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 10MATH OF CURVED SPACE THE METRIC
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Overview

• Distance and metric
• chapter 10 (this one)
• Vectors
• chapter 12
• Differentiation
• chapter 13
• Curvature
• chapter 16

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Objectives

• Tell curvature form distances
• Write down metric in examples
• General line element
• Embedding (can be omitted)
• sphere
• general

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Objectives

• Homogeneous manifolds
• closed, flat, open
• Schwarzschild
• Weak field

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Introduction

• Gravity spacetime curvature
• How do we describe curvature

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Extrinsic

• We can see that the 2D surface of the earth is
  curved
• By going out of it to 3D and looking down

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Extrinsic

• We can discuss an N dimensional object (e.g. N
  2)
• By embedding it in a larger M dimensional flat
  object, M gt N (e.g. M 3)
• And discussing the curvature in M dimensions

Embedding
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Extrinsic

• In relativity N 4 (ie. 31)
• The extra M ? N dimensions are fictitious
• Make sure physics is independent of these
  fictitious dimensions

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Intrinsic

• Can we tell surface of earth is curved
• While staying on the surface?

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Intrinsic

• Can we tell curvature intrinsically?
• By measuring distances
• Reduce to infinitesimal distances
• Assume quadratic form
• Riemannian geometries
• All these can be discussed more physically (but
  less rigorously) by embedding

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Embedding vs Intrinsic
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Terminology

• Euclidean / Minkowski

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Terminology

• Manifold

• Manifold ? "space" Not to be confused with
  space in spacetime

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Coordinates xm

• Need not all have same units

Grid

• Not a vector
• Upper indices
• Lower indices xm do not exist

• xm (3, 9)

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• Many coordinate systems possible
• Independence of coordinates

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• Coordinate grid
• Coordinate ? distance
• e.g. polar coordinate Df ? distance

• later Dt ? time

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Central question

• How is distance Ds
• related to the change in coordinates Dxm ?

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Examples ofCoordinates and Distances
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Examples

• A flat space x, y
• B flat space r, f
• C sphere q, f
• D sphere r, f
• E tilted axes

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A (x, y)
Ds2 Dx2 Dy2
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Ds2 Dx2 Dy2 Ds2 is a linear function of the
quadratics Dx2, Dy2
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B (r, f)
x r cosf y r sinf
Ds2 Dr2 r2Df 2
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DeriveD s2 D r2 r2Df 2
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Method 1
Ds2 Ds12 Ds22 Dr2 r2Df 2
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Method 2
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Ds2 Dr2 r2Df 2 Ds2 is a linear function of
the quadratics Dr2, Df2 with non-constant
coefficients
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C sphere of radius a
Ds2 a2Dq 2 a2 sin2q Df 2
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DeriveD s2 a2Dq 2 a2 sin2q Df 2
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Method 1
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Method 1
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• NS distance Ds1 aDq
• EW distance Ds2 rDf a sinq Df
• Ds2 Ds12 Ds22
• a2Dq 2 a2 sin2q Df 2

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Method 2
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Ds2 a2Dq 2 a2 sin2q Df 2 Ds2 is a linear
function of the quadratics Dq 2, Df 2 with
non-constant coefficients
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D (r, f)
r is NOT radial distance
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Two Definitions of r
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1. Radial definition
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2. Circumferential definition
C r ? 2p
Ds r ? Df
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Radial circumferential definitions of r are
consistent iff space is flat
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E tilted axes

• x u a v bcosg
• y v bsing
• (x1, x2) (u, v)

Ds2 ...
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Ds2 a2du2 2abcosg dudv b2dv2 Ds2 is a
linear function of the quadratics Du2, Dv2 and
DuDv with cross term
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Distances Curvature(Qualitative)
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Distances and curvature (qualitative)

• Can tell curvature from distances
• Axially symmetric case
• Sphere

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Axaially symmetric
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D sphere
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• Measure distances on the surface (intrinsic)
• Determine curvature
• Central idea in differential geometry

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Riemannian Geometry
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Riemannian geometry

• Metric

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Embedding Sphere
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Embedding General
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Cartesian coordinates
Example
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Distance
Example
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Curvilinear coordinates
Example
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Example
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Example
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Example
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Embedding General
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Homogeneous Manifolds
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Homogeneous manifolds

• Why homogeneous
• 3 types
• closed
• flat
• open
• Robertson-Walker metric

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Recall 2D homogeneous (surface of sphere)

• Advantage
• reduce to flat space if a ? ?
• Intuitive for 2D "people" one r, one f

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Flat 2D
Homogeneous curved 2D
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Flat 3D
Homogeneous curved 3D
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Cosmology
Spacetime (4)
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Cosmology(Space only)

• Three types
• closed
• flat
• open

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Closed
Flat
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Together
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Together

• 1/a2 k gt 0 closed
• k 0 flat

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Open
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General
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Most general 3D homogeneous space is describe by

• One discrete parameter K (1, 0, ?1)
• One continuous parameter a "size of universe"

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Reduced radius
"Stuck to grid"
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F Robertson-Walker metric
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Questions

• 1. K ?

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Other Examples
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Other examples

• G. Schwarzschild metric
• H. Weak fields

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G Schwarzschild
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Will NOT derive till laterAccept for the moment
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• r ? ? flat
• r circumferential

• clocks at different rates
• gravitational redshift

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test particle
m
r
M

• ? ltlt 1 spacetime curvature is small GR NOT
  important
• ? 1 GR important
• Black holes

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H Weak fields

• Newtonian potential F, F ltlt 1

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Will NOT derive till laterAccept for the moment
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Exercise
?
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Objectives

• Tell curvature form distances
• Write down metric in examples
• General line element
• Embedding (can be omitted)
• sphere
• general

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Objectives

• Homogeneous manifolds
• closed, flat, open
• Schwarzschild
• Weak field

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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