The Universe is now thought to be curved in the three spatial dimensions. In effect, it is the three dimensional surface of a 4-sphere, all the points that are a constant distance R from the center of the sphere.

Consider two points on a four dimensional sphere of radius R. Those two points and the center of the 4-sphere determine a plane. In that plane the two points are separated by an angle θ in radians. The distance between the points in the 4 -sphere is then S=Rθ. The rate of change of S is given by (dS/dt)=(dR/dt)θ. The rate of change of S is perceived as the velocity v at which the other point is receding so v=(dR/dt)θ. But θ is equal to S/R so

v = (dR/dt)S/R = [(dR/dt)/R]S

This relationship is what is known as Hubble's Law and [(dR/dt)/R] is known as Hubble's constant and denoted as H. The dimension of H is inverse time and the reciprocal of its current estimate is about 14.4 billion years, somewhat larger than the estimated age of the Universe as 13.8 billion years.

Thus

v = HS

If H is constant over time then

H = (dR/dt)/R = d(ln(R)/dt
and hence
ln(R) − ln(R(0)) = Ht
or, equivalently
R(t) = R(0)exp(Ht)

But S=Rθ so

S(t) = S(0)exp(Ht)

And likewise

v(t) = HS(t) = HS(0)exp(Ht) = v(0)exp(Ht)

The Masses of Galaxies

It is estimated that our Milky Way contains 300±100 billion stars. Our Sun has a mass of 2×10³⁰ kilograms (kg). If it is a typical (average) star then the mass of the Milky Way is about 6×10⁴¹ kg. The Andromeda Galaxy has about three times as many stars as the Milky Way so its mass is about 1.8×10⁴² kg.

The Radiant Energies of Galaxies

The total radiation from our Sun is about 4.6×10²⁶ joules per second. If this is typical of stars then the radiation from all the stars in the Milky Way is 1.38×10³⁸ joules per second and that of the Andromeda three times this, or 4.14×10³⁸ joules per second.

Potential Energies

The potential energy due to the gravitational attraction between two bodies of masses m₁ and m₂ which are separated by a distance s is

PE = −Gm₁m₂/s

where G is the gravitational constant, which is equal to 6.674×10⁻¹¹ m³/(kg·s²).

The change in PE in an interval Δt is then

ΔPE = [G₁m₂/s²]vΔt
but v=Hs so
ΔPE = [Gm₁m₂/s]HΔt
or, more simply
ΔPE/PE = −HΔt

The Potential Energy between
the Milky Way and Andromeda

The separation distance between the Milky Way and Andromeda is about 2.5 million light-years, which is 2.37×10²² meters. The potential energy for them is then

PE = − 3.05×10⁵¹ joules

The change in gravitational potential energy in a time Δt due to the expansion of the Universe is then

ΔPE = PE·HΔt

Since H is equal to 2.2×10⁻¹⁸ per second, for Δt=1

ΔPE = 6.71×10³³ joules

This is an enormous amount of energy coming from who knows where, but it is infinitesimal compared with the energy the stars in the two galaxies radiate, 5.52×10³⁸ joules per second.

One possibility that would eliminate the generation of potential energy from the expansion of the Universe is that the gravitational constant is not constant but increases exponentially with time at a rate equal to the Hubble constant. Then

PE(t) = −G(t)m₁m₂/s(t) = −G₀exp(Ht)m₁m₂/(s₀exp(Ht)) = G₀m₁m₂/s₀ = PE(0)

The constants for all of the other forces would also have to increase exponentially at rate of H in order for all of the other sources of potential energy to be independent of the expansion of the Universe.

The Kinetic Energies of Galaxies
due to the Expansion of Space

If the recessional velocities due to the expansion of space were real there would be an enormous amount of kinetic energy involved. Take a galaxy the size of the Milky Way traveling at 500 km/sec. Then

KE = (0.5)(6×10⁴¹)(5×10⁵)² = 7.5×10⁵²joules

The rate at which that would be increasing is

(dKE/dt) = mv(dv/dt) = mv(H(ds/dt)) = mv²H
or
(dKE/dt)/KE = 2H

The increase in KE in one second would be 3.3×10³⁵ joules. This is small compared with the energy involved in the radiation from the stars but about 50 times greater than the increase in potential energy per second due to expansion of the Universe. However as the diagram below shows there is no real motion of the galaxies within the Universe due to the expansion.

There are no movements of the two galaxies within the circles due to the expansion of the universe. It is the same as points on an inflating balloon. The points on the balloon are moving away from each other but they have no motion with respect to the balloon.

There are, however, real motions of galaxies with respect to each other. For example, the Andromeda Galaxy is moving toward the Milky Way Galaxy and they are projected to collide in 2.5 billion years.

The expansion of the universe does create kinetic energies for the galaxies in the fourth dimension, in a direction which is perpendicular to each of the three spatial dimension. However this motion and its kinetic energy is irrelevant for the physics within the universe.

Rotations within the Universe

Systems of bodies revolve about their centers of mass. The rate of revolution ω is governed by the conservation of angular momentum L, where L=Jω with J being the moment of inertia of the system. The moment of inertia of a system is proportional to the square of its scale. Thus the moments of inertia of stronomical systems are growing at a rate of 2H and the rates of revolution (ω=L/J) are decreasing at a rate of 2H. The tangential velocities (rω) are then decreasing at a rate of H. This means that the rotational kinetic energies are decreasing at a rate of 2H. That decrease is counterbalanced by an increase in the potential energies of the systems