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Home , Covariant derivative, Dual basis, Metric tensor

Change of

We have made no restrictions up on our choice of basis vectors e Our basis vectors are

simply a linearly indep endent set of four vector elds allowing us to express any vector as

a linear combination of basis vectors The choice of basis is not unique we may transform

from one basis to another simply by dening four linearly indep endent combinations of

0

our original basis vectors Given one basis fe g we may dene another basis fe g

X X

distinguished by placing a prime on the lab els as follows

0

e e

0

X X

X

The prime is placed on the index rather than on the vector only for notational conve

nience we do not imply that the new basis vectors are a p ermutation of the old ones

Any linearly indep endent linear combination of old basis vectors may b e selected as

the new basis vectors That is any nonsingular fourbyfour may b e used for

The transformation is not restricted to b eing a the lo cal

0

reference frames dened by the bases are not required to b e inertial or in the presence of

gravity freelyfalling Because the transformation matrix is assumed to b e nonsingular

the transformation may b e inverted

0

0

0

e e

0

X X

X

X X

Comparing equations and note that in our notation the inverse matrix places

the prime on the other index Primed indices are never summed together with unprimed

indices

If we change basis vectors we must also transform the basis oneforms so as to

preserve the condition equation The reader may verify that given the

transformations of equations and the new oneforms are

0 0

e e

X

X X

We may also write the transformation matrix and its inverse by pro ducts of the

old and new basis vectors and oneforms dropping the subscript X for clarity

0 0

0

he e i he e i

0

Apart from the basis vectors and oneforms a vector A and a oneform P are by

denition invariant under a Their comp onents are not For example

using equation or we nd

0 0 0 0

0

A A A A e A e he Ai

The vector comp onents transform opp ositely to the basis vectors eq Oneform

e basis vectors as suggested by the fact that b oth are lab eled comp onents transform lik

with a subscript

0

0 0 0

A A e i A h A A e A e

0

Note that if the comp onents of two vectors or two oneforms are equal in one basis they

are equal in any basis

Tensor comp onents also transform under a change of basis The new comp onents may

b e found by recalling that a m n is a function of m vectors and n oneforms

and that its comp onents are gotten by evaluating the tensor using the basis vectors and

oneforms eg eq For example the comp onents are transformed under the

change of basis of equation to

0 0 0 0 0 0

e e g g e e e g e g

0 0

Recall that evaluating a oneform or vector means using the scalar pro duct eq

We see that the covariant comp onents of the metric ie the lower indices transform

exactly like oneform comp onents Not surprisingly the comp onents of a tensor of rank

m n transform like the pro duct of m vector comp onents and n oneform comp onents

If the comp onents of two of the same rank are equal in one basis they are equal

in any basis

Co ordinate bases

We have made no restrictions up on our choice of basis vectors e Before concluding

our formal introduction to tensors we intro duce one more idea a co ordinate system A

co ordinate system is simply a set of four dierentiable scalar elds x not one vector

X

eld note that lab els the co ordinates and not vector comp onents that attach a

unique set of lab els to each p oint x That is no two p oints are allowed to

have identical values of all four scalar elds and the co ordinates must vary smo othly

throughout spacetime although we will tolerate o ccasional aws like the co ordinate

singularities at r and in spherical p olar co ordinates Note that we imp ose

no other restrictions on the co ordinates The freedom to choose dierent co ordinate

vailable to us even in a Euclidean there is nothing sacred ab out systems is a

Cartesian co ordinates This is even more true in a non where Cartesian

co ordinates covering the whole space do not exist

Co ordinate systems are useful for three reasons First and most obvious they allow

us to lab el each spacetime p oint by a set of numbers x x x x The second and

more imp ortant use is in providing a sp ecial set of basis vectors called a co ordinate

basis Supp ose that two innitesimally close spacetime p oints have co ordinates x and

x dx The innitesimal dierence vector b etween the two p oints denoted dx is a

vector dened at x We dene the co ordinate basis as the set of four basis vectors e

X

such that the comp onents of dx are dx

dx dx e denes e in a co ordinate basis

From the trivial app earance of this equation the reader may incorrectly think that we

wever that is not so According to have imp osed no constraints on the basis vectors Ho

equation the basis vector e for example must p oint in the direction of increasing

X

x at p oint x This corresp onds to a unique direction in fourdimensional spacetime just

as the direction of increasing latitude corresp onds to a unique direction north at a given

p oint on the earth In more mathematical treatments eg Walk e is asso ciated

at x with the directional x

It is worth noting the transformation matrix b etween two co ordinate bases

x

x

Note that not all bases are co ordinate bases If we wanted to b e p erverse we could

dene a nonco ordinate basis by for example p ermuting the lab els on the basis vectors

but not those on the co ordinates which after all are not the comp onents of a vector In

this case he dxi the comp onent of dx for basis vector e would not equal the co ordinate

This would violate nothing we have written so far except equation dierential dx

Later we will discover more natural ways that nonco ordinate bases may arise

The co ordinate basis fe g dened by equation has a dual basis of oneforms fe g

dened by equation The dual basis of oneforms is related to the We

obtain this relation as follows Consider any scalar eld f Treating f as a function of

X

the co ordinates the dierence in f between two innitesimally close p oints is

f

dx f dx df

x

Equation may b e taken as the denition of the comp onents of the gradient with an

alternative brief notation for the However partial dep end

on the co ordinates while the gradient should not What then is

the gradient is it a vector or a oneform

From equation b ecause df is a scalar and dx is a vector comp onent f x

must b e the comp onent of a oneform not a vector The notation with its covariant

subscript index reinforces our view that the partial derivative is the comp onent of a

oneform and not a vector We denote the gradient oneform by r Like all oneforms

Using equation the gradient may b e decomp osed into a sum over basis oneforms e

and equation as the requirement for a dual basis we conclude that the gradient

is

e in a co ordinate basis r

Note that we must write the basis oneform to the left of the partial derivative op erator

We will return to this p oint in for the basis oneform itself may dep end on p osition

Section when we discuss the covariant derivative In the present case it is clear from

equation that we must let the derivative act only on the function f We can now

rewrite equation in the co ordinatefree manner

df hrf dxi

If we want the of f along any particular direction we simply replace

dx by a vector p ointing in the desired direction eg the to some curve

equal one of the co ordinates using equation the gradient gives us Also if we let f

X

the corresp onding basis oneform

rx e in a co ordinate basis

The third use of co ordinates is that they can b e used to describ e the b etween

two p oints of spacetime However co ordinates alone are not enough We also need the

We write the squared distance b etween two spacetime p oints as

ds jdx j g dx dx dx dx

This equation true in any basis b ecause it is a scalar equation that makes no reference

to comp onents is taken as the denition of the metric tensor Up to now the metric

could have b een any symmetric tensor But if we insist on b eing able to measure

given an innitesimal dierence vector dx only one tensor can give the

squared distance We dene the metric tensor to b e that tensor Indeed the squared

magnitude of any vector A is jA j gA A

Now we sp ecialize to a co ordinate basis using equation for dx In a co ordinate

basis and only in a co ordinate basis the squared distance is called the and

takes the form

ds g dx dx in a co ordinate basis

X

We have used equation to get the metric comp onents

If we transform co ordinates we will have to change our vector and oneform bases

0

Supp ose that we transform from x to x with a prime indicating the new co ordinates

X X

For example in the Euclidean plane we could transform from Cartesian co ordinate x

0 0

x x y to p olar co ordinates x r x x r cos y r sin A onetoone

mapping is given from the old to new co ordinates allowing us to dene the Jacobian

0 0 0

Vector comp onents transform x matrix x x and its inverse x

0

0 0

like dx x x dx Transforming the basis vectors basis oneforms and tensor

comp onents is straightforward using equations The reader should verify that

equations and remain valid after a co ordinate transformation

We have now intro duced many of the basic ingredients of that we will

need in Before moving on to more advanced concepts let us reect on

our treatment of vectors oneforms and tensors The mathematics and notation while

straightforward are complicated Can we simplify the notation without sacricing rigor

One way to mo dify our notation would b e to abandon ths basis vectors and oneforms

and to work only with comp onents of tensors We could have dened vectors oneforms

and tensors from the outset in terms of the transformation prop erties of their comp onents

However the reader should appreciate the clarity of the geometrical approach that we

have adopted Our notation has forced us to distinguish physical ob jects like vectors from

basisdep endent ones like vector comp onents As long as the denition of a tensor is not

forgotten computations are straightforward and unambiguous Moreover adopting a

basis did not force us to abandon geometrical concepts On the contrary computations

are made easier and clearer by retaining the notation and meaning of basis vectors and

oneforms

Isomorphism of vectors and oneforms

Although vectors and oneforms are distinct ob jects there is a strong relationship b e

tween them In fact the linear space of vectors is isomorphic to the dual

of oneforms Wald Every equation or op eration in one space has an equivalent

equation or op eration in the other space This isomorphism can b e used to hide the

distinction b etween oneforms and vectors in a way that simplies the notation This

approach is unusual I havent seen it published anywhere and is not recommended in

formal work but it may b e p edagogically useful

As we saw in equations and the link b etween the vector and dual vector

;

is provided by g and g If A B comp onents A B then A B

and B g B So why do we b other with comp onents A B where A g A

oneforms when vectors are sucient The answer is that tensors may b e functions of

ever there is also an isomorphism among tensors of b oth oneforms and vectors How

dierent rank We have just argued that the tensor spaces of rank vectors and

mn

tensor spaces of rank m n with xed m n are isomorphic In fact all

are isomorphic The metric and inverse metric tensors link together these spaces as

exemplied by equations and

The isomorphism of dierent tensor spaces allows us to intro duce a notation that

unies them We could eect such a unication by discarding basis vectors and oneforms

and working only with comp onents using the comp onents of the metric tensor and its

inverse to relate comp onents of dierent typ es of tensors as in equations and

However this would require sacricing the co ordinatefree geometrical interpretation of

vectors Instead we intro duce a notation that replaces oneforms with vectors and m n

tensors with m n tensors in general We do this by replacing the basis oneforms

with a set of vectors dened as in equation e

;

e g e g e

as a dual basis vector to contrast it from b oth the basis vector e e will refer to e W

The dots are present in equation to remind us that a and the basis oneform e

oneform may b e inserted to give a scalar However we no longer need to use oneforms

we may form the Using equation given the comp onents A of any oneform A

vector A dened by equation as follows

A A e A g e A e

The reader should verify that A A e is invariant under a change of basis b ecause e

transforms like a basis oneform

The isomorphism of oneforms and vectors means that we can replace all oneforms

with vectors in any tensor equation Tildes may b e replaced with arrows The scalar

pro duct b etween a oneform and a vector is replaced by the dot pro duct using the metric

eq or The only rule is that we must treat a dual basis vector with an upp er

index like a basis oneform

e e g e e he e i e e e e g

The reader should verify equations using equations and Now if we need

the contravariant comp onent of a vector we can get it from the dot pro duct with the

dual basis vector instead of from the scalar pro duct with the basis oneform

A e A he Ai

t oneform r eq to a vector We may also apply this recip e to convert the gradien

though we must not allow the dual basis vector to b e dierentiated

r e g e in a co ordinate basis

It follows at once that the dual basis vector in a co ordinate basis is the vector gradient

of the co ordinate e rx This equation is isomorphic to equation

The basis vectors and dual basis vectors through their tensor pro ducts also give a

basis for higherrank tensors Again the rule is to replace the basis oneforms with the

corresp onding dual basis vectors Thus for example we may write the rank metric

tensor in any of four ways

e g g e e g e e g e e g e

In fact by comparing this with equation the reader will see that what we have

;

which is isomorphic to g through the written is actually the inverse metric tensor g

with e But what are the mixed comp onents of the metric g and replacement of e

g From equations and we see that they b oth equal the

Consequently the metric tensor is isomorphic to the identity tensor as well as to its

inverse However this is no miracle it was guaranteed by our denition of the dual basis

;

vectors and by the fact we dened g to invert the mapping from vectors to oneforms

implied by g The reader may fear that we have dened away the metric by showing it to

b e isomorphic to the identity tensor However this is not the case We need the metric

from A We cannot take advantage of A from e or tensor comp onents to obtain e

the isomorphism of dierent tensor spaces without the metric Moreover as we showed

in equation the metric plays a fundamental role in giving the squared magnitude

of a vector In fact as we will see later the metric contains all of the information ab out

the geometrical prop erties of spacetime Clearly the metric must play a fundamental

role in general relativity

Example Euclidean plane

We close this by applying tensor concepts to a simple example the Euclidean

plane This at twodimensional space can b e covered by Cartesian co ordinates x y

with line element and metric comp onents

ds dx dy g g g g

xx yy xy yx

We prefer to use the co ordinate names themselves as comp onent lab els rather than using

numbers eg g rather than g The basis vectors are denoted e and e and their use

xx x y

in plane geometry and is standard Basis oneforms app ear unnecessary

b ecause the metric tensor is just the identity tensor in this basis Consequently the

x y

e e e and no distinction is needed b etween dual basis vectors eq are e

x y

sup erscripts and subscripts

However there is nothing sacred ab out Cartesian co ordinates Consider p olar co or

y sin A simple exercise dinates dened by the transformation x cos

in partial derivatives yields the line element in p olar co ordinates

ds d d g g g g

This app ears eminently reasonable until p erhaps one considers the basis vectors e and

e is e recalling that g e e Then while e e and e e e e

not a unit vector The new basis vectors are easily found in terms of the Cartesian basis

vectors and comp onents using equation

y x

p p

e e e y e x e e

x y x y

x y x y

;

The p olar unit vectors are e and e

Why do es our formalism give us nonunit vectors The answer is b ecause we insisted

that our basis vectors b e a coordinate basis eqs and In terms of the

orthonormal unit vectors the dierence vector b etween p oints and d d is

dx d d In the co ordinate basis this takes the simpler form dx e d e d

ve to worry ab out normalizing our vectors e In the co ordinate basis we dont ha dx

all information ab out lengths is carried instead by the metric In the nonco ordinate

basis of orthonormal vectors f g we have to make a separate note that the distance

elements are d and d

In the nonco ordinate basis we can no longer use equation for the line element

We must instead use equation The metric comp onents in the nonco ordinate basis

f g are

g g g g

The reader may also verify this result by transforming the comp onents of the metric

;

from the basis fe e g to f g using equation with Now

equation still gives the distance squared but we are resp onsible for remembering

dx d d In a nonco ordinate basis the metric will not tell us how to measure

distances in terms of co ordinate dierentials

With a nonco ordinate basis we must sacrice equations and Nonetheless

for some applications it proves convenient to intro duce an orthonormal nonco ordinate

basis called a tetrad basis Tetrads are discussed by Wald and Misner et al

The use of nonco ordinate bases also complicates the gradient eqs and

In our p olar co ordinate basis eq the inverse metric comp onents are

;

g g g g

The matrix g is diagonal so its inverse is also diagonal with entries given by the

e g They are isomorphic to recipro cals The basis oneforms ob ey the rules e

; ;

the dual basis vectors e g e eq Thus e e e e

Equation gives the gradient oneform as r e e Expressing this

as a vector eq we get

e r e

The gradient is simpler in the co ordinate basis The co ordinate basis has the added

advantage that we can get the dual basis vectors or the basis oneforms by applying

the gradient to the co ordinates eq e r e r

From now on unless otherwise noted we will assume that our basis vectors are

a co ordinate basis We will use oneforms and vectors interchangeably through the

mapping provided by the metric and inverse metric eqs and Readers who

dislike oneforms may convert the tildes to arrows and use equations to obtain

scalars from scalar pro ducts and dot pro ducts