An infinite ensemble of particles, such as the universe, cannot be monolithically characterized as either contracting or expanding. A complete description must include both phenomena. They manifest differently, however. Contraction appears as radiant energy; expansion shows more subtly, as increasing distance between remote parts.
These two phenomena must be measured by the same yardstick. Contraction energy is measured in watts-per-kilogram (w/kg), the observable quantity radiated “into space,” divided by the inferred mass doing the radiating. Expansion energy must be calculated, using the expansion rate, H, and other observables. By weighing expansion and contraction on the same scale, w/kg, no energy deficiency is found. The accelerating expansion adds up.
CONTRACTION ENERGY ESTIMATE
The unit-power output of visible matter is estimates by dividing the radiant power output of a “typical galaxy” by its mass. The Milky Way galaxy has a visible mass of roughly 10^11 suns, with a dark component about 10 times that. It has a radiant energy output of about 10^11 suns. Using the figure of 4 x 10^26 watts for the sun and a mass of 2 x 10^30 kg, we have:
Relative power output of Milky Way = (10^11 suns)*(4 x 10^26 watts/sun)/([10^12 suns]*[2 x 10^30 kg/sun]) = 2 x10^(-5) w/kg
Thus, on the assumption that the Milky Way is “typical,” the observed radiant power output of the universe, including dark matter, is roughly 2 x10^(-5) w/kg. The universe is contracting, locally, at an average rate of 2 x10^(-5) w/kg.
EXPANSION ENERGY ESTIMATE
An infinite, dualistic model of the universe is shown at two different times, “close-up,” in Figure 2. The sides of the cube are of length R, which represents the scale at which local gravitational contraction gives way to cosmic expansion. Its magnitude is 10^23 meters (3 - 5 Mpc). It is the distance beyond which all sources are red-shifted. Local velocity of galaxies relative to their (gravitational-bound) neighbors is +/- 200 km/s. Thus at 5 megaparsecs (Mpc), the Hubble-expansion velocity is 360 km/s, well exceeding any local motion. Therefore, 5 Mpc is used as a conservative scale-distance for the model. The actual value may be closer to 3 Mpc. In the model, each cube is repeated in all directions to infinity, as suggested by the “wide-angle” view of infinity provided by Figure 3.
Figure 2 Dualistic Model of Infinity, Close-up (side of cube magnitude 10^23 meters)
Figure 3 Artist's Impression of Infinity, Wide Angle (length of rods magnitude 10^23 m)
Expansion power is estimated by assuming all matter in each cube exists as a point-mass in the center. The dark cubes in Figure 3 represent the central clumps of matter in the lower half of Figure 2, repeated ad nausea. The rods represent the gravitational attraction between nearest neighbors, and are also of length R. The relative power required to expand such an infinite system is given by:
P(rel_infinite-body) = 8(pi)(HGM/R) (w6)
where H is Hubble’s constant, G is Newton’s constant, M is the mass of the central clump, and R is the distance beyond which the Hubble relation is valid, or distance between clumps.
The clump in the middle of Figure 2 represents a typical group of galaxies at present time, in on-going, local freefall. The dimension of cube is 5 Mpc, the distance at which Hubble expansion overtakes local motion. Density of matter is cosmic value, 3x10^-27 kg/m^3.
Side of the cube in meters:
R = 5 Mpc*3.3x10^6 ly/Mpc*9.5x10^15 m/ly = 1.6x10^23 meters.
The amount of matter in cube is:
M = (1.6x10^23 m)^3*3x10^-27 kg/m^3 = 1.1x10^43 kg.
Summarizing the inputs to the model:
H = Hubble’s constant = 2.3x10^-18 /s G = Newton’s constant = 6.7x10^-11 N-m^2/kg^2 M = local mass not participating in Hubble expansion = 1.1x10^43 kg R = distance beyond which all matter is receding = 1.6x10^23 M density = M/R^3 = 3x10^-27 kg/m^3
Using these numbers in w6:
P(rel_universe) = 2.7x10^-7 w/kg
Thus, by the same measure, the universe is contracting at a rate (2 x10^-5 w/kg) about 100 times faster than it is expanding (2.7x10^-7 w/kg). We live in a contracting universe; the expansion is a small, recoil effect.
In visualizing this, it is helpful to convert these numbers into more familiar terms. Since these numbers are relative--per kilogram--they can be converted into elevation at Earth’s surface. This involves converting watts-per-kilogram to watts-per-pound, then to joules-per-second-per-pound, then to joules-per-year-per-pound, then to foot-pounds-per-year-per-pound, which is just feet-per-year.
Expressed this way, the universe is “falling” about 500 feet per year due to local gravitational contraction; it is “rising” about 5 ft/yr due to cosmic expansion. The net result is that the universe is going downhill. You probably already knew that. Now you have a number to attach to your feeling: things are going downhill, at a net rate of about 495 ft/yr.
THE MEANING OF EQUATION W6
P(rel_infinite-body) = 8(pi)(HGM/R) (w6)
The equation expresses the power, per kilogram, required to expand the universe at the observed rate. It has the same form as the 2-body equation (using equal-masses):
P(rel_2-body) = 0.5(HGM/R) (w2)
They differ only by a numerical factor. The relative energy gain in the 2-body and the infinite-body system are both proportional to HGM/R. In other words, H is a measure of energy-gain, not age.
This is not recognized by the mainstream. The usual interpretation of H is that its inverse, about 13 billion years, represents the age of the universe. This is incorrect. The error can be seen by considering the 2-body problem. The Earth-Moon system is expanding at a rate of 1 part in 10 billion per year. Clearly, this is not telling us the Earth-Moon system is 10 billion years old. The expansion rate of the Earth-Moon system is telling us how fast Earth’s rotational energy is being converted to gravitational energy. It tells us nothing about how long this has been happening. Whether it is an elevator going up, a bullet fired skywards, expansion of the Moon’s orbit or the universe: the expansion rate of a gravitational system says how fast it is gaining energy, not how long it has been doing so.
THE ANSWER IN REVIEW
In the Background section, it was stated that the behavior of an infinite ensemble of particles is the same as the behavior of a 2-body system…only different. How do they differ?
=> The 2-body system either contracts or expands; the infinite system does both.
=> Conservation of angular momentum prevents the 2-body system from immediately collapsing; the uniform pull of matter from all directions prevents the infinite system from doing so.
=> Tidal interaction is the primary driving force for change in the 2-body system; radiation is the primary driving force for change in the infinite.
=> It takes about 50 times more energy, pound-for-pound, to expand the infinite system, relative to the 2-body…all other considerations being equal.
How are they the same?
=> The shape of the energy-vs-distance relation (below) is identical, because the infinity-problem is a sum of 2-body problems.
=> In both the 2-body and infinite-body systems, the rate of expansion is a measure of energy gain, not age.
=> In both the 2-body and infinite-body: if the system is fed energy at a constant rate, expansion will accelerate, due to the flattening of the energy-curve with distance.
The universe is contracting. It is contracting under the influence of gravity at every scale, from dust motes to galaxy collisions. It is also contracting at nuclear-scales, due to fusion reactions, in the cores of stars. The observable net effect of all this contraction is local heating, and hence the generation of radiant energy, lost “to space,” at a rate of some 2 x10^(-5) w/kg, which we see as stars shining in the night.
This energy does not exit the universe, however. Less than 1% of it, estimated at 2.7x10^-7 w/kg, is coupled back into space, causing its expansion, in a recoil effect. The situation can be likened to a waterfall, where a small fraction of the water gets carried back up as spray or mist. No mysterious energy is appearing from empty space. Some of the energy radiated into space returns as gravitational energy, but the source of the energy no mystery. It is coming from the very matter that is shining.
The universe is expanding and contracting; yet the net effect is contraction. It is running “downhill,“ in the ordinary sense. In a manner of speaking, the universe is falling together (contracting) many times faster than it is falling apart (expanding).
Incidentally, this ratio explains the mysterious “k” factor of one-millionth, introduced in the Math section, to calculate lambda. In a pure gravitational system, at least three variables must be specified. When electromagnetism is added, however, a fourth variable is required, as described. In the Newtonian model, the fourth parameter is the fraction k, in %, of radiant energy that gets converted back into expansion energy.
In GR, density is the primary input, instead of mass. The mathematics used to handle density instead of mass are opaque to most mortals. In this strange accounting scheme, a source of energy appears as per-square-meter, which is why “dark energy” has the mysterious units of per-square-meter. Also, GR rolls contraction and expansion into one equation. Therefore, the fraction of radiant energy turned into gravitational energy in Newtonian gravity becomes lambda in GR. Lambda represents the distance-between in Newtonian gravity, plus the coupling from electromagnetism. These two factors combine to yield the simple equation: lambda = k/R^2.
In both models, it is impossible to calculate k on first principles. The observations and measurements must be plugged into the equation, and then you read out the ratios. Using Newtonian math and crude estimates and approximations, this ratio is about 1%; in GR, it is about 0.0001%, or about 10,000 times smaller.
The universe is expanding and contracting; yet the net effect is contraction. It is running “downhill,“ in the ordinary sense. The universe is falling together (contracting) roughly 1,000,000 times faster than it is falling apart (expanding).
In the simplest possible Newtonian model, four parameters are required: the rate of expansion (H); characteristic mass (M); characteristic distance (R); and percent contraction-energy coupled to expansion (k). Lambda in GR represents these last two: distance-between, and fraction of contraction-energy converted to expansion energy.
In both models, it is impossible to calculate this ratio on first principles. The observations and measurements must be plugged into the equation, and then you read out the ratios. Using Newtonian math and my own guesses, estimates and approximations here, this ratio is about 1%; in GR, it is about 0.0001%, or about 10,000 times smaller.
In any case, these ratios are much less than 1 in both models, therefore there is no energy deficit, any way you figure it. Both the current General Relativity model and the simple Newtonian one presented here predict accelerating expansion of the cosmos.
The difference is that the proponents of GR have no idea what the third variable, lambda, represents in their equations. Here, we have said exactly what it is. Lambda represents the last two variables in the Newtonian model: the distance between clumps, R, and the fraction of contraction energy (radiation) that is coupled to expansion, k. Using the simplest possible Newtonian model, k is estimated at 1%; using similar assumptions in GR, it comes out to 0.0001%. The smaller number from GR is intuitively more believable, for reasons beyond discussion here.
The mainstream observed the acceleration first, then fudged their model to fit the observations. Here, we have built the model from the beginning with the required minimum number of inputs, four, to show exactly what is going on. The universe is contracting. Every action has a reaction. There is a tiny recoil effect to all this gravitational contraction. For every million parts of local contraction, there is one partcosmic expansion. For every million watts of energy the sun radiates into space, one watt goes into driving the expansion of it.
Accelerating expansion of the cosmos, based on the known laws of physics, is exactly what we should expect to see.