Vincent Rodgers © 2005 . Vincent Rodgers © 2005 A Very Brief Intro to Tensor Calculus Two...


A Very Brief Intro to Tensor Calculus

Two important concepts: Covariant Derivatives and Tensors

Familiar objects but dressed up a little differently


What are tensors?

Two important objects in elementary Calculus

Derivative Operators

Differentials

Recall Calculus 101


START WITH COORDINATE TRANSFORMATION:

Functions transforms as:

Then the derivative operator transforms like:

The differentials transform as:

COVARIANTTRANSFORMATION

CONTRAVARIANT

SCALAR1


We can build coordinate invariantsby using the covariant and contravariant tensors.

This is invariant under coordinate transformations.

This is called a scalar or tensor of rank zero.

contravariant

covariant

scalar

scalar


Transforms covariantly

Transforms contravariantly

Transformations in more than one dimension:

Transforms covariantly

Transforms contravariantly

Inve

rses

of

eac

h ot

her

Einstein Implied Sum Rule is Used. Also we always use these definition to define the

fundamental raised and lowered indices.


Simple Example: Consider a rectangular to polar coordinate transformation

where

Notation


x

y

rLen

gth of

A

sin( ) cos( )

cos( ) sin( )

dx r dr

dy r d


2 22 2

2 22 2

( )( )

1sin( ) cos( )

( )( )

1cos( ) sin( )

r

x x r x

x y

x r x yx y

x r r

r

y y r y

y x

y r x yx y

y r r

sin( ) cos( )

1 1cos( ) sin( )x y rr r


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 coordinatesystem. This is the essence of Tensors.


SOME COMMON EXAMPLES OF TENSORS

Covariant

Contravariant

Mixed

Tensor Product


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).

dx

dy

xF x S F


rLen

gth of

A

The Metric Tensor: used to measure distance and to map contravariant tensors into covariant tensors

x

yIn the (x,y) coordinate system

In the (r, ) coordinate system

THE METRIC AS WELL AS ALL TENSORS HAVE MEANING INDEPENDENTLY OF A

COORDINATE SYSTEM. THE COORDINATESYSTEM IS ONLY REPRESENTING THE

METRIC!ds


Not all metrics are the same.Here are two metrics that cannot be related by a smooth coordinate

transformationA metric on a

flat sheet of paper

A metric on a basketball


Two other favorites but in four dimensions

Minkowski Space metric using Cartesian coordinates

A black hole metric using spherical coordinates


HOW DO WE TAKE DERIVATIVES OF TENSORS?

BIG PROBLEM! Ordinary derivative of tensors are not tensors!

Derivatives are supposed to measure the difference betweenthe 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.

xx + dxx


INTRODUCE COVARIANT DERIVATIVES

tensor nontensor nontensor

Christoffel Symbol


Christoffel Symbol parallel transports the tails of tensors together!

x + dxx

Tails are together so now we can compute the

difference.

a cbcP


GEOMETRP

Distances are determined by the metric. Here it is clearly different on these two dimensional surfaces

PARALLELL TRANSPORT IDEA ON A BALL AND FLAT SHEET OF PAPER


Metric gives us the geometry through theCovariant derivative.

Little loops can be measured through the commutator of the Covariant derivative operator.

Riemann Curvature Tensor

Ricci Tensors

Ricci Scalar

Metric tensor2

( ) da b b a c abc d

cdab acbd

abab

a bab

C R C

R R g

R R g

ds g dx dx


Explicitly “geometry” is defined by the Riemann Curvature Tensor. Explicitly we write:

12 ( )

a a a q a q abcp b pc c pb pc qb pb qc

a adbc c db b dc d bc

a a

R

g g g g

x


GEOMETRY

R=0

These manifolds have different Ricci scalars that is how we know they have 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 ifthey are related by a gauge transformation. You get different E and B if they are not.

R=1/r2


THE GEODESIC EQUATION AS A FORCE LAW

Velocities (the time components are the same gamma factorsseen in special relativity).

Acceleration Force

Proper time

Shortest Distance is defined by the differential equation called the Geodesic Equation.


Maxwell Theory in terms of Tensors

• Start with the Vector Potential

( , ) ( / , , , )x y zA x t V c A A A

Scalar potential

Vector Potential


Gauge transformations

AB

At

E

AA

xixg

giggAgA

t

'

'

))(exp()(

11'

These Change

These Remainthe same


y

x

z

z

y

x

BAAF

BAAF

BAAF

EAAF

EAAF

EAAF

AAF

133113

323223

122112

033003

022002

011001

The Covariant Relationship to E and B

ANTISYMMETRIC RANK 2 TENSOR


0

0

0

0

0

0

0

0

xyz

xzy

yzx

zyx

xyz

xzy

yzx

zyx

EEcB

EEcB

EEcB

cB

cB

cB

G

BBcE

BBcE

BBcE

cE

cE

cE

F

The Maxwell Field Tensor and its Dual Tensor

FG 2

1


0

/

),(;0

000

0

0

B

JEB

E

BE

JcJGJF

t

t

MAXWELL’S EQUATIONS IN A COVARIANT NUTSHELL (SI units)


THE COVARIANT DERIVATIVE

FAA

AAAA

A

))(())((,

Curvature of a gauge theory!


Electricity and Magnetism’s Energy-Momentum Tensor

2

14

00

0

00

1

; ( )

0

0

0; 1,2,3

ab ca b cd abc cd

i i

aba

j jkkc

F F F F g

E E B B

S S c E B

St

S jt x

A Conservation Law

Vincent Rodgers © 2005 . Vincent Rodgers © 2005 A Very Brief Intro to Tensor Calculus Two...

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