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tensor calculus

Theory of relativity

Special theory of relativity   |   General theory of relativity   |   Tensor calculus

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Tensor calculus

Tensor calculus was first developed to aid the development of Einstein's general theory of relativity. Since then it has been a separate branch of applied mathematics. Tensors are multi-linear maps. That is to say, they are linear in many variables and are functions of them just like vectors are also functions. They are functions of coordinates variable. As an example, metric tensor is a function of coordinates. Riemann tensor is also function of coordinates. Tensors transform in special way under change of coordinates. Application of calculus to tensor gives different resutls than scalars and vectors. Tensor calculus not only involves calculus but also linear algebra. Tensor field is tensor valued function defined on a general manifold. Manifold is a generalized space of which Euclidean space is a special case. In each point in a manifold tensor field assigns a tensor quantity like vector field assigns a vector at each point in space. A special example of tensor field is metric tensor which assigns a tensor to measure distance at each point in a manifold. For example on the surface of a sphere a metric tensor has different value at different point.

The method of tensors

The methos of tensors contains the answer to the question which is rendered urgent by the arbitrary character of our co-ordinates. How can we know whether a formula expressed in terms of coordinates expresses something which describes the physical occuraences, and do not merely the particular coordinate system which we happen to be employing? A striking example of the mistakes that are possible in this respect is afforded by simultaneity. Suppose we have two events, whose coordinates , in the system we are employing , are (x,y,z,t) and (x`,y`,z`,t) - i.e their time coordinates are the same. Before the special theory of relativity everybody would have asserted that this represented a physical fact about the two events-namely, that they are simultaneous. But this is to speak the language od special theory of relativity. In general theory, our coordinates may have no important physical significance, and a pair of events which have one coordinate identical need not have any intrinsic physical property not possessed by any other pair of events. In practice, there must be some principle on which coordinates are assigned, and this principle must have some physical significance. But we might, for instance ,measure time by the worst clock ever made, provided it only went wrong and did not stop. And we might use a certain worm as our unit of length, disregarding the "Fritzerald contraction" to which motion subjects him. In that case, if we say that there was unit distance between two events which both occurred at a certain instant, we shall be making complicated comparison between the events, a bad clock and the worm- that is to say, we shall be making a statement which depends opon our coordinate system. We want to discover a sufficient, if not necessary, condition , if fulfilled, then statements expressed in terms of coordinates have meaning independent of them. This is what method of tensors is concerned with. This convenience of specifying coordiantes has made it possible to express all general laws of physics in the language of tensors. If it can not be done the laws of physics must be wrong, and must require such correction as will enable it to be expressed as a tensor equation. The law of gravitation is a noteworthy example of this; but perhaps the conservation of energy is scarcely less noteworthy. It seems natual to suppose that it would be possible to develop a less direct method of expressing physical laws that afforded by the method of tensors, which is perhaps a consequence of the historical development of physics.
It is also natural to ask could we dispense with coordinates altogether, since they have become little more than conventional names systematically assigned? Perhaps this will become possible in time, but at present the necessary mathematics is lacking. We sish , for example, to be able to differentiate a function, and we can not differential a function unless its arguments and values are numbers. This is not dueto what might seem the more difficult parts of the definition of a differential. We can define for a non-numerical function the limit( if it exists) of a function for a given arguments, and also the four limits which exist more frequently- viz, the maximum and the minimum for approaches from above and from below; we can also define a "continuous" non-numerical function. What , so far, has not been developed , except for numbers, is a fraction. Now dy/dx is the limit of a fraction; thus although we can generate the notion of a limit, we cannot at present generalize dy/dx because we can not generalize the notion of a fraction. It seems clear a priori that, since differentiation of co-ordinates is physically useful even when the quantitative value of the coordinates is conventional, there must be some process, of which differentiation is a special numerical form, which can be applied wherever we have continuous functions, even when they are non-numerical. To define such a process is a problem in mathematical logic; probably subtle, but hitherto unsolved. If it were solved, it might become possible to avoid the elaborate and round-about process of assigning coordinates and then treating almost all their properties as irrelevant, which is what is done when method of tensors is employed.
There are, it is true, , certain numbers which are important in the new geometry: they are those giving the measure of intervals. But , as we have already seen, two points at a finite distance apart do not have an unambiguous interval; and any two points are at a finite distance apart. The numbers derivable from the sisteen coefficients g(uv) involved in the formula for ds^2 in the general theory of relativity. These coefficient themselves depend upon the coordinate system but ds^2 does not. We cannot develop this theme until we have considered geodesics: it is from then that we must derive the numbers which have, in the new geometry the same sort of physical importance as co-ordinates were supposed to have in the old. These numbers will be the integals of ds taken along certain geodesics. But unlike lengths in the old metrical geometry, they are geometrically insufficient. To avoid irrelevant complications, we may illustrate this insufficiency by considering the special theory.
Special theory
The most obvious example of the failure of interval to constitute geometry is derived from the consideration of light rays. The interval between two events which are parts of the same light rays is zero. Suppose now that a light-ray starts from an event A, and then arrives at an event B; at the same moment when it reaches B, another light-ray starts from B and reaches C. Then the interval between A and B is zero, that between B and C is zero; but that between A and C may have any time-like magnitude. Euclid proved that two sides of a triangle ar together greater than the third side, and was criticized on the ground that this proposition was evident even to asses. But in relativity geometry this proposition is false. In our triangle ABC , AB and BC are zero, while AC may have any finite magnitude.
Again events which are parts of a single light ray have a definite time-order , in spite ofthe fact that the interval between any two of them is zero. This appears as follows: Suppose a light-ray proceeds from the sun to the moon and is thence reflected to the earth: it reaches the earth later than a direct ray which left the sun at the same time. There is therefore a definite sense in saying that the ray reached the moon later than it left the sun-i.e we can say that the ray went from the sun to the moon, not from the moon to the sun. Generalizing , we may say: if A and B are part of one light-ray, and light-rays from A and Bm distinct from the previous light ray, contains event C, C` is the the same whatever these new light-rays may be-i.e we shall have always C before C`, or always C` before C. In the first case, we say that the "sense" of the ray is from A to B; in the second from B to A. This illustrates the difficulties which would arise if we were to attempt to found our geometry on interval alone. We must also take account of the purely ordinal properties of the space-time manifold. These properties give a wide separation between the departure of a light ray from the sun and its arrival on the earth, although the "interval" between these tiwo events is zero.
Reverting now to the method of tensors and its possible eventual simplification, it seems probable that we have an example of a general tendency to over-emphasize numbers, which has existed in mathematics ever since the time of Pythagoras , though it was temporarily less prominent in later Greek geometry as exemplified in euclid. Euclid's theory of proportion does not , of course, dispense with numbers, since it uses "equimultiples"; but at any rate it requires only integers, not irrationals. Owing to the fact that arithmetic is easy, Greek methods in geometry have been in the back-ground since Descartes, and coordinates have come to seem indespensable. But mathematical logic has shown that number is logically irrelevant in many problems where it formerly seemed essential, notably mathematical induction, limits and continuity. A new technique, which seems difficult because it is unfamiliar, is required when numbers are not used; but there is a compensating gain in logical purity. It should be possible to apply a similar process of purification to physics. The method of tensors first assigns coordinates , and then shows how to obtain results which , though expressed in the language of tensors, not such as depend on the choice of co-ordinates. I do not say that such a method, if discovered , would be preferable in practice, but I do say that it would give a better expression of the essential relations, and greatly facilitate the task of the philosopher. In the meantime, the method of tensors is technically delightful, and suffices for mathematical needs.

Tensor as a function

Vectors and tensors are functions. Tensors take several vetors as inputs and outputs a scalar.

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Tensor analysis

Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro. It was used by Albert Einstein to develop his theory of general relativity. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensors are matrices but they can be multi-dimensional array of numbers with an additional property that these numbers obey certain transformation rules. The role of tensor lies in the fact that physical laws written in tensor language remains valid everywhere. Laws of physics should not change from place to place in the physical universe. That is if some quantities are equal in one coordinate frame they should also be equal in any other coordinate frame. For example vectors are quantities which retains its properties (magnitude and direction) no matter which coordinate system is used. We can have expressions for them , which are free of coordinates but when we have relations between any two such expressions, coordinates need to be specified. The dependence of coordinates can not be avoided. These relations are called tensors.
Tensor can be explained in another way : we saw that coordinates describe certain relations appearing in the expressions of vectors and tensors. But the way coordinates enter into them makes the truth of the expressions independent of them. The difference is comparable to that between linguistic statements and statements about what the words mean. Suppose you want to translate the sentence "human is the most intellectual species on earth" into French or German. You can easily do it without altering the truth of the sentence. But if you have a sentence like "strength is a word which contains seven syllables and one vowel" and you want to translate it into French , you can not do it easily without altering the truth of the sentence. In tensor analysis coordinates play the role of words except that linguistic statements are harder to distinguish than others. Tensor accomplishes this task.
Coordinates are numbers that are needed to fix a point in space or event in spacetime. A collection of coordinates called 'n-tuple' forms what is known as coordinate system. Almost all the elementary concepts of physics are dependent on coordinates system. Theory of electromagnetism, Newtonian physics are examples of those. As I mentioned tensors , a simple mathematical representation of tensors can be given :

tensor calculus


Covariant and contravariant tensor

The components of tensor (A[rs]) in the left hand side are referred to a certain reference frame and in right hand side the components(A[jk]) in another reference frame are related to the components of left hand side by certain transformation rules. The coordinates on the numerator are functions of coordinates on the denominator. These components (A[rs]) form the tensor of rank two. It has dimension 3 indicated by the value of the indices j and k. In relativity theory j and k has 4 values (1,2,3,4). Corresponding coordinates are x, y, x and t. The repeated indices j and k on the right hand side is summed over by convention. This tensor is called covariant tensor. Tensors played important role in formulating natural laws. Everybody has to agree on the results obtained by tensor equations. This is called general covariance which motivated Einstein to formulate his theory of General Relativity. All the equations of general relativity are generally covariant.
It seems natural to suppose that it would be possible to develop a less indirect method of expressing physical lawsthat that afforded by the method of tensors, which is perhaps a consequences of historical development of physics. Originally , in physics, the coordinates were intended to express physical relations between the events concerned and the origin. Three of the coordinates were intended to express physical length , which it was thought, could be ascertained by measurement with a rigid rod. The fourth , time can be measured by a chronometer. There were difficulties , which the progress of physics made evident. So long as the earth could be regarded motionless , axes fixed relatively to the earth and clocks which remains on the surface of the earth seem to suffice. It was possible to disregard the fact that nobody is quite rigid and no clock is quite accurate , because the system of physical laws suggested by the choice of most rigid bodies and most accurate clock can be used to estimate the departure from these instruments from strict constancy , and the results were on the whole self-consistent. But in astronomical problems, including that of the tides , the earth could not be treated as fixed. It was necessary to Newtonian dynamics that the axes should not have any acceleration, but it resulted from law of gravity that any material axes must have some acceleration. The axes, thus, become ideal structures in absolute space; actual measurement carried out by any rigid road could only approximate to the results which would have if we used any unaccelerated axes. This difficulty was not the most serious ; the worst trouble was concerned with absolute acceleration. Then came the experimental discovery of the facts which led to the special theory of relativity: the variation of length and mass with velocity, and the constancy of the velocity of light in vacua no matter what body was used to define the co-ordinates. This set of difficulties was solved by special theory of relativity, which showed that equivalent results come from employing as reference-body any set of bodies in uniform rectilinear motion.

tensors


The tensor above is a contra -variant rank two tensor. It has special transformation properties as given in the equation. The differentials in the denominator and numerator are reversed compared to that in the covariant one described before. We can also form a mixed tensor with both covariant and contravariant components. Different component in such tensor will transform differently. An example is given below:

tensor calculus

There is no physical distinction between covariant and contravariant tensors. They both contain the same physical content. It is only the interpretation that are different. As the transformation is defined as the infinitesimal displacement , we need not worry about curvature of space. In case of two displacement, the difference can be spotted easily. When the axes do not intersect orthogonally, there is only two way to project the vector onto the axes : either parallel or vertical. Covariant and contravariant component of a vectore can be geometrically interpreted as the component that is parallel projection to axis and component that is perpendicular projection to axis.


covariant tensors

When the axes are orthogonal, contra-variant and covariant projections becomes identical.

The above tensor is type (1,2) tensor. So why are tensors so important in physics or mathematics ? It is actually an entity studied under linear algebra. In linear algebra , it can have any rank and thus can be more generalized form of vectors or scalars. In physics , it has became a necessity to use tensor. Many physical quantities needs more than one direction to be specified. For example stress is a tensor quantity which has two directions : one is normal to a surface and other is tangential to the surface. The former is called normal stress and the latter is called shear stress. Mechanical engineers are used to it more than others. It is this tensor that attracted Einstein to use it in his theory. He just included a time dimension to it and make it relativistic. We will come to that when discussing the theory of relativity in more details.

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The notion of covariant derivative can be developed by taking the derivative of vector function which takes one or more value as argument and returns a vector.


tensor calculus
The covariant derivative is the summation of usual derivative and a Christoffel symbol coupled term. If we assign a vector to each direction in space we get a tensor of second order. Such tensor's components are labelled by pair(i,j).
tensor calculus
Stress is a second rank tensor. In three dimensions it has nine components which Einstein generalized in four dimensions including time. The generalized tensor is known as Stress-energy tensor, which acts as a source of graviational field.
tensors

The tensor consists of nine components {σ{ij} that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n:

tensors

Tensor product is defined as the bilinear map that takes elements from two vector spaces and produces element of a third vector space. It does, in this process, generalises outer product of vectors.
tensor calculus

Tensor product can be formed with a vector space and its dual as well. In that case we get what is called mixed tensors each of which is itself an element of another vector space. So tensor product is generally , a composition of vector spaces over a field F. In short, it maps cartesian product of two vector space to another vector space, that generalizes vector outer product. Outer product of two vectors of rank n and m is the matrix nxm.


tensor calculus
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Levi cevita tensor

Levi cevita tensor is a rank two tensor. It is often called permutation symbol, alternative symbol. It is anti-symmetric tensor.
 le cevita tensors
so it is a collection of 2x2=4 number s in two dimensions. It can be generalized in higher dimensions also.

Invariant theory

Tensors and tensor equations are invariant quantity. Vector and tensor , as we have seen do not change when we change coordinates or coordinate system. The components of tensors change but a linear combination of them remain the same. We begin with simple invariant quantity ds^2.

ds^2 = ds.ds= g(11)dx(1)^2 + g(12)dx(1)dx(2) + ......+ g(44)dx(4)^2 .

The above quantity ds^2 is a dot product vector ds with itself. It is the spacetime interval which remain invariant or same no matter whichever coordinates are used. The components g(ik) change but ds^2 remains unchanged.

Generalizing divergence of a vector, we can form divergence of any tensor using the covariant derivative of it. Taking a tensor A(uv) , divegence of it has four components :
[A(1v)]1 + [A(2v)]2 + [A(3v)]3 + [A(4v)]4 where (v = 1, 2, 3, 4) and
[A(1v)]1 = d/dx(1) A(1v) + [a1,1]A(au) - [u1,a]A(1u) Where [ab,c] is the Christoffel symbol.
We can get Einstein's tensor from Riemann's curvature tensor.
Riemann curvature tensor is G(uv) = d/dx([uv,a]) - d/dx([uv,a]) + [uv,a][uv,a] - [uv,a][uv,a]
Divergence of Einstein tensor's is identically zero. This is known as the "fundamental theorem of mechanics". So we can say
[G(uv) - (1/2)g(uv)G],a = 0.
The mass and momentum copnservation law, on the other hand, says that divergence of Stress-energy tensor is identically zero. So
[T(uv)],a = 0 .
Equating to zero both quantities, we can get field equation of Einstein's theory of general relativity
G(uv) - (1/2)g(uv)G = KT(uv) [where K is 8π/c^4 ]

General relativity with tensor calculus

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric formulation of gravitation published by Albert Einstein in 1915 and the current explanation of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, affording a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly connected to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations. Without the method of tensor calculus GR formulation would not have been possible
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay. The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are warped in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is abundance evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; for example, microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holes, respectively. The bending of light ray by gravity can lead to the phenomenon of gravitational lensing, in which multiple mirror images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.


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Notes and additional comments

A field F is a set endowed(equip) with some operations(addition, multiplication). The difference between group and field is that group has one operation while field has two operations. Field has to satisfy certain axioms and so the groups. Much of the mathematics and its related concepts depend on the axioms and propositions. In the past mathematics was unable to answer much of quesions and properties regarding mathematical objects. Now mathematics is capable of giving answers. Mathematics is not only a language but it is reasoning itself. The essense of reasoning is logic. I can recite in Russell's word : Unlike we are very mistaken all mathematics are logical deduction from premises and axioms.

One of the most difficult matters in all controversy is to distinguish disputes about words from disputes about facts: it ought not to be difficult, but in practice it is. This is quite as true in physics as in other subjects. In the seventeenth century there was a terrific debate as to what 'force' is; to us now, it was obviously a debate as to how the word 'force' should be defined, but at the time it was thought to be much more. One of the purposes of the method of tensors, which is employed in the mathematics of relativity, is to eliminate what is purely verbal (in an extended sense) in physical laws. It is of course obvious that what depends on the choice of co-ordinates is 'verbal' in the sense concerned. A person punting walks along the boat, but keeps a constant position with reference to the river-bed so long as she or he does not pick up the pole. The Lilliputians might debate endlessly whether the punter is walking or standing still; the debate would be as to words, not as to facts. If we choose co-ordinates fixed relatively to the boat, the punter is walking; if we choose co-ordinates fixed relatively to the river-bed, the punter is standing still. We want to express physical laws in such a way that it shall be obvious when we are expressing the same law by reference to two different systems of co-ordinates, so that we shall not be misled into supposing we have different laws when we only have one law in different words. This may be accomplished by the method of tensors. Some laws which seem plausible in one language cannot be translated into another; these are impossible as laws of nature. The laws that can be translated into any co-ordinate language have certain characteristics: this is a substantial help in looking for such laws of nature as the theory of relativity can admit to be possible. Of the possible laws, we choose the simplest one which predicts the actual motion of bodies correctly: logic and experience combine in equal proportions in obtaining this expression. But the problem of arriving at genuine laws of nature is not to be solved by the method of tensors alone; a good deal of careful thought is wanted in addition. Some of this has been done; much remains to be done.
Laws of motion
The present chapter will adopt, for the moment, a naïve attitude towards Newton’s Laws. It will not examine whether they really hold, or whether there are other really ultimate laws applying to the ether; its problem is merely to give those laws a meaning. The first thing to be remembered is—what physicists now-a-days will scarcely deny—that force is a mathematical fiction, not a physical entity. The second point is that, in virtue of the philosophy of the calculus, acceleration is a mere mathematical limit, and does not itself express a definite state of an accelerated particle. It may be remembered that, in discussing derivatives, we inquired whether it was possible to regard them otherwise than as limits—whether, in fact, they could be treated as themselves fractions. This we found impossible. In this conclusion there was nothing new, but its application in Dynamics will yield much that is distinctly new. It has been customary to regard velocity and acceleration as physical facts, and thus to regard the laws of motion as connecting configuration and acceleration. This, however, as an ultimate account, is forbidden to us. It becomes necessary to seek a more integrated form for the laws of motion, and this form, as is evident, must be one connecting three configurations. 456. The first law of motion is regarded sometimes as a definition of equal times. This view is radically absurd. In the first place, equal times have no definition except as times whose magnitude is the same. In the second place, unless the first law told us when there is no acceleration (which it does not do), it would not enable us to discover what motions are uniform. In the third place, if it is always significant to say that a given motion is uniform, there can be no motion by which uniformity is defined. In the fourth place, science holds that no motion occurring in nature is uniform; hence there must be a meaning of uniformity independent of all actual motions—and this definition is, the description of equal absolute distances in equal absolute times. The first law, in Newton’s form, asserts that velocity is unchanged in the absence of causal action from some other piece of matter. As it stands, this law is wholly confused. It tells us nothing as to how we are to discover causal action, or as to the circumstances under which causal action occurs. But an important meaning may be found for it, by remembering that velocity is a fiction, and that the only events that occur in any material system are the various positions of its various particles. If we then assume (as all the laws of motion tacitly do) that there is to be some relation between different configurations, the law tells us that such a relation can only hold between three configurations, not between two. For two configurations are required for velocity, and another for change of velocity, which is what the law asserts to be relevant. Thus in any dynamical system, when the special laws (other than the laws of motion) which regulate that system are specified, the configuration at any given time can be inferred when two configurations at two given times are known.
The second and third laws introduce the new idea of mass; the third also gives one respect in which acceleration depends upon configuration. The second law as it stands is worthless. For we know nothing about the impressed force except that it produces change of motion, and thus the law might seem to be a mere tautology. But by relating the impressed force to the configuration, an important law may be discovered, which is as follows. In any material system consisting of n particles, there are certain constant coefficients (masses) m1, m2 . . . mn to be associated with these particles respectively; and when these coefficients are considered as forming part of the configuration, then m1 multiplied by the corresponding acceleration is a certain function of the momentary configuration; this is the same function for all times and all configurations. It is also a function dependent only upon the relative positions: the same configuration in another part of space will lead to the same accelerations. That is, if xr, yr, zr be the coordinates of mr at time t, we have xr = fr (t) etc., and m1 x¨1 = F (m1, m2, m3, . . . mn, x2 − x1, x3 − x1 . . . xn − x1, y2 − y1, . . .). This involves the assumption that x1 = f1(r) is a function having a second differential coefficient x¨1; the use of the equation involves the further assumption that x¨1 has a first and second integral. The above, however, is a very specialized form of the second law; in its general form, the function F may involve other coefficients than the masses, and velocities as well as positions.


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