Newtonian Gravity of Antimatter Particles and Gravitational Foundation Underlying Field Quantization through Quantum Spacetime Geometrization

3.2. Newtonian Force Between Point-Like Matter and Antimatter Particles

Given the negative energy values (1.14) of antimatter, also the energy tensor density (12) becomes negative leading to the repulsive antimatter Newtonian potential as given by (27)

$${\varphi }_{-}=G{\displaystyle \frac{m_{-}}{r-R_{-}}}.$$

(28)

Nonetheless, in order to determine the characteristics of the antimatter gravity, its kinematics must also be taken under consideration.

Given that the force applied to antimatter particle with mass $m_{-}$ (with negative valued energy), for energy conservation, generates an acceleration in the opposite direction, so that

$$F=-m_{-}\ddot{r}=-{\partial }_R{U}_{-},$$

(29)

it follows that the antimatter Newtonian force (3. 22) on an antimatter particle test $m_{-}$ results repulsive

$$F=-m_{-}{\partial }_r{\varphi }_{-}=G{\displaystyle \frac{m_{-}{m}_{-}}{{\left(r_{-}-R_{-}\right)}^2}},$$

(30)

but the acceleration of the antiparticles

$${\ddot{r}}_{-}=-G{\displaystyle \frac{m_{-}}{{\left(r_{-}-R_{-}\right)}^2}}={\partial }_r{\varphi }_{-}$$

(31)

results attractive (even if the force is repulsive).

Therefore, to determine the force between matter and antimatter, we must recognize that, in the matter-antimatter interaction, the force on the antiparticle does not comply with the third law of dynamics, which requires that it be equal in magnitude and opposite in direction to the force on the particle. Consequently, if we calculate the variation of the total energy with respect to the displacement of the particle and antiparticle, this variation is zero.

However, if we consider the apparent "kinematic" forces acting on the antimatter, expressed as $F_{kin}=-F=m_{-}\ddot{r}={\partial }_R{U}_{-}$ where $U_{-}$​ represents the gravitational field energy of the antiparticle, the validity of the third law is restored. Thus, the kinematic force derived from the variation in the spatial position of the matter and antimatter particles is given by:

$$\begin{array}{l}{F}_r=-{\partial }_R{U}_{+}+{\partial }_R{U}_{-}=-{\partial }_R{\displaystyle \sum _{i,j}\frac{1}{2}{{\displaystyle \int m_i\varphi }}_{j\ne i}dV}=-{\displaystyle \frac{1}{2}}{\partial }_R\left({\displaystyle \int {\displaystyle \int m_{-}{\varphi }_{+}}}dV-m_{+}{\varphi }_{-}dV\right)\\ =-{\displaystyle \frac{1}{2}}{\partial }_R\left(m_{-}{{\displaystyle \int \delta (r-R_{-})\varphi }}_{+}dV-m_{+}{{\displaystyle \int \delta (r-R_{+})\varphi }}_{-}dV\right)\\ ={\displaystyle \frac{G}{2}}{\partial }_R\left(m_{+}{m}_{-}{\displaystyle \int \delta (r-R_{-})\frac{1}{|r-R_{+}|}}dV+m_{-}{m}_{+}{\displaystyle \int \delta (r-R_{+})\frac{1}{|r-R_{-}|}}dV\right)\\ =G{\displaystyle \frac{m_{-}{m}_{R_{+}}}{2}}{\partial }_R\left({\displaystyle \frac{1}{|R_{+}-R_{-}|}}+{\displaystyle \frac{1}{|R_{-}-R_{+}|}}\right)=-G{\displaystyle \frac{m_{-}{m}_{+}}{R^2}}\end{array}$$

(32)

where $R=|R_{+}-R_{-}|$. Equation (32) demonstrates that the kinematic forces between matter and antimatter are attractive. Consequently, the antiparticles experience a repulsive force.

3.3. Newtonian Gravity at Short Distance Between Two Quantum Bodies

The results (29-32) are valid as far as the wave function localization produces a mass distribution that is satisfactory well described by the Dirac’s delta respect the scale of the problem so that the results are appropriate for sufficiently large distance between particles (typical of the classical macroscopic approach). On very short distance, when the physical length of the problem is of order of the quantum body mass distribution, the gravitational interaction is influenced by the quantum mass density distribution $|\psi {|}^2$.

Here we consider the case of quantum bodies (sufficiently large) to be described by continuous fields. Furthermore, the gravity interaction is derived by assuming the particle densities are much lighter than the Planck mass in a cube of planch length side, so that (at zero order of approximation) the spacetime can be assumed Minkowskian. Then, the gravitational force is derived by the first order curvature produced by such mass distributions.

Before deriving the gravity of a quantum mass distribution localized around a center of attractive force it must be observed that, in the classical acceptation, the gravitational force is that one experienced by a test particle that does not alter the gravitational potential. In the quantum case this is not possible since the presence of the test particle alters the gravity field of the reference particle due to their non-linear coupling in the QGE. Thence, the gravitational potential, as conceptualized in the Newtonian limit independent by the mass of the particle on which it acts, disregards this non-linearity of the QGE and derives by the gravitational fields generated by the mass density distribution of each single particle, disregarding the mutual, very small, perturbation.

Therefore, we are in the position to derive the gravitational potential of a particles as deriving by the curvature that its mass distributions $|\psi {|}^2$ alone produce through the gravity equation

$$\begin{array}{l}{R}_{\nu \mu }-{\displaystyle \frac{1}{2}}{g}_{\nu \mu }{R_{\alpha }}^{\alpha }={\displaystyle \frac{8\pi G}{c^4}}{\displaystyle \frac{m{c}^2|\psi {|}^2}{\gamma }}\left(\begin{array}{l}\left(\sqrt{1-{\displaystyle \frac{V_{qu}}{m{c}^2}}}-1\right)g_{\mu \nu }-{\Lambda }_Q{g}_{\mu \nu }\\ +{\sqrt{1-{\displaystyle \frac{V_{qu}}{m{c}^2}}}}^{-1}{\left({\displaystyle \frac{\hslash }{2mc}}\right)}^2{\partial }_{\mu }ln[{\displaystyle \frac{\psi }{\psi *}}]{\partial }^{\lambda }ln[{\displaystyle \frac{\psi }{\psi *}}]g_{\lambda \nu }\end{array}\right).\\ ={\displaystyle \frac{8\pi G}{c^4}}{\displaystyle \frac{m{c}^2|\psi {|}^2}{\gamma }}\left(\begin{array}{l}\left(\sqrt{1-{\displaystyle \frac{V_{qu}}{m{c}^2}}}-1\right)g_{\mu \nu }-{\Lambda }_Q{g}_{\mu \nu }\\ +{\sqrt{1-{\displaystyle \frac{V_{qu}}{m{c}^2}}}}^{-1}{\left({\displaystyle \frac{\hslash }{2mc}}\right)}^2{p}_{\mu }{p}^{\lambda }{g}_{\lambda \nu }\end{array}\right)\end{array}$$

(33)

Here, for sake of completeness, we also consider the contribution that can come from the quantum pressure $-{\Lambda }_Q{g}_{\mu \nu }$ where the cosmological-like term ${\Lambda }_Q$, whose expression is given in ref. [2], reduces to a small constant] in quasi-Minkowskian spacetime approximation [10]. Therefore, Equation (33) for the classical case ${\displaystyle \frac{V_{qu}}{m{c}^2}}\ll 1$ leads to

$$R_{\nu \mu }-{\displaystyle \frac{1}{2}}{g}_{\nu \mu }{R_{\alpha }}^{\alpha }={\displaystyle \frac{8\pi G}{c^4}}{\displaystyle \frac{m{c}^2|\psi {|}^2}{\gamma }}\left(\begin{array}{l}-\left({\displaystyle \frac{V_{qu}}{2m{c}^2}}+{\Lambda }_Q\right)g_{\mu \nu }\\ +\left(1+{\displaystyle \frac{V_{qu}}{2m{c}^2}}\right)u_{\mu }{u}_{\nu }\end{array}\right).$$

(34)

Furthermore, by using the identity

$$-\left({\displaystyle \frac{V_{qu}}{2m{c}^2}}+{\Lambda }_Q\right)\left({\delta }_{\alpha }^{\alpha }-u_{\alpha }{u}^{\alpha }\right)=-3\left({\displaystyle \frac{V_{qu}}{2m{c}^2}}+{\Lambda }_Q\right)$$

(35)

it follows that

$$-{R_{\alpha }}^{\alpha }={\displaystyle \frac{8\pi G}{c^4}}m{c}^2|\psi {|}^2\left(1-3\left({\displaystyle \frac{V_{qu}}{2m{c}^2}}+{\Lambda }_Q\right)\right)$$

(36)

leading to the gravitational potential

$${\partial }_r\varphi ={\displaystyle \frac{Gm}{{\left(r-R\right)}^2}}{\displaystyle \underset{0}{\overset{V}{\int }}|\psi {|}^2\left(1-3\left(\frac{V_{qu}}{2m{c}^2}+{\Lambda }_Q\right)\right)d^{3}V}$$

(37)

where, by utilizing (1.16), the classical and quantum parts read, respectively,

$${\partial }_r{\varphi }_{Class}={\displaystyle \frac{Gm}{{\left(r-R_j\right)}^2}}\left({\displaystyle \underset{0}{\overset{\left(r-R\right)}{\int }}{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}|\psi {{|}^2}_{\left(r-R,\vartheta ,\phi \right)}{\left(r-R\right)}^{2}d}}}\left(r-R\right)\cos \vartheta d\vartheta d\phi \right)$$

(38)

$${\partial }_r{\varphi }_Q={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\hslash }^2}{m{c}^2}}{\displaystyle \frac{G}{{\left(r-R_j\right)}^2}}\left({\displaystyle \underset{0}{\overset{\left(r-R\right)}{\int }}{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}|\psi {|}_{\left(r-R,\vartheta ,\phi \right)}{\left(r-R\right)}^2\left({\partial }_{\mu }{\partial }^{\mu }|\psi |+2m{c}^2|\psi |{\Lambda }_Q\right)d}}}\left(r-R\right)\cos \vartheta d\vartheta d\phi \right).$$

(39)

It is worth mentioning that the quantum contribution (39) becomes larger smaller the particle mass leading to the asymptotical expression

$${\lim }_{m\to 0}{\partial }_r{\varphi }_Q\approx {\displaystyle \frac{3}{2}}{\displaystyle \frac{{\hslash }^2}{m{c}^2}}{\displaystyle \frac{G}{{\left(r-R_j\right)}^2}}\left({\displaystyle \underset{0}{\overset{\left(r-R\right)}{\int }}{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}|\psi {|}_{\left(r-R,\vartheta ,\phi \right)}{\left(r-R\right)}^2{\partial }_{\mu }{\partial }^{\mu }|\psi |d}}}\left(r-R\right)\cos \vartheta d\vartheta d\phi \right)$$

(40)

From (40) it is important to highlight that, to prevent gravitational energies from diverging, the mass of particles cannot decrease continuously to zero but must be quantized with minimum values. This implies the existence of elementary particles and the necessity of quantizing their fields, pointing to a gravitational constraint that drives field quantization. Furthermore, this hypothesis supports the notion that only fields require quantization, while gravity itself, as defined by the left side of gravitational equation (1), becomes indirectly a quantum field through the equivalence to the quantized fields on the right side of (1) [11].