For a variety of reasons the universe is believed to be curved. It would have to be curved in another fourth dimension. (This would be separate from time as the fourth dimension.) Since it is impossible for a person knowing only a three dimensional world to visualize curvature in another dimension one must resort to an analysis of a two dimensional world curved in a third dimension. This would be the surface of a sphere.

The Two Dimensional Case

Let R be the radius of a spherical universe. Consider a particle in this universe radiating energy. This energy would be distributed over a circle as shown below.

If the pulse of energy has traveled a distance s within the universe the angle θ subtended by the pulse from its origin is given by

θ = s/R

The perimeter p of the circle is given by

p = 2πr
where
r = Rsin(θ)

In the two dimensional world the intensity of the radiation is inversely proportional to p. Likewise the intensity g of gravitation due to a particle of mass M in the two dimensional curved universe would be

g = G/p = GM/(2πRsin(θ)) = GM/(2πRsin(s/R)

where G is a constant.

For z close to zero, sin(z) is approxiately equal to z. Thus for s small relative to R, Rsin(s/R)=s. Hence when R is very large relative to s the intensity of the gravitation reduces to

g = GM/(2πs)

For z not so small sin(z)=z−z³/6 and thus Rsin(s/R) would be approximately s−s³/(6R²). Gravitational intensity would then be given by

g = GM/(2π(s−s³/(6R²))) = (GM/[2πs)(1−(1/6)(s/R)²)]
which, making use of 1/(1-z) being approximately (1+z)
means that g is approximately equal to
g = (GM/(2πs))(1+(1/6)(s/R)²)

Inhabitants of the curved universe who do not realize that it is curved would see mass as being somehow enhanced with distance or that there exists phantom mass associated with mass. It is important to note that for the inhabitants of the curved universe the arc lines appear to be straight, not just approximate straight but exactly straight. The arc lines however are not straight in terms of a Euclidean universe.

The Three Dimensional Case

In a three dimensional curved universe there would be a similar effect. The intensity of gravitation declines with distance because it is spread over a surface of area 4πr² where r is the Euclidean distance from the surface to its center. But the distance s measured within the curved universe is different from r; i.e., r=Rsin(s/R). Therefore gravitational intensity would be of the form

g = GM/(4π(Rsin(s/R))²)
or, equivalently,
g = GM/(4πR²sin²(s/R))

For s<<R this reduces to g=GM/(4πs²). For s not so small the first two terms of the Taylor's series for sin²(z) would apply. Those terms are z²−(1/3)z⁴. Thus R²sin²(s/R) is approximately equal to s²[1−(1/3)(s/R)²]. Therefore gravitational intensity is approximately equal to

g = (GM/(4πs²))[1+(1/3)(s/R)²]

This would mean that the mass apparently increases with distance according to the formula M_apparent=M[1+(1/3)(s/R)²] or that there exists phantom mass associated with mass. This phantom mass becomes greater on the periphery of an astronomical system, simply because of the quadratic effect. This of course sounds very much like the hypothesized dark matter halo around galaxies. The effect is shown below.

The Taylor's series approximation can be dispensed with. The formula for gravitational intensity can be expressed as

g = GM/(4πs²)(1/(R²sin²(s/R)/s²)
which may be reduced to
g = (G/(4πs²)[M/(sin²(s/R)/(s/R)²)
and further to
g = (G/(4πs²)(M/(sinc²(s/R))

where sinc(z)=sin(z)/z. Thus apparent mass is given by

M_apparent = M/sinc²(s/R)

The Anomalies in the Trajectories of Pioneer 10 and 11

The Pioneer 10 and 11 spacecrafts which were launched in 1972 and 1973 appear to experiencing a slowing down as the leave the solar system. Their trajectories do not agree precisely with those computed on the basis of theories of gravitation of Newton and Einstein. No discrepancies were apparent until the spacecraft were out beyong the orbit of Saturn. But by 2008 the deviation between their measure position and their computed positions is on the order of 400,000 kilometers. This is small compared to the immense distances involved but is measurable. Space scientists are searching for an explanation for the discrepancy, including a possible refinement of the law of gravitation.

The explanation given above is the curvature of space. The material above was stated in terms of a global curvature but the local curvature due to the mass of the Sun would be the most relevant. The quadratic nature of the adjustment means the effect would not be perspectable for small distances but would grow in magnitude at an increasing rate. This would lead to the notion of a halo of invisible matter at the edge of an astronomical system.

(To be continued.)