Vector and Scalar Energy
© 2003 David Barwacz
http://members.triton.net/daveb
Abstract
In this paper I will reconcile the vector energy and associated equations developed in the previous paper with the common notion of energy as a scalar quantity and show that the concepts are completely compatible. I will derive a scalar equivalent energy formula and show that is reduces to the classical Newtonian energy for small velocities.
Energy measurements
Figure 2 below depicts the measurement of energy. The space time diagram is converted to energy as described in the previous paper.
The measurement occurs at a time prior to the time the observer sees the objects interact.
The ruler line has been replaced with a “line of measurement”. This is simply a representation of the space line on which the observer places his measuring device. In the case of energy it might be an energy transfer device, or an optical device. Whatever it is, to the user, it lies on an axis perpendicular to his time axis.
In Figure 2 above the energy vectors are projected onto the observers’
energy line. As long as the observer time line remains the same (ie an inertial observer) the measurement can be considered
a scalar and involves two components: The 
component and the 
component.
We now write the vector and scalar representation for the total energy.
equation 1.1 from the vector representation and
equation 1.2 from the scalar representation
The vector magnitude equation is just a form of the well
known equation 
Let’s look at the terms in the scalar equation. is
the scalar measurable quantity of rest energy.
The scalar rest energy of an object decreases with velocity.
is
the scalar measurable quantity of 
. We can write in terms of 
and the angle.
Clearly =
but 
and hence we arrive at
the scalar energy equation:
Equation 1.3
The first term can be considered the dilated rest energy.
The magnitude of the second term is precisely the potential energy of an object
orbiting another object. Take an electron for example. We know that 
which give 
which is just the
potential energy.
It would appear that potential energy is just the component of total energy that we cannot directly measure. It must first be converted to kinetic to be measured. This theory predicts the existence of potential energy without the presumption of force fields. One could define force in terms of the immeasurable component of total energy which, as we have seen is the result of a time difference between actual measurement and perceived measurement.
It is not common to even attempt to measure rest energy
while an object is moving. We always measure it while the object is at rest and
then just assume that it remains the same. It is therefore instructive to
develop an equation in terms of 
.
and
which yields:
and it is clear that
Combining we get yet another exact scalar energy equation:
Equation 1.4
We can expand both the sin and cos functions and evaluate Equation 1.4 for small angles.
, 
and 
= 
But 
and 
. Substituting yields:
which
is the classical Newtonian energy equation.
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