Derivation of the Hubble “Constant” as a Function of Distance

Derivation of the Hubble “Constant” as a Function of Distance Lewis F. McIntyre CS-AAPT Fall Session, 2002 Radford University, VA

ABSTRACT This paper presents a derivation of the Hubble Constant using the Lorentz Velocity Angle, developed in previous presentations by the author. This paper shows that the Hubble “Constant” thus derived should in fact vary as a function of distance H=1/(cT-r), where T equals the age of the universe, and that T should be twice the maximum observable distance in the Universe. The geometry developed demonstrates an intuitive approach to the expanding universe, assuming only collocation at T=0.

OUTLINE • HISTORY OF THE HUBBLE CONSTANT • ALTERNATIVES TO DOPPLER SHIFT • FIRST PRINCIPLE DERIVATION OF HUBBLE DOPPLER FACTOR • VELOCITY ANGLE REVIEW • HUBBLE FACTOR AND VELOCITY ANGLE • SUMMARY

HISTORY OF THE HUBBLE CONSTANT • @ 1929 EDWIN HUBBLE DETERMINED A CONSTANT FACTOR RELATING REDSHIFT TO DISTANCE • Empirical, z=Δλ/λ=H*r • Based on Nearby Galaxies • SUBSEQUENTLY INTERPRETED BY GAMOW @ 1948 AS DOPPLER SHIFT, INDICATIVE OF EXPANDING UNIVERSE

Alternatives to Doppler • Hubble shift as a Compton scattering http://www.geocities.com/CapeCanaveral/9335/compton.html • Photon “Half-Life” http://www2.3dresearch.com/~alistolmar/Resolving.htm

Hubble Details

CLASSICAL HUBBLE DOES NOT SATISFY LORENTZ TRANSFORM r r, t cT cT-r

FIRST PRINCIPLE DERIVATION r r, t cT cT-r

FIRST PRINCIPLE • FOR SMALL r FIRST PRINCIPLE DERIVATION TENDS TOWARD CLASSICAL HUBBLE: • Classical Hubble Predicts v->c @ r->cT • First Principle Predicts v->c @ r->cT/2

COMPARSION OF z • RELATIVISTIC z DOPPLER MUST BE USED AS v->c

GRAPHICAL COMPARISON

REVIEW OF VELOCITY ANGLE • ALL OBJECTS TRAVEL AT PROPER VELOCITY c • THE DIRECTION OF TRAVEL DEFINES LOCAL TIME • RELATIVE VELOCITY EXISTS AS SPATIAL COMPONENT OF PROPER VELOCITY c BETWEEN TWO REFERENCE FRAMES • Relative Velocity DOES NOT EXIST Independent of Proper Velocity • Difference in Angular Direction λ (Lorentz Angle)

GRAPHICAL DERIVATION x, t World line: Locus of measurements In observer’s reference t t* Timeline: Locus of points of proper time λ

Hubble and Proper Time r r, t cT-r λ

HUBBLE FACTOR AND THE EXPANDING UNIVERSE-EXPANDING IN TIME r=0.28 t=0.72 z=0.5 v=0.38 Observer at t=T r=0.375 t=0.625 z=1.0 v=0.6 r=0.48 t=0.52 z=4 v=0.92 t*=0.66 t*=0.5 t*=0.2

Classical Does not fit Lorentz Transform to t*=0 First Principles Satisfies Lorentz Transform SUMMARY

Derivation of the Hubble “Constant” as a Function of Distance