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Chapter 9

Geometry and Gravitation

9.1 Introduction to Geometry

Geometry is one of the oldest branches of mathematics, competing withnumber theory for historical primacy. Like all good science, its origins werebased in observation and, with historical hindsight, we realize that the ev-ident truths discovered by early geometers were really a result of limitedperspective. But like them, for our discussion, we will take certain ideas asevident and as the basis for what we understand. The idea of the point andconnected sets of points and particularly the idea of the straight line. As isevident from our discussion of Special Relativity, see Sections 3.3.5 and ??,we take the straight line to be the shortest distance between two points inspace and the longest distance between two events in space-time.

Geometry developed from the need to measure land surfaces for agricul-tural purposes. The geometry that developed was what we now call planegeometry and the basis for it was first clearly articulated by Euclid and thusthe name Euclidean geometry. Euclid set the foundation for plane geometryby means of a set of axioms, evident truths. Modern formulations of geom-etry realize that there are consistent systems that do not have the same setof axioms. The question then becomes one of choice or appropriateness. Infact, if the early geometers had considered the geometry that is appropriateto large distances on the earth, they would have developed a geometry thatwas not Euclidean. This alternative geometry is well known and is calledspherical geometry. It differs from the Euclidean with the replacement ofone axiom, the axiom of parallels. In Euclidean geometry, the axiom ofparallels states that given a straight line and a point not on that line thatthere is one and only one straight line through that point that never touches

149

150 CHAPTER 9. GEOMETRY AND GRAVITATION

the the original line no matter how far the lines are extended and that lineis called parallel. In spherical geometry, the straight lines are the arcs ofgreat circles, circles on the surface whose center is the center of the sphere.A point to note is that the center of the sphere is not on the surface. Inthe case of the sphere, all straight lines through a point not on the originalline meet the original line, in fact twice. There is a line through a pointnot on the original line that requires the greatest distance to the nearestintersection of extension before meeting. This line at that point is said tobe locally parallel to the original line and this line is unique.

Because in spherical geometry the axiom of parallels is no longer valid,many of the usual rules of Euclidean geometry no longer hold. The sum ofthe interior angles of a triangle do not add to π but is always greater thanπ. Think of a triangle on the sphere of the earth formed by the equator andtwo lines of longitude. At the equator the two lines are locally parallel andthe angle between them and the equator is π

2 . They will meet at the northor south pole at some non-zero angle and thus the sum of all three anglesis greater than π. Make a square, a four sided figure of equal length sideswith all sides meeting at right angles, on the surface. In contrast to theEuclidean case, it does not stop and start at the same point but over-closes,two of the legs of the square meet before the full side length is achieved.A third test is to make a circle, a set of points that are equidistant fromsome point, on the earth. The ratio circumference

2r , where r is the distancefrom the point to the circle defined as the radius, is less than π. To mostpeople this is trivial. The problem is that we are measuring on the surface ofthe sphere. In the underlying three dimensional space in which the sphereis imbedded, the geometry is Euclidean and the world makes sense. Forinstance, if, instead of the distance as measured from the center on thesphere, the distance used, r′, is the distance to the axis that is perpendicularto the plane of the circle passing through the center, the usual result thatthe ratio circumference

2r′ is π. Because this first identification of a non-Euclideangeometry was on an imbedded sphere, these non-Euclidean geometries arenow called curved spaces. This is an unfortunate accident of history as wewill discuss shortly but it is so prevalent that everyone uses these terms andwe will continue to use this nomenclature. Geometries are flat, Euclidean, orcurved, non-Euclidean, with an example being a two dimensional sphericalsurface imbedded in a flat, Euclidean, three space.

9.2. GAUSSIAN CURVATURE 151

9.2 Gaussian Curvature

The next significant step in the development of modern geometry was takenby the great mathematical physicist Gauss. Gauss was interested in thegeneral problem of the shape of a two dimensional surfaces in our threedimensional space. Instead of a plane, the basis for Euclidean geometry, ora sphere the basis for spherical geometry, consider a two dimensional surfacein the shape of a pear imbedded in three space. At a point on the surfacethere are various curvatures, using an intuitive idea that will be articulatedwith greater care shortly. At the points near the bottom or top of the pearthe surface is much like that of a sphere while in the neck region there isa another type of bend. Also at any point, if the region of examination issmall enough, the geometry acts as if it is Euclidean or flat, i. e. for a smallenough triangle, the sum of the interior angles of triangles is π.

In order to proceed, Gauss needed a definition of curvature. It had tobe local, at a point, and agree with our intuitive notions about curvature.The basic idea is that, on a curved surface, as you move through nearbypoints on the surface, the normal to the surface changes direction. Thus heproduced the following construction: as you move over an element of areaon the surface, the tip of the unit normal will paint an area on the unitsphere, see Figure 9.1. the curvature at a point on the surface is the ratio

12

3

1 2

3As

An

Surface

Figure 9.1: Gauss’s Definition of Curvature Gauss defined curvature asthe ratio of the area generated by the tips of the unit normals, An, for anelement of area, As, on the surface as the area on the surface, As, goes tozero, KG ≡ limAs→0

AnAs

.

of the area generated by the tips of the unit normals, Arean, for an elementof area, Areas, on the surface as the area on the surface goes to zero,

KG ≡ limAreas→0

Arean

Areas. (9.1)


In order to appreciate the subtlety of this construction, let’s considerseveral examples. A flat surface has no curvature since the normal is alwaysthe same and thus the Arean that is generated is that of a point and thus theArean is zero. On a sphere of radius r, using the usual spherical coordinates,θ and φ, a patch of Areas = r2δθδφ and the normal which is the radius vectorgenerates an Arean = δθδφ. Thus the curvature is 1

r2 . This constructionshows that this idea of curvature makes sense and that the limit defining itexists for reasonably shaped surfaces. Also note that in the limit of larger the curvature is zero. Now consider a point on the neck of the pearmentioned above. Another example and probably easier to visualize is aPringle potato chip, see Figure 9.2.

12

3As 1

32

An

Figure 9.2: Curvature of a Pringle A Pringle is an example of a negativelycurved surface. The area, An, generated by the normals to the surface, As,at any point is not zero. The difference between this case and the spherethough is that the area, An, is oppositely oriented from that of the area onthe surface, As, i. e. a right hand coordinate plane on As generates a lefthanded coordinate system on An, see Section 9.3.

9.3 Example of negative curvature: the Pringle

I have no idea how Pringles are manufactured, but I will construct myPringle-like surface by taking a circle of radius R1 centered on the originin the two plane, (x, z), displacing it by R2, R2 > R1, and then makingthis circle a surface of revolution about the z axis. This generates a torusor donut shape. We can take a segment of the inner surface, the surfacetoward the z axis, as our Pringle.

The advantage of this construction is that the labeling of points on thesurface and the properties of the normal vector can be determined easily. Forexample, a point on the surface can be determined from the angle around

9.3. EXAMPLE OF NEGATIVE CURVATURE: THE PRINGLE 153

the original circle as measured from the top most point, θ, and the angle ofrotation of the circle around the z axis, φ both ranging from zero to 2π.

Using these coordinates, a point on the surface is at

x = [R2 −R1 sin θ] cos φ

y = [R2 −R1 sin θ] sinφ

z = R1 cos θ, (9.2)

and the area, As, generated by incrementing the two coordinates whichare orthogonal is [R2 − R1 sin θ]R1δθδφ. The unit normal vector is alongthe line from the center of the circle at φ and the point on the surface orn = (− sin θ cos φ)x + (− sin θ sinφ)y + cos θz. As the area As is swept out,the change in the unit normal is δn = (− cos θ cos φδθ + sin θ sinφδφ)x +(− cos θ sinφδθ − sin θ cos φδφ)y + (− sin θδθ)z. Again the lines swept outby the coordinate increments are orthogonal and the area, An, generated issin θδθδφ. The Gaussian curvature is |KG| = sin θ

(R2−R1 sin θ)R1.

I have put absolute value signs on this result because the curvature inthis case is actually negative. You should realize that, if we choose thecoordinate directions in As to be right handed in the sense that the normalis outward and generated by rotating directed lines at constant θ into linesof constant φ, then the area An is left handed in the sense that the imagetraces of constant θ and φ are now left handed. This change in orientationof the areas is the indicator that this curvature is negative and thus

KG = − sin θ

(R2 −R1 sin θ)R1. (9.3)

There are other features of this result that are worth commenting on.The obvious result that the curvature is independent of φ is expected. Moreintriguing is the θ dependence, KG(θ). Note that, had we done the analysisfor the region π < θ < 2π, the orientation of the image plane would havebeen the same as the original element of surface and thus, as given byEquation 9.3, the curvature is positive. At θ = π

2 , the curvature is KG(π2 ) =

1(R2−R1)R1

. The square root of the inverse of the curvature is the geometricmean radius of the two circles that make up the surface at this point, theradius of our original circle and the radius of the surface from the axis ofsymmetry, the z axis. This same observation is also valid for the θ = 3π

2 .This is a general result that we will deal with in more detail in the nextsection, Section 9.4. The other interesting set of points is at θ = 0 andθ = π. Here, the curvature is zero. This can be looked at in two ways.These points are the transition points from the region of negative curvature,


the inside of the torus, and the region of positive curvature, the outside ofthe torus. Since we expect the curvature to smooth, it is required that thecurvature vanish at these points. More significantly, This region really isflat in the sense that it is Euclidean.

Think of a cylinder. The curvature of a cylinder is zero – the normalmoves along a line as you move around the cylinder but does not change asyou move along the axis of the cylinder. Thus, the area, An is zero. It is alsoimportant to note that the geometry of the cylinder is the same as that ofa flat plane; you can unroll the cylinder onto a flat plane. You can do yourgeometry in the flat plane with the straight lines being the same as usualand the geometry is Euclidean, interior angles of triangles add to π. Thusthe cylinder can be covered entirely by a single flat map. You cannot covera curved surface entirely with a single flat map. You can cover it locallybut at some places the distortion caused by the mapping becomes so severethat points are mapped to lines and visa versa. Think of a map of the earth.The usual atlas projection treats the poles, points, as lines. If you excludethe anomalous points by restricting the range of the coordinates you do notcover the earth with a single map but need more than one flat map. This isalso a general property of non-Euclidean spaces. Is a cone flat or curved?

9.4 Curvature and Geodesics

In order to proceed further, we will have to examine the general issue ofcurves in the surface. An arbitrary path connecting two points in the surfacecan have lots of turns and bends. There are two sources of these, the bendsof the surface and the bends of the path within the surface. We can eliminatethe bends within the surface by considering only straight line paths betweenthe points. These, by definition, are the shortest distance paths between thepoints. Since these may be very curved instead of calling them straight linesa better name is geodesic. One of many theorems of the theory of surfacesis that these are unique. These geodesic paths thus contain the bends ofthe surface and only those bends. In Section 9.5, we will develop a specificdifferential condition for geodesics that is valid in any coordinate system.For now, we will continue with the more intuitive notions of their properties.

Remembering that our two surface is imbedded in a flat three space, wecan identify three directions at any point on the path, the direction alongthe local tangent to the path, the direction in the surface perpendicularto that direction (Don’t forget that, at a point on the surface for a smallenough region, the surface is flat and thus this direction is known. To find

9.4. CURVATURE AND GEODESICS 155

it, pick another point on the surface not on the original straight line anddraw another geodesic through it. These two paths determine a plane, thetangent plane. All geodesics through p share this tangent plane.), and thedirection that is perpendicular to these two. This last direction is locallyperpendicular to the surface in the sense that the two other directions havegenerated the tangent plane at the point. This direction is called the normaldirection. We already took advantage of these ideas in the identification ofthe normal to the surface in the previous section, Section 9.2, in which weconstructed the Gaussian curvature.

In the neighborhood of the point, the original geodesic is contained inthe plane formed from the normal direction and the tangent direction ofthe geodesic. In the neighborhood of the point p, pick two other pointson the original geodesic on opposite sides of p but near p, which will allbe in that plane. As is well known from analytic geometry, three pointsdetermine a circle. This circle is called the osculating circle. Osculating isfrom the latin word for “kissing.” In some sense, the idea of the osculatingcircle is the next step up from the tangent. The tangent is determined bytwo nearby points, determines a magnitude and a direction, and in the limitleads to the concept of the derivative. The osculating circle is determinedby three nearby points and utilizes the second derivative, the difference intwo tangents, the tangents formed from the original point and the othertwo points. The inverse of the radius of this osculating circle is called thecurvature of the original geodesic. Remember that by using geodesics, thereis no bending in the surface. All the bending is due to the surface. There isanother geodesic through p that is orthogonal. On that geodesic, constructan osculating circle. Thus at p, for a pair of orthogonal geodesics, there aretwo osculating circles, one for each of the mutually orthogonal geodesics. Asthe orientation of this orthogonal pair of geodesics is varied, there will bea direction in which the curvature for each of the orthogonal geodesics willbe an extremum. There is no other orientation of the geodesics that haveextremum curvatures except trivial variations on this orientation. This lastresult is called Euler’s Theorem. Gauss showed that the Gaussian curvatureof the surface as defined in Section 9.2 is the product of these two extremumcurvatures,

KG =1

R1 ×R2, (9.4)

where R1 and R2 are the radii of the osculating circles. In addition, thesign of the curvature is determined by the relationship of the two osculatingcircles. The curvature is positive if both the osculating circles are on thesame side of the surface. This is the case for the sphere as discussed earlier.


For the Pringle, Section 9.3, on the inner edge, the osculating circles are onopposite sides of the surface and this is the signature of negative curvature.

As is always the case with the Gaussian curvature, this curvature is anbasic property of the surface and does not depend on the coordinate systemthat we used to make the construction. Granted that the construction ofthe curvature is most readily done in a coordinate system that is based on asystem of orthogonal geodesics, it is still clear from the nature of the Gaussmap and Equation 9.10 that the coordinates make the construction possibleby staking out the grid but that the local value of the curvature is the sameregardless of the coordinate system used. In fact the coordinate system thatwas used for the torus, Equation 9.2, are not geodesic coordinates; the linesof constant φ are geodesics but the lines of constant θ are not. This issuewill be discussed in much greater detail later, see Sections 9.5, 9.6, 9.7 andAppendix C.

9.5 The Theorema Egregium and the Line Ele-ment

As is clear from Section 9.4, Gauss made an extensive study of the natureof surfaces imbedded in a Euclidean three space. He is responsible for manyof the insights and theorems that govern understanding of these surfaces.He was, of course, interested in two surfaces imbedded into the larger threespace. He recognized the important role of curvature in defining the natureof the surface; to within an orientation and a translation, the surface is deter-mined by its curvature. His most famous theorem in the theory of surfaceswas so striking to him that when he recognized its implications he gave it thetitle of the Theorema Egregium. A direct translation of the latin would callthis the egregius theorem. The modern sense of egregius: outstandingly badis not the original meaning. The original use of the word was in the sense ofoutstandingly good and is what is intended in the latin. It was later usagethat lead to the current interpretation of egregious as outstandingly bad,see [OED 1971]. It seems that modern young people are not the first onesto reverse the meaning of bad and good when describing things. Regardless,the point of Gauss’ name for the theorem was in the sense of outstandinglygood. Maybe a better translation would be the Extraordinary Theorem.

This theorem proved that all the important properties of the surfacecould be developed from information that is intrinsic to the surface anddid not need to use properties that were determined by the imbedding ofthe surface in a Euclidean three space or the coordinate system that was

9.5. THE THEOREMA EGREGIUM AND THE LINE ELEMENT 157

used to do the construction of the Gauss map. The only element that isneeded to construct curvature is the length of the line element in whatevercoordinate system is being used. In other words, if when you begin tolabel points on the surface with some set of coordinate labels and, if at thesame time, you determined the actual lengths separating nearby coordinatepoints, you would have all the information that you need to determine thecurvature. The other amazing fact is the realization of Riemann that thesetechniques developed by Gauss carry over to manifolds of any number ofdimensions, Section 9.7. The theorem’s proof is rather tedious and notreally enlightening except in its use of intermediate elements that are veryimportant in our later study of geometry in higher dimensions and thus isworth the effort here for providing and intuitive understanding of the natureof these seemingly abstract quantities.

9.5.1 Proof of Theorema Egregium

Because he was studying two surfaces imbedded in a Euclidean three space,Gauss knew immediately the distances between nearby coordinate labeledpoints. Given a two surface in the usual three space with a coordinatesystem, (q1, q2), on it, the distance between nearby coordinate labeled pointsis

dl2 = ∆~x ·∆~x =2∑

i,j=1

∂~x

∂qi· ∂~x

∂qjδqiδqj . (9.5)

This form is called the line element in that coordinate system. The element,∂~x∂qi · ∂~x

∂qj can be written

gij(q1, q2) ≡ ∂~x

∂qi· ∂~x

∂qj(9.6)

is called the first fundamental form or more commonly the metric. Thisnotation is consistent with the Einstein convention, see Section 4.6, and isactually a second rank tensor as indicated, see Appendix C.

We had earlier examples of metrics in coordinates systems in space-timein Section 6.5. Basically, the Theorema Egregium is the statement thatthe metric is all that is needed for the construction of the curvature at anypoint on the surface. In other words, we will construct the Gauss map forany point using only the metric and its derivatives. The line element nowappears as

dl2 = gij(q1, q2)∆qi∆qj (9.7)


where I have also introduced a common notation called the Einstein con-vention. In any term in an equation, when an index is repeated lower onone variable and upper on another, it is summed over the range of values forthat index. There are deeper concepts implied by this notation as explainedin Appendix C but for now we are merely using it just as a summationconvention.

As a step in proving the theorem, note that ~x1 ≡ ∂~x∂q1 and ~x2 ≡ ∂~x

∂q2 aretangents to the surface at the point labeled (q1, q2). The vectors ~x1 and~x2 can be used to form a basis in the tangent plane at that point. Thesevectors and the normal,

~n =~x1 × ~x2

|~x1 × ~x2|, (9.8)

at that point can be used as a basis in the three space for vectors at thatpoint.

In order to construct the Gauss map, we will need the change in thenormal as the coordinate value increases. Define ~n1 ≡ ∂~n

∂q1 and similarly for~n2. Since ~n is of fixed length,

~n2 = 1,

these vectors lie in the tangent plane

~ni · ~n = 0.

These vectors can now be expanded in the tangent plane using the ~xi

basis,~ni = −bj

i~xj . (9.9)

The sign convention used here is for convenience later.In addition for small advances in the coordinate, these vectors generate

the area of the Gauss map via

Arean = |~n1 × ~n2| δq1δq2. (9.10)

Since the area of a patch for a small advance in the coordinates isAreas = |~x1 × ~x2| δq1δq2, the Gaussian curvature is

∣∣KG

(q1, q2

)∣∣ = limδq1,δq2→0

|~n1 × ~n2| δq1δq2

|~x1 × ~x2| δq1δq2=|~n1 × ~n2||~x1 × ~x2|

(9.11)

The sign of the curvature can be recovered from these cross products whenit is realized that each of the cross products is a vector in the Euclideanthree space and in the space of the Gauss map. If the orientation of the


coordinate advance is maintained in both spaces and the vectors of thecross product are the compared the sign of the curvature can be determined.Using Equation 9.9,

~n1 × ~n2 =(−bi

1~xi

)×(−bj

2~xj

)= bi

1bj2 (1− δij) (−1)j ~x1 × ~x2

=(b11b

22 − b2

1b12

)~x1 × ~x2

= det (b) ~x1 × ~x2 (9.12)

where~xi × ~xj = (1− δij)(−1)j ~x1 × ~x2 (9.13)

and

δij =

1 if i = j0 otherwise

(9.14)

is the usual Kronicker delta and the set of elements bji are treated as a

matrix. The factor det (b) carries the information of the relative orientationof the cross products. Thus,

KG

(q1, q2

)= det(b) (9.15)

Define ~xij ≡ ∂2~x∂qi∂qj

. This vector can be expanded in the three vectorbasis defined by ~x1, ~x2, and ~n at (q1, q2),

~xjk = Γijk~xi + βjk~n. (9.16)

It is worthwhile to point out that the expansion coefficients have indicesthat are meant to be consistent with our summation convention; they maynot be tensors of the indicated rank, see Appendix C. Since ~xjk = ~xkj ,

Γijk = Γi

kj (9.17)

andβjk = βkj . (9.18)

The first coefficient of the expansion, Γijk, is called the Christoffel symbol of

the second kind or connection. It is simply related to the Christoffel symbolof the first kind, Γjk,l since

Γjk,l ≡ ~xjk · ~xl =(Γi

jk~xi + βjk~n)· ~xl = Γi

jkgil. (9.19)


The latter term,

βij ≡∂2~x

∂qi∂qj· n (9.20)

is called the second fundamental form. It is an easy matter to connect theelements of the second fundamental form, βjk to the expansion coefficientsbij in Equation 9.9 by differentiating ~xj · ~n = 0.

0 =∂ (~xj · ~n)

∂qk

= ~xjk · ~n + ~xj · ~nk

= βjk + ~xj ·(−bi

k~xi

)= βjk − bi

kgji

orβjk = bi

kgij . (9.21)

Treating all the elements as matrices,

β≡(

β11 β12

β21 β22

)=(

g11 g12

g21 g22

)(b11 b1

2

b21 b2

2

)= Gb,

where the respective matrix definitions are obvious.Thus taking the determinant of both sides, det (β) = det (G) det (b) or,

using Equation 9.15,

KG

(q1, q2

)=

det (β)g

. (9.22)

where g ≡ det (G).Defining ~xijk ≡

∂~xjk

∂qi ,

~xjkl =∂Γi

jk

∂ql~xi + Γi

jk~xil +∂βjk

∂ql~n + βjk~nl.

The ~xijk can be expanded in terms of our vector basis at(q1, q2

). Using

Equations 9.9 in the fourth term and Equation 9.16 in the second,

~xjkl =∂Γi

jk

∂ql~xi + Γp

jk

(Γi

pl~xi + βpl~n)

+∂βjk

∂ql~n− βjkb

il~xi

=

(∂Γi

jk

∂ql+ Γp

jkΓipl − βjkb

il

)~xi +

(∂βjk

∂ql+ Γp

jkβpl

)~n. (9.23)


Since the order of the derivatives is irrelevant and we get a similar expressionfor ~xjlk, we can construct the null vector

~0 = ~xjkl − ~xjlk

=

(∂Γi

jk

∂ql−

∂Γijl

∂qk+ Γp

jkΓipl − Γp

jlΓipk − βjkb

il + βjlb

ik

)~xi

+(

∂βjk

∂ql−

∂βjl

∂qk+ Γp

jkβpl − Γpjlβpk

)~n.

Since the ~xi and ~n form a basis, the coefficient of each term must be zero.Defining the mixed Riemann curvature as

Rijkl ≡

∂Γijl

∂qk−

∂Γijk

∂ql+ Γp

jlΓipk − Γp

jkΓipl, (9.24)

we haveRi

jkl = βjlbik − βjkb

il. (9.25)

The Riemann curvature is defined as

Rhjkl ≡ gihRijkl, (9.26)

and therefore satisfies

Rhjkl = gih

(βjlb

ik − βjkb

il

)= βjlβkh − βjkβlh. (9.27)

Thus a particular component of the Riemann curvature is related to thecurvature since

R1212 = det (β) , (9.28)

orKG

(q1, q2

)=

R1212

g. (9.29)

It may seem strange that one particular component of the 24 compo-nents plays such a special role. This happens in the special case of a twodimensional manifold. I am now using a the more sophisticated title for aspace. A manifold is a space that is defined in such a way that the usualrules for calculus can be applied. Obviously, we want to deal with manifolds.The Riemann curvature enjoys a great deal of symmetry under interchangeof its indices. These will be discussed further in Section 9.7. Suffice it tosay here that in two dimensions, there is only one independent Riemann


curvature and that all the other can be written in terms of that one, chosento be R1212 as

R1212 = −R2112 = −R1221 = R2121

R1111 = R1122 = R2211 = R2222 = 0 (9.30)

orRhjkl = (ghkgjl − ghlgjk)

R1212

g. (9.31)

Despite the great progress in finding new and interesting ways to expressthe curvature, we have not delivered on our original promise to show thatthe construction of the curvature can be done intrinsically; that is by usingonly the metric and its derivatives. First, the relationship between theChristoffel symbol of the first kind and the Christoffel symbol of the secondkind, Equation 9.19, can be expressed as a matrix equation,

Γj ≡(

Γj1,1 Γj1,2

Γj2,1 Γj2,2

)=(

Γ1j1 Γ2

j1

Γ1j2 Γ2

j2

)(g11 g12

g21 g22

)= ΓjG.

This equation can be solved by using the inverse matrix for the metric.

Γj=ΓjG−1.

The inverse metric matrix can be written in our summation conventionnotation since

GG−1 = 1(g11 g12

g21 g22

)(g11 g12

g21 g22

)=

(1 00 1

)gikg

kj = δji (9.32)

or since the inverse of a two by two matrix is well known

G−1 ≡(

g11 g12

g21 g22

)=

1g

(g22 −g12

−g12 g11

). (9.33)

In the index notation, the Christoffel symbols are related as

Γljk = gliΓjk,i. (9.34)

Taking the derivative of the metric,

∂gik

∂qj=

∂ (~xi · ~xk)∂qj

= ~xij · ~xk + ~xi · ~xkj

= Γij,k + Γjk,i.


Similarly, calculating ∂gjk

∂qi and −∂gij

∂qk and adding the results,

Γij,k =12

(∂gik

∂qj+

∂gjk

∂qi− ∂gij

∂qk

). (9.35)

With the Christoffel symbols in terms of the derivatives of the metric, wehave completed the proof of the Theorema Egregium.

In addition, an important feature of the geometry of surfaces is thenature of the shortest lines or geodesics. The idea of the shortest line or“straight line” is an intuitive one and can be carried over to lines on a twosurface. Again, if the two surface is imbedded in a Euclidean three space, it iseasy to think of the line that bends only as much as required to remain in thetwo surface and that is the straightest line. That this line is also the shortestdistance between two points is again intuitively satisfying. This intuition isreinforced by the realization that a curve in three space can be generated bythe intersection of two surfaces. If in the case of our surface of interest wehave two points and we want of find the straightest line between them, wecan chose from the infinity of planes passing through the two points the onethat produces the shortest length line. This line will also be the straightest.The best known example of this is the ”straight lines” on the surface of theearth, see Section 9.6.2. In this case, these lines are the intersection of thesphere and the plane that passes through the two points and the center ofthe sphere, the great circle path. This construction is clearly and extrinsicconstruction and, if these were the only techniques at hand, we would alwayshave to deal with spaces only after imbedding them. Again, to reduce thisproblem to an analysis of the intrinsic properties of the space, we will needa procedure that deals with the metric and its derivatives. The geodesic isdefined as the curve in the surface between the points that has the least valuefor the integral of the line element over that curve. As we will see in theexamples, Section 9.6, this definition is more restrictive than the intuitiveone of straight line above but the geodesic curves are those that bend theleast.

Putting this idea into practice, define the curve dependent path lengthbetween two points as

S(q10, q

20; q

1f , q2

f

)[path] ≡

∫ q1f ,q2

f

q10 ,q2

0 ;pathds (9.36)

where ds is line element

ds =

√gij

dqi

ds

dqj

dsds, (9.37)


and the path is a set of parametric functions,(q1 (s) , q2 (s)

), of the path

length, s, which requires that

1 = gijdqi

ds

dqj

ds. (9.38)

Equations 9.37 and 9.38, look as if they are trivial but when Equation 9.37is substituted into Equation 9.36,

S(q10, q

20; q

1f , q2

f

)[path] ≡

∫ q1f ,q2

f

q10 ,q2

0 ;path

√gij

dqi

ds

dqj

dsds (9.39)

and s is interpreted as a time, this would be the action for a system with

two degrees of and a Lagrangian of the form√

gijdqi

dtdqj

dt . Then we can usethe analysis of Section A.2 to minimize the length. The two equations, theEuler-Lagrange equations, Equation ??, that result from the variation of thefunctions q1 (s) and q2 (s) are

d

ds

δ

δ(

dql

ds

) (√gijdqi

ds

dqj

ds

)− δ

δql

(√gij

dqi

ds

dqj

ds

)= 0 (9.40)

for l = 1, 2. These equations must be supplemented by the condition thats be the path length, Equation 9.38. This result accomplishes our goal ofexpressing all the important geometric issues intrinsically, in terms of gij

and its derivatives.These geodesic equations can be cast into a more geometric form. Car-

rying out the variational derivatives in Equation 9.40,

d

ds

122glj

dqj

ds√gij

dqi

dsdqj

ds

−12

∂gij

∂qldqi

dsdqj

ds√gij

dqi

dsdqj

ds

= 0.

Expanding the total s derivative and using the fact that Equation 9.38implies that d

ds

(gij

dqi

dsdqj

ds

)= 0,

1√gij

dqi

dsdqj

ds

∂glj

∂qm

dqm

ds

dqj

ds+ glj

d2qj

ds2

−

12

∂gij

∂qldqi

dsdqj

ds√gij

dqi

dsdqj

ds

= 0.

or

0 = gljd2qj

ds2+

∂glj

∂qm

dqm

ds

dqj

ds− 1

2∂gij

∂ql

dqi

ds

dqj

ds


= gljd2qj

ds2+

12

∂gil

∂qm+

∂glj

∂qm− ∂gij

∂ql

dqi

ds

dqj

ds

= gljd2qj

ds2+ Γij,l

dqi

ds

dqj

ds, (9.41)

where I have relabeled the summed indexes. Using the inverse metric,

d2ql

ds2+ Γl

ij

dqi

ds

dqj

ds= 0 (9.42)

which is the geodesic equation using the connection or Christoffel symbol.The important role of the connection in association with derivatives will beclarified in Section 9.7 and in Appendix C.

Two further points to note. Although we are now in a position to beable to find the curvature and geodesics from the line element in a givencoordinate system, does this knowledge tell us all that we need to knowabout the surface and, clearly, the choice of coordinates has to be irrelevantto these larger questions of what is important to know about the surface.Consider a surface imbedded in a Euclidean three space with a coordinatemesh defined. Any rigid shift in the position or rigid rotation of the surfacein the three space does not produce an important change. Since both thefirst fundamental form and second fundamental form are defined in termsof derivatives of the inner products of the three space coordinates, these areunchanged by these rigid body shifts. Thus, the curvature is not affected bythese changes and, in the sense that rigidly rotated and/or shifted surfacesare the same, the curvature is the determinant of the shape of the surface.

The problem of the coordinate dependence of intrinsic knowledge of thesurface is not as simple. Two coordinate systems require that the samepoint in three space have two valid designations, ~x

(q1, q2

)= ~x

(q′1, q′2

).

Consistency requires that there is a pair of functions, q′1 = q′1(q1, q2

)and

q′2 = q′2(q1, q2

). In either coordinate system the line element is the same,

ds2 = gijdqidqj

= g′ijdq′idq′j

= g′ij∂q′i

∂ql

∂q′j

∂qmdqldqm,

where labels of summed indices were changed appropriately. Thus the metricin the primed coordinate is related to the metric in the unprimed coordinateas

glm = g′ij∂q′i

∂ql

∂q′j

∂qm. (9.43)


This type of transformation is the defining nature of a tensor object, seeAppendix C; the metric is a second rank tensor. Not all indexed objectsare tensors. For example the Christoffel symbols are not, see Section 9.7and Appendix C. The quantity, g = det (G), looks like a scalar but is not.Equation 9.43 can be cast in the form of a matrix equation:

G = ∂q′

∂qG′ ∂q′

∂q

T

where∂q′

∂q≡

(∂q′1

∂q1∂q′2

∂q1

∂q′1

∂q2∂q′2

∂q2

). (9.44)

and ∂q′

∂q

Tis the transpose. Thus

g =(det(

∂q′

∂q

))2g′. (9.45)

The four indexed objects Rijkl are tensors of the indicated nature andthus change in different coordinate systems. The curvature on the otherhand is the same in all coordinate systems. This is most easily realizedby examining the Gauss map definition. Using intrinsic techniques, thecurvature is given by KG = R1212

g . Since both R1212 and g are coordinatedependent, it is worth the effort to show the coordinate independence of thecurvature:

R1212

g=

∂q′i

∂q1

∂q′m

∂q2

∂q′n

∂q1

∂q′o

∂q2

R′imno

g

=∂q′i

∂q1

∂q′m

∂q2

∂q′n

∂q1

∂q′o

∂q1

(g′ing′mo − g′iog

′mn

) R′1212

g′g

= (g11g22 − g12g21)R′

1212

g′g

=R′

1212

g′. (9.46)

There is an interesting and ultimately very important counting argumenthere. We have shown that the important features of the surface is thecurvature. It is also independent of the coordinate system used to computeit. In the Theorem Egrigium, we have shown that the we can constructthe curvature from the metric and its derivatives. The metric is composedof three independent functions, g11, g22 and g12 = g21 and is coordinatedependent. The coordinate information is contained in two functions. Thus

9.6. EXAMPLES OF TWO DIMENSIONAL GEOMETRY 167

it makes sense that there is one, 3−2, coordinate independent function, thecurvature, for a two dimensional surface. In higher dimensional cases thispattern will change but the counting idea will still apply, see Section 9.7.

9.6 Examples of Two Dimensional Geometry

In the following subsections, we will examine several examples of two ge-ometries. In all cases, we will attempt to understand the situation froman analysis of the line element alone as is consistent with the TheoremaEgregium, Section 9.5. When that gets too complicated, we will resort toour understanding of the space as it is imbedded into the larger three space.

9.6.1 Euclidean Two Plane in Polar Coordinates

The simplest case has to be the Euclidean two plane. In Cartesian coordi-nates it has the line element ds2 = dx2 + dy2. The metric is gxx = gyy = 1and gxy = 0. All the connections are zero. The curvature is zero. Thegeodesic equation is

d2x

ds2= 0

d2y

ds2= 0 (9.47)

with the general solution

x = as + b

y = cs + d. (9.48)

for all a, b, c, and d.Note that a geodesic is a more specific than just a straight line. The

geodesic in Equation 9.48 is a straight line but to be a geodesic the line mustalso advance linearly in the length element. In other words, the system

x = as2 + b

y = cs2 + d (9.49)

is a straight line but not a geodesic.This simplicity of the Euclidean two plane can be masked by going to

a different coordinate system. A common example is the polar coordinates.In this coordinate, the line element is ds2 = dr2 + r2dθ2. In order to remove


the tendency to identify the underlying space, relabel the coordinates, y →r, x → θ, so that the line element is expressed as

ds2 = y2dx2 + dy2, (9.50)

Actually, it would be more consistent with the spirit of this section to findthe line element from the coordinate transformation. In this set of labels,the coordinate transformation is

x = tan−1

[y0

x0

]y =

√x2

0 + y20, (9.51)

where x0 and y0 are the cartesian coordinates and some care must be exer-cised in the handling of the tan−1 function and the inverse,

x0 = y cos (x)y0 = y sin (x) . (9.52)

The transformation matrix is

∂x0

∂x≡

(∂x0∂x

∂y0

∂x∂x0∂y

∂y0

∂y

)=(−y sinx cos xy cos x sinx

).

Equation 9.44 then yields gxx = y2, gxy = gyx = 0, and gyy = 1, the metricin Equation 9.50.

The coordinate cover of the Euclidean two plane is shown in Figue 9.3.In the (x, y) coordinate, the non-trivial connections are

Γxxy =

1y

= Γxyx

Γyxx = −y. (9.53)

The geodesic equations are

d2x

ds2+

2y

dx

ds

dy

ds= 0

d2y

ds2− y

(dx

ds

)2

= 0. (9.54)

From Equation 9.54, the curves generated by dxds = 0 or x = c′ and y = a′s+b′

are geodesics for all a′, b′ and c′. These are the straight lines which would


Lines of constant x

Lines of constant yx=0

x=2π

y=0

Figure 9.3: Cover of the Plane in Polar Coordinates The coordinatesystem that covers the Euclidean two plane. y is what is commonly calledthe radius and its range is 0 ≤ y < ∞. x is θ and its range is 0 ≤ x < 2π.

project through the origin identified with θ = θ0 and the radius increasinglinearly in length. These are not all the geodesics but only the subset ofEquation 9.48 with a = 0 and 0 ≤ b ≤ 2π and 0 ≤ c, d. This set is a smallsubset of the the geodesics, Equation 9.48, since it admits only lines throughthe origin. That the set of curves in Equation 9.48 in the cartesian coordinateare solutions of Equation 9.54 when substituted into Equation 9.51 for alla, b, c and d can be seen by direct substitution.

It is also an interesting exercise to do some simple geometry in the Eu-clidean two plane in this coordinate. First, draw a fan of geodesics aboutsome point in the Cartesian plane. Figure 9.5 is a fan of equal openinggeodesics about a general point in the plane in the Cartesian plane and theimage fan in the polar plane. The rays in the polar plane are given by

x = tan−1

[sin (φi) s + y0i

cos (φi) s + x0i

]y =

√(cos (φi) s + x0i)

2 + (sin (φi) s + y0i)2, (9.55)

where φi is the opening angle to the ith ray in the cartesian plane and s isthe length parameter of the ray which ranges from 0 to 1. That this is thecorrect identification for the parameter s in the polar plot follows from thefact that we have the same line element in both coordinates or, saying the

same thing, that from Equation 9.55, y(s)2(

dxds

)2+(

dyds

)2= 1.

The rays of the fan in Figure 9.5 are all of unit length. The same raysare shown in the two coordinate systems in Figure 9.5. Since, in this case,the rays all have unit length, the tips of rays trace out a circle of unit radius.


-1 1 2 3

-1

1

2

3

1

2

3

4

5

6

7

8

1 2 3 4 5 6

0.5

1

1.5

2

2.5

3

3.5

4

123

4

5

67

8

Figure 9.4: Fan of Geodesics in the Euclidean Two Plane On the left isa fan of geodesics starting at the point (x0i , y0i) in the Cartesian coordinates.On the right is the same fan shown in the polar coordinates plane. Thecorresponding rays in each case are numbered. Since the rays all have unitlength, the tips also trace out the shape of the circle of radius 1, the locusof places that are unit distance from the point

(tan−1

(y0ix0i

),√

x20i

+ y20i

).

In this case, x0i = 1.5 and y0i = 2.

In other words, we can construct a circle of radius r in the polar plane byreplacing s in Equation 9.55 by r and allow φi to range continuously fromzero to 2π. Using this form for the circle in the polar plot, we can see thatthe angles between any two intersecting geodesics is the same in the polarcoordinates as in the Cartesian. Writing the line element as a function of φi

for fixed r, y(φi)2(

dxdφi

)2+(

dydφi

)2= r2. Thus the ratio of the arc length of

a circle to the radius is

∆θ ≡∫arc ds

r=∫

∆φi

dφi = ∆φi (9.56)

which, as stated above, is the angle between the corresponding geodesics inthe Cartesian plane.

Thus if, as discussed in Section 9.1, we do simple geometry in the polarcoordinate plane, we will get all the usual results that one obtains for aEuclidean two plane: the sum of the interior angles of a triangle is π, theratio of the circumference to the radius of a circle is 2π, a square will close.Although these figures can look rather bizarre in the polar coordinates (drawa triangle with one vertex on the origin and examine the sum of the interior


angles), this space is flat. The bizarre appearance of geometric figures is anaccident of the coordinate choice. It is not due to an intrinsic property ofthe space. This will be a problem in some of our examples in space-time,see Section ??, – a coordinate choice that is nice for some applications canhave a non-physical appearance for certain circumstances.

9.6.2 The Unit Two Sphere in Euclidean Three Space

θ

ϕ

Figure 9.5: Unit Two Sphere The coordinates θ and φ defined on the unittwo sphere.

Another well known example is the unit two sphere imbedded into aEuclidean three space. Using the usual spherical polar coordinates, the lineelement is well known to be

ds2 = sin2 (θ) dφ2 + dθ2, (9.57)

where 0 ≤ φ < 2π and 0 ≤ θ < π. From this, the non-vanishing connectionsare

Γφφθ = cot (θ)

Γθφφ = − cos (θ) sin (θ) , (9.58)

and the geodesic equations are

d2φ

ds2+ 2 cot (θ)

dθ

ds

dφ

ds= 0


d2θ

ds2− cos (θ) sin (θ)

(dφ

ds

)2

= 0. (9.59)

The curves φ = φ0, dφds = 0, and θ = as + θ0 are geodesics lines. By

requiring that s be given by Equation 9.57, a = 1. Of course, these are thewell known lines of longitude passing through the point (φ0, θ0). Thus withsuitable evolution in angle, θ, all lines of longitude are geodesics. Theselines all have length 2π. For θ = θ0, dθ

ds = 0, the only geodesics that areeasy to see in Equation 9.59, are at θ0 = 0, π, π

2 . From the line element,Equation 9.57, two of these, θ0 = 0 and π, have zero length and are thus notlines at all. The remaining case is the equatorial line and, with the properevolution of φ, φ = s + φ0, is a geodesic. Again, the length of this geodesicis 2π.

Of course, these are not all of the geodesics. There is an infinity ofthem through each point in the two plane. Instead of solving the geodesicequations, Equations 9.59, we will us the extrinsic properties of the sphereimbedded in a Euclidean three space. Given the results of the TheoremaEgregium, Section 9.5, we should not resort to such subterfuge but it is clearfrom the form of the geodesic equations that general solutions are not easyto obtain. On the other hand, the sphere is so symmetric that we can rotateabout any axis through the origin and generate additional geodesics from thesimple ones that we have. For example, rotating the sphere about the x axisby an angle α moves the equatorial geodesic to a new orientation. This isthe well known result that geodesics on a sphere are great circle paths. Thusfor any point, (φ0, θ0), we can find a pair of orthogonal geodesics passingthrough the point by using a line of longitude, (φ = φ0, θ = s + θ0) and theequatorial line,

(φ = s + φ0, θ = π

2

)with φ0 = π

2 rotated about the x axisby an angle of π

2 − θ0 and then rotating this sphere about the z axis by theangle φ0 − π

2 . This produces the geodesic

φ = tan−1

cos φ0 sin s + sinφ0 sin θ0 cos s

− sin φ0 sin s + cos φ0 sin θ0 cos s

θ = tan−1

√sin2 s + sin2 θ0 cos2 s

cos θ0 cos s

. (9.60)

That Equations 9.60 are geodesics can be confirmed by direct substi-tution into the geodesics equations, Equation 9.59. Again, of course, thisdevelopment of the geodesic equations is a violation of the spirit of the theTheorema Egregium, Section 9.5, since we are using the rotational symmetryof the sphere which at this point can only be inferred extrinsically. We will


θ=1

θ (ϕ=0)

ϕ(θ=0)

ϕ=π/3 ϕ=π/3, θ=5π/6)

ϕ=π/3, θ=2π/3)

ϕ=π/3, θ=π/2)

ϕ=π/3, θ=π/3)

ϕ=π/3, θ=π/6)

Figure 9.6: Geodesics, Fan, and Circle of Radius 0ne on the UnitSphere in the (φ, θ) Plane A series of orthogonal geodesics along theφ = π

3 geodesic. This sequence of geodesics shows important properties ofpositively curved non-Euclidean spaces. Parallel geodesics that are locallyparallel at a point curve toward each other. In this case, these parallelgeodesics actually meet. In addition, the figure show a fan emerging from thenorth pole, (φ = 0, θ = 0) with equal opening angles of π

4 between elementsand a circle centered on the origin of radius one.

need some measure of symmetry in a general coordinate or on the generalmanifold to use a trick like this to generate geodesics. How this can be donewill be addressed later later in Section 9.7. For now, we can proceed usingthese geodesics to do some simple geometry. In a (φ θ) plot, we can showthe nature of the geodesics for some φ0, actually φ0 = π

3 , and several θ seeFigure 9.6. Along the line of longitude geodesic, the orthogonal geodesicsare parallel. Yet all these geodesics move toward each other and, in this case,even meet. This is the signature of the positively curved non-Euclidean twospace. This is a direct consequence of the abandonment of Euclid’s axiomof parallels – For a line and an external point, there is one and only oneline through the external point parallel to the original line and that linewhen extended never intersects the original line . For example for our case,given any of the geodesics, for example the simplest one, θ = π

2 , and anexternal point, (φ0, θ0), the geodesic given by Equation 9.60 produces theunique geodesic that is locally parallel to the original geodesic. As this linethrough (φ0, θ0) is extended along the geodesic it moves closer to the origi-nal line and ultimately intersects it. The other feature of this homogeneouspositively curved non-Euclidean space is that since they are generated byrotation of special geodesics, all these geodesics have the same length and


this length is finite and equals 2π. Since the geodesic equations are invariantunder changes of coordinate choices, these properties are independent of thecoordinate choice and reflect the intrinsic geometry of the manifold.

Figure 9.6, also shows a fan of equal length rays, length equal to one,emerging from the north pole in the imbedded unit sphere and the enclosingcircle. Again, we can use the homogeneity of the imbedded sphere to drawconclusions about a circle and fan drawn anywhere on the sphere and thiscircle, although it does not look natural in the (φ, θ) plane, is particularlysimple to analyze. If we now make a fan with arbitrary ray length, r, anddo an analysis similar to that of the flat plane example in Equation 9.56,

∆θ?≡∫arc ds

r=

1r

∫∆φi

sin (r) dφi =sin (r)

r∆φi, (9.61)

where ∆φi is the difference in the labels of the adjacent rays. In the caseof this definition, the angle would depend on the size of the radius used.Again, this is a characteristic feature of non-Euclidean spaces. The correctdefinition of angle for any space is thus

∆θ ≡ limr→0

∫arc ds

r(9.62)

which for our case, yields a result consistent with our intuition. This defini-tion will always work since, for all non-singular regions, the geometry of anyRiemann manifold is Euclidean for a sufficiently small region. The idea ofsufficiently small will be clarified when we discuss manifolds more generallyin Section 9.7.

9.6.3 The Two Torus

As a third example, consider the case of the torus discussed in Section 9.3.Using the coordinates of that section, Equations 9.2, the line element is

ds2 = [R2 −R1 sin θ]2 dφ2 + R21dθ2. (9.63)

The non-zero connections are

Γφφθ = Γφ

θφ = − R1 cos (θ)R2 −R1 sin (θ)

Γθφφ =

cos (θ) [R2 −R1 sin (θ)]R1

. (9.64)


Using these connections, the Riemann tensor is, R1212 = −R1 [R2 −R1 sin (θ)],and curvature is KG = sin(θ)

R1[R2−R1 sin(θ)] which agrees with Equation 9.3 whichwe derived using the Gauss map.

Again using the connections of Equation 9.64, the geodesic equations are

d2φ

ds2− 2

R1 cos (θ)R2 −R1 sin (θ)

dθ

ds

dφ

ds= 0

d2θ

ds2+

cos (θ) [R2 −R1 sin (θ)]R1

(dφ

ds

)2

= 0. (9.65)

The case dφds = 0 or φ = a has a solution for θ of θ = bs + c. In other

words the lines of constant φ are geodesics if the θ advance is appropriate.Requiring that s be the path length parameter, R2

1

(dθds

)2= 1, makes b =

1R1

. The more interesting case is lines of constant θ which would requireφ = bs + c. These are geodesics only for θ = π

2 and 3π2 .

The exact solution of the geodesic equations, Equations 9.65, are notobvious and instead of relying on an extrinsic solution, it will be sufficientfor our purposes to find an approximate solution. Given the differentialform of the geodesic equations, it is reasonable to assume that from a point(φ0, θ0), the geodesics can be expressed as a smooth function of the pathlength. Using the dimensionless expansion parameter s

R2

φ (s) = φ0 + as

R2+ b

(s

R2

)2

+ O

((s

R2

)3)

θ (s) = θ0 + cs

R2+ d

(s

R2

)2

+ O

((s

R2

)3)

, (9.66)

There are four constants, a through d, and at s = 0 four conditions:

1. That s is the path length or from Equation 9.63, [R2 −R1 sin θ]2(

dφds

)2+

R21

(dθds

)2= 1.

2. The angle α relative to the φ axis of the geodesic,dθdsdφds

= tan (α).

3. The two geodesic equations, Equation 9.65.

We can most easily find an orthogonal pair of geodesics using the alreadyknown one, φ = φ0 and θ = θ0 + 1

R1s and one with emergent angle α = 0.

These will be the closest that we can come to the constant φ, constant θ lines


as geodesics. At s = 0, the second condition above requires that dθds |s=0 = 0

or c = 0. Since dθds |s=0 = 0, the first condition requires that a = 1

[R2−R1 sin θ]

or φ (s) = φ0 + s[R2−R1 sin θ] + bs2 + O

(s3). Plugging this into the φ geodesic

equation at s = 0, requires that b = 0. The θ geodesic similarly requiresrequires d = − cos(θ0)

2R1[R2−R1 sin(θ0)] . Thus the two parametric equations for thesecond geodesic are

φ (s) = φ0 +s

[R2 −R1 sin (θ0)]+ O

((s

R2

)3)

θ (s) = θ0 −cos (θ0)

2R1 [R2 −R1 sin (θ0)]s2 + O

((s

R2

)3)

(9.67)

Using the first of these geodesic equations to solve for s,

s ≈ [R2 −R1 sin (θ)] (φ− φ0) ,

it is easy to find the shape of these geodesics,

θ − θ0 ≈ −cos (θ0) [R2 −R1 sin (θ0)]

2R1(φ− φ0)

2 . (9.68)

This is an interesting result. Figure 9.7 shows a set of these geodesicsalong the geodesic φ = π

3 . From Equation 9.68, these geodesics are parabolaswith concavity up when the initial θ is between π

2 and 3π2 and negative

otherwise. The concavity is zero when θ0 is π2 and 3π

2 . This produces apattern of initially parallel geodesics that are approaching each other inregions of positive curvature and receding from each other in regions ofnegative curvature. This approach and recession pattern is, of course, thesignature of the sign of the curvature. It was this pattern that will allowus to identify the sign of the curvature in the space-time two planes for thefree fall bodies in the vicinity of the earth, see Section 8.5.2.

Figure 9.7, also shows two fans one located in the region of positivecurvature and one in the region of negative curvature. The rays of the fansare all of unit length and thus the tips sketch out the shape of the circle inthe (φ, θ) plane.

9.6.4 The Accelerated Reference Frame Revisited

In Section 6.5, we developed a coordinate system based on the constructionby an accelerated observer. The metric is given in Equation 6.36 or

gαβ =

e

“2gx

c2

”0

0 −e

“2gx

c2

” . (9.69)


1 2 3 4 5 6

1

2

3

4

5

6

θ

ϕ

Positive Curvature

Negative Curvature

(ϕ0=π/3,θ0=2π)(ϕ0=π/3,θ0=7π/4)(ϕ0=π/3,θ0=3π/2)(ϕ0=π/3,θ0=5π/4)(ϕ0=π/3,θ0=π)

(ϕ0=π/3,θ0=3π/4)(ϕ0=π/3,θ0=π/2)

(ϕ0=π/3,θ0=π/4)

Figure 9.7: Geodesics and Fans on the Two Torus in the (φ, θ) PlaneA series of orthogonal geodesics on the two torus along the φ = π

3 geodesic.This sequence of geodesics shows important differences between positivelycurved and negatively curved non-Euclidean spaces. Parallel geodesics thatare locally parallel at a point curve toward each other when the curvatureis positive and away when the curvature is negative. In addition, the figureshows two fans with rays of unit length emerging from points located in thepositively curved region and in the negatively curved region. These fans arecentered on

(φ = 3π

2 , θ = 5π4

)and on

(φ = 3π

2 , θ = π4

)respectively.

The inverse metric is

(gαβ)−1 = gαβ =

e

“−2gx

c2

”0

0 −e

“−2gx

c2

” (9.70)

9.6.5 The Hyperbolic Space-Time Two Surface

Let’s conclude with a case in space-time. Our underlying space is the usualMinkowski space with 2 spacelike directions, x and y, and one timelikedirection,t, see Section 4.3. If we take the invariant curve for Lorentz trans-formation in the x − t plane, see Section 4.5, and generate a surface ofrevolution about the t axis, we will generate a non-trivial two surface thathas both time-like and space-like trajectories. This will provide us with aninteresting non-trivial two dimensional space-time. Using as coordinates,the proper time along the original invariant curve and the rotation angle,the metric, see Section ?? is

There is an alternative method for generating hyperbolic surfaces and


it points out an important feature of these surfaces. The hyperbolic twosurface embedded in a three space is the envelope of the straight lines with agiven inclination and these are the only straight lines in the embedding threespace that are in the two space. Any other curve in the surface is curved inthe three space. Figure 9.8 shows the surface as generated by these straightlines. When I was young, many trash receptacles were constructed in thisfashion.

Figure 9.8: Hyperboloid

9.7 Geometry in Four or More Dimensions

9.8 Some notation and nomenclature:

The metric is gij is called the first fundamental form and is defined as

gµν ≡∂xγ

∂qµ

∂xγ

∂qν(9.71)

The second fundamental form is bµν is

bµν =∂2xγ

∂qµ∂qνnγ (9.72)

The mixed Riemann curvature tensor is

Rµνγρ =

∂Γµνρ

∂qγ− ∂Γµ

νγ

∂qρ+ Γη

νρΓµηγ − Γη

νγΓµηρ (9.73)

9.9. COORDINATE LABELS IN GENERAL RELATIVITY 179

whereΓµ

νγ = gµηΓνγ,η (9.74)

where

Γνγ,η =12

∂gηγ

∂qν+

∂gην

∂qγ− ∂gνγ

∂qη

(9.75)

and the Riemann curvature tensor is

Rµνργ = Rηνργgηµ (9.76)

The Ricci tensor isRµν ≡ Rρ

µρν (9.77)

The symmetries of the Riemann curvature tensor are:

Rλµνκ = Rνκλµ (9.78)

Rλµνκ = −Rµλνκ = −Rλµκν = Rµλκν (9.79)

The cyclicity condition:

Rλµνκ + Rλκµν + Rλνκµ = 0 (9.80)

9.9 Coordinate Labels in General Relativity

9.10 Distances and Time Intervals

At a fixed spatial coordinate, the time interval that lapses for an incrementin the local coordinate time is given by

c2dτ2 = g00c2dt2. (9.81)

Thus the lapsed time is

τ (t) =1c

∫ x0

x0i

√g00dx′0 =

∫ t

t0

√g00dt′ (9.82)

Note that this implies that g00 > 0. This is not the same condition thatthe metric must have principal eigenvalues such the there is one positiveand three negative. That is a condition that the metric be that of a realgravitational field – a metric for space-time. The condition g00 > 0 is onethat says that the system of reference could not be realized by material


bodies. An example is the coordinate system rotating at a rate Ω. Usingcylindrical coordinates,

c2dτ2 = c2dt2 − dr2 − r2dφ2 − dz2, (9.83)

and the coordinate transformation

r = r′

φ = φ′ − Ωt′

z = z′

t = t′, (9.84)

the interval becomes

c2dτ2 =(c2 − r′2Ω2

)dt′2 − dr′2 − r′2dφ′2 − dz′2 + 2r′2Ωdφ′dt′ (9.85)

The condition g00 > 0 is violated when a rigid merry-go-round could beutilized as the basis for the coordinate transformation. The metric conditionis the condition that the merry-go-round could rotate at the rate Ω and beextended to distance D > c

Ω .

xo+dxo2

xo+dxo1

xo light rays

x x+dx

Figure 9.9: Distance in a General Relativity Given a metric on thecoordinates, the spatial distance between two nearby coordinate points isdetermined by the radar method, see Section 3.1, as the speed of light timesthe time between transmission and response to the nearby point divided bytwo.

Spatial distances must be measured by the radar method, see Section 3.1.For two places infinitesimally close, a light ray traveling from one coordinateplace to another and back is as shown in Figure 9.9. The trajectory of thelight ray is given by

c2dτ2 = 0 = g00dx02 + 2g0adx0dxa + gabdxadxb. (9.86)

9.10. DISTANCES AND TIME INTERVALS 181

This is a quadratic for the coordinate time increment and thus there are thetwo solutions represented in Figure 9.9.

dx01 = − 1

g00

g0adxa −

√(g0ag0b − g00gab) dxadxb

(9.87)

and

dx02 = − 1

g00

g0adxa +

√(g0ag0b − g00gab) dxadxb

(9.88)

The difference in the coordinate time is

dx02 − dx0

1 =2

g00

√(g0ag0b − g00gab) dxadxb, (9.89)

and, thus, the spatial distances are

dl2 =(

gab −g0ag0b

g00

)dxadxb. (9.90)

Thus, the three space metric, γab, is

γab =(

gab −g0ag0b

g00

). (9.91)

The inverse of this three space metric, γab, is gab, and it is three space partof original four space metric, gµν . This comes about since gµνgνα = δµ

α and,therefore, gabgbc + ga0g0c = δa

c and gabgb0 + ga0g00 = 0. Solving for ga0 fromthe second equation and substituting into the first, we have gabγbc = δa

c .Once the line element has been determined in any given coordinate, we

can address the important issue of simultaneity. Obviously, in a general co-ordinate system, the surfaces of constant coordinate time will not necessarilyand, most likely, will not be surfaces of simultaneity. Like the problems oftime and distance above there is really no global construction of the sur-faces of simultaneity. Again, we use the radar method, see Section 3.1, toestablish simultaneity at nearby spatial coordinates. In that case, the eventat the trajectory associated with the position ~x that is simultaneous withthe event labeled x0 on the trajectory with position ~x + d~x is x0 + dx0

1+dx02

2 .From Equation 9.87 and Equation 9.88,

∆x0 = −g0adxa

g00. (9.92)

We have already seen an application of this result in Section 6.5. Had westuck with the classic confederate scheme for coordinates using the local


clocks, Equation 6.20, the lines of constant τ ′ would not have been linesof simultaneity. This is indicated by the line element, which, from Equa-tion 6.20, is

c2dτ2proper = c2dτ ′2 − 2

(gτ ′

c

)(1 + gh

c2

)cdτ ′dh−

1−

(gτ ′

c

)2

(1 + gh

c2

)2

dh2. (9.93)

Using this line element and Equation 9.92, we have that the event labeled

by (h, τ ′) is simultaneous with the nearby event(

h + ∆h, τ ′ +gτ ′c

1+ gh

c2

∆h

).

This is consistent with our previous result that the time coordinate, τ ≡τ ′

1+ gh

c2

, generates lines of simultaneity for the accelerated coordinate system

of Section 6.5 in that the line between these events has the same slope, orin this case hangle, as the line from the magic point

(− c2

g , 0)

to the event(h, τ ′) and, obviously, both lines pass through that event.

As in the previous cases, this is a local rule and its use for more thaninfinitesimal separations is somewhat problematic. In the case above ofthe accelerated system, there is no problem with extending the result tolarge separations. This will not be the case generally. Consider the caseof the merry-go-round. From the line element, Equation 9.85, the shift incoordinate time to synchronize clocks for a shift in angle, φ′ at fixed r′, z′,and t′ is ∆x0 = −

∫ φ′

φ′0

r′2Ωc2−r′2Ω

dφ′′. A closed curve then yields and inconsistentresult that the clock at at fixed r′, φ′0, z′, and t′ has to be shifted by theamount − 2πr′2Ω

c2−r′2Ωto be brought into synchronization with itself. Of course,

in this coordinate system, the ghh metric element is not the appropriatemeasure of the spatial length of the coordinate change ∆h. It is instead,Equation 9.91,

γhh =(

ghh −gτ ′hgτ ′h

gτ ′τ ′

)= −1. (9.94)

While we are dealing with these issues of the import of the line element,note how nicely the event horizon for the accelerated system appears in theline element, Equation 9.93. At fixed τ ′, changes in the coordinate h produceno change in length at the places that1−

(gτ ′

c

)2

(1 + gh

c2

)2

= 0. (9.95)

9.11. EINSTEIN EQUATIONS 183

This is the eventsc2

g+ h0 = ±cτ ′0, (9.96)

which are the light-like asymptotes through the magic point.

9.11 Einstein Equations

Rµν − 12gµνR = −8πG

c4Tµν (9.97)

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