## Wednesday, May 7, 2014

### Center of Mass of the Universe: More thoughts on Symmetries and Conservation Laws

I was re-reading Feyman's "The Character of Physics Law" when I stumbled upon some interesting sentences. (Chapter 3, Pg 82 starting just above Figure 22.)
"In this way the conservation of angular momentum implies the conservation of momentum. This in turn implies something else, the conservation of another item which is so closely connected that I did not put it in the table. This is a principle about the centre of gravity...The point is that of all the stuff in the world, the centre of mass, the average of all the mass, is still right where it was before."

If you take a large collection of particles, the forces of interaction between the particles is not capable of changing the center of mass. The position of the center of mass only changes if the center of mass was already moving with a certain velocity, V. This velocity, V, is unaffected by the forces of repulsion and attraction between the particles. If this velocity, V, were zero to start, then it would always stay that way, unless acted upon by particles that weren't included in the original set when calculating the center of mass. But what if we take the collection of particles to be all of the particles in the universe?

In my understanding of our universe (which is different than most astrophysicists and physicists), the universe is a wrinkled, expanding surface of a 4D sphere,where the radius of the sphere is the time dimension. The center of mass of such a surface is the center of the sphere. The center of the 4D sphere is the location in space-time of the Big Bang  (r=t=0.) So, while the surface my be wrinkled (due to local variations in the density of matter), the variations have to cancel exactly on average, so that the center of the mass of the universe is still exactly the the center of the 4D sphere. If there were any fluctuations (i.e. fluctuations such that there is an increase in mass/energy that is not exactly balanced on the opposite side of the sphere), then the location of the center of mass would be offset from the center of the sphere.

One can hopefully see that this has major implication for quantum gravity as well as R-type quantum theories that require a collapse of the wavefunction. The introduction of uncertainty into the location of massive objects might cause us to have to drop an important conservation law (i.e. the conservation of the momentum...which leads to the conservation of the center of mass in specific circumstances.) For example, if the mass inside of a blackhole were to randomly fluctuate, then this could cause the center of mass of the blackhole to change, and then would in turn cause the center of mass of the universe to go off center.

There are clearly some problem reconciling quantum mechanics with certain conservation laws, such as the conservation of momentum, because if the location of an electron in an atom were truly random (i.e. stochastic) before we measure it, then this could cause the center of mass of the universe to be ever so slightly off center (unless there were an completely symmetry fluctuation on the opposite side of the universe that kept the center of the mass constant. Of course, such a symmetric fluctuation would imply that electrons on opposite sides of the universe can communicate with each other at speeds much greater than the speed of light.) This is one of the many reasons why I'm skeptical of introducing stochastic (i.e. probabilistic) processes into the laws of physics. In order to introduce stochastic processes into nature, you run into potential problems of either (a) communication at infinite speed in order for the fluctuations to cancel out, or (b) throw out the conservation of momentum...i.e. allow of the center of mass of the universe to move stochastically around the origin as particles randomly appear here and disappear there. (Note U-type Quantum mechanics is deterministic, it's only R-type Quantum mechanics that is stochastic. For more discussion of U vs. R QM, see Chapter 22 of Penrose's The Road to Reality.)

Another interesting question is:  what is the total angular momentum of the universe? Is there an axis about which the 4D sphere is rotating?
I think that this question is similar in nature to following questions:  what is the total electrical charge of the universe? what is the total weak charge and total color charge of the universe? It is entirely possible that the answer to all of these questions is zero.

In the remainder of this post, I'll be summarizing the concepts discussed in \Figure 14 in Chapter 3 of Feyman's "The Character of Physical Law."  The ideas that Feynman discuss here can be found on any website (such as wiki) that covers Emmy Noether's Theorem that Conservation Laws imply continuous symmetries of differential equations (and vice versa.)

Energy/Mass is conserved locally and is the source of a field (gravity). The underlying cause of this conservation law is the symmetry of all the laws of physics to time translation. The total energy of the universe is (likely) a positive value, and the energy of individual system can only be positive (i.e. there are likely no negative mass particles that cause gravitational repulsion.) Adding mass is like counting positive real numbers (or perhaps adding positive integers if space-time is discrete.)

Electric Charge is quantizied, is conserved locally and is the source of a field (E&M field).  The underlying cause of this conservation law is the symmetry of changing the phase of the QM gauge locally at every place in space-time. The total charge of the universe is likely zero (although it's important to note that I have no way of proving that it is exactly zero, if the universe is finite, then the net charge is likely zero so that electric field lines begin and end on charged particles.) Electrical charge (unlike mass) can be positive or negative. Since charge can be positive or negative, checking this conservation laws is like addition over the field of integers.

Angular Momentum is quantizied, is conserved locally, and is not known to be a source of a field. The underlying cause of this conservation law is the symmetry of all the laws of physics to angular rotations. In other words, no known law of physics depends upon knowing the absolute rotational angle of the universe. We don't need to know our longitude, latitude, etc... on the surface of the 4D sphere in order to do calculations. In any equation where angles appear, the variable theta, omega, etc. show up at derivatives with respect to the angle. Angular momentum can be positive or negative, and the axis of rotation is a three dimensional vector. This is already much more complicated than the cases above. One question is:   is the angular momentum of the universe equal to (0,0,0)?
And on a side point, while I mentioned that angular momentum is not known to be a source of field, there is an indirect "force" that is associated with spin. For example, Fermions can exert a 'force' that keep them from occupying the same quantum state, and hence from gravitationally collapsing on each other. However, this is not a field because the force doesn't increase as you increase the spin of the particle.

Color Charge is quantized, is conserved locally, and is the source of a field (the strong nuclear force). The underlying cause of this conservation law is an SU(3) gauge invariance. What I want to point out here is that this conservation law is more complicated than the conservation of mass or conservation of electric charge. In this case, you can arrive at zero net color charge buy at least two different way. First, like electric charge, the value of color charge can be positive or negative. So, you could combine a red quark with an anti-red quark, and then would be no net color charge (similar to combining a proton and an electron to form a neutral Hydrogen atom.) Second, unlike electric charge, there are three components to color charge (just as there are three components to angular momentum.) The three components, however, are not completely independent of each other. It's not like having 3 new forces of nature. For example, imagine that the three charges are like rotations of a 3D sphere. You can rotation a sphere 90 degrees in one direction, then rotate by 90 degrees in a supposedly perpendicular direction, and then rotate 90 degrees in a third supposedly perpendicular direction. If you do so, you can end up right back where you started from. This means that you can combine a red quark, a green quark, and a blue quark, and you can obtain a net zero color charge. This means that the 3 color charges are not completely independent of each other.

Weak Charge is quantized, is conserved locally and is the source of a field (the weak nuclear force). The underlying cause of this conservation law is an SU(2) gauge invariance. In E&M, there's one type of charge, and it can be positive, zero, or negative. In the Weak Nuclear force, there's two types of charge, and as we saw above, in the Strong Nuclear force, there are three types of charge (red charge, green charge, and blue charge...however, they are not completely independent because quarks appear to have only type of charge at any given time.)  In the Weak nuclear force, there is a quantity called weak isospin. It is similar (though not exactly similar) to the concept of electrical charge. Left-handed fundamental particles have weak isospin values of +/- 0.5, whereas right-handed fundamental particles have weak isospin values of 0. The W bosons (i.e. force carriers) carry weak isospin of 1. This is unlike Electromagnetism, where the photon does not itself carry any electrical charge. The weak nuclear force is weird indeed. Also. none of the other forces of nature care about the spin of the particle. For example, a right-handed electron still interacts via gravity and the electromagnetic force, and a right-handed quark still interacts via gravity, electromagnetism, and the strong nuclear force.

There's still a lot we don't know about the weak nuclear force. Luckily, there are experimental physicists and astrophysicists devoting their careers to answering some important questions about the weak nuclear force.
I'll end this post on a completely unrelated note my showing a picture of some fo the strongest evidence yet for dark matter (and against Modifications of Newtonian Dynamics at large distances. i.e.  MOND.) This is likely be the topic of a post in the near future.