Conservation of Mass
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If we want to make an analogy between the regularities of the physical world and the laws that we make to govern ourselves, then we must rightly regard the conservation laws as comprising the Constitution of the Universe. In particular, we must require that all other laws conform to the conservation laws, the latter set taking precedence over all of the former. Indeed, we have an exact analogy: just as Congress cannot make a law that conflicts with the Constitution, so physicists accept no proposed law that conflicts with the conservation laws. We can then turn that requirement around and use the conservation laws to deduce the forms that the other laws must necessarily have.
What new law can we derive from conservation of linear momentum, just to take one example? How shall we parley these fundamental laws into further laws of nature? I start by asking questions about the concepts that I have brought up in my most recent derivation.
We have discovered an emergent property of matter - mass - and discovered that it participates in the conservation law pertaining to linear momentum. Does the structure of Reality also conserve mass? More specifically, I ask whether the mass of some body can change spontaneously; whether the mass of that body could go from M to M+N, for example. What consequences follow that assumption?
If the body moves past me at the velocity V, then its linear momentum has increased spontaneously from MV to MV +NV, in blatant violation of the conservation law. I must dismiss that possibility from consideration.
Perhaps the body's velocity changes in a way that compensates the increase in mass? In this example the velocity must change from V to (M/(M+N))V. If that happens, then the body's linear momentum won't change, at least for me. But for an observer past whom the body initially moves with velocity V+W, the body's linear momentum increases spontaneously by (N/(M+N))W. That fact means that such a second observer could use the body to create motion in a suitable mechanism and yet I would see that very same mechanism not changing its motion at all. That kind of contradiction necessitates that I dismiss the premise that led to it, so I must say that the velocity of a body cannot change in response to a spontaneous change in the body's mass.
Perhaps space exerts a force that compensates the increase in linear momentum? If so, then space must exert a different force for different observers if those observers move at different velocities. Further, space itself must have mass. But we deduced mass as an emergent property of bodies in space, so we must ask whether we may regard space itself as a body in space. That idea may seem absurd, but we have no a priori reason to dismiss it as such, so let's press on.
Certainly space cannot act as a rigid body. In order to do so it would have to have the possibility of moving as a rigid body. But space as a whole cannot move, for the very reason that led us to infer the conservation law pertaining to linear momentum in the first place. If we must regard space as a body, then we must conceive it as a fluid body, one in which the parts may flow and swirl while the whole remains motionless.
Let's assume that when a body changes its mass spontaneously some parcel of space that we can mark for observation accelerates in a way that compensates the body's change of linear momentum in accordance with the conservation law. If one observer sees the body not moving, then that observer must also find the parcel of space not accelerating when the body's mass changes. But if another observer sees the body moving with some velocity, then that observer must find the parcel of space exhibiting a non-zero acceleration when the body's mass changes. That contradiction, one observer seeing a parcel of space accelerating and another observer seeing the same parcel not accelerating, obliges us to dismiss the specific premise that led us to it. Thus we declare the proposition that parcels of space exert reaction forces upon bodies whose masses change spontaneously false to Reality and dismiss it from further consideration.
Perhaps the body's increase in mass is compensated by the decrease in the mass of some other body? Can we have a body with mass M gaining a mass N if some other body loses mass? In that case we know that the lost mass takes some linear momentum with it and the gained mass brings some linear momentum to the body it joins. Those two momenta must be equal and oriented in the same direction if Newton's third law is to be properly upheld. We don't know whether the structure of Reality allows that kind of direct teleportation, but we do know that it does allow mass to move from one body to another so long as the transferred mass carries its own linear momentum with it and confers it upon the new body that it joins. So we can change the masses of bodies by breaking pieces off some of them and sticking the pieces onto others. But that necessarily means that the total mass of all of the bodies under consideration does not change, it merely gets shifted around.
We must thus declare as true to Reality the proposition that mass cannot increase or decrease spontaneously, but can only be transferred among bodies. Algebraically we have then
But we can describe a mass as the density of the matter involved (represented by the Greek letter rho) integrated over the volume occupied by the body in question; that is, dM = Ddxdydz. Substituting that equation into Equation 1 gives us
which gives us, when we represent the pseudo-infinitesimal element of volume by dxdydz = dV and carry out the obvious integrations,
in which vx, vy, and vz represent the components of the velocity with which the matter moves through the element of volume. Calculating the product of the matter's density and its velocity gives us the mass current, kilograms passing through a given square meter per second. If we have Dv = j, then Equation 3 becomes
That equation of continuity tells us that the net rate at which mass increases within a given volume equals the negative of the net flow of mass across the surface bounding that volume. Now we know that an equation of continuity expresses a conservation law pertaining to a property of matter. In addition Green's Theorem tells us that integrating the mass current j over the entire surface area a enclosing the volume V yields the same number as does integrating the divergence of the current,L·j over the volume so enclosed, so we can express Equation 4 in differential form as
We thus have a choice of using an integral equation or a differential equation to express the law of conservation of mass.
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