# Massless particles and the Neutrino

 Resource type: this resource may consist of Fringe science.

For the four potential theory of gravitation, cf. Four potential theory of gravitation, a quantum particle, the 'gravon', can be found. It is massless and is defined by the following four vector wave function ${\displaystyle \mathbf {\Phi } }$

${\displaystyle (1)~~~\mathbf {\Phi } =(\Phi _{0},{\vec {\Phi }})=\left(\Phi _{0},\Phi _{x},\Phi _{y},\Phi _{z}\right)=\left(-E,k_{x},k_{y},k_{z}\right)\exp \left(i(Et-k_{x}x-k_{y}y-k_{z}z)\right)}$

wherein ${\displaystyle \mathbf {\Phi } }$ is the gravitational four potential [1]. The gravon has obviously the spin 0.

With (1) we obtain ${\displaystyle (2)~~~\partial _{t}\Phi _{0}={\vec {\nabla }}{\vec {\Phi }}}$

This is a linear wave equation for the gravon and also a continuity equation; it is furthermore the third Maxwell type field equation for the four potential gravitation[2].

In analogy to (1) a photon ${\displaystyle \phi }$ can be defined by

${\displaystyle (3)~~~\phi =(1,s_{x},s_{y},s_{z})\exp \left(i(-Et+k_{x}x+k_{y}y+k_{z}z)\right)}$

wherein ${\displaystyle (s_{x},s_{y},s_{z})}$ is the spin vector for the spin 1 of the photon; the spin vector behaves like an ordinary spatial vector and is directed along the momentum ${\displaystyle (k_{x},k_{y},k_{z})}$ of the photon. The time like component ${\displaystyle s_{0}}$ of the complete spinor ${\displaystyle (1,s_{x},s_{y},s_{z})}$ in (3) is ${\displaystyle 1}$. The linear wave equation for this photon is ${\displaystyle (4)~~~\partial _{t}1\exp \left(i(-Et+k_{x}x+k_{y}y+k_{z}z)\right)=-(s_{x},s_{y},s_{z}){\vec {\nabla }}\exp \left(i(-Et+k_{x}x+k_{y}y+k_{z}z)\right)}$

That equation (4) allows for example to apply the four potential of gravitation ${\displaystyle (\Phi ^{0},{\vec {\Phi }})}$ in analogy to the influence of the common electromagnetic four potential onto the Dirac electron[3]

${\displaystyle (5)~~~i\,\hbar {\frac {\partial \psi }{\partial t}}=\left[c\,{\vec {\alpha }}\,\left({\vec {p}}-m_{0}\,{\vec {\Phi }}\,\right)+m_{0}\,c^{2}\,\Phi ^{0}+\,c^{2}\beta \,m_{0}\right]\psi }$

to the propagation of the photon

${\displaystyle (6)~~~{\frac {\partial \exp \left(i(-E_{photon}t+k_{x}x+k_{y}y+k_{z}z)\right)}{\partial t}}s_{0}=\left[\left({\vec {p}}-{\frac {E_{photon}}{c^{2}}}\,{\vec {\Phi }}\,\right)(s_{x},s_{y},s_{z})+E_{photon}\,\Phi ^{0}\right]\exp \left(i(-E_{photon}t+k_{x}x+k_{y}y+k_{z}z)\right)}$

When we suppose that ${\displaystyle {\vec {\Phi }}}$ vanishes, if the source of gravitation is virtually at rest and if we consider for simplicity a photon which propagates along the ${\displaystyle z}$-axis with the momentum ${\displaystyle k_{z}}$ we obtain eventually

${\displaystyle (7)~~~E_{photon}=k_{z}(1+\Phi ^{0})}$

and thus for the respective energies of the photon at positions ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$ an energy difference ${\displaystyle \Delta E}$

${\displaystyle (8)~~~\Delta E=\Phi ^{0}(z_{1})-\Phi ^{0}(z_{2})}$

This result is already known from the general theory of relativity [4].

1. 'Massless particles and the Neutrino', from Hartwig Poth, published on 11.01.2022 under ISBN 9783844243321
2. 'Four Potential Gravitation and its Quantization', from Hartwig Poth, published on 05.04.2014 under ISBN 978-3-8442-9165-0
3. 'Four Potential Gravitation and its Quantization', chapter 13 on page 29 equation (191), from Hartwig Poth, published on 05.04.2014 under ISBN 978-3-8442-9165-0
4. Torsten Fließbach, Allgemeine Relativitätstheorie (Spektrum Akademischer Verlag, Heidelberg Berlin, 4. Auflage ISBN3-8274-1356-7), 12 Photonen im Gravitationsfeld p.62