PHY 4460 RELATIVITY

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PHY 4460RELATIVITY

• K Young, Physics Department, CUHK
• ?The Chinese University of Hong Kong

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CHAPTER 12MATH OF CURVED SPACE VECTORS
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Motivation
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Motivation

• We want to understand dynamics of point
  particles
• In Newtonian physics

• How momentum changes

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In relativity we also want
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• Understand vectors (Chapter 12)

• Understand differentiation (Chapter 13)

• Give law of motion (Chapter 14)

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How Did You Learn Vectorsin High School?
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How did you learn vectorsin high school?

• Start with displacement
• Other objects with "same property" are also
  vectors
• Same strategy here!

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Objectives

• Displacement vector
• Tangent plane
• Embedding in flat space
• Basis vectors
• Transformation of vectors
• Gradient of scalar
• Local cartesian system physical components

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Displacement Vector
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Displacement vector

• "Usual" definition
• Displacement is prototype vector
• From A to B
• Like a "straight arrow"

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Displacement vector

• On a curved surface, such a displacement is not a
  "straight arrow"

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Displacement vector

• Unless we talk about an infinitesimal displacement

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Displacement vector

• x is not a vector (x1, , xN)
• Dx is not a vector (Dx1, , DxN)
• dx is a vector (dx1, , dxN)

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Displacement vector

• The different components need not have the same
  units
• e.g. x (r, q, j)
• dx (dr, dq, dj)

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Displacement vector

• dx lives on tangent plane

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• Prototype vector is an infinitesimal displacement
• like a straight arrow
• lives on tangent plane Tp(x)

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Tangent Plane
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Tangent plane

• Vectors "live" on tangent planes
• Vectors do not "live" on the manifold itself

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Tangent planes

• Vectors live on different tangent planes

Tp(x) ? Tp(y)

• Vectors on Tp(x) and Tp(y) cannot be simply added
  or subtracted
• Differentiation subtraction of 2 vectors in
  nearby tangent planes, may therefore be nontrivial

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Example
Surface of a 2-sphere of radius a (x1, x2) (q,
j)

• On the tangent plane Tp (q, j)
• This is the vector pointing from (q, j) to (q
  dq, j dj)
• It is a "straight arrow"

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Embedding in Flat Space
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Embedding in flat space

• General formalism
• Examples
• Meaning of dx (dx1, dx2, )

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Example Polar coordinates on a plane
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Example Polar coordinates on a plane
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Example
Define basis vectors
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Example Surface of a sphere
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Example Surface of a sphere
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Example
Define basis vectors
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Embeddding in flat space

• Cartesian coordinates in M-D

• Corresponding unit vectors

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• Change to new coordinates

• Reduce to manifold in N-D (N ? M) by setting

• xN1 cN1, , xM cM
• e.g. R a

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• So is its differential

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• Change to new coordinates

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• m 1, , M ? m 1, , N if stay on manifold

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• m 1, , N

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Basis Vectors
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Basis vectors

• Definition
• Dot product and relation to metric

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Basis vectors

• Continue with Example

Natural to define basis vectors
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Basis vectors

• In general
• Not unit vectors
• Not orthogonal
• Meaning

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General Definition ofBasis Vectors
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I. Embedding definition
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II. Intrinsic definition
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• in the direction of increasing xm, keeping all
  other components fixed
• length displacement per unit change in xm

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• e.g. a point on the surface of the sphere
• It is a vector in the embedding point of view
• e.g. regarded in 3-D space

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• But it is not a vector on the manifold, only a
  general point P
• Therefore write as

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• Change xm to xm dxm (dxn 0, n ? m)
• Point P moves by displacement (vector) dP
• Take ratio

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• Mathematicians sometimes drop P

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Length of Basis Vectors
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I. Example

• Basis vectors are not unit vectors
• In general they are not even ?

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II. Embedding derivation
From Chapter 10
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III. Intrinsic derivation
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Compare
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Example
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Velocity and Momentum
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Velocity and momentum

• Curved space (not spacetime) or curved coordinates

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Velocity and momentum
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Velocity and momentum
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Transformation of Vectors
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Transformation of vectors

• Previously
• We considered only linear transformation of
  coordinates

• where amn are constants

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• Now
• We considered general transformation of
  coordinates

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Under general transformation

• The vectors at a point P transform linearly

Constant on a given TP Different on different TP's
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General Definitionof a Vector
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General definition of a vector
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• Upper index

Lower index
Contraction
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Higher rank tensors
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Compare Lorentz transformation
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Lowering of indices
Raising of indices
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• Metric tensor

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• dxm has two meanings

• change in xm, i.e. d(xm)

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dxm has only one meaning

• Not a change in xm
• No such thing as xm
• xm is not a vector, its indices cannot be lowered.

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Gradient of Scalar
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Gradient of scalar
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Example

• One surface of a sphere

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Objectives

• Displacement vector
• Tangent plane
• Embedding in flat space
• Basis vectors
• Transformation of vectors
• Gradient of scalar
• Local cartesian system physical components

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Acknowledgment

• This project is supported in part by the Hong
  Kong University Grants Committee (UGC) Teaching
  Development Grants (TDG) 3203005 and 3201032
• I thank Prof. S.C.Liew for software
• I thank Prof. M.C.Chu and Dr. S.S.Tong for advice
• I thank Miss H.Y.Shik for design