NonEuclides Geometries 
    The notion of a geometry other than that of Euclideses, where the 
cosmos is plane and flat, arose in the 1780s. The first nonEuclides 
geometry was a hyperbolic space and this concept entertained 
mathematicians thru the early 1800s. About 1820 Gauss and separately 
Lobachevsky tried to demonstrate the reality of hyperbolic space. 
Gauss surveyed several mountain peaks and Lobachevsky studied stellar 
parallaxes. No deviation from Euclides space was found. Gauss simply 
took far too small a piece of the cosmos to measure out and 
Lobachevsky suffered from the spurious parallaxes then reported from 
    About 1850 Riemann developed a spherical geometry and astronomers 
tried to demonstrate it by parallaxes. In a spherical universe the 
parallaxes of very remote stars are negative. But the negative 
parallaxes turned in by astronomers proved to be infected by 
instrumental errors. 
    Einstein was the first to definitely apply Riemann geometry to the 
cosmos in his original model. He simply declared that the geometry is 

Geometric Curvature 
    The curvature in geometry at a given point of a surface is defined 
by scribing two lines at the point delL1 and delL2. They are at right 
angle and lie in the surface. Each line, in general, is curved such 
that it forms an arc of a circle. Let the radius of the arc for delL1 
be R1 and that for delL2 be R2. 
    The two delL are rotated in the surface as a unit so that either 
R1 or R2 is a minimum value. When this situation is attained, the 
curvature C is defined to be C = 1/(R1*R2). 

 |                        | 
 | C = 1 / (R1 * R2)      | 

If the two R are the same, the curvature of the surface is 1/(R^2). 
But they do not have to be equal. Furthermore, the two radii may be on 
the same side of the point or on opposite sides. If we walk around the 
point anticlockwise, a positive R points up; negative R, down. Which 
way they do point depends on the way the delL are curved, whether we 
stand on the 'inside' edge of their tangent circle or the 'outside'. 
Hence a surface may have a positive or a negative C at a given point. 
    If R1 or R2 is infinite, the curvature is zero, describing a flat 
surface. If BOTH are infy, we have the flat surface of Euclides, a 
plane. Being that only ONE of R1 and R2 need be infy for 1/(R1*R2) to 
be zero we can have an other kind of flat surface, that of Gauss. 
    The surface of a cylinder is such a Gauss flat surface. Let delL1  
be a circumferential arc and delL2 a line parallel to the axis. R1 is 
finite and R2 is infinite. C = 1/(R1*R2) = 1/(R1*infy) = 1/infy = 0. 

Angles of a Triangle 
    If we draw on the surface a triangle we derive a remarkable 
feature of curvature. We look at the sphere, being that home 
astronomers are well familiar with its properties. From ordinary 
spherical geometry the area of a triangle whose sides are great 
circles and with angles A, B, C (in radians) is 

 S = (A + B + C - pi) * R^2 

As simple as this formula is, few home astronomers know it! 
   But the curvature of the sphere (R1 = R2) is C = 1/R^2, so 

 C * S  = S / R^2 
        = ((A + B + C - pi) * R^2) / R^2 
        = A + B + C - pi 
        = sum(theta) - pi 

 |                         | 
 | C * S = sum(theta) - pi | 

 C*S, the curvature times the area of the triangle, is equal to the 
excess of S,  the angle sum, over pi! It's easy to check this for a 
plane and a sphere. On a plane R1 = R2 = infy and C = 1/infy^2 = 1/infy 
= 0. 

 C = 0 
   = sum(theta) - pi 

 sum(theta) = pi 

This is the ordinary rule that the sum of the angles in a plane 
triangle is pi, or 180 degrees. 
   On the sphere let a triangle be drawn along the 0 deg longitude 
circle, the 120 deg longitude circle, and the 0 deg latitude circle. 
This triangle by construction occupies 1/6 of the entire area of the 
sphere (4*pi*R^2)/6. The angles are 2*pi/3 at the north pole, and two 
pi/2 angles on the equator. The sum is 

 sum(theta) = ( 2 *pi/ 3 ) + (pi / 2) + (pi / 2) 
            = (4 * pi / 6) + (3 * pi / 6) + (3 * pi / 6) 
            = 10 * pi / 6 

This excedes pi by 

 sum(theta) - pi = (10 * pi / 6) - pi 
                 = (10 * pi / 6) - (6 * pi / 6) 
               = 4 * pi / 6 


 C * S = S / R^2 
       = (4 * pi * R^2 / 6) / R^2 
       = 4*pi/6 
       = sum(theta) - pi 

    The hyperbolic case with one of the R being negative is not as easy 
to see, but it can be shown that the above does apply to it. 
    According as the geometry, the excess carries the same sign as the 
curvature. Positive C has positive excess (sum(theta) > pi), zero C 
has excess = 0, negative C has negative excess (sum(theta) < pi). 

Curvature of the Universe 
    The factor k/R^2 is a measure of the 'curvature' of the universe in 
that it is identical to C with k = 1. We consider only symmetrical 
universes where R1 = R2, by the isotropic condition of cosmology. 
(There are other models that allow for nonisotropic shapes with R1 <> 
    For a plane R = infy and k/R^2 = 0. The surface is flat and the 
geometry in it is that of Euclides. For a k/R^2 > 0 the surface is a 
sphere and the geometry is spherical. It is not easy to visualize a 
k/R^2 < 0 but such is the curvature of a hyperbolic surface, 
resembling a horse's saddle. 
    Note that the sphere is bounded, occupying a definite region of 
[3D] space. The plane and hyperbolic are infinite in extent and have 
no natural boundaries. The sphere is a closed space; the plane and 
hyperbolic, open. The table here summarizes these cases. 

 k/R^2  geometry    discoverer  boundary  triangle  thru a point 
 -----  ----------  ----------  --------  --------  ------------
  < 0   hyperbolic  Lobachevsky open      < 180deg  many || 
  = 0   plane       Euclides    open      = 180deg  only one || 
  > 0   spherical   Riemann     closed    > 180deg  no || 

    The last two columns compare two common geometric features of 
space. One is the sum of the angles in a triangle scribed in the 
space. The other is the number of lines that can be scribed thru a 
given point parallel to a given line away from the point In Euclides 
geometry the triangle sums to 180 degrees, a straight line, a half 
circle. There is only one line thru a given point that can be parallel 
to an outside line. While we derived the triangle relation above, the 
bit about the parallel lines we accept as is. 
    On a sphere there are no parallel lines: all 'lines' are great 
circles that intersect at both ends of the sphere's diameter. Any 
triangle, made of segments of three great circles, sum to more than 
180 degrees (and less than 360 degrees). 
    On the hyperbolic thru a given point there are an infinity of 
lines that can be parallel to an outside line. They all diverge from 
each other in 'curves' -- as compared to lines in a plane -- that 
never intersect. A triangle, made of three such 'curves', sums to less 
than 180 degrees (and more than 0 degrees). 
    By the behavior of k/R^2 in models of the universe we can type the 
geometry as 'spherical and closed', 'planar and open', or 'hyperbolic 
and open'. 

NonZero k 
    Altho we can cope with a notion that R1 and R2, while equal, may 
have opposite signa, this upsets the principle of isotropy. One 'side' 
of the universe hs curvature differently from the other 'side'. We 
allow R1 = R2 identicly and use k to modulate the sign of the 
    In all the treatment so far we deliberately set k = 0 and LAMBDA = 
0. This is the special case of the Friedmann model, developed in an 
early form by Einstein & deSitter. While still keeping LAMBDA = 0 (no 
lambda force) there are the cases for k = -1 and k = +1 to explore. 
These force the curvature to be positive or negative. 
    When we asserted k = 0, this forced rho0 to concord with H0 and 
forced q = 1/2. We saw that this leads to the missing mass problem 
because the OBSERVED rho0 seems orders less than the CONCORDANT rho0, 
denoted rho@. 
    Releasing the requirement of k = 0 removes the problem of rho0 <> 
rho@. And q <> 1/2. That is, the q can not be hidden as 1/2 in the 
equations but must be set out explicitly. 

 | FRIEDMANN EQUATIONS OF THE UNIVERSE                       | 
 |                                                           | 
 | 2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2)            | 
 |                                                           | 
 | 1der(R,t)^2/2 = 4 * pi * gamma * rho0 / (3 * R) - k * c^2 | 

where we allow for a nonzero k. Recall, also, 

 | DEFINITION OF H AND q            | 
 |                                  | 
 | H = 1der(r,t) / R                | 
 |                                  | 
 | q = -R * 2der(R,t) / 1der(R,t)^2 | 
 |   = -2der(R,t) / (R * H^2)       | 

Substituting the H and q into the Friedmann equations, we get equations 
in k, H, and q. First for q 

    2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2) 

    2der(R,t) * (R * H^2) / (R * H^2) 
        = -4 * pi * gamma * rho0 / (3 * R^2) 

    -q * R * H^2 = -4 * pi * gamma * rho0 / (3 * R^2) 

    q = 4 * pi * gamma * rho0 / (3 * R * H^2) 

And for k 

    1der(R,t)^2 / 2 = 4 * pi * gamma * rho0 / (3 * R) - k * c^2 

    1der(R,t)^2 / 2 = 4 * pi*gamma*rho0*H^2/(3*R*H^2)-k*c^2 
                 = q * H^2 - k * c^2 

 1der(R,t)^2 = 2 * q * H^2 - 2 * k * c^2 

    1der(R,t)^2 / R^2 = 2 * q * H^2 / R^2 - 2 * k * c^2 / R^2 

    H^2 = 2 * q * H^2 / R^2 - 2 * k * c^2 / R^2 

    2 * k * c^2 / R^2 + H^2 = 2 * q * H^2 

    2 * k * c^2 / R^2 = 2 * q * H^2 - H^2 
                   = (2 * q - 1) * H^2 
 | FRIEDMANN MODEL IN q, H, AND        k | 
 |                                       | 
 | LAMBDA = 0                            | 
 |                                       | 
 | 2 * k * c^2 / R^2 = (2 * q - 1) * H^2 | 

    Look at the k equation again, noting that only (2*q-1) can 
modulate k 

 2 * k * c^2 / R^2 = (2 * q - 1) * H^2 

 k = (2 * q - 1) * H^2 * R^2 / (2 * c^2) 

By setting k = -1, 0, +1 we see that the only factor on the right that 
satisfies is (2*q-1) = -1, 0, +1. All the other factors are positive. 
The cases are laid out here 
    k   k/R^2  2*q-1    q     rho0   geometry    boundary 
   --   -----  -----  -----  ------  ----------  --------
   -1    < 0    < 0   < 1/2  < rho@  hyperbolic  open 
    0    = 0    = 0   = 1/2  = rho@  plane       open 
   +1    > 0    > 0   > 1/2  > rho@  spherical   closed 

    The case for k = 0, q = 1/2 is the Einstein-deSitter, or 
classical, model. The universe starts from a bigbang, expands at 
decreasing H, and ends at an infinite R and zero energy. This case 
separates q into two ranges, that less than 1/2 and that more than 
    We cannot do a rigorous treatment of the k <> 0 cases. The maths 
get too deep. We approach them from an analysis of limits and 
inequations, techniques very powerful, simple, yet almost unknown 
among home astronomers. 

Gravitational Curvature
    That spacetime can be curved in geometry is rather alien to the 
ordinary senses. We normally perceive our surrounds as a Euclides 
space where the angles of a triangle sum to 180 degrees and where 
there is only one line thru a pint that is parallel to an outside 
line. But this world is only an approximation in Einstein physics. It 
prevails exactly only in the absence of gravitational mass.  gravity-
free space is indeed Euclides in geometry. 
    Einstein showed that the presence of mass causes the spacetime to 
take on a nonEuclides geometry. In fact, in a static universe, with no 
Hubble expansion, its geometry must be nonEuclides. It is the balance 
between the mass (better, the mass density) and the Hubble expansion 
that together determine the geometry of the universe. 
    We can derive in a surprisingly simple way the curvature of the
spacetime due to mass. Consider a mass M and a test particle in 
[circular] orbit around it. The orbit is sustained by the balance 
between the kinetic energy of the particle and the potential energy of 
the gravity field around M.

     (V^2) / 2 = gamma * M / R 
    V^2 = 2 * gamma * M / R 

    R = 2 * gamma * M / V^2 
    This curvature is the 'space' part, the part describing the motion 
of a mass particle. There is in Einstein physics a 'time' component 
which describes the motion of a energy particle (a photon) in gravity. 
We can not speak of the curvature of just 'space' but that of 
    We have to be approximate here and allow that a photon moves with 
the speed of light near mass, V = c. We will see later that a photon 
moves slower near mass than in a gravity-free world. If this is odd, 
it's because the rule is usually stated 'light travels at the same 
speed c for all observers'. This is not true! Photons do travel at the 
one speed c only for observers in gravity-free spacetime. When 
observers are embedded in mass-filled space, photons travel slower 
than c. This is why effects like gravitational redshift and blackhole 
behavior are not symmetrical between observers. One observer is near 
the mass and the other is in gravity-free space. The local environs are 
actually different for the two observers. 
    The better rule is that within a given gravity regime c is the 
same for all observers. If c is observed from a weaker gravity domain, 
it is sen to be slower; from a stronger gravity regime, faster. 
    If the gravity fields are small, the speed of light is still very 
close to c. It does take a very strong gravity field -- the neutron 
star or blackhole -- to really slow up photons. We allow that near 
small masses, yp to many Suns, photons travel at V = c. 

    c^2 = (2 * gamma * M / R^2) * R 

    1 = (2 * gamma * M / c^2) * (1 / R^2) * R 

We can not reduce the (1/R)^2*R to 1/R because the R in (1/R^2) is NOT 
the same as the R in R. The former is the R for gravity acting on mass 
particles and the latter is the radius of curvature for a photon. We 
distinguish them by Rm and Rp 

    1 = (2 * gam ma * M / c^2) * (1 / Rm^2) * Rp 

    Rp = Rm^2 / (2 * gamma * M / c^2) 
       = Rm^2 * c^2 / (2 * gamma * M) 

 |                                   | 
 | Rm = 2 * gamma * M / V^2          | 
 |                                   | 
 | Rp = Rm^2 * c^2 / (2 * gamma * M) | 
 |                                   | 
 | C = 1 / (Rm * Rp)                 | 
 |   = 2 * gamma * M / (c^2 * Rm^3)  |  

    As an example consider a photon grazing the photosphere of the 
Sun, Rm = 696,000km. The Sun's mass is still too small to 
significantly slow down C as seen from gravity-weak space at Earth. 

    Rp = Rm^2 * c^2 / (2 * gamma * M) 
       = (6.96e8m)^2 * (2.998e8m/s)^2
        / (2 * (6.672e-11m^3.s^2 / Kg) * (1.989e30kg)) 
       = (6.96e8m)^2 / (2.953e3m) 
       = 1.640e11m 
   -> 1,096 AU 

    Any short piece of an arc of this huge radius any where in the 
solar system looks pretty straight. It is generally safe to say that 
photons within the solar system travel in Euclides straight lines. 
Note that the radius of curvature is a function of Rm^2 so that it 
increases rapidly with distance from the central mass. At even only a 
tenth AU away, the radius Rp is so large that it is for all purposes 
infinite. The path is a true Euclides straight line. 

Bending of Starlight 
    When Einstein worked all this out he was asked for some evidence 
of its truth. Being that the most massive handy thing around was the 
Sun he figured out how light would 'bend' around the Sun from a star 
on the solar limb. In his day, the 1910s, we could not see stars in 
the vicinity of the Sun.
    In May 1918 there was an eclipse of the Sun which offered a test. 
Eddington organized teams all along the eclipse path, to maximize 
chance of clear viewing, to photograph the sky against the Sun, thru 
the corona. The Sun was in a field of bright stars in Taurus during 
the eclipse to provide many candidates to impress on the photos.
  . THe arranged for this area to be carefully photographed before the 
eclipse, at night,when the stars were not disturbed by the Sun's 
    The places of the stars around the eclipsed Sun were compared with 
those in the earlier photos. 
    Despite the horrors of errors there was a definite bias of stars 
displaced radially away from the Sun during the eclipse. This was 
strong, tho rather delicate, evidence in favor of the new Einstein 
    Eddington was converted to Einstein. He became a widely-read 
popularizer of the Einstein relativity and a respected cosmologist. 
    A simplified estimate of the amount of deviation of starlight next 
to the Sun can be made from the Rp. Because Rp increases steeply with 
distance from the Sun, we take a short segment of the ight path one 
solar diameter long, centered on the point of tangency. Farther out 
parts of the path have radii so large they are for our example 
    We have an arc of  1.392e9m standing 1.640e11m from its center. 
The angle this arc spans, which is the angle the arc bends, or the 
light is deflected away from the Sun, is 

    deflection = (length of arc) / (radius of arc) 
               = 1.392e9m / 1.640e11m 
               = 8.4878 radian 
               = 1.7422 seconds 

surprisingly close to the 1.75 second deflection predicted by Einstein 
and observed by Eddington. This defelction is routinely demonstrated 
by radio asronomers when the Sun occults a radio source, being that 
light and radio signals are both electromagnetic radiation. 
    The starlight bending was the one way to show Einstein's 
relativity theory to the public to being easy to picture and describe. 
The other two were too much beyond the public's apprehension, and for 
the most part they still are today. 
Curvature in Strong Gravity 
    When the gravity field is strong the curvature increases until we
can not ignore it. Disregarding the curvature of spacetime leads to 
nonsensical results having no relevance to physical reality. Until the 
1960s, we astronomers had no good tests of spacetime curvature. The 
bending of starlight was a truly subtile test flecked with errors. The 
migration of Mercury's line of apsides and redshift of light from 
white dwarfs were also subtile evidences. 
    In the 1960s space-based communications and observations began. We 
could demonstrate spacetime curvature between the Earth and a 
spaceprobe by monitoring its radio signals. 
    We discovered new objects in the universe with enormous gravity 
fields that severely curve spacetime. These include symbiotic binary 
stars, X-ray sources, quasars, active galaxy cores, pulsars and 
neutron stars. We also entertained the prospect of blackholes, perhaps 
in binary stars like Cygnus X-1. Many phaenomena observed at the 
objects required relativity and spacetime curvature to describe. 
    All these offered intriguing new tests for Einstein physics some 
impossible to try on Earth or near it. We look here at one effect of 
spacetime curvature in the neighborhood of a blackhole.
    Starting with the orbital motion of a particle, like we did above 

    V^2 / 2 = gamma * M / Rm 

    V^2 = 2 * gamma * M / Rm 

     V^2 / c^2 = 2 * gamma * M / (Rm * c^2) 
               = (2 * gamma * M / c^2) * (1 / Rm) 
               = R| / Rm 

R| is the Schwarzschild radius of a blackhole, the radial distance 
where the gravitational escape speed equals the local observer's speed 
of light. 

 |                         | 
 | R| = 2 *gamma * M / c^2 | 

     (V / c)^2 = R| / Rm 

    This is a remarkable identity. One of the most common factors in 
Einstein relativity is sqrt(1-(V/c)^2), so common that it has the 
special symbol 'beta'. We can directly substitute R|/Rm for the 

     beta = sqrt(1- (V / c)^2) 
          = sqrt(1- ( R| / Rm)) 

and use this beta in all the relativity formulae. The result is that 
space and time -- spacetime -- are distorted not only with high relative
speeds between observers, (V/c), but also by strong gravity, (R|/Rm). 
   For example we have from special relativity that 

    delL' = delL * beta 

    delT' = delT / beta 

which are the standard length contraction and time dilation.
    The prime values are the stationary frame's experience of the 
moving frame's plain values. Or they are the length and time of the 
moving frames as measured by the stationary frame's rod and clock. 
    They are NOT, as so commonly and erroneously explained, the size 
of the plain values as seen by the one and same moving frame, as if 
the observer in that frame somehow can notice his clocks slowing down 
and his meterstick shrinking. The effect between the moving and 
stationary frames is entirely one of intercomparison. 
    Velocities are, too, distorted. We see that

    V' = delL' / delT' 
       = (delL * beta) / (delT / beta) 
       = (delL / delT) * beta^2 
        = V * beta^2 

Let V be c

     c' = c * beta^2 
        = c * (1 - R| / Rm)) 
        = c * (<1) 
        < c 

for Rm < R|, outside a blackhole. Light travels SLOWER in a gravity 
field as measured from a no-gravity observer! This is a feature of 
relativity that is missed out from the usual treatments and leads to 
grotesque miscalculations and ridiculous conclusions. The constancy of 
c prevails only in gravity-free spacetime.

 Inside the Blackhole 
    In a very elegant manner we can appreciate that ordinary 
(including Einstein) physics breaks down inside the blackhole. The 
boundary of the blackhole is the Schwarzschild radius.The surface 
around the blackhole at this radius is the event horizon. For the Sun 
this is quite 3km. 
    Let a body emitting light approach a blackhole. As it closes in, 
Rm decreases and c' also decreases. At the Schwarzschild radius Rm = 
R|, beta = sqrt(1-R|/Rm) = sqrt(1-R|/R|) = sqrt(1-1) = sqrt(0) = 0. 
The speed of light as emitted from the body at the blackhole frontier 
is seen by the remote, gravity-free, observer to dwindle to zero! 
    As the body falls thru the event horizon to the interior of the 
blackhole, Rm < R| and beta < 0 and we have a mathematicly negative 
squareroot. Such an animal is a complex or imaginary number with as 
yet no physical interpretation. The physics inside the blackhole as 
experienced by us on the outside is radicly different from any we 

The R,t Graph
    A usual way to track the evolution of the universe is the R,t 
graph. It plots scalefactor R against time t. The present value of R , 
R0, and of time, T0, are at the present moment in the life of the 
universe. A sketch is given here for a simple Friedmann model.

 R |                                                   o    
   |                                             o    
   |                                       o      
   |                         /       o                                      
   |                       /    o          
   |                     /   o   
 R0|------------------ X       
   |                /o | 
   |             /o    | 
   |           /       | 
   |         /   o    | 
   |                  |
   |           o      |
             BB        t0                            t 

    The 'p' are points on the evolution curve from R = 0, marked 'BB' 
for 'bigbang', into some future larger R. 'X' is the present time, R = 
R0, r = T0, The '/' ae the tangent to the curve at X.  Its slope 
delR/delT is the present rate of expansion, the Hubble factor H0. It 
is also the first derivative, 1der(R,t), of the curve's equation R(t). 
By observation we live now in an expanding universe, H0 > 0. 
    The bow or sweep of the curve is the second derivative, 2der(R,t). 
    Left of T0 is past time, observed thru the look-back effect. Right 
of T0 is the future, projected by a model prolonging past evolution. 
    Below R0 is past scalefactor, which in almost all  models was 
smaller than now. The standard model puts R = 0 at an creation or 
birth of the universe. Above R0 most models forecast increasing 
scalefactor in an expanding universe. Some models predict a final 
maximum R or a peak R with a decline than after. 

Preparing the Scenarios 
    Starting with the Friedmann equations we examine the behavior of 
2der(R,t) and 1der(R,t) for various R. Then we compare them to the 
cases for k = -1 and k = +1. No detailed solutions are offered here 
due to the massive maths involved. The trends developed here by 
inequations and limits will yield useful understanding. 

 | FRIEDMANN EQUATIONS OF THE UNIVERSE                        | 
 |                                                            | 
 | 2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2)             | 
 |                                                            | 
 | 1der(R,t)^2 = 8 * pi * gamma * rho0 / (3 * R)- 2 * k * c^2 | 
 |                                       | 
 | 2 * k * c^2 / R^2 = (2 * q - 1) * H^2 | 

    Recall that the 1der(R,t) is the slope of the tangent to the curve 
R(t) at any t. A positive 1der(R,t) means the tangent slopes lowerleft 
to upperright; negative, upperleft to lowerright; zero, left to right 
(horizontal). An infinite 1der(R,t) means the tangent is vertical. 
    The 2der(R,t) is the arc or bow of the R(t) curve. A negative 
2der(R,t) means the curve bows downward or it is concave as seen from 
underneath; positive, upward or convex; zero, straight. For 2der(R,t) 
= infinity there is no defined bow or arc and the curve hits a 
singularity point. 
    An other property of 2der(R,t) is that for positive 2der the 
tangent, 1der(R,t), rotates clockwise with increasing t; negative, 
anticlockwise; zero, no rotation. 
    The second Friedmann equation is the energy equation multiplied 
thru by 2. 1der(R,t)^2 is twice the kinetic energy of the universe 
(per unit mass). Hence the behavior of 1der(R,t) in the maths implies 
a corresponding behavior of the kinetic energy in the cosmos. 
    Also recall that for the standard model the point R = 0 of the 
R(t) curve, the 'beginning' of the universe, occurs at t0 = 2*T0/3. 
    For ease of comparison we impose the requirement that for all 
three cases the R(t) curve passes thru the same point t = 0, R0 = 1 
and H = H0. This is NOT a tenet of theory! It merely helps to compare 
the differences between the scenarios. 

k = 0, q = 1/2, rho0 = rho@ 
    In the first Friedmann equation 2der(R,t) is negative for all 
positive R. Because in this equation k is absent, this conclusion 
holds true for all three cases. 

    2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2)  
              = -4 * pi * gamma * rho@ / (3 * R^2)  

    Let A = -4 * pi * gamma * rho@ / 3  as a constant 

    2der(R,t) = -A / R^2 
              = A / (> 0) 
              = A * (< 0) 
           = (<0) 

 The R curve bows downward for all positive R. The tangent rotates 
clockwise. In addition the R curve must somewhere intersect the t 
axis. This is the point R = 0, t = t0 = 2*T0/3. 
    In the second Friedmann equation 1der(R,t) is positive for 
positive R 

    1der(R,t)^2  = 8 * pi * gamma * rho0 / (3 * R) - 2 *k*c^2 
                = 8 * pi * gamma * rho@ / (3 * R) - 2 * k *c^2 
                = 8 * pi * gamma * rho@ / (3 * R) - 0 
                = 8 * pi * gamma * rho@ / (3 * R) 
                = A / R

shere we again collected the numerator into a single constant 

    1der(R,t)^2 = A / (>= 0) 
For R going to infinity 1der(R,t) goes to zero. The tangent is horizontal 
and the R curve flattens out. The kinetic energy vanishes. The 
expansion stops. 

    1der(R,t)^2 = A / (>= 0) 
                = A / infy 
             = 0 

    The universe in the standard model starts from R = 0 at t = t0 = 
2*T0/3, expands at ever decreasing rate, until at R = infinity the 
expansion stops. 
    We refer below to this R(t) curve for k = 0 as the R@ curve. 

k = -1, q < 1/2, rho0 < rho@ 
    The 2der(R,t) is still negative for all positive R but less so 
than in the k = 0 case. This is because rho0 < rho@. 

    2der(R,t) = - 4 * pi * gamma * rho0 / (3 * R^2)    
              = A * rho0 
              = A * (< rho@) 
              < 2der(R,t)@ 

By placing this R curve tangent to the R@ curve at t = 0 we see that 
this R curve bows less and lies above the R@ curve. It intersects the 
t-axis earlier than the R@ curve. Thus t0 > 2*T0/3. 
    In the limit for q = 0. 2der(R,t) = 0 and the R curve has no arc; 
it is a straight line. The slope of this line is H0 and the intersect 
at R = 0 is at t = t0 = 1/H0 = T0 . Hence over the range of rho0 from 
0 up to rho@ T0 >= t0 > 2*T0/3. 

    2der(R,t) = A * (< rho@) 
              = A * 0 
              = 0 
    The 1der(R,t) is positive for all positive R. 

    1der(R,t)^2 = 8 * pi * gamma * rho0 / (3 * R) -2 * k * c^2 
                = 8 * pi * gamma * rho0 / (3 * R) - 2 *(-1) * c^2 
                = 8 * pi * gamma * rho0 / (3 * R) + 2 * c^2 
                = A / (>= 0) + 2 * c^2 
For R = infinity 1der(R,t) attains 2*c^2, a positive value. The R curve 
never flattens out; it continuously rises with increasing R. The 
kinetic energy is still positive and expansion continues thru R = 
    1der(R,t)^2 = A / (>= 0) + 2 * c^2 
                = A/infy+2*c^2 
             = 0 +  2 *c^2 
             = 2 * c^2 

    The universe in the k = -1 model starts from R = 0 at T0 >= t0 > 
2*T0/3, then swells at ever decreasing rate, until at R = infinity the 
expansion still continues. 

k = +1, q > 1/2, rho0 > rho@
    The 2der(R,t) is still negative for all positive R but more so 
than in the k = 0 case. This is because rho0 > rho@ 

    2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2)    
              = A * (> rho@) 
               > 2der(R,t)@ 

Placing this R curve tangent to the R@ curve, this R curve bows more 
than and lies below the R@ curve. It intersects the t axis later than 
the R@ curve. t0 < 2*T0/3. 
    In the limit as q tends to infinity, the universe has infinite 
density, the 2der(R,t) goes to infinity. This is a reasonable result. 
There is so much matter that the deceleration is immense. The universe 
can never overcome the gravity and is stillborn. 

    2der(R,t) = A * (> rho@) 
              =  A * infy 
              = infy 

    For positive R the 1der(R,t) can be EITHER positive OR negative 
due to the antagonism of the two terms on the right. If R is small A/( 
>0) > 2*c^2 and 1der(R,t) > 0. For large R A/(>= 0) < 2*c^2 and 
1der(R,t) < 0. 
    This means, combined with 2der(R,t) < 0, that the R curve must 
cross R = 0 TWICE! The left crossing is at t0 and is the bigbang 
moment. The right one is at some future time and represents a colossal 

    1der(R,t)^2 = 8 * pi * gamma * rho0 / (3 * R) - 2 * k * c^2 
                = 8 * pi * gamma * rho0 / (3 * R) - 2 * (=1) * c^2 
                = 8 * pi * gamma * rho0 / (3 * R) - 2 * c^2 
                = A / R - 2 * c^2            
                = A / (>= 0) - 2 * c^2 
                > 0 for small R 
                < 0 for large R 

 For R = infinity 1der(R,t) attains 

    1der(R,t)^2 = A / (>= 0) - 2 * c^2 
                = A / infy - 2 * c^2 
                = -2 * c^2 

In this k = +1 case R CAN NOT go to infinity! It has a finite maximum. 
This occurs at the point where 1der(R,t) is itself zero. 
    1der(R,t)^2 = 8 * pi * gamma * rho0 / (3 * R) - 2 * c^2  
                = 0 
    2 * c^2 = 8 * pi * gamma * rho0 / (3 * R) 

    6 * R*c^2 = 8 * pi * gamma * rho0 
    R = 8 * pi * gamma * rho0 / (6 * c^2) 
      = 4 * pi * gamma * rho0 / (3 * c^2) 

Note that with 1der(R,t) = 0 and 2der(R,t) < 0 the R curve for later t 
continues bending down and must intersect the t axis a SECOND time. 
This second R = 0 point occurs in future time. According to some 
cosmologists this point represents the destiny of the universe, a 
colossal collapse where all the world condenses onto itself in a 
reverse-cinema of the bigbang. 
     This R[max] is midway between the two points R = 0. We presently 
in this k = +1 case are on the rising part of the R curve -- the 
universe is expanding now, not collapsing -- and we will reach the 
R[max] in some far off future. Thereafter, in the farther off future, 
we self-destruct in the colossal collapse. 
    The universe in the k = +1 model starts from R = 0 at t < 2*T0/3, 
at first expands, then  STOPS SWELLING, begins to CONTRACT, until it 
ends at R = 0 in some future epoch. 
    This destiny of the world 'with a bang' gave rise to the 
speculation of regeneration. There is a second bigbang which produces 
an allnew universe, which itself goes thru the swell-shrink cycle. 
This continues, supposedly, indefinitely. This is the oscillating 

    We collect here the results of the three cases

    k     q        bigbang       1der[ult]  R[ult] 
   ---  -----  ----------------  ---------  ------
   -1   < 1/2  T0 > t0 > 2*T0/3    2*c^2     infy 
    0   = 1/2    t0 = 2*T0/3         0       infy 
   +1   > 1/2  2*T0/3 > t0 > 0   undefined  R[max] =