eBook ISBN:  9780821890127 
Product Code:  MEMO/218/1025.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $34.80 
eBook ISBN:  9780821890127 
Product Code:  MEMO/218/1025.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $34.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 218; 2012; 69 ppMSC: Primary 34; 76; 83; 85
The authors prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of selfsimilar expansion waves, and the critical (\(k=0\)) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. Removing a scaling law and imposing regularity at the center, they prove that the family reduces to an implicitly defined oneparameter family of distinct spacetimes determined by the value of a new acceleration parameter \(a\), such that \(a=1\) corresponds to the Standard Model.
The authors prove that all of the selfsimilar spacetimes in the family are distinct from the noncritical \(k\neq0\) Friedmann spacetimes, thereby characterizing the critical \(k=0\) Friedmann universe as the unique spacetime lying at the intersection of these two oneparameter families. They then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, they use these rigorous estimates to calculate the exact leading order quadratic and cubic corrections to the redshift vs luminosity relation for an observer at the center.

Table of Contents

Chapters

1. Introduction

2. SelfSimilar Coordinates for the $k=0$ FRW Spacetime

3. The Expanding Wave Equations

4. Canonical Comoving Coordinates and Comparison with the $k\neq 0$ FRW Spacetimes

5. Leading Order Corrections to the Standard Model Induced by the Expanding Waves

6. A Foliation of the Expanding Wave Spacetimes into Flat Spacelike Hypersurfaces with Modified Scale Factor $R(t)=t^{a}$.

7. Expanding Wave Corrections to the Standard Model in Approximate Comoving Coordinates

8. Redshift vs Luminosity Relations and the Anomalous Acceleration

9. Appendix: The Mirror Problem

10. Concluding Remarks


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The authors prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of selfsimilar expansion waves, and the critical (\(k=0\)) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. Removing a scaling law and imposing regularity at the center, they prove that the family reduces to an implicitly defined oneparameter family of distinct spacetimes determined by the value of a new acceleration parameter \(a\), such that \(a=1\) corresponds to the Standard Model.
The authors prove that all of the selfsimilar spacetimes in the family are distinct from the noncritical \(k\neq0\) Friedmann spacetimes, thereby characterizing the critical \(k=0\) Friedmann universe as the unique spacetime lying at the intersection of these two oneparameter families. They then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, they use these rigorous estimates to calculate the exact leading order quadratic and cubic corrections to the redshift vs luminosity relation for an observer at the center.

Chapters

1. Introduction

2. SelfSimilar Coordinates for the $k=0$ FRW Spacetime

3. The Expanding Wave Equations

4. Canonical Comoving Coordinates and Comparison with the $k\neq 0$ FRW Spacetimes

5. Leading Order Corrections to the Standard Model Induced by the Expanding Waves

6. A Foliation of the Expanding Wave Spacetimes into Flat Spacelike Hypersurfaces with Modified Scale Factor $R(t)=t^{a}$.

7. Expanding Wave Corrections to the Standard Model in Approximate Comoving Coordinates

8. Redshift vs Luminosity Relations and the Anomalous Acceleration

9. Appendix: The Mirror Problem

10. Concluding Remarks