Physics education research, electronic materials, and the musings of Christopher Moore, Ph.D.

Astro · 2010-07-31 02:23:15

A Mathematical-Epistemological Discrepency in one of Einsteins Assumptions concerning 4-Vector Tensor Algebra
A conceptually-induced problem of the mathematics found in one of Einsteins derivations

In this study, i will show that one of Einsteins propositions concerning the interpretation he inspired from 4-vector analysis on his relativistic equations are based on falty premises due to strict underlying rules of QM.

Einstein concluded within in his equations that the energy of the moving particle is defined by the classical definition:

$\frac{d \mathbf{K}}{dt}= \mathbf{f} . \mathbf{u}$

With some expanding of the $\mathbf{P}^{\mu} \mathbf{P}_{\mu}=0$ term you can yield from his formulas an equation which satisfies:

$K=\gamma Mc^2+S$

Where $S$ is for some constant, and as we shall see, the constant is mistruded under a conceptual misunderstanding between the wedding of classical and modern physics concerning the world of relativity.

However, Einstein does further in his equations that $K$ can reach the value of zero, under the ''which would seem reasonable'' assumption of the system being at inertial rest $\mathbf{u}=0$ .

This would then continuate to provide a mathematical definition of a negative energy solution of the form of $S$ .

An illogical mathematical inconsistency arises when Einstein made the postulate that $K$ almost certainly can and does in his presentation equal zero for a particle at rest, but the incongruity lies in the incompleteness of relativity to be comprehended in terms of the quantum world dynamics. In the case about to be explained, since quantum mechanics rules the show, Einstein's equation simply cannot allow a value for $K$ to be $K=0$ .

Even though the mathematical presentation only highlights the fact that relativity remains a classical theory because it does not incorporate the Uncertainty Principle, and this is very much the same reason why the mathematical concept that his equation can present $K=0$ iff $\mathbf{u}=0$ is completely forbidden by $\frac{\hbar}{2}= \Delta x \Delta p$ .It is, probably very obvious now that the equation pretty much stops there, without being able to derive the value of $S=-Mc^2$ . However, i also found that this may not necessrily be seen in light of the Uncertainty inherent within the commutative variables of position and momenta, but rather one dogmatically-related to the Zero-Point Energy Field. For any quantum harmonic oscillator, even such as $i\hbar \partial_t(e^{i\eta_{\mu}x^{\mu}})$ never has a zero energy bound state in the spatial and temporal freedom of the dimensions. Instead, if you try and freeze a particle to satisfy Einsteins use of $\mathbf{u}=0$ then you will inexorably still find a sum of $S=\frac{1}{2}\hbar \omega$ , which is absolutely equivalent to the kinetic energy Einstein denoted as $K=\gamma Mc^2 + S$ . Whilst mathematically it is consistent, it is not equally consistent to measurement - and it's interesting he did not notice this, since it was he who wrote about the existence of a ZPF with another scientist in the 1920's. Einsteins equation then, must be modified to satisfy;

There are two ways to re-write the Equations that Einstein wrote. For the kinetic energy to equal zero, then we can say that the energy term on the right is reduced to zero by setting $\gamma=0$ . This leaves the Lorentz Transformation on the equation to treat $K$ , the kinetic energy as being non-zero, implying that $\mathbf{u}$ cannot be an absolute observable quantity in the field of quantum mechanics.

$K=\gamma Mc^2+S$

where $S = {\frac{1}{2}( \hbar \omega)}$

The second way to modify the equation so that there are no inconsistencies with measurement, is by alteration of simply:

$K+\frac{1}{2}\hbar \omega = \gamma Mc^2 +S$ .

Where we are allowing Einstein to have $Mc^2=0$ but taking into recongition there is still a magnitude of half the systems energy retained due to $\mathbf{u} \ne 0$ because of $\Delta x \Delta p = \frac{\hbar}{2}$

And viola!

Conclusions

I believe that because Einstein was too wrapped up in his belief that deterministic qualities can be measurable which led him to ignore the implications of the Heisenberg Uncertainty Principle; in short, it's shown that trying to obtain the total energy of a system from the formulations he mathematically conducted cannot by default of quantum mechanical laws derive a true expression of $\Delta E = \gamma Mc^2$ .