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Introduction to Tensor Calculus for General Relativity

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Introduction to Tensor Calculus for General Relativity Massachusetts Institute of Technology Department of Physics Physics Spring Introduction to Tensor Calculus for General Relativity c Edmund Bertschinger Introduction There are three essential ideas underlying general relativity GR The rst is that space time may b e describ ed as a curved fourdimensional mathematical structure called a pseudoRiemannian manifold In brief time and space together comprise a curved four dimensional nonEuclidean geometry Consequently the practitioner of GR must b e familiar with the fundamental geometrical prop erties of curved spacetime In particu lar the laws of physics must b e expressed in a form that is valid indep endently of any co ordinate system used to lab el p oints in spacetime The second essential idea underlying GR is that at every spacetime p oint there exist lo cally inertial reference frames corresp onding to lo cally at co ordinates carried by freely falling observers in which the physics of GR is lo cally indistinguishable from that of sp ecial relativity This is Einsteins famous strong equivalence principle and it makes general relativity an extension of sp ecial relativity to a curved spacetime The third key idea is that mass as well as mass and momentum ux curves spacetime in a manner describ ed by the tensor eld equations of Einstein These three ideas are exemplied by contrasting GR with Newtonian gravity In the Newtonian view gravity is a force accelerating particles through Euclidean space while time is absolute From the viewp oint of GR as a theory of curved spacetime there is no gravitational force Rather in the absence of electromagnetic and other forces particles ed by mass follow the straightest p ossible paths geo desics through a spacetime curv Freely falling particles dene lo cally inertial reference frames Time and space are not absolute but are combined into the fourdimensional manifold called spacetime In sp ecial relativity there exist global inertial frames This is no longer true in the presence of gravity However there are local inertial frames in GR such that within a suitably small spacetime volume around an event just how small is discussed eg in MTW Chapter one may cho ose co ordinates corresp onding to a nearlyat spacetime Thus the local prop erties of sp ecial relativity carry over to GR The mathematics of vectors and tensors applies in GR much as it do es in SR with the restriction that vectors and tensors are dened indep endently at each spacetime event or within a suciently small neighborho o d so that the spacetime is sensibly at Working with GR particularly with the Einstein eld equations requires some un derstanding of dierential geometry In these notes we will develop the essential math ematics needed to describ e physics in curved spacetime Many physicists receive their introduction to this mathematics in the excellent book of Weinb erg Weinberg minimizes the geometrical content of the equations by representing tensors using com p onent notation We b elieve that it is equally easy to work with a more geometrical description with the additional b enet that geometrical notation makes it easier to dis tinguish physical results that are true in any co ordinate system eg those expressible using vectors from those that are dep endent on the co ordinates Because the geometry of spacetime is so intimately related to physics we b elieve that it is b etter to highlight the geometry from the outset In fact using a geometrical approach allows us to develop the essential dierential geometry as an extension of vector calculus Our treatment is closer to that Wald and closer still to Misner Thorne and Wheeler MTW These b o oks are rather advanced For the newcomer to general relativity we utz Our notation and presentation is patterned largely warmly recommend Sch after Schutz It expands on MTW Chapters and The student wishing addi tional practice problems in GR should consult Lightman et al A slightly more advanced mathematical treatment is provided in the excellent notes of Carroll These notes assume familiarity with sp ecial relativity We will adopt units in which the sp eed of light c Greek indices etc which take the range f g will b e used to represent comp onents of tensors The Einstein summation convention is assumed rep eated upp er and lower indices are to b e summed over their ranges B A B A B A B A B Fourvectors will b e represented with eg A an arrow over the symb ol eg A while oneforms will b e represented using a tilde Spacetime p oints will b e denoted in b oldface type eg x refers to a p oint eg B Our metric has signature the at spacetime Minkowski metric with co ordinates x comp onents are diag Vectors and oneforms The essential mathematics of general relativity is dierential geometry the branch of mathematics dealing with smo othly curved surfaces dierentiable manifolds The physicist do es not need to master all of the subtleties of dierential geometry in order to use general relativity For those readers who want a deep er exp osure to dierential geometry see the introductory texts of Lovelo ck and Rund Bishop and Goldb erg or Schutz It is sucient to develop the needed dierential geometry as a straightforward extension of linear algebra and vector calculus However it is imp ortant to keep in mind the geometrical interpretation of physical quantities For this reason we will not shy from using abstract concepts like p oints curves and vectors and we will distinguish b etween a vector A and its comp onents A Unlike some other authors eg Weinberg we will introduce geometrical ob jects in a co ordinatefree manner only later introducing co ordinates for the purp ose of simplifying calculations This approach requires that we distinguish vectors from the related ob jects called oneforms Once the dierences and similarities b etween vectors oneforms and tensors are clear we will adopt a unied notation that makes computations easy Vectors We b egin with vectors A vector is a quantity with a magnitude and a direction This primitive concept familiar from undergraduate physics and mathematics applies equally in general relativity An example of a vector is dx the dierence vector b etween two Vectors form a linear algebra ie a vector innitesimally close p oints of spacetime space If A is a vector and a is a real numb er scalar then aA is a vector with the same direction or the opp osite direction if a whose length is multiplied by jaj If A and B are vectors then so is A B These results are as valid for vectors in a curved fourdimensional spacetime as they are for vectors in threedimensional Euclidean space Note that we have intro duced vectors without mentioning co ordinates or co ordinate transformations Scalars and vectors are invariant under co ordinate transformations vector comp onents are not The whole p oint of writing the laws of physics eg F ma using scalars and vectors is that these laws do not dep end on the co ordinate system imp osed by the physicist We denote a spacetime p oint using a b oldface symbol x This notation is not meant to imply co ordinates Note that x refers to a p oint not a vector In a curved spacetime x is not useful the concept of a radius vector x p ointing from some origin to each p oint b ecause vectors dened at two dierent p oints cannot b e added straightforwardly as they can in Euclidean space For example consider a sphere embedded in ordinary threedimensional Euclidean space ie a twosphere A vector p ointing east at one p oint on the equator is seen to p oint radially outward at another p oint on the equator � The radially outward direction is undened on the whose longitude is greater by sphere Technically we are discussing tangent vectors that lie in the tangent space of the manifold at each p oint For example a sphere may b e emb edded in a threedimensional Euclidean space into which may b e placed a plane tangent to the sphere at a p oint A two dimensional vector space exists at the p oint of tangency However such an emb edding is not required to dene the tangent space of a manifold Wald As long as the space is smo oth as assumed in the formal denition of a manifold the dierence vector dx between two innitesimally close p oints may b e dened The set of all dx denes the tangent space at x By assigning a tangent vector to every spacetime p oint we can recover the usual concept of a vector eld However without additional preparation one cannot compare vectors at dierent spacetime p oints b ecause they lie in dierent tangent spaces In later notes we intro duce will parallel transp ort as a means of making this comparison Until then we consider only tangent vectors at x To emphasize the status of a tangent vector we will o ccasionally use a subscript notation A X Oneforms and dual vector space Next we intro duce oneforms A oneform is dened as a linear scalar function of a vector That is a oneform takes a vector as input and outputs a scalar For the oneform P V is also called the scalar pro duct and may b e denoted using angle brackets P V i P P V h The oneform is a linear function meaning that for all scalars a and b and vectors V and W the oneform P satises the following relations P aV bW hP aV bW i ahP V i bhP W i aP V bP W Just as we may consider any function f as a mathematical entity indep endently of indep endently of any particular any particular argument we may consider the oneform P e may also asso ciate a oneform with each spacetime p oint resulting in a vector V W oneform
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