains a card that is either round and blank or marked and multiply edged but the properties of an individual card are not discernible until its envelope is opened.

 

This description of quantum entanglement, whereby qubits of information travel instantaneously since when one card has a property determined the other card’s other property is known immediately, is incomplete since it is not proven that the “blankness” and “shape” of cards are hermitian properties.

 

M-theory includes that objects are instantiated when branes collide.

 

Parallel universes: At each and every stage in a process, every possible decision is made and each option continues as a universe within its own brane.

 

The predictions of string theory for a Dirichlet brane with 10 dimensions (a D^p-brane, where p=10), include the existence of symmetric counterparts to the current list of known particles e.g. leptinos, gravitinos etcetera and the existence of tachyons that possess negative mass squared and travel backwards in time along flux tubes reflecting leaky branes connected with bits of string.

 

M-theory provides a description of the geometries of branes within the super-symmetric bulk. A suggestion is that the location at which the observable universe began is a reflection of an event when an open string connected branes across the bulk. There is a painting by Leonardo da Vinci(?) that has the index fingers of ‘God’ and ‘Adam’ separated by a spark. Commentators state that Leonardo was ahead of his time – it is difficult to disagree.

 

 

Currently

 

When we stand on the shoulders of giants at the bottom of the deepest bores we are not at the centre of the earth, the earth is not at the centre of the solar system, that is not at the centre of the galaxy, that is not at the centre of the universe, that has no centre.

 

A computer may connect to another such that a communication path is established. If either of these were connected to a third and so on, it is imaginable that the lattice of connected paths extends in all directions[32] to the extent that the “boundary condition” is that there is no boundary.

 

The upper speed limit of material interchange is that of light in a vacuum: if information exchange is an energetic process then the same limit applies.

 

Networks exist independently of the brains that build them.

 

 

String theory UK, 1992+

 

 

String Theory Seminar at Imperial College, London

 

elsewhere

 

Ian E. Consterdine

School House,

Holwick,

DL12 0NW

UK

 

ianschool@yahoo.co.uk