lorentzian manifolds
TRANSCRIPT
Abstract
In this essay, the concept of Lorentzian metrics is studied. Webegin by giving an introduction to general Riemannian and pseudo-Riemannian metrics in the first section. In the second section we in-troduce the concept of Lorentzian metric along with some terminologyand notation. The third section deals with the Alexandrov topologythat may be constructed on a manifold when given a Lorentzian met-ric. We mainly discuss when this topology is the same as the givenmanifold topology.
Following the introduction to Lorentzian metrics, we enter intothe Čech cohomology theory and use the first Stiefel-Whitney class todetermine if a Lorentzian manifold is time and/or space orientable.This is done in the fourth and fifth section.
Finally, in the last section, we briefly look at an application ofLorentzian geometry, namely the theory of general relativity.
To fully understand this essay, the reader should be at least some-what familiar with differential geometry and cohomology theory.
Throughout this essay, we will mainly use the notation of Beemand Ehrlich [1] for Lorentzian geometry and the notation of Nakahara[2] for general differential geometry.
Contents
1 Riemannian and pseudo-Riemannian metric 4
2 Introducing Lorentzian metrics 7
3 The Alexandrov topology 12
4 Čech cohomology 14
4.1 The first Stiefel-Whitney class . . . . . . . . . . . . . . . . . . 15
5 Orientability of Lorentzian manifolds 18
6 Lorentzian metrics applied to physics 21
6.1 Time and space orientability . . . . . . . . . . . . . . . . . . . 22
6.2 The Schwarzschild solution . . . . . . . . . . . . . . . . . . . . 22
6.2.1 The perihelion precession . . . . . . . . . . . . . . . . . 23
6.2.2 The Schwarzschild black hole . . . . . . . . . . . . . . 23
6.3 The Robertson-Walker space-times . . . . . . . . . . . . . . . 24
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1 Riemannian and pseudo-Riemannian metric
Given a smooth manifold M of dimension m, we introduce the concept of ametric by the following definition.
Definition 1.1 A Riemannian metric is a tensor field g of type (0, 2) whichsatisfies the following conditions
gp(X,Y ) = gp(Y,X) (1.1)gp(X,X) ≥ 0 , gp(X,X) = 0 ⇐⇒ X = 0 (1.2)
for any p ∈ M and vectors X,Y ∈ TpM.A pseudo-Riemannian metric satisfies
gp(X,Y ) = 0 ∀ Y ⇐⇒ X = 0 (1.3)
instead of equation (1.2).
Given a chart (Ui, ϕi) of M with local coordinates xµ and some metric g onM, we may express g in local coordinates as
gp = gµν(p)dxµ ⊗ dxν (1.4)
where gµν(p) = gp(∂
∂xµ ,∂
∂xν ). Obviously gµν(p) constitutes the elements of anon-degenerate m×m matrix for any p ∈ M.
By construction, the matrix gµν(p) is diagonalizable with real eigenvaluessince it is symmetric. The non-degeneracy also states that all eigenvaluesare non-zero. From this follows that if there are n+ positive eigenvalues andn− negative eigenvalues, then n+ + n− = m. The combination of numbers(n−, n+) is said to be the signature of the metric g (we may also denotethis by (−, . . . ,−,+, . . . ,+), with n− minus signs and n+ plus signs). Thesignature is clearly independent of the chart on which it is computed.
The signature of a metric is also independent on the point p of the manifoldsince a change of signature would imply that the metric is degenerate atsome point (the eigenvalues may not change discontinuously). Thus, it makessense to speak about the signature of a metric. By definition, a metric isRiemannian iff it has signature (0,m).
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Example 1.2 The manifold M = Rm equipped with the usual euclidean
metric.
For Rm we need only the trivial chart given by the euclidean coordinates tocover Rm. In these coordinates the metric tensor is given by
gµν(p) = δµν . (1.5)
For any vectors X,Y ∈ TpRm = R
m, the mapping g : TpRm × TpR
m → R isthe usual inner product between vectors.
We now show that an embedding of a manifold N into a manifold Mequipped with a Riemannian metric g naturally induces a Riemannian metricon N .
Theorem 1.3 If M and N are manifolds, g is a Riemannian metric on Mand f : N → M is an embedding, then h = f ∗g is a Riemannian metric onN .
Proof: We need to show that equations 1.1-1.2 holds for h. First, fix a pointp ∈ N and take any vectors X,Y ∈ TpN , by definition of h and the conditionimposed on g by equation (1.1)
hp(X,Y ) = gf(p)(f∗X, f∗Y ) = gf(p)(f∗Y, f∗X) = hp(Y,X) (1.6)
which states that h also fulfills equation (1.1). That hp(X,X) ≥ 0 follows inthe same manner.Since f is an embedding, the map f∗ : TpN → Tf(p)M is injective. Thusf∗X = 0 ⇐⇒ X = 0 and it follows that
h(X,X) = g(f∗X, f∗X) ≥ 0 , h(X,X) = 0 ⇐⇒ X = 0 (1.7)
and thus h is a Riemannian metric on N .
Since any manifold M of dimension m may be embedded into a higher di-mensional Euclidean space, we have the following corollary of Theorem 1.3.
Corollary 1.4 Any manifold M allows a Riemannian metric.
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In general, Theorem 1.3 does not hold for pseudo-Riemannian metrics as thepullback of g by the embedding f is not necessarily non-degenerate and mayhave different signature at distinct points, in fact there are trivial counterexamples (see Example 1.5). However, if the pullback has a globally definedsignature, it is a (pseudo-Riemannian) metric.
Example 1.5 The circle S1 embedded into R2 by the natural embeddingf(θ) = (cos θ, sin θ) (with the obvious interpretation of θ as a local coordin-ate). If we equip R2 with the pseudo-Riemannian metric g = dx⊗dx−dy⊗dy,the pullback h = f ∗g is not a metric on S1.
Calculating the pullback h = f ∗g we get
h = (sin2 θ − cos2 θ)dθ ⊗ dθ = − cos(2θ)dθ ⊗ dθ (1.8)
which is degenerate at θ ∈ π4, 3π
4, 5π
4, 7π
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where it also changes signature.
Example 1.6 An example of a Riemannian metric induced by an embeddingis the induced metric on S2 from its natural embedding into R3.
We equipR3 with the Riemannian metric g = δµνdxµ⊗dxν and embed S2 into
R3 by f(θ, ϕ) = (sin θ cosϕ, sin θ sinϕ, cos θ). Straightforward calculations
give the induced metric tensor on S2 to be
h = dθ ⊗ dθ + sin2 θdϕ⊗ dϕ (1.9)
where θ and ϕ are interpreted as the spherical coordinates restricted to somelocal chart.
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2 Introducing Lorentzian metrics
In this section we introduce the concept of Lorentzian metrics and give ashort overview of one of the fields in physics where it is applied, the theoryof relativity. We start by giving a definition.
Definition 2.1 (Lorentzian metric) Given a manifold M of dimensionm+ 1, a pseudo-Riemannian metric g is called a Lorentzian metric if it hassignature (1,m). The pair (M, g) is called a Lorentzian manifold.
Lorentzian geometry turns out to be the tool perfectly suited for describ-ing the theory of relativity. As we will notice, Lorentzian geometry and thetheory of relativity are so closely related that the terminology of Lorentziangeometry has complete correspondence with the physical interpretations.
Example 2.2 Lorentzian geometry applied to the theory of general relativity.
General relativity is a theory describing the phenomena known as gravitation.The curvature of the physical space time is given through the Einstein fieldequations
Ric− 1
2Rg + Λg = 8πT (2.1)
where Ric is the Ricci tensor, R the Ricci scalar, g the metric tensor, Λthe cosmological constant and T is the energy-momentum tensor. We haveassumed units where c = G = 1 (the speed of light and the gravitationalconstant are set to unity).
A torsion free affine connection is given by the Levi-Civita connection. Thetrajectories of objects (particles, planets, etc) are given as geodesics. Fromthis, one can predict the perihelion precession of Mercury, the gravitationalredshift of light and other physical phenomena. More on this subject can befound in the last section.
We now introduce some terminology. Given a Lorentzian manifold (M, g)there is a natural decomposition of vectors into three different classes. Thisdecomposition is provided by the Lorentzian metric g.
Definition 2.3 A vector X ∈ TpM is timelike if gp(X,X) < 0, spacelike ifgp(X,X) > 0 and null if gp(X,X) = 0.
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There is one more important class of vectors, namely the non-spacelike vec-tors. This set of vectors is the union of timelike and null vectors.
In much the same way, we may talk about timelike, spacelike, null and non-spacelike vector fields. A vector field X (a section of TM) is said to betimelike if it is timelike at every point, that is
gp(X(p), X(p)) < 0 ∀ p ∈ M. (2.2)
Spacelike, null and non-spacelike vector fields are defined in the same manner.
One also defines the timelike and non-spacelike curves. Given a smooth curveγ : I −→ M (where I is some interval of R), we say that it is timelike if thetangent vector γ(t) is timelike for all t ∈ I. From here we may continue withpiecewise smooth curves and say that a piecewise smooth curve is timelike ifevery piece of the curve is timelike. Non-spacelike curves are defined similarlyto timelike curves. Finally, a continuous curve is timelike if for every t0 ∈ I,there exists a convex normal neighbourhood U(γ(t0)) of γ(t0) and an ε > 0such that γ(t) ∈ U(γ(t0)) for all t ∈ (t0− ε, t0 + ε) = I ′ and for any t1, t2 ∈ I ′
(t1 < t2) there exists a smooth timelike curve from γ(t1) to γ(t2) withinU(γ(t0)).
A convex neighbourhood is a neighbourhood such that any two points ofthe neighbourhood can be connected by a unique geodesic segment which isentirely within the neighbourhood. For any point p ∈ M, we may select abasis vi of TpM R
n. If x ∈ Rn and |x| is sufficiently small, then the map
x → expp
(n∑
i=1
xivi
)(2.3)
where the exponentiation map is defined as usual1, is a diffeomorphism froma neighbourhood of the origin in Rn to a neighbourhood U(p) of p in M.The coordinates x are said to be normal coordinates for U(p) based at p. Aconvex normal neighbourhood U(p) is a convex neighbourhood such that forany point q ∈ U(p), there are normal coordinates for U(p) based at q.
We are now ready to define an important concept in the study of Lorentzianmanifolds, namely the property of being time orientable.
1Let cv(t) be the unique geodesic in M such that cv(0) = p and cv(0) = v. Thenexpp(v) = cv(1) defines the exponentiation.
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Definition 2.4 A Lorentzian manifold (M, g) is said to be time orientableif there exists a global timelike vector field (this field is everywhere non-vanishing by definition). Given such a vector field X, non-spacelike vectorsare divided into two classes. A non-spacelike vector Y ∈ TpM is future dir-ected if gp(X(p), Y ) < 0 and past directed if gp(X(p), Y ) > 0. A Lorentzianmanifold along with a time orientation is called a space-time.
It is about time to consider an example of a Lorentzian manifold. Thesimplest example is the Minkowski space-time which plays the same rolein Lorentzian geometry as Euclidean space does in Riemannian geometry.
Example 2.5 The n+ 1-dimensional Minkowski space-time is given by
(M, g) =
(R
n+1,−dx0 ⊗ dx0 +n∑
k=1
dxk ⊗ dxk
)(2.4)
where xµ are the usual Cartesian coordinates on Rn+1. The time orientation(hence “space-time”), is provided by the vector field X = ∂
∂x0 .
The Minkowski space-time is important for both mathematical and physicalreasons. As stated above, it plays the same role in Lorentzian geometry asEuclidean space does in Riemannian geometry. It is also the staging groundof the theory of special relativity describing a universe where there is nogravitation.
Given a space-time (M, g), we may define chronological and causal relationsbetween points in M.
Definition 2.6 Given two points p, q ∈ M, we say that q is in the chronolo-gical future of p (or p is in the chronological past of q) if there is a timelikefuture directed (the tangent vector is future directed at each point for smoothcurves, etc) curve γ : [0, 1] −→ M such that
γ(0) = p , γ(1) = q. (2.5)
We denote this by p q. In much the same way, we say that q is in thecausal future of p if there is a non-spacelike curve γ : [0, 1] −→ M fulfillingEquation (2.5). This we denote by p ≤ q.
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The chronological and causal future (resp. past) of a point p ∈ M are denotedby I+(p) and J+(p) (resp. I−(p) and J−(p)). That is
I+(p) = q ∈ M : p qI−(p) = q ∈ M : q p (2.6)J+(p) = q ∈ M : p ≤ qJ−(p) = q ∈ M : q ≤ p .
It is important to note that ≤ is not necessarily a partial order on M as wemay well have p ≤ q and q ≤ p for distinct points p, q ∈ M. In fact, there arespace-times for which p q and q p for all p, q ∈ M. Such a space-timeis said to be totally vicious. If, however, ≤ is a partial order, the space-timeis said to be causal . Spacetimes for which p /∈ I+(p) for all p ∈ M are saidto be chronological .
In Riemannian geometry, there is the concept of distance between points.Given a piecewise smooth curve γ : I → M, the length of the curve isdefined as
L0(γ) =
∫I
√g(γ, γ)dt. (2.7)
The distance between points p and q is defined through
d0(p, q) = infγ∈Γ
(L0(γ)) (2.8)
where Γ = γ : [0, 1] −→ M|γ(0) = p, γ(1) = q.
In Lorentzian geometry, the above definition of distance does not make anysense. As a matter of fact, the expression g(γ(t), γ(t)) is negative if γ istimelike at t. Instead, we use the following definition of Lorentzian distance.
Definition 2.7 Given a smooth non-spacelike curve γ : I −→ M in a space-time (M, g), we define the length of the curve as
L(γ) =
∫I
√−g(γ, γ)dt. (2.9)
The length of a piecewise smooth curve is defined as the sum of the lengthsof the pieces. For a spacelike curve we define L(γ) = 0. We also define thedistance between two points p, q ∈ M as
d(p, q) = supγ∈Γ
(L(γ)) (2.10)
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where Γ is the set of future directed non-spacelike curves from p to q. If notp ≤ q then we define d(p, q) = 0.
The Lorentzian distance has many properties which are similar to the proper-ties of Riemannian distance. However, some properties are fundamentally dif-ferent. For instance, in Riemannian geometry, the identity d0(p, q) = d0(q, p)always holds which is not the case in Lorentzian goemetry.
Since Riemannian distance is defined as an infimum and the Lorentzian as asupremum, there is often a correspondence between the words “minimal” inRiemannian and “maximal” in Lorentzian geometry. For example, in Rieman-nian geometry we have the triangle inequality
d0(p, q) ≤ d0(p, r) + d0(r, q) (2.11)
for any points p, q and r. The corresponding inequality in Lorentzian geo-metry is the inverse triangle inequality
d(p, q) ≥ d(p, r) + d(r, q) (2.12)
which holds if r is in the causal future of p and the causal past of q.
Even if the sets I±(p) are open (they are the interior of the sets J±(p)),the sets J±(p) are not necessarily closed, in fact there are trivial counterexamples, see Figure 2.1.
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R2
J+(p)
K
p
Figure 2.1: The two dimensional Minkowski space with the set K removed isa typical example of a space-time where J+(p) is not closed for all p.
3 The Alexandrov topology
Given a space-time (M, g), there is a number of ways to construct new to-pologies on M. The most straightforward of these topologies is the so calledAlexandrov topology.
Definition 3.1 (Alexandrov topology) The basis of the Alexandrov to-pology on a space-time (M, g) are the sets
Ipq = I+(p) ∩ I−(q) (3.1)
for all p, q ∈ M.
Since I±(p) are open sets in the manifold topology, Ipq are open sets in themanifold topology as they are finite intersections of open sets. It follows thatthe original manifold topology is at least as fine as the Alexandrov topology.We may ask ourselves what is required for the Alexandrov topology to beequivalent to the original manifold topology.
Definition 3.2 In a space-time (M, g), the open set U is said to be causallyconvex if no non-spacelike curve intersects U more than once. (M, g) is said
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to be strongly causal at p ∈ M if there exists arbitrarily small causally convexneighbourhoods of p. A space-time is strongly causal if it is strongly causal ateach point.
Being strongly causal turns out to be exactly the condition a space-timemust fulfill for the Alexandrov topology to coincide with the given manifoldtopology.
Theorem 3.3 For any space-time (M, g), the Alexandrov topology agreeswith the original manifold topology iff (M, g) is strongly causal.
Proof: We first assume that (M, g) is strongly causal and prove that theAlexandrov topology agrees with the given manifold topology.
Let U be any open set in M. Since (M, g) is strongly causal, for any pointp ∈ U , for which there exists a causally convex set U1 such that p ∈ U1 ⊂ U .We may now choose points p1, p2 ∈ U1 such that p1 p p2 and Ip1p2 ⊂U1 (this is possible since U1 is causally convex). For each point of U wecan perform this procedure and assign an open set I(m) in the Alexandrovtopology to m such that I(m) ⊂ U . It follows that U is an open set in theAlexandrov topology as it is a union of open sets.
Now suppose that (M, g) is not strongly causal, that is, there is a pointp ∈ M such that there exists a convex normal neighbourhood V (p) suchthat for any neighbourhoodW (p) ⊂ V (p), there exists a non-spacelike curvesstarting in W (p), leaves V (p) and then returns to W (p). It follows that allneighbourhoods of p contain points outside of V (p) and thus V (p) is not anopen set in the Alexandrov topology. Thus, the Alexandrov topology doesnot agree with the original manifold topology.
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4 Čech cohomology
In this section we will deal mainly with the Čech cohomology. In the nextsection we will use the Čech cohomology to deduce whether a Lorentzianmanifold is time (or space) orientable or not. We will be concerned withthe group Z2 and choose to represent it as Z2 = 1,−1 so that the groupoperation is the “ordinary” mulitplication. We start by defining the Čechcochains.
Definition 4.1 (Čech cochain) Let M be a manifold and Ui a simpleopen covering (that is, all Ui and intersections of Ui are contractible) of M.A function f(i0, . . . , ir) ∈ Z2, defined on Ui0 ∩ Ui1 ∩ . . . ∩ Uir = ∅ which istotally symmetric under permutation of the arguments, that is
f(i0, . . . , ir) = f(iP (0), . . . , iP (r)), (4.1)
is a Čech r-cochain. We denote the multiplicative group of Čech r-cochainsby Cr(M,Z2).
As usual in cohomology theory, we introduce a coboundary operator δ whichmaps r-chains to r + 1-chains with the property that δ2 maps everything tothe trivial element. In this case, the coboundary operator is defined through
(δf)(i0, . . . , ir+1) =r+1∏k=0
f(i0, . . . , ik, . . . , ir+1) (4.2)
where f is a Čech r-cochain and where the entry under the is omitted.Example 4.2 For r = 1, a cochain assigns the value ±1 to the overlap ofUi and Uj. If f(i, j) is a 1-cochain, its coboundary is given by
(δf)(i, j, k) = f(i, j)f(i, k)f(j, k) (4.3)
The above definition automatically gives δ2f = 1 since
(δ2f)(i0, . . . , ir+2) =∏k =m
f(i0, . . . , ik, . . . , im, . . . , ir+2), (4.4)
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f(i0, . . . , ik, . . . , im, . . . , ir+2) = f(i0, . . . , im, . . . , ik, . . . , ir+2) and a2 = 1 forany element a ∈ Z2. This gives rise to the Čech complex
. . . Cr+1(M,Z2)δ Cr(M,Z2)
δ Cr−1(M,Z2)δ . . .δ . (4.5)
We may now define the groups of Čech cocycles and Čech coboundaries as
Zr(M,Z2) = x ∈ Cr(M,Z2) : δx = 1 = ker δr (4.6)Br(M,Z2) =
x ∈ Cr(M,Z2) : x = δy, y ∈ Cr−1(M,Z2)
= Im δr−1 (4.7)
and we use this to define the Čech cohomology groups.
Definition 4.3 (Čech cohomology groups) The r-th Čech cohomologygroup is defined as
Hr(M,Z2) =Zr(M,Z2)
Br(M,Z2)=
ker δr
Im δr−1
. (4.8)
Suppose we have two simple coverings Ui and Vi of M that contain thesame open sets apart from that two sets in Ui are replaced by their unionin Vi. It is easy to see that this does not change the Čech cohomologygroups. It follows that the Čech cohomology groups are independent of thespecific choice of simple open covering since for two different coverings A andB. If we start with a simple cover C with arbitrarily small open sets suchthat both A and B can be formed by replacing elements of C with unionsof elements in C, then it follows that the Čech cohomology groups are thesame for both A and B. It follows that the Čech cohomology groups are welldefined for any manifold.
4.1 The first Stiefel-Whitney class
Let E be a vector bundle over the manifold M and let E have some innerproduct (which is, in general, dependent on the point p ∈ M) on the fibre.The r-th Stiefel-Whitney class wr assigns an element (wr(E)) of Hr(M,Z2)to the vector bundle E . Here we consider the first Stiefel-Whitney class.
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If we have some simple covering Ui of the base manifold M, an elementof E may be uniqely defined by an element of Ui × F , where F is the fibre.Also, given an element x ∈ E such that the projection of x on the basemanifold is in Ui, there is a unique element of Ui × F associated to x. Onthe intersection Ui ∩ Uj, any element x ∈ E may be represented by a uniqueelement (p, fi) ∈ Ui × F or a unique element (p, fj) ∈ Uj × F . Clearly,the elements of fi and fj need not be the same. The transition functionstij : Ui∩Uj → G (where G is the structure group of the vector bundle), relatefi to fj by fj = tjifi.
The transition functions must obey certain consistency conditions, namely
tij(p) = tji(p)−1 (4.9)
tij(p)tjk(p)tki(p) = 1 (p ∈ Ui ∩ Uj ∩ Uk) (4.10)
Conversely, given a base manifold M, a simple open covering Ui, a vectorspace F , a group G acting on F and transition functions tij : Ui ∩Uj → G, avector bundle is uniquely defined by taking the union of the sets Vi = Ui ×Fand identifying (p, fi) ∈ Vi with (p, tjifi) ∈ Vj for p ∈ Ui ∩ Uj.
Since E is equipped with an inner product on the fibre, it makes sense tospeak about normalized vectors and the structure group of E is G = O(n)where n is the dimension of the fibre.
Definition 4.4 (The first Stiefel-Whitney class) Given a vector bundleE as above and denoting the transition function from Uj to Ui by tij ∈ G, wedefine a Čech 1-cochain f by
f(i, j) = det tij. (4.11)
The first Stiefel-Whitney class w1 is defined by
w1(E) = [f ] ∈ H1(M,Z2). (4.12)
We need to check that this definition is consistent, namely that [f ] really isan element of the first Čech cohomology group (that is, f is an element ofthe first Čech cocycle group), and that the element does not depend on the
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particular choice of the transition functions tij. This is just a simple matterof computation as
(δf)(i, j, k) = det tij det tik det tjk = det(tijtjktki) = det(1) = 1 (4.13)
by the consistency condition imposed on the transition functions and ele-mentary properties of the determinant. That f(i, k) ∈ Z2 is obvious astij ∈ O(n) =⇒ det tij = ±1.
As for the element [f ] to be independent of the particular transition functions,the most general change of transition function is given by t′ij = hitijh
−1j
(where hi : Ui −→ G corresponds to a change of fibre basis in Ui) which leadsto
f ′(i, j) = det t′ij = dethi dethj det tij = (δf0)(i, j)f(i, j) (4.14)
where f0(i) = dethi is a Čech 0-cochain. Since f and f ′ differ by the cobound-ary of f0, they represent the same cohomology class and the first Stiefel-Whitney class is well defined.
As we will see in the next section, the first Stiefel-Whintey class can be usedto determine if a vector bundle E is orientable or not.
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5 Orientability of Lorentzian manifolds
It is now time to determine when a Lorentzian manifold is time and spaceorientable.
Let (M, g) be a Lorentzian manifold of dimension m+ 1. Since the metric gis symmetric, it is diagonalizable with real eigenvalues and real eigenvectorsat each point p ∈ M. Thus, at a given point p there are m + 1-eigenvectorsei(p) of the metric tensor such that g(ei, ej) = λ(i)δij, where λ(i) = −1 fori = 0 and λ(i) = 1 otherwise. The set ei(p) form a basis for TpM. Anyvector X ∈ TpM is naturally split into two parts
X = X− +X+ = Xµ0 e0µ +
m∑i=1
Xµi eiµ. (5.1)
Thus, the tangent space TpM is split into two subspaces, the subspacespanned by e0 and the subspace spanned by ei : 1 ≤ i ≤ m. We denotethese subspaces by T−
p M and T+p M respectively.
Making the above splitting at every point p ∈ M, any vector field X ∈ TMsplits as
X(p) = X−(p) +X+(p) (5.2)
where X± are vector fields with the property that X±(p) ∈ T±p M. This gives
a splitting of the tangent bundle TM into a real line bundle T−M and anm-dimensional vector bundle T+M. A natural inner product on both thesebundles is induced through the metric tensor g.
Definition 5.1 A vector bundle E of fibre dimension n with some innerproduct on the fibre is orientable if it is possible to choose all transitionfunctions tij such that tij ∈ SO(n). A manifoldM is orientable if its tangentbundle is orientable.
The above definition implies that there is some global notion of positiveorientation for a fibre basis. In fact, we could skip the requirement for aninner product on the fibre and demand that the transition functions are inGL+(n,R) instead of SO(n), how this generalization affects the reasoningbelow should be quite clear.
From the splitting of the tangent bundle TM of a Lorentzian manifold (M, g)into TM = T−M ⊕ T+M as well as Definitions 2.4 and 5.1, it should be
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clear that (M, g) is time orientable iff the real line bundle T−M is orientable(there exists a global timelike vector field iff there is a global non-zero sectionof T−M). Analogously, it also makes sense to define (M, g) to be spaceorientable if the vector bundle T+M is orientable.
Theorem 5.2 A vector bundle E with some metric is orientable iff the firstStiefel-Whitney class of E is trivial.
Proof: Suppose that the first Stiefel-Whitney is class trivial, that is f = δf0
for some Čech 0-cochain f0. We may always perform a local change of fibrebasis for each Ui by applying hi : Ui −→ O(n). It is, of course, possible tochoose hi such that dethi = f0(i). The new transition functions t′ij are givenby t′ij = hitijh
−1j leading to
det t′ij = dethi dethjf(i, j) = f0(i)2f0(j)
2 = 1. (5.3)
It follows that t′ij ∈ SO(n).
Now suppose E is orientable. By definition of orientability, we may choose thetransition functions tij such that tij ∈ SO(n) where n is the fibre dimension.It follows that
f(i, j) = det tij = 1 (5.4)
and thus w1(E) = [f ] is trivial.
Example 5.3 The cylinder E = S1×R is orientable while the infinitly broad
(not needed but comparison to E more obvious) Möbius strip M is not.
Both E and M can be simply covered by three open sets Vi = Ui ×R whereUi are open sets of S1. We may choose Ui = (2πi
3, 2π(2+i)
3) , i ∈ 0, 1, 2 with
the obvious interpretation of θ ∈ Ui as the polar angle. As a metric on thetangent bundle, we may choose g = dθ ⊗ dθ + dx⊗ dx.
The transition functions on TE may all be taken to be the identity matrixand by the calculations in the proof of Theorem 5.2 and the first Stiefel-Whitney class is trivial. When considering the tangent bundle TM, it is notpossible to choose all transition functions to be in SO(2) and we must havean odd number of transition functions in O(2) \ SO(2). Let us assume thatt20 /∈ SO(2) and t01, t12 ∈ SO(2). This gives
f(i, j) =
−1 i, j = 0, 21 otherwise . (5.5)
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=
=E M
Figure 5.1: The cylinder is orientable while the Möbius strip is not.
However, if f is a Čech coboundary, then f(i, j) = f0(i)f0(j) for some Čechcochain f0. But f(0, 1) = f0(0)f0(1) = 1 and f(1, 2) = f0(1)f0(2) = 1 impliesthat f(0, 2) = f0(0)f0(2) = f0(0)f0(1)
2f0(2) = f(0, 1)f(1, 2) = 1. Thus the fin Equation (5.5) is not a couboundary and w1(TM) is not trivial. See Figure5.1
According to our previous discussion, Theorem 5.2 has the following corollary:
Corollary 5.4 A Lorentzian manifold (M, g) is time orientable iff w1(T−M)
is trivial and space orientable iff w1(T+M) is trivial.
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6 Lorentzian metrics applied to physics
Like we briefly noted in the beginning of section 2, Lorentzian manifolds arethe best tool for describing the general theory of relativity. The curvature andmetric tensor are related to the energy momentum tensor through the Ein-stein field equations (Equation (2.1)) and the Levi-Civita connection providesa torsion free affine connection on the manifold.
In the theory of relativity, we are mainly concerned about four dimensionalLorentzian manifolds for obvious reasons. However, we may often disregardcertain dimensions when solving problems (when the solution in that dimen-sion is trivial), and one might always use other dimensions in examples andso on.
As a special case of the theory of general relativity, we get the theory ofspecial relativity which is described by the Minkowski space-time. Most ofthe predictions that has come out of general relativity are based on particularsolutions to the Einstein field equations when the energy momentum tensorhas some simple form. In this section we discuss some of those solutions andthe predictions they bring.
The solutions we will study all have the cosmological constant Λ set to zero.Einstein introduced this constant to prevent his theory from predicting thatthe universe should either contract or expand. When the expansion of theuniverse was found empirically, Einstein referred to the cosmological constantas his worst mistake. In recent days it has turned out that the cosmologicalconstant might not have been such a bad mistake, even if its value is not suchthat it gives the universe a steady state. Never the less, it does have somemathematical interest since the extra term does not prevent the existence ofsolutions to the field equations.
The trajectories of objects in space-time which are not affected by otherforces is given by future directed non-spacelike geodesics (timelike for massiveobjects and null for massless objects). The proper time between two events foran object (the time elapsed as measured by a “clock” following the object) isthe Lorentzian length of the trajectory taking the object from the first eventto the second.
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6.1 Time and space orientability
In the theory of general relativity, we often assum that the manifold wework with is a space-time. Never the less, we may of course think about thephysical consequences if the universe is not time orientable.
The first notable effect is that there is no such thing as a global time direction(by definition). As a result, there is no distinction of future and past, thechronology and causality of events is obviously ill defined and it makes littlesense to assume global causality in physical theories (we may still talk aboutlocal causality after defining a local time direction).
If the universe is not space orientable, there are also a number of interestingphysical consequences. For example, if we send out a right hand glove in theuniverse and at some later point reunite with it, it might well have becomea left hand glove.
Non orientability of time and space also have great consequences on particlephysics where we work with time and space reflections as possible symmet-ries (the time and parity symmetries). For example, it seems like the weakinteraction is not symmetric under space reflection and to get a symmetrywe need also exchange all particles for anti particles (the charge conjugationsymmetry), however, non-orientability of space should indicate that there isno distinction between the different parities.
Finally, if the universe is not time orientable, there is no way of globallydefining some of the most important statements of physics, namely the con-servation laws like conservation of energy, charge, momentum and so on.
6.2 The Schwarzschild solution
One of the first and most interesting solutions to the field equations is the socalled Schwarzschild solution. It is valid outside of a spherically symmetricuncharged and non-rotating mass distribution in space. The Schwarzschildsolution is most easily expressed in spherical coordinates for the spatial partoutside of r = 2M (for r < 2M , r becomes the timelike coordinate) and isgiven by
g = −(1− 2M
r
)dt⊗ dt+
(1− 2M
r
)−1
dr ⊗ dr + r2h (6.1)
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where h = dθ ⊗ dθ + sin2 θdϕ⊗ dϕ is the Riemannian metric on S2 inducedby the natural embedding in R3 (see Example 1.6).
6.2.1 The perihelion precession
In Newtonian mechanics, the gravitational field outside of a spherically sym-metric object placed in the origin is given by
g(r) = −M
r3r (6.2)
where M is the mass of the object and r = |r|. This predicts the orbit ofany object moving in the potential to be elliptic (this is true even whenconsidering the gravitational effect of the moving object has on the originalobject, we just have to consider the reduced mass). The perihelion is thepoint of the orbit which has the least value of r. According to Newtonianmechanics, this point does not change.
Even before the theory of general relativity and the Schwarzschild solution,a perihelion precession of the planet Mercury was observed. That is, theperihelion preceeding around the sun for each Mercury year.
Using the Schwarzschild solution to describe the gravitational effect of thesun, the perihelion precession of Mercury is predicted to be about 5.02 · 10−7
radians per orbit. Taking into account the effect of other planets, this is inremarkably good correspondence to the observed effect. Of course, there is aperihelion precession for the other planets as well. However, this effect is sosmall that it is barely noticable.
6.2.2 The Schwarzschild black hole
One of the most interesting phenomena in physics is the concept of a blackhole, a region of space from which not even light might escape. One assumesthat all the mass of a spherically symmetric object is gathered within theSchwarzschild radius r = 2M . The metric tensor is now apperently singularat the Schwarzschild radius. However, this apparent singularity is simply dueto a poor choice of coordinates and can be removed. This was done by Kruskal
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and Szekeres by a coordinate transformation from t and r to u and v through
uv = (2M − r) exp
(r − 2M
2M
)(6.3)
t = 2M ln(−v/u). (6.4)
In these coordinates, which are valid for uv < 2M , the Kruskal-Szekeresmetric becomes
g = −8M2
rexp
(r − 2M
2M
)(du⊗ dv + dv ⊗ du) + r2h. (6.5)
The future directed null geodesics of this metric restricted to the uv-planehave one of the coordinates increasing and the other coordinate constant. Thespace is also transformed into four distinct regions which are the quadrants ofthe uv-plane. The first quadrant corresponds to the Schwarzschild black hole,the region with r < 2M and the second quadrant corresponds to r > 2M .The fourth quadrant is in some sense a copy of the second and correspondsto r > 2M for a sligtly different choice of the coordinate transformation tothe Schwarzschild time t. The third quadrant is called a “white hole” and hasno correspondence in the Schwarzschild solution.
It is not possible to find a future directed non-spacelike geodesic from theblack hole region to any of the other regions.
6.3 The Robertson-Walker space-times
Robertson-Walker space-times are space-times formed by Lorentzian warpedproducts between open intervals of the real line and isotropic Riemannianmanifolds. That is:
Definition 6.1 A Robertson-Walker space-time is a space-time (M0 ×f
H, g) such that
(M0 ×f H, g) = (M0 ×H,−dt⊗ dt+ f(t)h) (6.6)
where M0 is an interval of the real line, f : M0 −→ R+ and (H, h) is an
isotropic Riemannian manifold.
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In four space-time dimensions, the Robertson-Walker metric becomes
−dt⊗ dt+ S(t)2(dr ⊗ dr
1− kr2+ r2dΩ2
)(6.7)
where S =√f and dΩ2 is the Riemannian metric induced on S2 by the
natural embedding into R3. k is a constant which by scaling might be set to±1 or 0.
In physics, this is a solution to an isotropic universe. From this solution, onemay deduce that an expanding universe indicates that the universe has beenexpanding for all earlier values of the global time t. The different values of kcorrespond to a universe dense enough to contract, a universe which is notdense enough to contract but where the expansion rate tends to zero andfinally a universe where the expansion rate does not tend to zero and theexpansion goes on forever.
There are, of course, many other interesting aspects and predictions of gen-eral relativity such as the gravitational and cosmological redshifts of light.However, to keep this section brief we leave out these effects which can bestudied in practically any book on the subject of general relativity.
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References
[1] Global Lorentzian Geometry, J.K. Beem, P.E. Ehrlich, Marcel Dek-ker, Inc. 1981, ISBN 0-8247-1369-9
[2] Geometry, Topology and Physics, M. Nakahara, Institute of physicspublishing 1990, ISBN 0-85274-095-6
[3] Anomalies in Quantum Field Theory, R.A. Bertlmann, OxfordUniversity press 2001, ISBN 0-19850-762-3
[4] Relativity Theory, J. Mickelsson, T. Ohlsson, H. Snellman, Instituteof Physics, Royal Institute of Technology 2001
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