Energy Conservation Deduced
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The Universe enacts its existence in a setting of absolute nothingness. As a consequence, the Universe has no motion. But things move within the Universe, so all of the motions in the Universe must always add up to a net zero. If some object changes its motion by some amount, then, in order to preserve the zero, some other object must change its motion by the same amount in the opposite direction. That is a statement of the law of conservation of linear momentum with an outline of how we deduce it as a necessary feature of Reality.
The linear momentum of an object consists of the quantity of matter (mass) in the object combined with the object’s velocity. That is a vector quantity and the rate at which it changes measures the net force exerted upon the object. Imagine an observer lined up on a straight path and an inert body of mass m0 set into uniform motion at some speed w on a straight line perpendicular to the observer’s path. The observer accelerates to a speed v and sees the inert body moving slower than before the acceleration. The Lorentz Transformation indicates that the distance that the inert body moves between route markers doesn’t change for the observer, but that the elapsed time does change, due to time dilation. The observer calculates
But the inert body’s linear momentum has not changed. The acceleration of the observer in one direction does not produce a force in any direction perpendicular to the direction of the acceleration, so the product mw does not change. In light of Equation 1, then, we must infer that
which gives us the formula describing the relativistic increase in a body’s mass.
The observer has another small inert body of rest mass m0 in their possession and a second observer remains unaccelerated as the first observer applies a force to their vehicle, accelerating straight in the direction that we identify as the x-direction. We have the force acting on the small body as
Making the appropriate substitution from Equation 2 and integrating the result over the distance the body moves while being accelerated gives us the work that the first observer’s vehicle must do in accelerating that small body,
We take the indefinite integral as representing the energy content of the body relative to some arbitrary state of zero velocity and get the familiar
Substitute from Equation 2 into Equation 5, multiply by the square root, and square the result. We get
We can divide out the square of lightspeed and rewrite that equation as
The expression on the right side of Equation 6 comes equal to a constant (on the left side) and the square of the momentum is just the dot product of the three-vector [px,py,pz]. Those facts imply that we have a Lorentz invariant and that momentum and energy constitute the components of a proper four-vector, as expressed in the second line of Equation 7. That four-vector is subject to a Lorentz Transformation of the form:
The second and third of those equations just encode the proposition on which we based our derivation of Equation 2. The first and the fourth of those equations tell us that momentum manifested in one inertial frame appears, in part, as energy in other frames and vice versa. Linear momentum and energy are interchangeable.
We have deduced a conservation law for linear momentum (and we can do the same for rotary motion to get conservation of angular momentum) and that law applies to linear momentum component by component: the vector components of linear momentum are all conserved separately. Energy is a component of linear momentum, albeit a strange one; therefore, energy must conform to a conservation law. Energy can only be transferred or transformed; it cannot be created or annihilated.
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