Qualitative arguments against theory of singular origin of univers

Harrison, E. R.

English

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?E. R. Harrison 
Laboratory f o r  Theoretical Studies 
National Aeronautics and Space Administration 
Goddard Space Flight Center 
Greenbelt, Maryland 
THE NONSINGW ORIGIN OF THE UNIVERSE 
bY 
?National Academy of Science Resident Research Associate 
On leave from the Rutherford High Energy Laboratory, 
Didcot, Berkshire, England 
https://ntrs.nasa.gov/search.jsp?R=19670008864 2020-03-24T02:20:20+00:00Z
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I n  an expanding universe the average energy density diminishes with time. 
The fur ther  we go back i n  time the greater was the density, and according t o  
current theory the universe originated from (or passed through) a space-time 
s ingular i ty  of i n f in i t e  density some lolo years ago'. 
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/ Attempts t o  quantize the gravitational fie1d;encounter both formal and 
conceptual d i f f icu l t ies2 .  Failing a ra t ional  theory we adopt quali tative 
arguments and show t ha t  it is unlikely that  the uqiverse originated as a singu- 
l a r i t y  . --. 
C. 
Consider first par t ic les  of mass m fo r  which the gravi ta t ional  coupling 
constant i s  unity: 
Their Compton wavelength A = a/mc equals t h e i r  gravi ta t ional  length G m/c2, 
or  
where hn = fi/%c and % is  the nucleon mass. 
arbi t rary s i z e .  
Sa far the par t ic les  are of 
If now t h e i r  ,density approaches p = m/h3, o r  
p = c5/G2B - los5 g ~ m - ~  (4) 
- & 
the  quantum fluctuations of the metric assume t h e i r  maximum value3J4. According 
t o  (1) they should be strongly interacting; but a;t the  density (4)  they are 
"ghost" par t ic les  enclosed i n  their own metric, and a t  the most interact  only 
feebly with each other. 
Consider now what happens when the universe has the  very high density 
(4 ) .  For simplicity l e t  a l l  par t ic les  be identical ,  of energy mc2. (The 
main conclusions are l d t t l e  affected for a spectrum of fermions and bosoqs 
i n  which the  Fermi and thermal energies are  comparable and equal t o  mc2.) 
m e  ages, t of the  universe at t h i s  :a twe i s  T - ( p G ) - l l 2 .  But,  from (4) 
and therefore the age 
T - A / C  - sec (6 1 
i s  the time required for  l igh t  t o  t r ave l  the in te rpar t ic le  distance A .  
parbibles recede from each other at the veloci ty  (almost) of l i gh t ,  and the  
observable universe' has a radius of A - 
Neighboring 
cm. 
When the age t of the universe is  large compared with T it i s  possible 
t o  construct i n  the usual way cosmological models of nested hypersurfaces 
orthogond t o  time. 
permit A t  << t. 
cast  i n to  the form 
This i s  because the unavoidable uncertainties nevertheless 
But at time t = T the uncertainty principle4 A E A t  - h can be 
A E A t  - ET 
- 2 -  
' where E = mc2, 7 - (Gp)-1/2 = %/E. For AE - E, it follows tha t  A t  - 7, and 
the s p a t i a l  hypersurface is  smeared throughout the age of the universe. The 
universe now verges on the edge of indescribable conditions i n  which t h e  notion 
of a $pace-time manifold i s  untenable. Thus, st is  impossible t o  t race  the 
or igin of the universe back t o  a space-time s ingular i ty  of i n f in i t e  density 
at t = 0, f o r  at time 7 the  universe dissolves in to  a "phantom" &kr;Lc~.of 
ghost par t ic les .  
- 3 -  
-., 
'H. P. Robertson, Rev. Mod. Phys. 2, 62 (1935). For more recent work and 
references see S. W. Hawking, Phys. Rev. Letters 2, 689 (1965). 
See J. L. Andersoq, Gravitation and Relativitx (ea. E. Y. Chiu and W. E. 2 
Hoffmann, Benjamin, New York, 196k) p. 279, fo r  a discusqion and references. 
3J. A. Wheeler, Geometrodynamics. (Academic Press, New York, 1962). Also, 
L. D. Landau, Niels Bohr and the Development of Physics (ea. W. Pauli, 
Pergamon, London, 195) suggests a cut-off energy PP mc2 i n  electrodynamics. 
If gClv a re  the metric coefficients, one might hazard t h a t  the uncertainty 4 
re la t ions  are of the  form 
.but Agoo - -Ag= - Gm/c2h = 1, and for  AE - E, A t  - t the u s u a l  relations:  
A E A t  - h, ApXAx - h follow. 
and h/c i n  time. 
The extent of the f luctuat ion i s  h i n  space 
5We have: R2 = 811GpR2/3 (the curvature constant i s  negligible),  and for  
p a R'4: t = (3211Gp/3)-1/2. 
'Assuming the present density i s  g cm'3 and R - 1(f8 cm, and a l so  p a Rq3 
f o r  p C m,/A3, - 10l6 g ~ m - ~ ,  and p a R-4 for  p > mn/A3n, i% follows tha t  
at  the  density (4)  the radius of curvature of a closed universe i s  R - lom7 cm. 


The nonsingular origin of the universe

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