Introduction to Differential Geometry & General Relativity

128 Pages·2002·0.607 MB·English

by  Waner S.

Introduction
to
Differential Geometry
&
General Relativity
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Departments of Mathematics and Physics, Hofstra University
Introduction to Differential Geometry and General Relativity
Lecture Notes by Stefan Waner,
with a Special Guest Lecture by Gregory C. Levine
Department of Mathematics, Hofstra University
These notes are dedicated to the memory of Hanno Rund.
TABLE OF CONTENTS
1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions
2. Smooth Manifolds and Scalar Fields
3. Tangent Vectors and the Tangent Space
4. Contravariant and Covariant Vector Fields
5. Tensor Fields
6. Riemannian Manifolds
7. Locally Minkowskian Manifolds: An Introduction to Relativity
8. Covariant Differentiation
9. Geodesics and Local Inertial Frames
10. The Riemann Curvature Tensor
11. A Little More Relativity: Comoving Frames and Proper Time
12. The Stress Tensor and the Relativistic Stress-Energy Tensor
13. Two Basic Premises of General Relativity
14. The Einstein Field Equations and Derivation of Newton's Law
15. The Schwarzschild Metric and Event Horizons
16. White Dwarfs, Neutron Stars and Black Holes, by Gregory C. Levine
2
1. Preliminaries
Distance and Open Sets
Here, we do just enough topology so as to be able to talk about smooth manifolds. We
begin with n-dimensional Euclidean space
E = {(y , y , . . . , y ) | y é R}.
n 1 2 n i
Thus, E is just the real line, E is the Euclidean plane, and E is 3-dimensional Euclidean
1 2 3
space.
The magnitude, or norm, ||yyyy|| of yyyy = (y , y , . . . , y ) in E is defined to be
1 2 n n
||yyyy|| = y 2(cid:19)+(cid:19)y 2(cid:19)+(cid:19).(cid:19).(cid:19).(cid:19)+(cid:19)y 2 ,
1 2 n
which we think of as its distance from the origin. Thus, the distance between two points yyyy
= (y , y , . . . , y ) and zzzz = (z , z , . . . , z ) in E is defined as the norm of zzzz - yyyy:
1 2 n 1 2 n n
Distance Formula
Distance between yyyy and zzzz = ||zzzz ---- yyyy|| = (z (cid:19)-(cid:19)y )2(cid:19)+(cid:19)(z (cid:19)-(cid:19)y )2(cid:19)+(cid:19).(cid:19).(cid:19).(cid:19)+(cid:19)(z (cid:19)-(cid:19)y )2 .
1 1 2 2 n n
Proposition 1.1 (Properties of the norm)
The norm satisfies the following:
(a) ||yyyy|| ≥ 0, and ||yyyy|| = 0 iff yyyy = 0 (positive definite)
(b) ||¬yyyy|| = |¬|||yyyy|| for every ¬ é R and yyyy é E .
n
(c) ||yyyy + zzzz|| ≤ ||yyyy|| + ||zzzz|| for every yyyy, zzzz é E (triangle inequality 1)
n
(d) ||yyyy - zzzz|| ≤ ||yyyy ---- wwww|| + ||wwww ---- zzzz|| for every yyyy,,,, zzzz,,,, wwww é E (triangle inequality 2)
n
The proof of Proposition 1.1 is an exercise which may require reference to a linear algebra
text (see “inner products”).
Definition 1.2 A Subset U of E is called open if, for every yyyy in U, all points of E within
n n
some positive distance r of yyyy are also in U. (The size of r may depend on the point yyyy
chosen. Illustration in class).
Intuitively, an open set is a solid region minus its boundary. If we include the boundary,
we get a closed set, which formally is defined as the complement of an open set.
Examples 1.3
(a) If a é E , then the open ball with center aaaa and radius rrrr is the subset
n
B(aaaa, r) = {x é E | ||xxxx-aaaa|| < r}.
n
3
Open balls are open sets: If xxxx é B(aaaa, r), then, with s = r - ||xxxx----aaaa||, one has B(xxxx, s) ¯
B(aaaa, r).
(b) E is open.
n
(c) Ø is open.
(d) Unions of open sets are open.
(e) Open sets are unions of open balls. (Proof in class)
Definition 1.4 Now let M ¯ E . A subset V ¯ M is called open in MMMM (or relatively open)
s
if, for every yyyy in V, all points of M within some positive distance r of yyyy are also in V.
Examples 1.5
(a) Open balls in MMMM
If M ¯ E , mmmm é M, and r > 0, define
s
B (mmmm, r) = {x é M | ||xxxx-mmmm|| < r}.
M
Then
B (mmmm, r) = B(mmmm, r) Ú M,
M
and so B (mmmm, r) is open in M.
M
(b) M is open in M.
(c) Ø is open in M.
(d) Unions of open sets in M are open in M.
(e) Open sets in M are unions of open balls in M.
Parametric Paths and Surfaces in EEEE
3333
From now on, the three coordinates of 3-space will be referred to as y , y , and y .
1 2 3
Definition 1.6 A smooth path in E is a set of three smooth (infinitely differentiable) real-
3
valued functions of a single real variable t:
y = y (t), y = y (t), y = y (t).
1 1 2 2 3 3
The variable t is called the parameter of the curve. The path is non-singular if the vector
dy dy dy
( 1 , 2 , 3 ) is nowhere zero.
dt dt dt
Notes
(a) Instead of writing y = y (t), y = y (t), y = y (t), we shall simply write y = y(t).
1 1 2 2 3 3 i i
(b) Since there is nothing special about three dimensions, we define a smooth path in EEEE
nnnn
in exactly the same way: as a collection of smooth functions y = y(t), where this time i
i i
goes from 1 to n.
4
Examples 1.7
(a) Straight lines in E
3
(b) Curves in E (circles, etc.)
3
Definition 1.8 A smooth surface immersed in EEEE is a collection of three smooth real-
3333
valued functions of two variables x1 and x2 (notice that x finally makes a debut).
y = y (x1, x2)
1 1
y = y (x1, x2)
2 2
y = y (x1, x2),
3 3
or just
y = y(x1, x2) (i = 1, 2, 3).
i i
∂y
We also require that the 3¿2 matrix whose ij entry is i has rank two. We call x1 and x2
∂xj
the parameters or local coordinates.
Examples 1.9
(a) Planes in E
3
(b) The paraboloid y = y 2 + y 2
3 1 2
(c) The sphere y 2 + y 2 + y 2 = 1, using spherical polar coordinates:
1 2 3
y1 = sin x1 cos x2
y2 = sin x1 sin x2
y3 = cos x1
y 2 y 2 y 2
(d) The ellipsoid 1 + 2 + 3 = 1, where a, b and c are positive constants.
a2 b2 c2
(e) We calculate the rank of the Jacobean matrix for spherical polar coordinates.
(f) The torus with radii a > b:
y = (a+bcos x2)cos x1
1
y = (a+bcos x2)sin x1
2
y = bsin x2
3
Question The parametric equations of a surface show us how to obtain a point on the
surface once we know the two local coordinates (parameters). In other words, we have
specified a function E ’E . How do we obtain the local coordinates from the Cartesian
2 3
coordinates y , y , y ?
1 2 3
Answer We need to solve for the local coordinates xi as functions of y. This we do in one
j
or two examples in class. For instance, in the case of a sphere, we get
x1 = cos-1(y )
3
5

cos-1(y (cid:19)/(cid:19) y 2+y 2(cid:19)) if y (cid:19)≥(cid:19)0
x2 =  1 1 2 2 .
2π(cid:19)-(cid:19)cos-1(y (cid:19)/(cid:19) y 2+y 2(cid:19)) if(cid:19)y (cid:19)<(cid:19)0
1 1 2 2
This allows us to give each point on much of the sphere two unique coordinates, x1, and
x2. There is a problem with continuity when y = 0, since then x2 switches from 0 to 2π.
2
There is also a problem at the poles (y = y = 0), since then the above functions are not
1 2
even defined. Thus, we restrict to the portion of the sphere given by
0 < x1 < π
0 < x2 < 2π,
which is an open subset U of the sphere. (Think of it as the surface of the earth with the
Greenwich Meridian removed.) We call x1 and x2 the coordinate functions. They are
functions
x1:::: U’E
1
and
x2:::: U’E .
1
We can put them together to obtain a single function xxxx:::: U’E given by
2
xxxx(y , y , y ) = (x1(y , y , y ), x2(y , y , y ))
1 2 3 1 2 3 1 2 3
  
 cos-1(y (cid:19)/(cid:19) y 2+y 2(cid:19)) if y (cid:19)≥(cid:19)0 
= cos-1(y ),(cid:19) 1 1 2 2 
 3 2π(cid:19)-(cid:19)cos-1(y (cid:19)/(cid:19) y 2+y 2(cid:19)) if(cid:19)y (cid:19)<(cid:19)0 
1 1 2 2
as specified by the above formulas, as a chart.
Definition 1.10 A chart of a surface S is a pair of functions xxxx = (x1(y , y , y ), x2(y , y ,
1 2 3 1 2
y )) which specify each of the local coordinates (parameters) x1 and x2 as smooth functions
3
of a general point (global or ambient coordinates) (y , y , y ) on the surface.
1 2 3
Question Why are these functions called a chart?
Answer The chart above assigns to each point on the sphere (away from the meridian) two
coordinates. So, we can think of it as giving a two-dimensional map of the surface of the
sphere, just like a geographic chart.
Question Our chart for the sphere is very nice, but is only appears to chart a portion of the
sphere. What about the missing meridian?
Answer We can use another chart to get those by using different paramaterization that
places the poles on the equator. (Diagram in class.)
6
In general, we chart an entire manifold M by “covering” it with open sets U which become
the domains of coordinate charts.
Exercise Set 1
1. Prove Proposition 1.1.(Consult a linear algebra text.)
2. Prove the claim in Example 1.3 (d).
3. Prove that finite intersection of open sets in E are open.
n
4. Parametrize the following curves in E .
3
(a) a circle with center (1, 2, 3) and radius 4
(b) the curve x = y2; z = 3
(c) the intersection of the planes 3x-3y+z=0 and 4x+y+z=1.
5. Express the following planes parametrically:
(a) y + y - 2y = 0.
1 2 3
(b) 2y + y - y = 12.
1 2 3
6. Express the following quadratic surfaces parametrically: [Hint. For the hyperboloids,
refer to parameterizations of the ellipsoid, and use the identity cosh2x - sinh2x = 1. For the
double cone, use y = cx1, and x1 as a factor of y and y .]
3 1 2
y 2 y 2 y 2
(a) Hyperboloid of One Sheet: 1 + 2 - 3 = 1.
a2 b2 c2
y 2 y 2 y 2
(b) Hyperboloid of Two Sheets: 1 - 2 - 3 = 1
a2 b2 c2
y 2 y 2 y 2
(c) Cone: 3 = 1 + 2 .
c2 a2 b2
y y 2 y 2
(d) Hyperbolic Paraboloid: 3 = 1 - 2
c a2 b2
7. Solve the parametric equations you obtained in 5(a) and 6(b) for x1 and x2 as smooth
functions of a general point (y , y , y ) on the surface in question.
1 2 3
2. Smooth Manifolds and Scalar Fields
We now formalize the above ideas.
Definition 2.1 An open cover of M(cid:19)¯ E is a collection {U } of open sets in M such that
s å
M = Æ U .
å å
Examples
(a) E can be covered by open balls.
s
(b) E can be covered by the single set E .
s s
(c) The unit sphere in E can be covered by the collection {U , U } where
s 1 2
U = {(y , y , y ) | y > -1/2}
1 1 2 3 3
U = {(y , y , y ) | y < 1/2}.
2 1 2 3 3
7
Definition 2.2 A subset M of E is called an nnnn-dimensional smooth manifold if we are
s
given a collection {U ; x 1, x 2, . . ., x n} where:
å å å å
(a) The U form an open cover of M.
å
(b) Each x r is a CÏ real-valued function defined on U (that is, x r: U ’E ), and
å å å 1
extending to an open set of E , called the rrrr-th coordinate, such that the map x: U ’E
s å n
given by x(u) = (x 1(u), x 2(u), . . . , x n(u)) is one-to-one. (That is, to each point in U ,
å å å å
we are assigned a unique set of n coordinates.) The tuple (U ; x 1, x 2, . . ., x n) is called a
å å å å
local chart of MMMM. The collection of all charts is called a smooth atlas of MMMM. Further, U is
å
called a coordinate neighborhood.
(c) If (U, xi), and (V, xj–) are two local charts of M, and if UÚV ≠ Ø, then we can write
xi = xi(xj–)
with inverse
xk– = xk– (xl)
for each i and k, where all functions in sight are CÏ. These functions are called the change-
of-coordinates transformations.
By the way, we call the “big” space E in which the manifold M is embedded the ambient
s
space.
Notes
1. Always think of the xi as the local coordinates (or parameters) of the manifold. We can
paramaterize each of the open sets U by using the inverse function x-1 of x, which assigns
to each point in some neighborhood of E a corresponding point in the manifold.
n
2. Condition (c) implies that
∂xi– 
det   ≠ 0,
∂xj 
and
∂xi 
det   ≠ 0,
∂xj– 
since the associated matrices must be invertible.
3. The ambient space need not be present in the general theory of manifolds; that is, it is
possible to define a smooth manifold M without any reference to an ambient space at
all—see any text on differential topology or differential geometry (or look at Rund's
appendix).
4. More terminology: We shall sometimes refer to the xi as the local coordinates, and to the
yj as the ambient coordinates. Thus, a point in an n-dimensional manifold M in E has n
s
local coordinates, but s ambient coordinates.
Examples 2.3
(a) E is an n-dimensional manifold, with the single identity chart defined by
n
xi(y , . . . , y ) = y.
1 n i
8
(b) S1, the unit circle, with the exponential map, is a 1-dimensional manifold. Here is a
possible structure:with two charts as show in in the following figure.
One has
x: S1-{(1, 0)}’E
1
x–: S1-{(-1, 0)}’E ,
1
with 0 < x, x – < 2π, and the change-of-coordinate maps are given by
 x+π if x(cid:19)<(cid:19)π
x – =  (See the figure for the two cases. )
 x-π if(cid:19)x(cid:19)>(cid:19)π
and
 x–+π if x–(cid:19)<(cid:19)π
x =  .
 x–-π if(cid:19)x(cid:19)– >(cid:19)π
Notice the symmetry between x and x.– Also notice that these change-of-coordinate
functions are only defined when ø ≠ 0, π. Further,
∂x/– ∂x = ∂x/∂x – = 1.
Note also that, in terms of complex numbers, we can write, for a point p = eiz é S1,
x = arg(z), x – = arg(-z).
(c) Generalized Polar Coordinates
Let us take M = Sn, the unit n-sphere,
Sn = {(y , y , … , y , y ) é E | £y2 = 1},
1 2 n n+1 n+1 i i
with coordinates (x1, x2, . . . , xn) with
0 < x1, x2, . . . , xn-1 < π
and
9
0 < xn < 2π,
given by
y = cos x1
1
y = sin x1 cos x2
2
y = sin x1 sin x2 cos x3
3
…
y = sin x1 sin x2 sin x3 sin x4 … cos xn-1
n-1
y = sin x1 sin x2 sin x3 sin x4 … sin xn-1 cos xn
n
y = sin x1 sin x2 sin x3 sin x4 … sin xn-1 sin xn
n+1
In the homework, you will be asked to obtain the associated chart by solving for the xi.
Note that if the sphere has radius r, then we can multiply all the above expressions by r,
getting
y1 = r cos x1
y2 = r sin x1 cos x2
y3 = r sin x1 sin x2 cos x3
…
yn-1 = r sin x1 sin x2 sin x3 sin x4 … cos xn-1
yn = r sin x1 sin x2 sin x3 sin x4 … sin xn-1 cos xn
yn+1 = r sin x1 sin x2 sin x3 sin x4 … sin xn-1 sin xn.
(d) The torus T = S1¿S1, with the following four charts:
xxxx: (S1-{(1, 0)})¿(S1-{(1, 0)})’E , given by
2
x1((cosø, sinø), (cos˙, sin˙)) = ø
x2((cosø, sinø), (cos˙, sin˙)) = ˙.
The remaining charts are defined similarly, and the change-of-coordinate maps are omitted.
(e) The cylinder (homework)
(f) Sn, with (again) stereographic projection, is an n-manifold; the two charts are given as
follows. Let P be the point (0, 0, . . , 0, 1) and let Q be the point (0, 0, . . . , 0, -1). Then
define two charts (Sn-P, xi) and (Sn-Q, xi–) as follows. (See the figure.)
10