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A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fraction of the speed of light. As the force or acceleration increases, the particles’ velocity asymptotically approaches but never achieves the speed of light obeying relativity. The asymptotic increase in the particles’ velocity toward the speed of light as acceleration increasingly surpasses the speed of light per unit time does not compensate for the momentum value produced on the particles at sub-light velocities. Hence, the particles’ inertial mass value must increase as acceleration increases. This increase in the particles’ inertial mass as the particles are accelerated produce a gravitational field which is believed to occur in the oscillation of quarks achieving velocities close to the speed of light. The increased inertial mass of the density of accelerated charged particles becomes the source mass (or Big “M”) in Newton’s equation for gravitational force. This implies that a space-time curve is generated by the accelerated particles. Thus, it is shown that the acceleration number (or multiple of the speed of light greater than 1 per unit of time) and the number of charged particles in the cloud density are surjectively mapped to points on a differential manifold or space-time curved surface. Two aspects of Einstein’s field equations are used to describe the correspondence between the gravitational field produced by the accelerated particles and the resultant space-time curve. The two aspects are the Schwarzchild metric and the stress energy tensor. Lastly, the possibility of producing a sufficient acceleration or electromagnetic force on the charged particles to produce a gravitational field is shown through the Lorentz force equation. Moreover, it is shown that a sufficient voltage can be generated to produce an acceleration/force on the particles that is multiples greater than the speed of light per unit time thereby generating gravity.

It has been shown that a gravitational field can be generated by the oscillation of a quark in a paper written by author Eli Peter Manor published in 2016 in the Journal of Modern physics [

In describing the assertion of this paper in more detail; a gravitational field is generated when a cloud of charged particles is accelerated to the precipice of the speed of light. The acceleration enacted on the particles exceed the speed of light per unit time, however massive particles cannot exceed the speed of light as is well known. Resultantly, as the acceleration increasingly exceeds the speed of light per unit time, the particles’ velocities approaches but never achieves a luminous velocity. Mathematically, the particles’ velocities asymptotically approaching the speed of light will not compensate for the amount of force or acceleration exerted on the particles; the inertial mass value of each particle must increase to compensate for the increasing acceleration or force. In this assertion, each charged particles’ velocity is approximated to a constant 99% of the speed of light (

All mass values correspond to a density value; even if the unit volume is infinitesimally small. The sum total of increasing individual inertial mass values of each particle in the density of accelerated particles is set equal to source mass M in the gravitational force

In using electromagnetic force or Lorentz force to accelerate the cloud of charged particles, acceleration

Acceleration

The momentum value

Keep in mind that momentum value

As acceleration

Mass value

To avoid confusion, it must be noted that relativistic mass dilation is different from the variation of variable mass or inertial

This can alternatively be expressed such that [

Equation (1.06) implies that the Lorentz factor

Thus inertial mass

This equivalence can be expressed as:

Equation (1.10) represents an important aspect of the assertion. The charged particles of the accelerated mass cannot exceed the speed of light, thus, variable inertial mass

Equation (1.11) can be expressed such that:

The premise of the assertion is the correlation of the acceleration of a cloud of charged particles and gravitation, therefore Newtonian gravitational force

The density of the accelerated cloud of charged particles is denoted

This implies that:

where M is the source mass in gravitational force

where

Mass value

Therefore, the accelerating force

The gravitational field generated by the cloud of accelerated particles on the verge of the speed of light inherently produces a space-time curve. Therefore the mathematical description of the space-time curve produced by the accelerated charged particles is given by Einstein’s field equation. Consider the function of expression (2.0) below.

The function

The symbol

The equivalence of function

Thus, for every value

where A and B are arbitrary values,

The Einstein tensor is given such that [

[where

, (2.07)

As a second example, function

The Stress-energy tensor is given such that [

[where

, (2.10)

Due to the equivalence of function

Therefore the components of the Einstein tensor and the stress-energy tensor reside in the codomain of function

Section 3 will introduce two formulations linking the number of particles

The Schwarzchild descritption

In reference to this hypothetical description, accelerated charged particles traveling at velocities bordering the speed of light generate a gravitational field on a spherically symmetric body, hence, the need to formulate a description using the Schwarzchild metric. The Schwarzchild radius is given such that [

Gravitational force

Gravitational potential energy

The maximum value of kinetic energy

As is conventionally performed, kinetic energy

This equivalence can be expressed as [

Solving for the Schwarzchild radius

The Schwarzchild metric is given such that [

The functions

[where r is the radius of the spherically symmetric body

Expressing the value of metric tensor

Therefore, the Einstein tensor

where the Ricci tensor

The requirement of the equivalence of

The stress-energy tensor description

A cloud of charged particles are again accelerated via an electromagnetic force (of any given source i.e. particle accelerator or subatomic charged particles emitted from a star) to the verge of the speed of light producing a gravitational field that is exerting on a fluid of particles of mass

where

The fluid 4-velocity denoted

The geodesic rule is acknowledged as shown below [

Hence the appropriate use of the Christoffel symbol

In substituting fluid velocity

Consider dynamic pressure

At this juncture, fluid velocity

Or alternatively,

Let dynamic pressure

This implies that:

Isolating the partial derivatives

Substituting Equation (3.26) into Equation (3.20) (or the stress tensor), one obtains:

Consider the unit vector u in

Gravitational force

Using the classical equation of pressure equal to force per unit area (

The sums of components of pressure

Substituting the value of Equation (3.31) into Equation (3.27) gives the stress energy tensor

The stress energy tensor is set equal to the Einstein tensor

Conclusively, the stress-energy tensor describing the pressure exerted by the gravitational field produced by the accelerated charged particles on a perfect fluid correspond to the surjective map of

It is of great importance to show the possibility and feasibility of accelerating a cloud of charged particles to an extent to where they actually produce a gravitational field in the real world. Thus, the Lorentz equation of electromagnetic force is applied to show this possibility. Lorentz force

The velocity vector

The vector value for the magnetic field is given such that:

The vector value for the electric field is given such that:

Carrying out the cross product of velocity vector

The value of Lorentz force vector

The magnitude of electromagnetic force vector

The magnitude of electromagnetic force

Recall that acceleration

The value of Equation (4.07) then becomes:

Equation (4.09) can be expressed such that:

The task is to obtain the required voltage at a given acceleration number

Recall that velocity

The value of electrical field E is equal to the negative partial derivative of voltage V in respect to length x [

Substituting this value (Equation (4.13)) into Equation (4.12) gives the differential equation shown below.

This can be rearranged such that:

The corresponding integrals in respect to voltage V and length x are expressed such that [

where

Length

Voltage

Equations ((4.18) and (4.19)) show the required voltage

Voltage

where gravitational force

The force associated with the Casimir effect describing vacuum energy was confirmed by an experiment conducted by physicist Steven Lamoreaux in 1996 [

Edward A. Walker, (2016) Gravitational Space-Time Curve Generation via Accelerated Charged Particles. Journal of Modern Physics,07,863-874. doi: 10.4236/jmp.2016.79078