CAPÍTULO II: BASE TEÓRICA
2.3 Base legal
2.3.1 Consideraciones normativas:
Quite often we deal with the divergence of vectors. Given an arbitrary vector fieldVα, its divergence is defined by Eq. (5.53),
Vα;α =Vα,α+αμαVμ. (6.36) This formula involves a sum in the Christoffel symbol, which, from Eq. (6.32), is
αμα =1 2g αβ(gβμ ,α+gβα,μ−gμα,β) =1 2g αβ(gβμ ,α−gμα,β)+ 1 2g αβgαβ ,μ. (6.37)
This has had its terms rearranged to simplify it: notice that the term in parentheses is antisymmetric inαandβ, while it is contracted onαandβwithgαβ, which is symmetric. The first term therefore vanishes (see Exer. 26(a), §3.10) and we find
αμα= 1
2g
αβgαβ
,μ. (6.38)
Since (gαβ) is the inverse matrix of (gαβ), it can be shown (see Exer. 7, § 6.9) that the derivative of the determinantgof the matrix (gαβ) is
g,μ=ggαβgβα,μ. (6.39) Using this in Eq. (6.38), we find
αμα =(√−g),μ/√−g. (6.40) Then we can write the divergence, Eq. (6.36), as
Vα;α=Vα,α+ 1
√−
gV
153 6.4 Parallel-transport , geodesics, and curvature
t
or Vα;α = 1 √− g( √− gVα),α. (6.42)This is a very much easier formula to use than Eq. (6.36). It is also important for Gauss’ law, where we integrate the divergence over a volume (using, of course, the proper volume element):
'
Vα;α√−gd4x= '
(√−gVα),αd4x. (6.43) Since the final term involves simple partial derivatives, the mathematics of Gauss’ law applies to it, just as in SR (§4.8):
' (√−gVα),αd4x= ( Vαnα√−gd3S. (6.44) This means ' Vα;α√−gd4x= ( Vαnα√−gd3S. (6.45)
So Gauss’ law does apply on a curved manifold, in the form given by Eq. (6.45). We need to integrate the divergence over proper volume and to use theproper surface element, nα√−gd3S, in the surface integral.
6.4 P a r a l l e l - t r a n s p o r t , g e o d e s i c s , a n d c u r v a t u r e
Until now, we have used the local-flatness theorem to develop as much mathematics on curved manifolds as possible without considering the curvature explicitly. Indeed, we have yet to give a precise mathematical definition of curvature. It is important to distinguish two different kinds of curvature: intrinsic and extrinsic. Consider, for example, a cylinder. Since a cylinder is round in one direction, we think of it as curved. This is itsextrinsic curvature: the curvature it has in relation to the flat three-dimensional space it is part of. On the other hand, a cylinder can be made by rolling a flat piece of paper without tearing or crumpling it, so theintrinsicgeometry is that of the original paper: it is flat. This means that the distance in the surface of the cylinder between any two points is the same as it was in the original paper; parallel lines remain parallel when continued; in fact,allof Euclid’s axioms hold for the surface of a cylinder. A two-dimensional ‘ant’ confined to that surface would decide it was flat; only its global topology is funny, in that going in a certain direc- tion in a straight line brings him back to where he started. The intrinsicgeometry of an n-dimensional manifold considers only the relationships between its points on paths that
154 Curved manifolds
t
remain in the manifold (for the cylinder, in the two-dimensional surface). Theextrinsiccur- vature of the cylinder comes from considering it as a surface in a space of higher dimension, and asking about the curvature of lines that stay in the surface compared with ‘straight’ lines that go off it. Soextrinsic curvature relies on the notion of a higher-dimensional space. In this book, when we talk about the curvature of spacetime, we talk about its intrinsiccurvature, since it is clear that all world lines are confined to remain in space- time. Whether or not there is a higher-dimensional space in which our four-dimensional space is an open question that is becoming more and more a subject of discussion within the framework of string theory. The only thing of interest in GR is the intrinsic geometry of spacetime.
The cylinder, as we have just seen, is intrinsically flat; a sphere, on the other hand, has an intrinsically curved surface. To see this, consider Fig.6.1, in which two neighboring lines begin atAandBperpendicular to the equator, and hence are parallel. When continued as locally straight lines they follow the arc of great circles, and the two lines meet at the pole P. Parallel lines, when continued, do not remain parallel, so the space is not flat.
There is an even more striking illustration of the curvature of the sphere. Consider, first, flat space. In Fig.6.2a closed path in flat space is drawn, and, starting atA, at each point a vector is drawn parallel to the one at the previous point. This construction is carried around the loop fromAtoBtoCand back toA. The vector finally drawn atAis, of course, parallel to the original one. A completely different thing happens on a sphere! Consider the path shown in Fig.6.3. Remember, we are drawing the vector as it is seen to a two- dimensional ant on the sphere, so it must always be tangent to the sphere. Aside from that, each vector is drawn as parallel as possible to the previous one. In this loop,AandCare on the equator 90◦ apart, andB is at the pole. Each arc is the arc of a great circle, and each is 90◦long. AtAwe choose the vector parallel to the equator. As we move up toward B, each new vector is therefore drawn perpendicular to the arc AB. When we get to B,
P
B A
t
Figure 6.1 A spherical triangleAPB.A
C B
155 6.4 Parallel-transport , geodesics, and curvature
t
B
C A
t
Figure 6.3 Parallel transport around a spherical triangle.V
→
V
→
U→ U→
t
Figure 6.4 Parallel transport ofValongU.the vectors are tangent toBC. So, going fromBtoC, we keep drawing tangents toBC. These are perpendicular to the equator atC, and so fromCtoAthe new vectors remain perpendicular to the equator. Thus the vector field has rotated 90◦ in this construction! Despite the fact that each vector is drawn parallel to its neighbor, the closed loop has caused a discrepancy. Since this doesn’t happen in flat space, it must be an effect of the sphere’s curvature.
This result has radical implications: on a curved manifold it simply isn’t possible to define globally parallel vector fields. We can still define local parallelism, for instance how to move a vector from one point to another, keeping it parallel and of the same length. But the result of such ‘parallel transport’ from pointAto pointBdepends on the path taken. We therefore cannot assert that a vector atAis or is not parallel to (or the same as) a certain vector atB.