vectors in physics and the einstein convention

7/30/2019 Vectors in physics and the Einstein convention

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Vectors in Physics and The Einstein Summation Convention

Gerardo Urrutia*

24 August 2013

*Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, Mexico D.F. 04510, Mexico*[email protected]

Abstract: In this note is explained the Einstein convention for repeated indices in algebraic expressions. I

also explain the physical significance of vectors in covariant and contravariant form along with the respective

rules for good algebraic treatment.

Introduction

In previous courses of algebra, we developed a sumwith n components of the following form

a1x1 + a2x2 + a3x3 + + anxn =n

i=1

aixi (1)

Einstein proposed a convention in the notation forremove sigma symbol in the algebraic expressions,that is to say; when appears the expression aixi itshould be understood that

aixi = a1x1 + a2x2 + a3x3 + + anxn (2)

only when appear repeated indices in a algebraic ex-pression. The next example sample clearly this idea,with aiixk and aijxj do so over the respective ranges

1 i n and 1 j n. Ifn = 4 then

aiixk = a11xk + a22xk + a33xk + a44xk (3)

aijxj = ai1x1 + ai2x2 + ai3x3 + ai4x4

In general, the expressions (1), (2) and (3) are notsuitable for the physicists.

Vectors in Physics

The typical vector is the displacement vector, whichpoints from one event to another and has compo-

nents equal to the coordinate differences:

x O (t,x,y, z) (4)

x is a vector having nothing particular to do withthe coordinate x, the arrow after x means hascomponents, and the O underneath it means inthe frame O; the components will always be in theorder t ,x,y,z (equivalently, indices in the order 0, 1,2, 3). The notation O is used in order to emp-hasize the distinction between the vector and itscomponents. The vector x is an arrow between two

events, while the collection of components is a set offour coordinate-dependent numbers. We shall alwaysemphasize the notion of a vector (and, later, any ten-sor) as a geometrical object: something which can bedefined and (sometimes) visualized without referringto a specific coordinate system. Another importantnotation is

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x O {x} (5)

where by {x} we mean all of x0, x1, x2,x3, If we ask for this vectors components in anot-her coordinate system, say the frame O, we write

x O {x} (6)

That is, we put a bar over the index to denote thenew coordinates. The vector x is the same, andno new notation is needed for it when the frame ischanged. Only the components of it change.The vector structure do not depend of the

coordinates, in this sense, they are similar to

physical laws. These mathematical objects are in-variant under linear transformations and it is alwayspossible to find such a transformation

x =

3

=0

x (7)

with the Einstein convention only write

x = x (8)

where is the rule of transformation. The simbol not is the only that may represent a linear trans-formation.

Basis Vectors

In any frame O there are four special vectors, definedby giving their components:

e0 O (1, 0, 0, 0)

e1 O (0, 1, 0, 0) (9)

e2 O (0, 0, 1, 0)

e3 O (0, 0, 0, 3)

These definitions define the basis vectors of the fra-me O. Similarly, O has basis vectors

e0 O (1, 0, 0, 0) etc.

Generaly e0 = e0 since they are defined in differentframes. The definition of the basis vectors is equiva-lent to

(e)

= (10)

That is, the component of e is the Kronockerdelta: 1 if= and 0 if=

Any vector can be expressed in terms of the basisvectors. If

A OA0, A1, A2, A3

then

A = A0e0 + A1e1 + A

2e2 + A3e3 (11)

A = Ae (12)

For transformation of basis vectors

e = e (13)

A development with more detail is available in [2].

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Covariance and contravariance of vec-

tors

Covariance and contravariance describe how thequantitative description of certain geometric or phy-sical entities changes with a change of basis. For ho-lonomic bases, this is determined by a change fromone coordinate system to another. When an orthogo-nal basis is rotated into another orthogonal basis, thedistinction between co- and contravariance is invisi-ble. However, when considering more general coor-dinate systems such as skew coordinates, curvilinearcoordinates, and coordinate systems on differentia-ble manifolds, the distinction is significant.For a vector (such as a direction vector or velocity

vector) to be basis-independent, the components ofthe vector must contra-vary with a change of basisto compensate. That is, the components must varywith the inverse transformation to that of the changeof basis. The components of vectors (as opposed tothose of dual vectors) are said to be contravariant.Examples of vectors with contravariant componentsinclude the position of an object relative to an ob-server

v = ve (14)

For a dual vector (also called a covector) to be basis-independent, the components of the dual vector mustco-vary with a change of basis to remain representingthe same covector. That is, the components mustvary by the same transformation as the change ofbasis. The components of dual vectors (as opposedto those of vectors) are said to be covariant.

v = ve (15)

Examples of covariant vectors generally appear whentaking a gradient of a function. In physics, vectorsoften have units of distance or distance times someother unit (such as the velocity), whereas covectorshave units the inverse of distance or the inverse ofdistance times some other unit.

More about Einstein Convention

With the previous considerations the la expression

(3) becomes

ax = a00x

+ a11x + a22x

+ a33x (16)

ax = a0x

0 + a1x1 + a2x

2 + a3x3

Free and Dummy Indices

In the equations (16), the expression ax involves

two sorts of indices. The index of summation, ,which ranges over the integers (0, 1, 2, 3) , cannot bepreempted. But at the same time, it is clear that theuse of the particular character is inessential; e.g.the expressions ax

and ax represent exactly

the same sum as ax does. For this reason, is

called a dummy index. The index , which may ta-ke on particular value independently, is called a freeindex. Note that, although we call the index freein the expression ax

, that freedom is limited inthe sense that generaly, unless =

ax = ax (17)

Example Write down explicity the equations repre-sented by the expression y = ax

. Holding fixed and summing over yields

y = a0x0 + a1x

1 + a2x2 + a3x

3 (18)

Next, setting the free index leads to three separateequations :

y0 = a00x0

+ a01x1

+ a02x2

+ a03x3

y1 = a10x0 + a11x

1 + a12x2 + a13x

3 (19)

y2 = a20x0 + a21x

1 + a22x2 + a23x

3

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y3 = a30x0 + a31x

1 + a32x2 + a33x

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Remarks

Any expression involving a twice-repeat index (oc-curing twice as subscript , twice as a superscript oronce as a subscript and once once a superscript)shall automatically stand for its sum over the va-lues (0, 1, 2, 3) of the repeated index. The range ofthis equations is 4.Remark 1: Any free index in a expression shall behave the same range as summation indices, unlessstated otherwise.Remark 2: No index may occur more than twice inany given expression.

Example (a) Acoording to Remark 2, an expressionlike ax

is without meaning. (b) The meaninglessexpression ax

x might be presumed to represent

a (x)

2, which is meaningfull. (c) An expression of

the form a (x + y) or a (x + y) is considered

well-defined, for it is obtained by composition of themeaningful expressions az

and z = x + y. Inother words, the index is regarded as occuring oncein the term (x + y).

Double sums

An expression can involve more than one summa-tion index. For example, ax

y indicates a sum-mation taking place on both and simultaneously.If an expression has two summation (dummy) indi-ces, there will be a total of 42 or in general n2 termsin the sum; if there are three indices, there will n3

terms; and so on. The expansion ofaxy can be

arrived at logically by first summing over , thenover

axy = a0x

y0 + a3xy3 (20)

before

axy =

a00x

0y0 + a03x0y3

+ . . . (21)

+ . . .

+a03x

3y0 + a33x3y3

Substitutions

Suppose it is re quired to substitute y = ax

in the equation Q = byx. Disregard of Remark

2 above would lead to an absurd expression likeQ = bax

x. The correct procedure is first toidentify any dummy indices in the expression to besubstituted that coincide with indices ocurring in

the main expression. Changing these dummy indi-ces to characteres not found in the main expression,one may the carry out the substitution in the usualfashion.

Step 1 Q = byx and y = ax

dummy index is duplicated.Step 2 y = ax

Change the dummy index from to .Step 3 Q = b (ax

)x = abxx

Kronecker Delta

= 1 (22)

= 0 if =

Now for example

xx = xx =

x12 +

x22 +

x32 (23)

Then we deduce the next rules

xx = xx

(24)

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ax = ax

(25)

Example Suppose that T = gay and y =bx . If further ab

= find T in terms of the

x . First write y = bx. Then, by substitution

T = gabx = g

x = gx (26)

Rules of basic algebra

The following nonidentities should be carefully no-ted:

a (x + x) = ax

+ ay

axy = ay

x

(a + a)xy = 2ay

x

Listed below are several valid identities; they, andothers like them, will be used repeatedly from nowon.

a(x + x) = ax

+ ay (27)

axy = ay

x (28)

axx = ax

x (29)

(a + a)xx = 2xx (30)

(a a)xx = 0 (31)

Referencias

[1] David C. Kay, Tensor calculus, Mc Graw Hill, USA, 1998

[2] Bernard Schutz, A first course in General Relativity, 2 ed, Cambridge University Press, UK,2009

[3] W. Hauser, Introduction to the Principles of Mechanics, Addison-Wesley Publishing Com-pany, USA, 2007

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vectors in physics and the einstein convention

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