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General Relativity and Gravitational Waves: Session 2. General Coordinates Thomas A

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General Relativity and Gravitational Waves: Session 2. General Coordinates Thomas A General Relativity and Gravitational Waves: Session 2. General Coordinates Thomas A. Moore — Les Houches — July 4, 2018 Overview of this session: 2.2 Definition of a Coordinate Basis 2.3 Tensors in a Coordinate Basis 2.4 The Tensor Gradient 2.5 The Geodesic Equation 2.6 Schwarzschild Geodesics 2.7 LOFs and LIFs u=–1 u=0 u=1 u=2 ew Q w=5 w=4+dw ds eu dw ew P w=4 P due w=4 u w=3 u=1 u=1+du Figure 1: This drawing shows an arbitrary coordinate system in a possibly curved space, a point , the basis P vectors eu, ew at that point, and a close-up view of how we define the basis vectors so that an infinitesimal displacement ds is the vector sum of the basis vectors times simply du and dw respectively. I am not going to emphasize rigor in what follows, but rather an intuitive approach that I think will help you quickly develop a working understanding of the mathematics. Some might complain about all of the assumptions I am leaving unstated and the hands that I am waving, but my time with you is too short. No matter how we construct our coordinate system, the di↵erential distance ds between two infinitesimally separated points in a two-dimensional space (or the spacetime separation between two events in a spacetime) is a coordinate-independent quantity, because we can measure it directly with a ruler (or a clock in spacetime) without having to define a coordinate system at all. The fundamental way that we connect arbitrary coordinates to physical reality is by specifying how the spacetime separation between two infinitesimally- separated points (or events) depends on their coordinate separations. Let’s first consider a two-dimensional flat space. In such a space, a cartesian xy coordinate system is one in which the distance ds between two infinitesimally-separated points is given by ds2 = dx2 + dy2 at all points in the space. A curvilinear coordinate system is any non-cartesian coordinate system where this simple pythagorean relationship is not true. How can we connect the coordinate-independent distance between two points with their coordinate separations in such a case? Consider arbitrary coordinatesCoordinateu, w for a 2D space. Basis When using index notation, we will interpret dxu as being equivalent to du, and dxu as being equivalent to dw, and we will assume that Greek indices have two possible values u and w. (In the last session, in the context of cartesian coordinates in non-curved spacetime, u=–1 u=0 u=1 u=2 I stated that indices could represent either t, x, y, or z, but when we use arbitrary coordinates, the indices take on values corresponding to whatever namesew the coordinate have.) I willQ also represent vectors in this two-dimensional space with the same bold-face notation as wew=5 usedw=4+dw for four-vectorsds last time. This will keep eu dw ew the notation from changing when we generalizeP to 4D spacetimes.w=4 P due Now, no matter how our u, w coordinate system is defined,w at=4 each pointu in the space, we can define P a pair of basis vectors eu, ew such that w=3 u=1 u=1+du 1. eu points tangent to the w = constant curve toward increasing values of u. Define basis vectors eu, ew such that 2. ew points tangent to the 1.u =eu constant points tangent curve toward to the increasing w = constant values curve of w. in the direction of increasing u 3. Their lengths are defined so that the displacement vector ds between a point at coordinates u, w 2. e points tangent to the u = constant curve and an infinitesimally separatedw neighboring point at coordinates u + du, w +Pdw can be written in the direction of increasingQ w µ 3. ddss= =du dueeuu ++ dwdweww = dx eµ (See figure 1.) (2.1) (Note: the lower index on eµ tells us which basis vector, not which component, we are talking about.) Now, this only works for di↵erential separations, because the directions of the basis vectors change as we move significant distances. Moreover, vector addition as I have illustrated it in Figure 1 is really only defined in a flat space. In a curved space, the separation between and must be so small that we can treat the space as flat, just as we treat a city map as flat on the curvedP surfaceQ of the earth. (Technically, we are doing this vector addition in a flat space that is tangent to the curved space at point .) P 2 Now, ifNow, we define if we define basis vectors basis vectors this way, this then way,du thenanddudwandbecomedw become the components the components of ds in of thatds in basis that and basis and we callwe the call set the of set basis of vectors basis vectorseu, eweau,coordinateew a coordinate basis. basis.A coordinateA coordinate basis is basis generally is generally di↵erent di↵ thanerent than Now, if we define basis vectors this way, then du and dw become the components of ds in that basis and the cartesianthe cartesian coordinate coordinate basis vectors basis vectorsex, ey einx, thatey in (1) thateu (1)andeuewandmayew notmay be not perpendicular, be perpendicular, (2) eu (2)andeuewand ew we call the set of basis vectors eu, ew a coordinate basis. A coordinate basis is generally di↵erent than may notmay have not unit have length, unit length, and (3) andeu (3)and/oreu and/orew mayew changemay change in magnitude in magnitude and/or and/or direction direction as we move as we from move from the cartesianpoint coordinate topoint point. to basis point. It is vectors also It is importante alsox, ey importantin that to note (1) toe thatu noteand defining thatew may defining a not coordinate be a perpendicular, coordinate basis is basis not (2) is theeu notandonly theewwayonly toway define to define e e may not havea set unit ofa length, set basis of vectors basis and (3) vectors oru aand/or coordinate or a coordinatew may system. change system. For in magnitudeexample, For example, standard and/or standard direction polar coordinates polar as we coordinates move in from two-dimensional in two-dimensional point to point. It is also important to note that defining a coordinate basis is not the only way to define flat spaceflat and space spherical and spherical coordinates coordinates in a three-dimensional in a three-dimensional flat space flat do spacenot dousenot a coordinateuse a coordinate basis, becausebasis, because a set of basis vectors or a coordinate system. For example, standard polar coordinates in two-dimensional the basisthe vectors basis vectors in those in coordinate those coordinate systems systems always always have unit have magnitude. unit magnitude. Using Using a coordinate a coordinate basis makes basis makes flat space and spherical coordinates in a three-dimensional flat space do not use a coordinate basis, because componentscomponentsof theof displacement the displacement vector vectords simplerds simpler (at the (at expense the expense of complexity of complexity in the in basis the vectors), basis vectors), and and the basis vectors in those coordinate systems always have unit magnitude. Using a coordinate basis makes this turnsthis out turns to out be crucial to be crucial in what in follows. what follows. components of the displacement vector ds simpler (at the expense of complexity in the basis vectors), and Once weOnce have we defined have definedCoordinate a coordinate a coordinate basis for basis coordinates for Basis coordinatesu and uwand,thenwew,thenwedefine definethe componentsthe components of a of a this turns out to be crucial in what follows. four-vectorfour-vectorA at anyA at point any pointto be Atou,A bewAsuchu,Aw thatsuch that Once we have defined a coordinateP basisP for coordinates u and w,thenwedefine the components of a u w four-vector A at any point to be A ,A such that u u w w µ µ P Define a four-vector’sA = AAe=u components+AAeue+w A= Aew soe=µ thatA eµ (2.2) (2.2) A = Aue + Awe = Aµe (2.2) This ensuresThis ensures that the that components the components of dus and of dsAwandtransformA transformµ alike. alike. Now, the scalar product of ds with itself is the square of the physical distance between the endpoints of This ensures thatNow, the the components scalar product of ds ofandds Awithtransform itself is alike. the square of the physical distance between the endpoints of Now, thethe scalar displacementthe product displacement of itd represents:s with it represents:The itself metric is the in square a coordinate of the physical basis distanceis between the endpoints of the displacement it represents: 2 2 ds = dsds ⇧=dsd=(s ⇧ duds =(eu +dudweue+w)dw⇧ (eduw)e⇧u(+dudweue+w)dw ew) 2 2 2 2 2 ds = ds ⇧ ds =(=dudueue=u+du⇧dweuee+uw⇧du)e⇧u( dwdu+ dueeuu dw+⇧ edwwe+ue⇧wdwe)w dw+ dwew dw⇧ eue+w ⇧dweu +ewdw⇧ ewew ⇧ ew 2 µ ⌫ µ ⌫ µ ⌫ µ ⌫ 2 = du eu ⇧ eu=+dxdudx= dwdxeeuµdx⇧⇧ eew⌫e+µ dw⇧geµ⌫⌫ dwdx gedxwµ⌫⇧dxeu dx+ dw ew ⇧ ew (2.3) (2.3) µ ⌫ µ ⌘⌫ ⌘ = dx dx eµ ⇧ e⌫ gµ⌫ dx dx (2.3) The setThe of fourset of components four componentsgµ⌫ ⌘= geµµ⌫⇧=e⌫eµrepresents⇧ e⌫ represents the metric the metric tensor tensorof ourof two-dimensional our two-dimensional coordinate coordinate The set ofbasis. four componentsbasis. Note that Noteg becauseµ that⌫ = e becauseµ both⇧ e⌫ represents the both magnitudes the the magnitudesmetric and directions tensor and directionsof of our the two-dimensional of basis the vectors basis vectors depend coordinate depend on position, on position, the the basis.
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