﻿PHYSICS 
  OF 
  THE 
  UNIVERSE 
  JEANS 
  177 
  

  

  THE 
  END 
  OF 
  THE 
  UNIVERSE 
  

  

  The 
  final 
  state 
  of 
  the 
  universe 
  must 
  be 
  such 
  that 
  the 
  entropy 
  can 
  

   not 
  be 
  increased 
  even 
  by 
  transmuting 
  the 
  elements 
  or 
  changing 
  atoms 
  

   into 
  radiation. 
  It 
  could, 
  of 
  course, 
  be 
  calculated 
  readily 
  enough 
  

   if 
  the 
  necessary 
  new 
  and 
  enlarged 
  scheme 
  of 
  thermodynamics 
  were 
  

   available, 
  but 
  competing 
  schemes 
  are 
  in 
  the 
  field. 
  The 
  Bose-Ein- 
  

   stein 
  scheme 
  leads 
  to 
  one 
  result, 
  the 
  Fermi-Dirac 
  scheme 
  to 
  another; 
  

   the 
  results 
  on 
  both 
  schemes 
  have 
  been 
  worked 
  out 
  by 
  Jordan.* 
  

  

  The 
  two 
  schemes 
  lead 
  to 
  the 
  same 
  result 
  in 
  one 
  particular 
  limiting 
  

   case, 
  and 
  this 
  limiting 
  case 
  happens 
  to 
  give 
  a 
  wonderfully 
  close 
  ap- 
  

   proximation 
  to 
  the 
  state 
  of 
  the 
  universe 
  as 
  a 
  whole. 
  The 
  limiting 
  case 
  

   is 
  that 
  in 
  which 
  space 
  is 
  almost 
  empty 
  of 
  matter, 
  a 
  specification 
  

   which 
  sounds 
  like 
  nonsense 
  until 
  we 
  find 
  some 
  common 
  standard 
  by 
  

   which 
  an 
  amount 
  of 
  matter 
  may 
  be 
  compared 
  with 
  an 
  amount 
  of 
  

   space. 
  If 
  we 
  measure 
  an 
  amount 
  of 
  matter 
  by 
  the 
  amount 
  of 
  space 
  

   it 
  occupies, 
  then 
  the 
  "emptiness" 
  of 
  space 
  is 
  one 
  of 
  the 
  commonplaces 
  

   both 
  of 
  modern 
  physics 
  and 
  of 
  modern 
  astronomy. 
  It 
  is 
  not 
  merely 
  

   a 
  question 
  of 
  the 
  emptiness 
  of 
  the 
  atom, 
  which 
  has 
  already 
  been 
  no- 
  

   ticed. 
  Hubble^ 
  has 
  estimated 
  that 
  if 
  all 
  the 
  matter 
  within 
  about 
  

   100,000,000 
  light-years 
  of 
  the 
  sun 
  were 
  uniformly 
  spread 
  out, 
  it 
  

   would 
  have 
  a 
  mean 
  density 
  of 
  the 
  order 
  of 
  only 
  about 
  10"^' 
  gm. 
  per 
  

   cubic 
  centimeter, 
  so 
  that 
  even 
  the 
  very 
  empty 
  atoms 
  would 
  be 
  at 
  

   several 
  thousand 
  million 
  times 
  their 
  diameters 
  apart. 
  ^ 
  

  

  We 
  can 
  express 
  this 
  emptiness 
  of 
  space 
  in 
  a 
  more 
  fundamental 
  

   manner. 
  The 
  energy 
  set 
  free 
  by 
  the 
  total 
  annihilation 
  of 
  1 
  gm. 
  

   of 
  matter 
  is 
  equal 
  to 
  C^ 
  or 
  9X10^° 
  ergs, 
  so 
  that 
  the 
  total 
  annihila- 
  

   tion 
  of 
  all 
  the 
  matter 
  of 
  the 
  universe, 
  assuming 
  an 
  average 
  density 
  

   of 
  10~^^ 
  gm. 
  per 
  cubic 
  centimeter, 
  would 
  only 
  provide 
  an 
  energy- 
  

   density 
  of 
  9X10~^^ 
  ergs 
  per 
  cubic 
  centimeter, 
  which 
  would 
  raise 
  

   the 
  temperature 
  of 
  space 
  from 
  absolute 
  zero 
  to 
  about 
  10° 
  abs. 
  The 
  

   emptiness 
  of 
  space 
  is 
  indicated 
  by 
  the 
  lowness 
  of 
  this 
  temperature 
  

   in 
  comparison 
  with 
  the 
  temperatures, 
  as 
  shown 
  in 
  Table 
  III, 
  which 
  

   q,re 
  necessary 
  to 
  effect 
  atomic 
  and 
  subatomic 
  changes. 
  If 
  we 
  make 
  

   the 
  approximation 
  of 
  neglecting 
  10° 
  in 
  comparison 
  with 
  the 
  temper- 
  

   ature 
  of 
  2,200,000,000,000° 
  which 
  corresponds 
  to 
  the 
  annihilation 
  

   or 
  creation 
  of 
  electrons 
  and 
  protons, 
  the 
  various 
  schemes 
  of 
  statis- 
  

   tical 
  mechanics 
  give 
  the 
  same 
  result 
  for 
  the 
  number 
  of 
  electrons 
  and 
  

   protons 
  left 
  undissolved 
  into 
  radiation. 
  Independently 
  of 
  the 
  size 
  

   of 
  the 
  universe, 
  the 
  dominating 
  factor 
  in 
  this 
  number 
  is 
  e~™^^/^^- 
  

   and 
  as 
  the 
  index 
  of 
  the 
  exponential 
  is 
  the 
  ratio 
  of 
  the 
  two 
  tempera- 
  

   tures 
  just 
  considered, 
  the 
  number 
  is 
  entirely 
  negligible. 
  Thus 
  the 
  

   final 
  state 
  of 
  maximum 
  entropy 
  is 
  one 
  in 
  which 
  every 
  atom 
  has 
  

  

  ' 
  Zeitsch.f. 
  Physik., 
  41, 
  711; 
  1927. 
  

   » 
  Astrophys. 
  Journ., 
  64, 
  368; 
  1926. 
  

  

  