Transcript Document

Vincent Rodgers © 2005
www.physics.uiowa.edu
A Very Brief Intro to
Tensor Calculus
Two important concepts: Covariant
Derivatives and Tensors
Familiar objects but dressed up a
little differently
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Recall Calculus 101
What are tensors?
Two important objects in elementary Calculus
Derivative Operators
Differentials
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START WITH COORDINATE TRANSFORMATION:
Then the derivative operator transforms like:
COVARIANT
TRANSFORMATION
The differentials transform as:
CONTRAVARIANT
Functions transforms as:
1
SCALAR
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This is called a scalar or tensor of rank zero.
We can build coordinate invariants
by using the covariant and
contravariant tensors.
covariant
scalar
contravariant
scalar
This is invariant under coordinate transformations.
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Transformations in more than one dimension:
Transforms covariantly
Transforms contravariantly
Transforms covariantly
Transforms contravariantly
Einstein Implied Sum Rule is Used.
Also we always use these definition to define the
fundamental raised and lowered indices.
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Simple Example: Consider a rectangular to polar coordinate transformation
where
Notation
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 dx   sin( ) r cos( )  dr 
 
 
dy
cos(

)

r
sin(

)
  
 d 
y
r
x
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 r   


x x r x 

x

y


 2
2
x
( x 2  y 2 ) r ( x  y ) 

 1

 sin( )  cos( )
x
r r

 r   


y y r y 

y

x


 2
2
y
( x 2  y 2 ) r ( x  y ) 
 

 x
  

y   r


cos( ) 
 sin( )


1
 1
sin( ) 
   cos( )
r
 r



1 
 cos( )  sin( )
y
r
r 
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So motion in two dimensions is independent of the coordinate chart.
Between the different coordinate systems there is a “dictionary”
the transformation laws, that tell one observer how a different observer
perceives some event.
Physics should be independent of the coordinate system.
GENERAL COORDINATE INVARIANCE
Build physical theories out of quantities that can be translated
to another coordinate without depending on a particular coordinate
system. This is the essence of Tensors.
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SOME COMMON EXAMPLES OF TENSORS
Covariant
Contravariant
Mixed
Tensor Product
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Example: Stress-Strain Tensor
Stress Tensor and Strain Tensor
Stress-Strain relationship represents how a body is
distorted in the y direction (say) due to a force
applied in the x direction (say).
Fx

x  S

F
dx
dy
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The Metric Tensor: used to measure distance and
to map contravariant tensors into covariant tensors
THE METRIC AS WELL AS ALL TENSORS
HAVE MEANING INDEPENDENTLY OF A
COORDINATE SYSTEM. THE COORDINATE
SYSTEM IS ONLY REPRESENTING THE
METRIC!
In the (x,y) coordinate system
ds
y
In the (r, ) coordinate system
r
x
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Not all metrics are the same.
Here are two metrics that cannot
be related by a smooth coordinate
transformation
A metric on a
flat sheet of paper
A metric on a basketball
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Two other favorites but in four dimensions
Minkowski Space metric
using Cartesian coordinates
A black hole metric using spherical coordinates
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HOW DO WE TAKE DERIVATIVES OF TENSORS?
BIG PROBLEM!
x
Ordinary derivative of tensors are not tensors!
x
x + dx
Derivatives are supposed to measure the difference between
the tops of the tensors but here the tails are not at the same place.
We need to figure out how to get the tails to touch.
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INTRODUCE COVARIANT DERIVATIVES
tensor
nontensor
nontensor
Christoffel Symbol
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Christoffel Symbol parallel transports the tails of tensors together!

x + dx
a
c
bc
P
x
Tails are together so
now we can compute the
difference.
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PARALLELL TRANSPORT IDEA
ON A BALL AND FLAT SHEET OF PAPER
GEOMETRP
Distances are determined by the metric. Here it is clearly different on these
two dimensional surfaces
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Metric gives us the geometry through the
Covariant derivative.
Little loops can be measured through the commutator of the
Covariant derivative operator.
( a b  b a ) Cc  R
d
abc
Rab  Racbd g
R  Rab g
cd
Ricci Tensors
Ricci Scalar
ab
ds  g ab dx dx
2
Cd
Riemann Curvature Tensor
a
b
Metric tensor
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Explicitly “geometry” is defined by the Riemann Curvature
Tensor. Explicitly we write:
Rbcpa   b  apc   c  apb   qpc  aqb   qpb  aqc
  g ( c g db   b g dc   d gbc )
a
bc
1
2
ad

a  a
x
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These manifolds have different Ricci scalars that is how we know they have
GEOMETRY
Different geometry and not just look different because of a choice of coordinates.
Recall gauge symmetry in Electrodynamics, different potentials give same E and B if
they are related by a gauge transformation. You get different E and B if they are not.
R=1/r2
R=0
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THE GEODESIC EQUATION AS A FORCE LAW
Shortest Distance is defined by the differential
equation called the Geodesic Equation.
Acceleration
Proper time
Force
Velocities (the time components are the same gamma factors
seen in special relativity).
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Maxwell Theory in terms of Tensors
• Start with the Vector
Potential
A ( x, t )  (V / c, Ax , Ay , Az )
Scalar potential
Vector Potential
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Gauge transformations
1
1
A  g A g  ig  g
'
g ( x)  exp(i ( x))
 '    t 
These Change
A '  A  

E    A
t
B   A
These Remain
the same
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The Covariant Relationship to E and B
F    A   A
ANTISYMMETRIC
RANK 2 TENSOR
F01   0 A1  1 A0   E x
F02   0 A2   2 A0   E y
F03   0 A3   3 A0   E z
F12  1 A2   2 A1  Bz
F23   2 A3   2 A3  Bx
F13  1 A3   3 A1   B y
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The Maxwell Field Tensor and its Dual Tensor

 0
  Ex
c
F   
 Ey

c
 E
 z c
Ex
Ey
c
0
c
Bz
 Bz
0
By
 Bx

Ez 
 0
c

  Bx
 By   
c
G


 B y
Bx 

c

 B
z
0 

c


G    F
1
2
Bx
By
c
0
c
 Ez
Ez
0
 Ey
Ex
Bz 
c

Ey 

 Ex 

0 


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MAXWELL’S EQUATIONS
IN A COVARIANT NUTSHELL (SI units)
 F

 0 J


G  0;

J  (c , J )
  E   t B
  E   / 0
  B   0 0  t E   0 J
B  0
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THE COVARIANT DERIVATIVE
      A
 ,    (



 A )(  A )  (  A )(   A )
   A   A  F
Curvature of a gauge theory!
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Electricity and Magnetism’s Energy-Momentum Tensor
  F F  F Fcd g
ab
ca
b
c
1
4
cd
ab
00  E  E  B  B
0i  S i ; S  c ( E  B )
 a ab  0 
A Conservation Law
00
S  0
t

j
jk
1 
S


 0; j  1, 2,3
k
c2
t
x
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