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3.4 Change of Basis

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3.4 Change of Basis Change of basis 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 matrix may b e used for The transformation is not restricted to b eing a Lorentz transformation 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 duality condition equation The reader may verify that given the transformations of equations and the new dual basis oneforms are 0 0 e e X X X We may also write the transformation matrix and its inverse by scalar 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 change of basis 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 tensor 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 metric 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 tensors 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 spacetime 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 space there is nothing sacred ab out systems is a Cartesian co ordinates This is even more true in a nonEuclidean space 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 derivative 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 gradient 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 partial derivative However partial derivatives dep end on the co ordinates while the gradient covariant derivative 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 directional derivative of f along any particular direction we simply replace dx by a vector p ointing in the desired direction eg the tangent vector 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 distance b etween two p oints of spacetime However co ordinates alone are not enough We also need the metric tensor 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 distances 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 line element 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
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