Analogies are often useful but do not always reveal the entire picture.
Consider a droplet of water falling through empty space.
The arrangement of molecules of water in this droplet is determined by various forces acting upon them.
These forces comprise repulsive forces arising from the positive (kinetic) energies of motion of the molecules, and attractive forces arising from the negative electrical (potential) energies of the molecules, respectively.
The repulsive forces give rise to a positive pressure which tends to push the molecules apart.
The attractive forces, on the other hand, give rise to a negative pressure which tends to pull the molecules together (which, at the surface, gives rise to the phenomenon known as surface tension).
By Newton’s law of action and reaction, these forces must exactly balance each other.
Therefore the net pressure in a droplet of water is given by
Net pressure = positive pressure + negative pressure = 0
Likewise, because the positive and negative pressures derive from the positive (kinetic) and negative (potential) energies, respectively, the positive and negative energies must also balance.
Therefore the net energy of a droplet of water is given by
Net energy = positive kinetic energy + negative potential energy = 0
This result is not meant to be taken literally, because obviously, in trying to simplify the discussion I have deliberately ignored details which are, however, not directly relevant to the principles that I have tried to bring out here; details such as the binding energies of the molecules, the intrinsic mass-energies of the molecules of water and so on.
But despite these obvious defects, I hope you can appreciate the basic principles involved, namely, concepts like conservation of energy, conservation of momentum etc.
There is another principle which I ought to mention: it is based on the second law of thermodynamics, which says, among other things, that particles tend to arrange themselves whenever possible so as to maximize the total entropy.
Put another way, the second law says that, in general, particles will tend to arrange themselves so as to minimize their potential energies.
Now, the surface potential energy of a homogeneous fluid is proportional to the area of the fluid. The smallest possible area for a given volume of fluid is represented by sphere, which is why a droplet of water tends to assume a spherical shape.
OK, having explored some of the basic principles involved, let us now apply these principles to the universe at large.
The universe may be broadly described as an isolated collection of particles which are held together by gravitational force.
The forces acting on the particles may be described as being of one or other of two types: namely repulsive forces and attractive forces.
The repulsive forces derive from the positive (kinetic) energies of those particles. In this context the kinetic energy of a particle includes its intrinsic (relativistic) mass-energy and the ordinary energy of motion, which can be lumped together by Einstein’s mass-energy relation E = mc^{2}.
Such forces give rise to a positive pressure which tends to push the particles of matter apart.
The attractive forces on the other hand derive from the negative (potential) gravitational energies of those particles.
Such forces give rise to a negative pressure which tends to pull the particles of matter together.
By Newton’s law of action and reaction, the forces acting on all the particles of matter in the universe must be equal and opposite. If follows that the positive and negative pressure which arise from these forces must balance each other exactly.
And so we have
Net pressure = positive pressure + negative pressure = 0
And because the positive and negative pressures balance exactly, the positive (kinetic) and negative (potential) energies from which those pressures are derived must also balance exactly.
So we have
Net energy = positive (kinetic) energy + negative (gravitational potential) energy = 0
You can see from these fundamental relations based on Newton’s law of action and reaction and the conservation laws, that even though gravity is an extremely weak force at small distances as compared to the other forces, at distances comparable to the observable size of the universe gravity becomes as important as all the other forces combined.
As to the geometry of the universe, one might suppose that the geometry is simply the shape of the universe.
Well, not quite!
As in the case of the droplet of water, the shape of the universe is dictated by the second law of thermodynamics.
That means that the particles of matter in the universe, as you will no doubt have gathered by now, will tend to arrange themselves so as to maximize the entropy and hence minimize the surface (gravitational) potential energy of the universe.
And as you might also have guessed, the most likely shape of the universe is a sphere, because a sphere represents the lowest surface area (at least for an ideal fluid) and hence the lowest surface potential energy, all else being equal of course.
So, if we regard matter in the universe as an ideal fluid, and if the universe really is like the surface of a sphere then the obvious question is: why do we see the universe as being flat?
To see why, consider a photon propagating from a point A on the surface of this hypothetical sphere to a nearby point B on the surface.
We shall assume that this sphere is expanding, since evidently the universe is expanding.
You can see (I hope) that because the sphere is expanding the path traced by the photon will depend not only on the shape and size of the sphere but also on the rate of expansion.
There are three possibilities which we need to consider, which are:
1. The path traced by the photon is curved inwardly in a direction toward the centre of the sphere. In this case we say that the geometry of the sphere is positively curved.
2. The path traced by the photon is curved outwardly in a direction away from the centre of the sphere. In this case we say that the geometry of the sphere is negatively curved.
3. The path traced by the photon is neither curved inwardly nor outwardly but coincides with the straight line passing through point A. In this case we say that the geometry of the sphere is flat.
For the purpose of establishing a reference, it might help if we assume that point A is fixed in space and time. It may be helpful also if we imagine a fixed straight line which passes through point A and which is tangential to the surface of the sphere at point A.
After a certain time, depending upon the rate of expansion, point B will pass through the fixed line.
If the sphere is expanding very slowly the photon will obviously reach point B before point B reaches the line.
In that case it is clear that the path traced by the photon will be curved away from the fixed line in the direction toward the centre of the sphere. In other words the geometry of the sphere is positively curved in this case as perceived, for example, by a hypothetical observer at point B.
If, on the other hand the sphere is expanding very quickly, point B will already have passed through the fixed line before the photon reaches point B.
In that case the path traced by the photon will be curved in the direction pointing away from the centre of the sphere, so in this case the geometry is negatively curved.
It is not hard to see, I think, that between these two extremes there is a critical value for the rate of expansion such that the photon traces a perfectly straight line.
Therefore if the universe is expanding at the critical rate then the geometry ought to be flat, assuming of course that the arguments I have presented here are right.
In 1951 by an astronomer by the name of William McCrea showed mathematically that the critical rate of expansion for a flat geometry is one which increases exponentially with time.
Also, according to theory, an empty universe (known as a de Sitter universe) is both flat and expands exponentially with time.
[A possible definition of an empty universe is that the net energy density of such a universe is zero.]
On the basis of the foregoing I would suggest that a good test for whether or not the net energy density of the universe is zero is to measure both the geometry of the universe and the rate of expansion. If the geometry is flat and if the universe is expanding at an exponentially accelerating rate then I would argue that the net energy density of the universe must be zero
Consider the evidence:
Since 1998 astronomers have discovered increasing evidence that the universe is geometrically flat and, what is particularly noteworthy, that the universe is expanding at an accelerating, exponential rate.
I think the evidence speaks for itself.
I hope this helps
[PS: Primordial & Bullfox: Hi
Years ago when I began studying cosmology, a couple of major assumptions in the standard model of cosmology were that the universe is expanding adiabatically and that the expansion is isentropic (constant entropy).
In these models, we treated the expansion as boundary work, which altered the kinetic energy component of the universe but did not change the total energy. Neither did it change the total entropy of the universe because, in the absence of heat transfer, boundary work is a thermodynamically reversible process, which means that there is no net increase or decrease in total entropy.
We justified these assumptions on the basis that for an isentropic, adiabatically expanding universe, in which the only energy transfers of energy derived from boundary work, and provided that there were no other (irreversible) changes taking place, the temperature of the universe should vary precisely in inverse proportion to the scale factor (a measure of the relative expansion of the universe).
The available evidence at the time seemed to support those assumptions. Specifically, the evidence consisted of temperature of the cosmic microwave background, and measurements of cosmological red shift of light from distant stars and galaxies, which was always regarded as a measures of scale factor. In particular, the measurements of cosmic red shifts seemed to be consistent with observations of time dilation, which are exemplified by the stretched out luminosity-time curves of exploding stars.
As you can entropy of the universe does not depend upon energy, or temperature alone; it depends upon the scale factor as well, which in very rough terms can be translated into volume.
To illustrate with a very ordinary example: if we partition a chamber and fill half the chamber with a gas and leave the other half empty then the entropy of the gas in the filled half is relatively low (it is less than it otherwise might be) because the molecules of gas are crowded together, which restricts the number of available energy levels to the molecules. That number is a measure of the entropy of the gas.
If now we suddenly remove the partition then the gas will rapidly diffuse to fill the entire chamber. The molecules are not restricted so much as before, so there are more available energy states to the molecules and so the entropy is greater. There is no net change in total energy of the molecules, or of their temperature; only a change in the volume of the gas and the increase in entropy reflects that change.
However, if the partition is replaced by a piston and the gas is allowed to do work on the piston, by pushing a weight against the force of gravity, then, under ideal conditions (no friction, etc.), there would be no change in entropy. But the temperature of the gas would drop, reflecting the increase in volume, which means that the molecules have less energy than they started with and so despite the increase in volume there is no change to the number of energy states available to those molecules.
With the foregoing in mind, an isentropic expansion is consistent with the idea that the expansion is associated with boundary work.
As I intimated earlier, the evidence based on the CMB and the red-shift-time dilation relation seemed very strong evidence that the expansion of the universe is isentropic;
A related argument in support of that proposition is based on the fact that light from distant stars and galaxies is red shifted implies that the kinetic energies of distant stars and galaxies are diminished.which we attribute of the expansion of the universe. On face value, that what would appear to be a violation of the law of conservation of energy.
But if we suppose that the expansion results from boundary work, then the apparent loss in kinetic energy can be explained by a corresponding gain in potential energy, and so there is no violation of conservation of energy.
Following the discovery that the expansion of the universe is accelerating, there has been a lot of discussion as to whether the entropy of the universe is increasing and if the universe should expand forever will the entropy increase to infinity.
On the basis of the foregoing I would argue that the total entropy is not increasing at all, despite any perceived appearances to the contrary.
Of course it can be argued that evidence for increasing entropy can be seen everywhere: stars are dying, as are plants and animals, decay seems to be a universal phenomenon; clear evidence one might argue that entropy is increasing.
Drop a teacup onto a hard floor and what happens? Most likely the teacup will shatter into a thousand pieces. We never see the pieces spontaneously recombining —evidence of the second law of thermodynamics prediction that entropy can increase but can never decrease.
But appearances can be deceiving.
The association between entropy and second law of thermodynamics goes way back to the early days of the steam engine but of course a great deal of our understanding has changed with the development of quantum theory. But the basic idea that entropy is not a conserved quantity according to the classical interpretaton of the second law does not seem to have changed.
The problem as I see it with the classical interpretation of entropy is that entropy is related to information.
Increasing entropy implies increasing information. So if the entropy of the universe is increasing then so is information. Where is that information coming from?
Or consider a black hole, which grows by matter falling into the black hole. Matter falling into a black hole carries information so the black hole grows bigger, as evidenced by an increase in the area of the black hole.
But black holes also can lose mass as well as gain mass by Hawking radiation. When a black hole radiates away all its mass where does the information go?
Personally, I am of the opinion that deep down at some fundamental level entropy is a conserved quantity, like energy and momentum, but most people don’t see it that way, I guess.
But that’s science!
Cheers