Intro to Differential Geometry and General Relativity - S. Waner.pdf
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Introduction
to
Differential Geometry
&
General Relativity
4th Printing January 2005
Lecture Notes
by
Stefan Waner
with a Special Guest Lecture
by
Gregory C. Levine
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
......................................................................................................3
2. Smooth Manifolds and Scalar Fields
..............................................................7
3. Tangent Vectors and the Tangent Space
.......................................................14
4. Contravariant and Covariant Vector Fields
................................................24
5.
Tensor Fields
....................................................................................................35
6. Riemannian Manifolds
...................................................................................40
7. Locally Minkowskian Manifolds: An Introduction to Relativity
.............50
8. Covariant Differentiation
...............................................................................61
9. Geodesics and Local Inertial Frames
.............................................................69
10. The Riemann Curvature Tensor
..................................................................82
11. A Little More Relativity: Comoving Frames and Proper Time
...............94
12. The Stress Tensor and the Relativistic Stress-Energy Tensor
................100
13. Two Basic Premises of General Relativity
................................................109
14. The Einstein Field Equations and Derivation of Newton's Law
...........114
15. The Schwarzschild Metric and Event Horizons
......................................124
16. White Dwarfs, Neutron Stars and Black Holes, by Gregory C. Levine
131
References and Further Reading
................................................................138
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
n
= {(y
1
, y
2
, . . . , y
n
) | y
i
é
R
}
.
Thus,
E
1
is just the real line,
E
2
is the Euclidean plane, and
E
3
is
3-
dimensional Euclidean
space.
The
magnitude
, or
norm
,
||y||
of
y = (y
1
, y
2
, . . . , y
n
)
in
E
n
is defined to be
||y|| = y
1
2
+ y
2
2
+ . . . + y
n
2
,
which we think of as its distance from the origin. Thus, the
distance
between two points
y
= (y
1
, y
2
, . . . , y
n
)
and
z = (z
1
, z
2
, . . . , z
n
)
in
E
n
is defined as the norm of
z - y:
Distance Formula
Distance between
y
and
z
=
||z - y|| = (z
1
- y
1
)
2
+ (z
2
- y
2
)
2
+ . . . + (z
n
- y
n
)
2
.
Proposition 1.1 (Properties of the norm)
The norm satisfies the following:
(a)
||y|| ≥ 0
, and
||y|| = 0
iff
y = 0 (
positive definite)
(b)
||¬y|| = |¬|||y||
for every
¬ é
R and
y é E
n
.
(c)
||y + z|| ≤ ||y|| + ||z||
for every
y
,
z é E
n
(triangle inequality 1)
(d)
||y - z|| ≤ ||y - w|| + ||w - z||
for every
y, z, w é E
n
(triangle inequality 2)
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
n
is called
open
if, for every
y
in
U
, all points of
E
n
within
some positive distance
r
of
y
are also in
U
. (The size of
r
may depend on the point
y
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
n
, then the
open ball with center
a
and radius
r
is the subset
B(a, r) = {x é E
n
| ||x-a|| < r}.
3
Open balls are open sets: If
x é B(a, r)
, then, with
s = r - ||x-a||
, one has
B(x, s) ¯ B(a,
r)
.
(b)
E
n
is open.
(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
s
. A subset
V ¯ M
is called
open in
M
(or
relatively
open
) if, for every
y
in
V
, all points of
M
within some positive distance
r
of
y
are also in
V
.
Examples 1.5
(a) Open balls in
M
If
M ¯ E
s
,
m é M
, and
r >
0, define
Then
B
M
(m, r) = {x é M | ||x-m|| < r}.
B
M
(m, r) = B(m, r) Ú M
,
and so
B
M
(m, r)
is open in
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
E
3
From now on, the three coordinates of 3-space will be referred to as
y
1
, y
2
,
and
y
3
.
Definition 1.6
A smooth
path
in
E
3
is a set of three smooth (infinitely differentiable) real-
valued functions of a single real variable
t
:
y
1
= y
1
(t), y
2
= y
2
(t), y
3
= y
3
(t).
The variable
t
is called the
parameter
of the curve. The path is
non-singular
if the vector
(
dy
1
dt
,
dy
2
dt
,
dy
3
dt
)
is nowhere zero.
Notes
(a)
Instead of writing
y
1
= y
1
(t), y
2
= y
2
(t), y
3
= y
3
(t)
, we shall simply write
y
i
= y
i
(t)
.
(b)
Since there is nothing special about three dimensions, we define a
smooth path in
E
n
in exactly the same way: as a collection of smooth functions
y
i
= y
i
(t)
, where this time
i
goes
from 1 to
n
.
4
Examples 1.7
(a)
Straight lines in
E
3
(b)
Curves in
E
3
(circles, etc.)
Definition 1.8
A
smooth surface embedded in
E
3
is a collection of three smooth real-
valued functions of
two
variables
x
1
and
x
2
(notice that
x
finally makes a debut).
or just
y
1
= y
1
(x
1
, x
2
)
y
2
= y
2
(x
1
, x
2
)
y
3
= y
3
(x
1
, x
2
)
,
y
i
= y
i
(x
1
, x
2
) (i = 1, 2, 3)
.
∂x
j
has rank two.
(b) The associated function
E
2
→
E
3
is a one-to-one map (that is, distinct points
(x
1
, x
2
)
in
“parameter space”
E
2
give different points
(y
1
, y
2
,
y
3
)
in
E
3
.
We call
x
1
and
x
2
the
parameters
or
local coordinates
.
Examples 1.9
(a)
Planes in
E
3
(b)
The paraboloid
y
3
= y
1
2
+ y
2
2
(c)
The sphere
y
1
2
+ y
2
2
+ y
3
2
= 1
, using spherical polar coordinates:
y
1
=
sin
x
1
cos
x
2
y
2
=
sin
x
1
sin
x
2
y
3
=
cos
x
1
Note that condition (a) fails at
x
1
= 0
and
π
.
c
2
= 1, where
a, b
and
c
are positive constants.
(e)
We calculate the rank of the Jacobean matrix for spherical polar coordinates.
(f)
The torus with radii
a > b:
y
1
= (a+b
cos
x
2
)
cos
x
1
y
2
= (a+b
cos
x
2
)
sin
x
1
y
3
= b
sin
x
2
(Note that if
a ≤ b
this torus is not embedded.)
(g)
The functions
y
1
= x
1
+ x
2
y
2
= x
1
+ x
2
y
3
= x
1
+ x
2
a
2
+
b
2
+
y
3
2
5
We also require that:
(a) The
3¿2
matrix whose
ij
entry is
∂y
i
(d)
The ellipsoid
y
1
2
y
2
2
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