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precession of the perihelion of mercury


General theory of relativity

Precession of the permission of mercury is similar to the precession of perihelion of electron in the eccentric orbit around the nucleus. It was found that the velocity increases when electron gets close to the nearest point from the focus of the ellipse. It is also evident from the kepler's planetary motion. But special theory of relativity tells that mass should also increase as the velocity of the electron increases. So electron's orbit does not stays fixed but shifts by an angle. Electron has to go further when it reaches the next perihelion . But this effects is purely special relativistic. Some complicated mathematics is needed to prove this phenomena of perihelion shift, which at the moment is left for time shortage. On the other hand precession of the perihelion of mercury is the effect which is related to General relativity.
Since almost two centuries earlier astronomers had been aware of a small flaw in Mercury's orbit around the Sun, as predicted by Newton's laws. As the closest planet to the Sun, Mercury orbits a region in the solar system where spacetime is disturbed by t he Sun's mass. Mercury's elliptical path around the Sun shifts slightly with each orbit such that its closest point to the Sun (or "perihelion") shifts forward with each pass. Newton's theory had predicted an advance only half as large as the one actually observed. Einstein's predictions exactly matched the observation.
Mercury's orbit is elliptical. The orientation of this ellipse's long axis slowly rotates around the sun. This process is known as the "precession of the perihelion of Mercury" in astronomical jargon. It's a total of 5600 arcseconds of rotation per century. The precession is mostly a result of totally classical behavior; almost all of the movement of the perihelion (about 5030 arcseconds per century) is present in a two-body system with point masses for the Sun and Mercury. Another 530 arcseconds per century are caused by gravitational effects of the other planets.
That leaves 40 arcseconds per century of unexplained movement. The observed value of 5599.7 arcseconds per century is measured very accurately, to within 0.04 arcseconds per century, so this is a significant deviation. It turns out that 43 arcseconds per century are expected to result from general relativity. One hand-wavey way of explaining this is that the curvature of spacetime itself by the two bodies (Sun and Mercury) causes some changes to the gravitational potential, so it isn't really exactly GMm/r.


General theory of relativity

The picture shown above deescribes the phenomena of mecury's orbit as it shifts.
To find the solution of the orbit we need the Swardchild's metric from general relativity. In terms of coordinates φ and R, which represent the sperical angle and radius of the sphere respectively. So a change (δ)in the angle φ will give the required solution. This motion of the planet according to this solution is still periodic but the period is not exactly 2 (π) as in Newton's unperturbed gravity. Newton explained this perihelion by the perturbation of the sun's gravity by other planets.


perihelion of mercury
The required solution in terms of mass M, gravitational constant and velocity of light c has thus been worked out. Although the derivation is quite lengthy , the final result implies slight deviation from Newtonian potential.
perihelion of mercury
Which can be the easy explanation of the phenomena of perihelion shift? The physical interpretation is that space-time is dragged by the sun like a vortex drags everything around it. Mathematical interpretation is related to the period of revolution. If the perihelion would not shift around the ellipse, the total period would be exactly 2π. If the period exceeds two π the perihelion must go past the previous one. That is exactly what happens in case of the mercury's orbit.

Arcsecond

Arcseconds is a measure of angular displacement. One (1) arcsecond is 1/60 of an arcminute or 1/3600 of a degree.

Mathematics of general relativity

To understand the perihelion shift of mercury we need to understand mathematics of general relativity. It is somewhat complex and abstruse. However there are few fundamental concepts like metric tensor and curvature which play vital role in general relativity. Covariant derivative is useful concept which is defined in this way

precession of perihelion of mercury
Metric tensor is the most useful concept which has the following form:

precession of perihelion of mercury

Frame Dragging

Frame dragging is the gravitation analog of electromagnetism. In electromagnetism a moving charge creates electromagetic waves and in general relativity a moving or accelerated mass creates gravitational waves. Frame dragging is a particular example of this effect. When a test mass is placed around a massive rotating body then the test mass accompany the rotation of the massive object. In other words the orbit of the test body will precess. This does not happen in Newtonian gravity as the gravitational field does not depend on the rotation. The equations of gravito-magnetism can be easily developed using analogy of Maxwell's equations.


precession of perihelion of mercury

There is also a quantity called geodetic effect which measures the distortion of spacetime due to heavy mass like the earth. So measurement of spacetime curvature includes both the frame-dragging effect and geodetic effect. A spinning gyroscope can measure both effects around the earth.


precession of perihelion of mercury
In particular, Kerr developed a metric for a spinning body using Einstein's field equation. Using this equation the effect of frame dragging can be computed.

precession of perihelion of mercury
The general equation of motion when accounted for these effects of frame dragging and geodetic precession will be :

precession of perihelion of mercury
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"Every search for a hero begins with something that the hero requires- a monster or a villain.."

General theory of relativity has made a great impact in science and mathematics. But what is the significance of the theory from philosophical point of view?
General theory of relativity makes space-time truly dynamic. Geometry of space time is not static. It evolves with time and can be curved. Geometry has its own history. Time and space are totally flexible. According to this theory time can bend , stretch and warp. Based on this principle many scientific speculations have been made. Time can even form a closed loop. So journey into the past is possible. In the movie "intersteller" many technical things about worm hole and black hole have been shown. Worm hole is a short-cut way in spacetime to connect two distant regions. Time travel is possible through worm hole. Worm hole is a solution of Einstein field equation. This solution is also known as Einstein-Rosen bridge. The existence of worm hole has not been proven yet. The hypothetical picture of a worm hole or Einstein-Rosen bride would look something like this :

Einstein - Rosen Bridge
Newton's universal gravity does not account for all the phenomena. General theory of relativity is much more elegant than Newton's law. Here is a comparison:

Einstein - Newton comparison

The concept of the theory is extraordinarily simple. Here is how :
Suppose two person are at different hights from the ground. There will be space difference between them. This account for the curvature of spacetime. Spacetime diagram given below depicts the time difference in two reference frames as to the differences in height above the ground.
Spacetime curvature
The more the height is the more time deviates. The horizontan axis is time coordinate. This arguements not only works for places above the ground but also works for places below the ground. We should be careful when we talk about the age of the earth. The difference in time at the center of the earth and the surface can be quite significant in this case.
Theory of General relativity has given a concrete causal structure of the world. It is an example of great causal law. According to this theory, every region of space-time can be ascribed a metrical property which can be found by superposing different structures which are symmetric about the centers: centers being portions of the pieces of matter and given these structures, every piece of matter move in geodesics or rather are geodesics.
Now we come to space-time of electron. Every electron is associated with a crinkle which spread throughout the space around the electron. That is to say, the crinkle becomes less marked as we move away from the electron but theoretically extends throughout the space. The metrical structure of any region of space-time can be found by (roughly speaking) superposing these crinkles. The metrical properties( distance measurement) is nothing but a way to state causal laws. How an electron will move depends on the positions of other electrons. We must then suppose that the expression for interval in a place is the superposition of various spherically symmetrical formulae, each of which corresponds to an electron its central region. The whole theory reduces to these
1) we can recognize peculiar regions of the space-time as those which can be regarded as in the immediate neighborhood of matter
2) the interval is the function of geodesic distances from the place to the pieces of matter.
3) all pieces of matter move in geodesics.
Concerning geometry, the old geometry of Euclid assumed spac ewas static, because space and time were supposed to be separable. It is natural to think of motion as following a path in space which is there before and after the motion: a tram moves along pre-existing tram-lines. This view of motion, however , is no longer tenable. A moving point cannot pursue the "same" course , since its time coordinate is always changing, which means that , in any other equally legitimate system of coordinates, its space coordinates also will be different. We think of a tram as performing the same journey every day, because we think of the earth as fixed; but form the sun's point of view the tram never repeats a former journey. "we cannot step twice into the same river" as Heraclitus says. It is obvious that in place of Euclid'd static straight line, we shall have to substitute a movement having some special property defined in terms of space-time, not of space. The movement required is geodesic.

Deductive theory

The world of elementary physics is semi-abstract while that of deductive relativity theory is wholly abstract. The apearance of deducing actual phenomena from mathematics is very illusive, what really happens is that the phenomena afford inductive verification of the general principles from which our mathematics starts. Every observed fact retains its full evidential value, but now it confirms not only a particular law but general law from which the deductive system starts. There is no logical necessity for one fact to folow given another, or a number of others , because there is no logical necessity about our fundamental principles.
The question of interpretation , it must be admitted , is somewhat difficult when physics is concieved in this very abstract manner. What , for example, is ds? We start from a view which is , to a certain extent, intelligible in terms of observation. In the case of a time-like interval, it is the time which elapses between the two events according to a clock, with suitablle precautions, from the visual perception of an observer. In the case with space-like interval, the ds^(2) is the distance measured by an observer who is present at both events, and for which the two events appear simultaneous. The elementary operation of measurement is here supposed possible. But when we pass from this initial view to the abstract view which is required by the general theory of relativity, the intrval can only be actually estimated by using rather elaborate physicsto make deductions from what can actually observed by means of clocks and footrules. Fr logical theory the nterval is primitive, but from the point of view of empirical verification , it is a complicated function of empirical data, deduced from physics from its semi abstract form. The unity and simplicity of deductive edifice , therefore, must not blind us to the complexicity of empirical physics, or to the logical independence of its various parts.
What is novel and interesting in the point of view we are considering is the relation between deductive and empirical physics. But there is no real diminution of the need of empirical observation. I do not for a moment suggest that anything in the above is a criticism of Professor Eddington; indeed , I imagine he would regard it as a string of truisms. I have been concerned only to guard against a possible misunderstaanding on the part of those who do not feel for Mathematics the contempt which is bred of familiarity.

Abstractness of physics

There are abstracted in physics, which modern physics have proven repeatedly. What is matter? We can explain it using theory of relativity or Quantum mechanics. This will lead us to conclude that matter is an attribution an of certain abstract properties. If we continue to divide matter we will see that what remains is formless and immaterial substance. Electron is made up of quarks , quarks are also made up with other sub atomic particles which can be devided further. According to string theory quarks are made up with strings or vibrating filament. This certainly obliges us to accept that matter ultimately consists of things which should not be regarded as matter.
Quantum field theory states there are fields which are everywhere and endless. This fields are the description of sub atomic particles. There are electron field, quarks field, neutrino field and other particle fields. The oscillation of quantum field creates this kind of particles. There is higgs field which permeates everywhere. Higgs field provides mass to particles by interaction. Field is nothing but some property of spacetime.
Mathematics that describes the particles in standard model is very abstract. There is group theory which characterizes the symmetry of the law governing particle physics. Sub atomic particles can be grouped together with certain properties like isospin and hypercharge. This group describes certain transformation laws. This is called symmetry group. This symmetry implies certain kind of conservation of physical properties. For example a phase shift in the quantum wave function correspond to the conservation of charge. In short , symmetry is the properties of physical system which leaves the system unchanged under certain kinds of transformations. Thus the standard model can be modeled with SU(3)xSU(2)xU(1) , which leave the symmetry invariant. This unified model is the combination of three symmetry groups SU(3) , SU(2) and U(1) which describe strong interaction, weak interaction and electromagnetic forces respectively. The symmetry shown by strong interaction can be explained as follows:
The strong force is carried by three quarks of different colors (red, yellow and green). The interaction between two identical quarks (red to red or yellow to yellow) are the same. On the other hand the interaction between two different colors are also the same. In addition if we shift the three different quark (from red, yellow, green to orange, violet, blue), even if the shift involves a change in space and a change in time , everything remains the same. Thus strong interaction shows some kind of symmetry. Physicists call it the "gauge symmetry" of particle physics. The word "gauge" came from the concept "gauge transformations" coined by Hermann Weyl. You can follow here for more explanation about well theory.
SU(n) is special unitary group whose elements are matrices. Each of these matrices is a representation of certain other group elements. This special unitary group elements represent certain transformation laws. The determinant of this SU(n) is one. That is why its name is special unitary. More explanation needs analysis of group theory and representation theory.
A particle can be an element of a vector space, upon which an element of the above group acts. Through this action another element of the same vector space can be found, which can describe another particle. This is the core idea behind developing standard model.
Physics can be categorized in three different branches: one - corpuscular physics, touch physics and sight physics. Touch physics is what we regard as those events happening to the billiard balls. From the common sense notion it could be noticed that the billiard balls touch each other when they collide or come into contact. But from the standpoint of physics those billiard balls never touch each other. The repulsive force between the electrons prevent them from touching. According to new quantum theory this force, however , is interpreted as exchange of other sub atomic particles.


Reference materials:


Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking

Russell's account

Addition and Multiplication

In most mathematical accounts of arithmetical operations we find the error of endeavouring to give at once a definition which shall be applicable to rationals, or even to real numbers, without dwelling at sufficient length upon the theory of integers. For the present, integers alone will occupy us. The definition of integers, given in the preceding chapter, obviously does not admit of extension to fractions; and in fact the absolute difference between integers and fractions, even between integers and fractions whose denominator is unity, cannot possibly be too strongly emphasized. What rational fractions are, and what real numbers are, I shall endeavour to explain at a later stage; positive and negative numbers also are at present excluded. The integers with which we are now concerned are not positive, but signless. And so the addition and multiplication to be defined in this chapter are only applicable to integers; but they have the merit of being equally applicable to finite and infinite integers. Indeed, for the present, I shall rigidly exclude all propositions which involve either the finitude or the infinity of the numbers considered.
113. There is only one fundamental kind of addition, namely the logical kind. All other kinds can be defined in terms of this and logical multiplication. In the present chapter the addition of integers is to be defined by its means. Logical addition, as was explained in Part I, is the same as disjunction; if p and q are propositions, their logical sum is the proposition “p or q”, and if u and v are classes, their logical sum is the class “u or v”, i.e. the class to which belongs every term which either belongs to u or belongs to v. The logical sum of two classes u and v may be defined in terms of the logical product of two propositions, as the class of terms belonging to every class in which both u and v are contained.* This definition is not essentially confined to two classes, but may be extended to a class of classes, whether finite or infinite. Thus if k be a class of classes, the logical sum of the classes composing k (called for short the sum of k) is the class of terms belonging to every class which contains every class which is a term of k. It is this notion which underlies arithmetical addition. If k be a class of classes no two of which have any common terms (called for short an exclusive class of classes), then the arithmetical sum of the numbers of the various classes of k is the number of terms in the logical sum of k. This definition is absolutely general, and applies equally whether k or any of its constituent classes be finite or infinite. In order to assure ourselves that the resulting number depends only upon the numbers of the various classes belonging to k, and not upon the particular class k that happens to be chosen, it is necessary to prove (as is easily done) that if k' be another exclusive class of classes, similar to k, and every member of k is similar to its correlate in k' , and vice versâ, then the number of terms in the sum of k is the same as the number in the sum of k' . Thus, for example, suppose k has only two terms, u and v, and suppose u and v have no common part. Then the number of terms in the logical sum of u and v is the sum of the number of terms in u and in v; and if u' be similar to u, and v' to v, and u' , v' have no common part, then the sum of u' and v' is similar to the sum of u and v. 114. With regard to this definition of a sum of numbers, it is to be observed that it cannot be freed from reference to classes which have the numbers in question. The number obtained by summation is essentially the number of the logical sum of a certain class of classes or of some similar class of similar classes. The necessity of this reference to classes emerges when one number occurs twice or more often in the summation. It is to be observed that the numbers concerned have no order of summation, so that we have no such proposition as the commutative law: this proposition, as introduced in Arithmetic, results only from a defective symbolism, which causes an order among the symbols which has no correlative order in what is symbolized. But owing to the absence of order, if one number occurs twice in a summation, we cannot distinguish a first and a second occurrence of the said number. If we exclude a reference to classes which have the said number, there is no sense in the supposition of its occurring twice: the summation of a class of numbers can be defined, but in that case, no number can be repeated. In the above definition of a sum, the numbers concerned are defined as the numbers of certain classes, and therefore it is not necessary to decide whether any number is repeated or not. But in order to define, without reference to particular classes, a sum of numbers of which some are repeated, it is necessary first to define multiplication.
This point may be made clearer by considering a special case, such as 1 + 1. It is plain that we cannot take the number 1 itself twice over, for there is one number 1, and there are not two instances of it. And if the logical addition of 1 to itself were in question, we should find that 1 and 1 is 1, according to the general principle of Symbolic Logic. Nor can we define 1 + 1 as the arithmetical sum of a certain class of numbers. This method can be employed as regards 1 + 2, or any sum in which no number is repeated; but as regards 1 + 1, the only class of numbers involved is the class whose only member is 1, and since this class has one member, not two, we cannot define 1 + 1 by its means. Thus the full definition of 1 + 1 is as follows: 1 + 1 is the number of a class w which is the logical sum of two classes u and v which have no common term and have each only one term. The chief point to be observed is, that logical addition of classes is the fundamental notion, while the arithmetical addition of numbers is wholly subsequent.

115. The general definition of multiplication is due to Mr A. N. Whitehead.* It is as follows. Let k be a class of classes, no two of which have any term in common. Form what is called the multiplicative class of k, i.e. the class each of whose terms is a class formed by choosing one and only one term from each of the classes belonging to k. Then the number of terms in the multiplicative class of k is the product of all the numbers of the various classes composing k. This definition, like that of addition given above, has two merits, which make it preferable to any other hitherto suggested. In the first place, it introduces no order among the numbers multiplied, so that there is no need of the commutative law, which, here as in the case of addition, is concerned rather with the symbols than with what is symbolized. In the second place, the above definition does not require us to decide, concerning any of the numbers involved, whether they are finite or infinite. Cantor has given† definitions of the sum and product of two numbers, which do not require a decision as to whether these numbers are finite or infinite. These definitions can be extended to the sum and product of any finite number of finite or infinite numbers; but they do not, as they stand, allow the definition of the sum or product of an infinite number of numbers. This grave defect is remedied in the above definitions, which enable us to pursue Arithmetic, as it ought to be pursued, without introducing the distinction of finite and infinite until we wish to study it. Cantor’s definitions have also the formal defect of introducing an order among the numbers summed or multiplied: but this is, in his case, a mere defect in the symbols chosen, not in the ideas which he symbolizes. Moreover it is not practically desirable, in the case of the sum or product of two numbers, to avoid this formal defect, since the resulting cumbrousness becomes intolerable. 116. It is easy to deduce from the above definitions the usual connection of addition and multiplication, which may be thus stated. If k be a class of b mutually exclusive classes, each of which contains a terms, then the logical sum of k contains a × b terms.* It is also easy to obtain the definition of ab, and to prove the associative and distributive laws, and the formal laws for powers, such as abac = ab + c. But it is to be observed that exponentiation is not to be regarded as a new independent operation, since it is merely an application of multiplication. It is true that exponentiation can be independently defined, as is done by Cantor,† but there is no advantage in so doing. Moreover exponentiation unavoidably introduces ordinal notions, since ab is not in general equal to ba. For this reason we cannot define the result of an infinite number of exponentiations. Powers, therefore, are to be regarded simply as abbreviations for products in which all the numbers multiplied together are equal. From the data which we now possess, all those propositions which hold equally of finite and infinite numbers can be deduced. The next step, therefore, is to consider the distinction between the finite and the infinite.

Vector fields and one-forms

There is a notion of ‘derivative’ of a function that is independent of the coordinate choice. A standard notation for this, as applied to the function φ defined on S, is dφ, where
vector one-form
When I write ‘dφ’ in the displayed formula above, on the other hand, I mean a certain kind of geometrical entity that is called a 1-form. 1-form is not an ‘infinitesimal’; it has a somewhat diVerent kind of interpretation, a type of interpretation that has grown in importance over the years, and I shall be coming to this in a moment. Remarkably, however, despite this signiWcant change of interpretation of ‘d’, the formal mathematical expressions—provided that we do not try to divide by things like dx—are not changed at all. There is also another issue of potential confusion in the above displayed formula, which arises from the fact that I have used F on the left-hand side and f on the right. I did this mainly because of the warnings about the distinction between φ and f that I issued above. The quantity φ is a function whose domain is the manifold S, whereas the domain of f is some (open) region in the (x, y)-plane that refers to a particular coordinate patch. If I am to apply the notion of ‘partial derivative with respect to x’, then I need to know what it means ‘to hold the remaining variable y constant’. It is for this reason that f is used on the right, rather than F, because f ‘knows’ what the coordinates x and y are, whereas F doesn’t.
Even so, there is a confusion in this displayed formula, because the arguments of the functions are not mentioned. The F on the left is applied to a particular point p of the 2-manifold S, while f is applied to the particular coordinate values (x, y) that the coordinate system assigns to the point p. Strictly speaking, this would have to be made explicit in order that the expression makes sense. However, it is a nuisance to have to keep saying this kind of thing, and it would be much more convenient to be able to write this formula as
vector one-form
I am going to try to make sense of these things. These formulae are instances of something referred to as the chain rule.
All this has probably caused the reader great confusion! However, my purpose is not to confuse, but to Wnd the right analytical form of a very basic geometrical notion. The diVerential operator j, which we have called a ‘vector Weld’, with its (consequent) very speciWc way of transforming, as we pass from patch to patch, has a clear geometrical interpretation, as
vector one-form
Taking this function to be φ, the action of ζ on φ, namely
vector one-form
, measures the rate of increase of φ in the direction of the arrows; We are now in a position to interpret the quantity dφ. This is called the gradient (or exterior derivative) of φ, and it carries the information of how φ is varying in all possible directions along S. A good geometrical way to think of dF is in terms of a system of contour lines on S. We can think of S as being like an ordinary map (where by ‘map’ here I mean the thing made of stiff paper that you take with you when you go hiking, not the mathematical notion of ‘map’), which might be a spherical globe, if we want to take into account that S might be a curved manifold.
vector one-form

vector one-form

What metric can do for you

Up to this point, we have been considering that the connection = has simply been assigned to our manifoldM. This providesMwith a certain type of structure. It is quite usual, however, to think of a connection more as a secondary structure arising from a metric defined on M. Recall from that a metric (or pseudometric) is a non-singular symmetric [ 0 2 ]-valent tensor g. We require that g be a smooth tensor field, so that g applies to the tangent spaces at the various points of M. A manifold with a metric assigned to it in this way is called Riemannian, or perhaps pseudo- Riemannian. He originated this concept of an n-dimensional manifold with a metric, following Gauss’s earlier study of ‘Riemannian’ 2-manifolds.) Normally, the term ‘Riemannian’ is reserved for the case when g is positive-definite . In this case there is a (positive) measure of distance along any smooth curve, defined by the integral of ds along it, where
line element
This is an appropriate thing to integrate along a curve to deWne a length for the curve—which is a ‘length’ in a familiar sense of the word when g is positive definite. Although ds is not a 1-form, it shares enough of the properties of a 1-form for it to be a legitimate quantity for integration along a curve. The length ‘ of a curve connecting a point A, to a point B is thus expressed as:
line element
It may be noted that, in the case of Euclidean space, this is precisely the ordinary definition of length of a curve, seen most easily in a Cartesian coordinate system, where the components gab take the standard ‘Kronecker delta’ form of (i.e. 1 if a = b, and 0 if a != b). The expression for ds is basically a reflection of the Pythagorean theorem (), but operating at the infinitesimal level. In a general Riemannian manifold, however, the measure of length of a curve, according to the above formula, provides us with a geometry which differs from that of Euclid. This reXects the failure of the Pythagorean theorem for Wnite (as opposed to infinitesimal) intervals. It is nevertheless remarkable how this ancient theorem still plays its fundamental part—now at the infinitesimal level.
precession of the perihelion of mercury
We shall be seeing that the case of signature +--- has particular importance in relativity, where the (pseudo) metric now directly measures time as registered by an ideal clock. Also, any vector V has a length |V|, defined |v|^2 = g(ab)V(a)V(b) ..
How does a non-singular (pseudo)metric g uniquely determine a torsion-free connection ∇? One way of expressing the requirement on ∇ is simply to say that the parallel transport of a vector must always preserve its length (a property that I asserted that for parallel transport on the sphere S2). Equivalently, we can express this requirement as
∇g = 0;
This condition (together with the vanishing of torsion) suffices to fix ∇ completely. This connection ∇ is variously termed the Riemannian, Christoffel, or Levi-Civita connection (after Bernhardt Riemann (1826–66), Elwin ChristoVel (1829–1900), and Tullio Levi-Civita (1873–1941), all of whom contributed important ideas in relation to this notion).

Mathematical Idea of a bundle

A bundle (or Wbre bundle) B is a manifold with some structure, which is deWned in terms of two other manifolds M and V, where M is called the base space (which is spacetime itself, in most physical applications), and where V is called the Wbre (the internal space, in most physical applications). The bundle B itself may be thought of as being completely made up of a whole family of Wbres V; in fact it is constituted as an ‘M’s worth of Vs’—see Figure below. The simplest kind of bundle is what is called a product space. This would be a trivial or ‘untwisted’ bundle, but more interesting are the twisted bundles. I shall be giving some examples of both of these in a moment. It is important that the space V also have some symmetries. For it is the presence of these symmetries that gives freedom for the twisting that makes the bundle concept interesting. The group G of symmetries of V that we are interested in is called the group of the bundle B. We often say that B is a G bundle over M. In many situations, V is taken to be a vector space, in which case we call the bundle a vector bundle. Then the group G is the general linear group of the relevant dimension, or a subgroup of it.


precession of the perihelion of mercury

to M, called the canonical projection from B to M, which collapses each entire Wbre V down to that particular point of M which it stands above.
The product space of M with V (trivial bundle of V over M) is written MXV. The points of MXV are the pairs of elements (a, b), where a belongs to M and b belongs to V; see Figure below. A more general ‘twisted’ bundle B, over M, resembles MXV locally, in the sense that the part of B that lies over any sufficiently small open region of M, is identical in structure with that part of MXV lying over that same open region of M. See Fig. 15.3b. But, as we move around inM, the Wbres above may twist around so that, as a whole, B is different (often topologically diVerent) from MXV. The dimension of B is always the sum of the dimensions of M and V, irrespective of the twisting


precession of the perihelion of mercury

(a) The particular case of a ‘trivial’ bundle, which is the product space MXV of M with V. The points of MXV can be interpreted as pairs of elements (a,b), with a in M and b in V. (b) The general ‘twisted’ bundle B, over M, with Wbre V, resembles MXV locally—i.e. the part of B over any suYciently small open region of M is identical to that part of MXV over same region of M. But the Wbres twist around, so that B is globally not the same as MXV.
All this may well be confusing, so get a better feeling for what a bundle is like, let me give an example. First, take our space M to be a circle S1, and the fibre V to be a 1-dimensional vector space (which we can picture topologically as a copy of the real line R, with the origin 0 marked). Such bundle is called a (real) line bundle over S1. Now MXV is a 2-dimensional cylinder; How can we construct a twisted bundle B, over M,


precession of the perihelion of mercury

perihelion of mercury by Feynman
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