Gardner. Differential geometry and relativity (lecture notes, web draft, 2004)(198s)_PGr_.pdf

(769 KB) Pobierz
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
Differential Geometry (and Relativity) - Summer
2000
Classnotes
Copies of the classnotes are on the internet in PDF, Postscript and DVI forms as given below. In order to
view the DVI files, you will need a copy of LaTeX and you will need to download the images separately.
Click here for a list of the images.
Chapter 1: Introduction. PDF. PS. DVI.
Section 1-1: Curves. PDF. PS. DVI.
Section 1-2: Gauss Curvature. PDF. PS. DVI.
Section 1-3: Surfaces in E 3 . PDF. PS. DVI.
Section 1-4: First Fundamental Form. PDF. PS. DVI.
Section 1-5: Second Fundamental Form. PDF. PS. DVI.
Section 1-6: The Gauss Curvature in Detail. PDF. PS. DVI.
Section 1-7: Geodesics. PDF. PS. DVI.
Section 1-8: The Curvature Tensor and the Theorema Egregium . PDF. PS. DVI.
Section 1-9: Manifolds. PDF. PS. DVI.
Chapter 2: Special Relativity: The Geometry of Flat Spacetime. PDF. PS. DVI.
Section 2-1: Inertial Frames of Reference. PDF. PS. DVI.
Section 2-2: The Michelson Morley Experiment. PDF. PS. DVI.
Section 2-3: The Postulates of Relativity. PDF. PS. DVI.
Section 2-4: Relativity of Simltaneity. PDF. PS. DVI.
Section 2-5: Coordinates. PDF. PS. DVI.
Section 2-6: Invariance of the Interval. PDF. PS. DVI.
Section 2-7: The Lorentz Transformation. PDF. PS. DVI.
Section 2-8: Spacetime Diagrams. PDF. PS. DVI.
Section 2-9: Lorentz Geometry. PDF. PS. DVI.
Section 2-10: The Twin Paradox. PDF. PS. DVI.
Section 2-11: Temporal order and Causality. PDF. PS. DVI.
Chapter 3: General Relativity: The Geometry of Curved Spacetime. PDF. PS. DVI.
Section 3-1: The Principle of Equivalence. PDF. PS. DVI.
Section 3-2: Gravity as Spacetime Curvature. PDF. PS. DVI.
Section 3-3: The Consequences of Einstein's Theory. PDF. PS. DVI.
Section 3-6: Geodesics. PDF. PS. DVI.
Section 3-7: The Field Equations. PDF. PS. DVI.
Section 3-8: The Schwarzschild Solution. PDF. PS. DVI.
Section 3-9: Orbits in General Relativity. PDF. PS. DVI.
Section 3-10: The Bending of Light. PDF. PS. DVI.
Black Holes. PDF. PS. DVI.
841698378.035.png 841698378.036.png 841698378.037.png 841698378.038.png 841698378.001.png 841698378.002.png
 
Chapter 1. Surfaces and the
Concept of Curvature
Notation. We shall denote the familiar three dimensional Euclidean
space (tradiationally denoted R 3 )as E 3 .
Recall. The Euclidean metric on E 3 is
x = ( x, y, z ) =
x 2 + y 2 + z 2 .
1
841698378.003.png
1.1 Curves
Definition. A curve in E 3 is a vector valued function of the parameter
t :
α ( t )=( x ( t ) ,y ( t ) ,z ( t )) .
Note. We assume the functions x ( t ), y ( t ), and z ( t ) have continuous
second derivatives.
Definition. The derivative vector of curve α is
α ( t )=( x ( t ) ,y ( t ) ,z ( t )) .
If α ( t ) is the position of a particle at time t ,then α ( t )isthe velocity
vector of the particle and α ( t )isthe acceleration vector of the particle.
The speed of the particle is the scalar function α ( t ) .
Note. According to Newton’s Second Law of motion, the force acting
on a particle of mass m and position α ( t )is F ( t )= mα ( t ) .
Definition. The length (or arclength )ofthecurve α ( t )for t ∈ [ a, b ]is
S = b
a α ( t ) dt.
Note. If β ( t )isacurvefor t ∈ [ a, b ], then β can be written as a
function of arclength (which we will denote α ( s )) as follows. First,
S ( t )= t
a β ( t ) dt
1
(that is, S ( t ) is an antiderivative of speed which satisfies S ( a )=0).
Therefore S is a one to one function and S − 1 exists. S − 1 gives the time
at which the particle has travelled along β ( t ) a (gross) distance s .So
we denote this as t = S − 1 ( s ). Second, we make the substitution for t :
β ( t )= β ( S − 1 ( s )) ≡ α ( s ) .
However, it may be algebraically impossible to calculate t = S − 1 ( s )
(see page 11, number 5).
Recall. If f is differentiable on an interval I and f is nonzero on I ,
then f − 1 exists (i.e. f is one-to-one on I )on f ( I )and f − 1 is differen-
tiable of I . In addition,
df − 1
dx
x = f ( a )
1
=
df
dx
x = a
1
or f − 1 ( f ( a )) =
f ( a ) .
Note. If β ( t ) is parameterized as α ( s )asabove,then
β ( t )= β ( S − 1 ( s )) = α ( s )
and
= dβ
β ( t )
β ( t ) .
dS − 1 dS − 1
dα
ds
1
1
S ( t ) =
= β ( S − 1 ( s ))
S ( S − 1 ( s )) = β ( t )
ds
Notice dα
ds
= α ( s ) is a unit vector in the direction of the velocity vector
of β ( t ) .
2
841698378.004.png 841698378.005.png 841698378.006.png 841698378.007.png 841698378.008.png 841698378.009.png 841698378.010.png 841698378.011.png 841698378.012.png 841698378.013.png 841698378.014.png
Definition. If α ( s ) is a curve parameterized in terms of arclength s ,
then the unit tangent vector of α ( s )is α ( s )= T ( s ). ( α ( s ) is called a
unit speed curve since α ( s ) =1.)
Example 3 (page 6). Consider the circular helix
β ( t )=( a cos t, a sin t, bt )
(see Figure I-3, page 6). Parameterize β ( t ) in terms of arclength α ( s )
and calculate T ( s ) .
Solution. We have β ( t )=( −a sin t, a cos t, b ) . With S ( t ) the total
arclength travelled by a particle along the helix at time t ,wehave
S ( t )= β ( t ) = √
a 2 + b 2 .
Therefore, S ( t )= t √ a 2 + b 2 (taking S (0) = 0). Hence
s
t = S − 1 ( s )=
√ a 2 + b 2
and
s
α ( s )= β ( t )= β ( S − 1 ( s )) = β
√ a 2 + b 2
s
s
bs
√ a 2 + b 2
√ a 2 + b 2
√ a 2 + b 2
a cos
,a sin
,
.
=
Also,
1
s
s
T ( s )= α ( s )=
√ a 2 + b 2
√ a 2 + b 2
√ a 2 + b 2
−a sin
,a cos
,b
.
Notice that
β ( t )
β ( t )
β ( S − 1 ( s ))
T =
β ( S − 1 ( s )) .
=
3
841698378.015.png 841698378.016.png 841698378.017.png 841698378.018.png 841698378.019.png 841698378.020.png 841698378.021.png 841698378.022.png 841698378.023.png 841698378.024.png 841698378.025.png 841698378.026.png 841698378.027.png 841698378.028.png 841698378.029.png 841698378.030.png 841698378.031.png 841698378.032.png 841698378.033.png 841698378.034.png

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin