An Alternative to the Alcubierre Theory: Warp Fields by the Gravitation via Accelerated Particles Assertion

A summarization of the Alcubierre metric is given in comparison to a new metric that has been formulated based on the theoretical assertion of a recently published paper entitled “gravitational space-time curve generation via accelerated particles”. The new metric mathematically describes a warp field where particle accelerators can theoretically generate gravitational space-time curves that compress or contract a volume of space-time toward a hypothetical vehicle traveling at a sub-light velocity contingent upon the amount of voltage generated. Einstein’s field equations are derived based on the new metric to show its compatibility to general relativity. The “time slowing” effects of relativistic gravitational time dilation inherent to the gravitational field generated by the particle accelerators is mathematically shown to be counteracted by a gravitational equilibrium point between an arrangement of two equal magnitude particle accelerators. The gravitational equilibrium point produces a volume of flat or linear space-time to which the hypothetical vehicle can traverse the region of contracted space-time without experiencing time slippage. The theoretical warp field possessing these attributes is referred to as the two gravity source warp field which is mathematically described by the new metric.


Introduction
A theoretical warp field and therefore a space-time metric is mathematically formulated based on the theoretical notion of the generation of gravitational space-time curves produced by massive charged particles accelerated to the precipice of the speed of light introduced by the Gravitational space-time curve generation via accelerated particles paper (Walker, 2016). Resultantly, It is asserted that particle accelerators with sufficient voltage values can generate a gravitational field and its corresponding space-time curve (Walker, 2016). The new theoretical warp field based on this assertion is referred to as the two gravity source warp field and is compared to the famous Alcubierre warp field theory.
Thus, the Alcubierre warp field assertion is summarized in comparison to the theoretical assertion of the two gravity source warp field in section 1. The velocity based gravitation by accelerated particles metric (VBGAP metric) which is used to described the two gravity source warp field is introduced and mathematically formulated based on the assertion of the Gravitational space-time curve generation via accelerated particles paper (Walker, 2016). The VBGAP metric describing the two gravity source warp field is formulated in accordance to Einstein's field equations. Therefore, Einstein's field equations are derived based on the VBGAP metric to show that the VBGAP metric is valid to general relativity. An arrangement of two particle accelerators generating two gravitational fields in close proximity to one another is theoretically shown to have the ability to compensate for relativistic gravitational time dilation within the gravitational fields and whose combined gravitational field can warp or contract space-time toward a hypothetical vehicle as it moves spatially with a sub light velocity (hence the two gravity source warp field). Therefore, the issue of gravitational time dilation resulting from the generation of the gravitational fields is mathematically shown to be resolved by an arrangement of two particle accelerators producing a gravitational equilibrium point. The notion of the two gravity source warp field is then mathematically incorporated into the VBGAP metric describing the motion of a hypothetical vehicle. The mathematical incorporation of the two gravity source warp field concept into the VBGAP metric allows the derivation of mathematical equations used to obtain hypothetical quantitative results and allow conclusive thoughts on the study.

The Alcubierre Warp Field vs. the Two Gravity Source Warp Field
In 1994, physicist Miguel Alcubierre formulated a space-time metric which is a unique solution to Einstein's field equations (Anderson, 2016). The Alcubierre metric describes a warp field to which space-time is compressed or contracted towards a hypothetical vehicle and expanded behind it (Anderson, 2016). In general relativity and/or cosmology the universe expands theoretically creating new space with gravity conversely having the ability to contract or compress it, the Alcubierre metric suggest that a similar phenomenon can transpire in a local sense (Cramer, 1996). This gives to the implication that both the expansion and contraction of space-time could be localized to a vehicle. The possibility for faster than light travel (FTL) in the Alcubierre theory comes from the theoretical ability to compress and expand space-time faster than the speed of light, which is permissible by general relativity (Cramer, 1996). Hence, this gives the possibility of contracting and expanding space-time to allow a vehicle within the region of warped space-time to cross distances in what equates to super luminous velocities relatively (Anderson, 2016). The area of warped space-time is referred to as a warp bubble (Anderson, 2016), within this warp bubble, the hypothetical vehicle travels along a world line or a geodesic embedded on a space-time manifold as required by Einstein's field equations of general relativity. A more detailed description of Alcubierre's warp concept is that it is constructed of hyperbolic tangent functions which create a very peculiar distortion of space at the edges of the flat-space volume (Cramer, 1996). The metric corresponding to this description (or the Alcubiere metric) is expressed such that (Anderson, 2016): Where ( ) is a velocity function (Anderson, 2016), the function ( ) is the function of the hyperbolic tangent functions describing the warped geometry of space-time encompassing the hypothetical vehicle (Anderson, 2016). Function ( ) is expressed such that (Anderson, 2016): (1.1) Where and are arbitrary parameters ( > 0, > 0) (Anderson, 2016). An important fact is that the hypothetical vehicle is not actually in motion but is interpreted by the Alcubierre metric to be in a state of "free fall" along a geodesic on the surface of the space-time manifold to which the warp bubble is defined on (Cramer, 1996). Hence, in a local sense, the vehicle is not actually moving at super luminous velocities (Anderson, 2016). An additional and important consideration is that the hypothetical vehicle at the center of the warp bubble experiences no relativistic mass increase or time dilation as specified by the Alcubierre theory (Cramer, 1996). However, there are issues prohibiting the Alcubierre theory from becoming a theoretical possibility. The first issue is that Alcubierre formulates a metric and applies it to Einstein's field equations (namely the Einstein tensor), mathematically the corresponding momentum-stress-energy tensor requires the generation of a form of exotic matter to whom cannot currently be produced with current technology (Anderson, 2016). Moreover, another issue is that, according to Serguei Krasnikov, it would be impossible to generate the bubble without being able to force the exotic matter to move at faster than light speeds locally, which would require the existence of theoretical tachyons (Anderson, 2016). Lastly, general relativity provides a procedure for determining how much energy density (energy per unit volume) is implicit in a given metric (Cramer, 1996), this suggested the requirement of the generation of high amounts of negative energy which is prevalent in vacuum energy (illustrated by the Casmir effect) on a cosmic scale but cannot be produced in vast amounts required to generate the Alcubierre warp bubble (Cramer, 1996).
A paper published entitled "Gravitational space-time curve generation via accelerated particles" (Walker, 2016) introduced a group of equations describing the theoretical generation of a gravitational space-time curve (and thus gravitational force) by particles accelerated to an approximate 99 percent of the speed of light. An acceleration equal to multiples of the speed of light greater than 1 per unit time is enacted on the particles whose velocities are at an approximate 99 percent of the speed of light (which in reality asymptotically approaches but never achieves the speed of light, thus .99 is simply an approximation for calculations). Resultantly, space-time (and therefore gravitational force) treats or reacts to the particles as if they are more massive. Therefore, the force of gravity and hence the curvature of space-time are proportional to an acceleration or force value which is multiples of the speed of light per unit time greater than 1. This implies that particle accelerators can potentially generate gravitational fields and space-time curves. This gives way to the formulation of a space-time metric in accordance to Einstein's field equations based on the theoretical notion of the gravitational space-time curves generated by accelerated particles assertion. Therefore, a space-time metric is formulated and proposed where a system or hypothetical vehicle produces a gravitational field via particle accelerators which can compress or contract space-time toward the vehicle. A difference to the metric proposed in this paper as compared to the Alcubierre metric is that the vehicle must have a velocity (a sub light velocity), essentially as the hypothetical vehicle travels through space-time at an arbitrary velocity, the generated gravitational fields contract or brings space-time closer to the vehicle permitting it to travel distances faster than it would in flat or linear space at the same velocity. In relating this to general relativity, the hypothetical vehicle is shown to travel on along a geodesic defined by the new metric. Due to the fact that a velocity is included to the gravitational space-time metric produced by the accelerated charged particles, the term "velocity based gravitation by accelerated particles" metric (VBGAP metric) is applied to the metric. However, gravitational time dilation is an issue to the system or vehicle as it transits space-time. The vehicle or system will travel slower through time as opposed to flat or linear space counteracting the spatial distance traveled by the vehicle as space-time is contracted toward the vehicle. An arrangement of two particle accelerators generating two theoretical gravitational fields will generate an equilibrium point between them which will be shown to generate an area of flat or linear space-time (see section 4 page 19). This equilibrium point between the two particle accelerators will compensate for the time slippage induced by gravitational time dilation. Resultantly, as the hypothetical vehicle travels through space-time at a sub light velocity, the combined effect of the gravitational fields generated by the two particle accelerators compress space-time toward the vehicle permitting spatial distance to be traversed without time slippage due to gravitational time dilation within the two gravitational fields. The gravitational field curving space-time around the system or vehicle generated by the two particle accelerators is referred to as the "two-gravity source" warp field as stated in the introduction. The VBGAP metric is then redefined in terms of the two-gravity source warp field. As compared to the Alcubierre metric, the generation of the warp field is contingent on the electrical energy(or voltage) powering the particle accelerators as opposed to the excessive amounts of negative energy and exotic matter. The two-gravity source warp field described by the VBGAP metric can be in theory generated using technology that is currently available. However, the issue hindering the practicality of the two-gravity source warp field is that exorbitant amounts of energy have to be expended to generate a warp field sufficient to have a substantial affect as shown by the quantitative examples in the conclusion. Lastly, the Alcubierre Warp field expands space-time in the direction opposite the direction of travel of the hypothetical vehicle where two-gravity source warp field contracts space-time in all directions, this issue is addressed in section 5.

The Velocity Based Gravitation by Accelerated Particles Metric (VBGAP Metric)
Section 2 introduces and formulates the velocity based gravitation by accelerated particles metric or the VBGAP metric. As aforementioned, this warp concept is based on the theoretical notion introduced in the "Gravitational space-time curve generation via accelerated particles" paper (Walker, 2016). Therefore, as presented in the "Gravitational space-time curve generation via accelerated particles" paper (Walker, 2016), a force value is generated on a total mass value (the mass or combined mass of charged particles) by an electromagnetic particle accelerator. Force is stated such that (Young & Freedman, 2004;Walker, 2016): The acceleration value has a value = ⁄ in Newton's equation of force (F=ma) above (Young & Freedman, 2004;Walker, 2016). The speed of light is denoted and variable (and its conditions expressed above) is referred to as the particle acceleration number. Thus momentum value is the integral of force value with respect to time as shown below (Young & Freedman, 2004;Walker, 2016).
Note, the particles of mass do not achieve super luminous velocities as dictated by relativity (Young & Freedman, 2004). A momentum value is given such that (Walker, 2016): = (.99)( ) (2.02) The particles' actual velocity is stated to be an approximated 99 percent of the speed of light (. 99 < ) as shown above. Mass value is stated to be the "variable inertial mass" (Walker, 2016) which will be referred to as variable mass in short further on. Momentum value is set equal to momentum value as shown below (Walker, 2016 (2.05) As derived in the "Gravitational space-time curve generation via accelerated particles" paper (Walker, 2016), this equation (2.05) can be stated as the "Variable mass increase equation". The actual interpretation of the variable mass equation that was not directly elucidated in the "Gravitational space-time curve generation via accelerated particles" paper is that variable mass is not a literal increase in inertial mass rather it is a description of how space-time treats or reacts to mass when it is accelerated to the verge of the speed of light having an acceleration value that is multiples greater than the speed of light acting on it (acceleration of equation 2.0). Thus as acceleration number increases to multiples greater than the speed of light , force value (and therefore acceleration of equation 2.0) and momentum value increase (Walker, 2016). However, the value of the particles' velocity remains below the speed of light as indicated by the equivalence to momentum value ( = (.99)( ) )(equation 2.03), resultantly variable inertial mass must mathematically increase as acceleration number increases (Walker, 2016).
To avoid confusion (as stated in the "Gravitational space-time curve generation via accelerated particles" paper (Walker, 2016), it must be noted that relativistic mass dilation is different from the variation of variable mass or inertial as the particle approaches the speed of light. This can be conveyed by setting variable inertial mass equal to the product of the variable inertial mass and the Lorentz factor as shown below (Young & Freedman, 2004;Walker, 2016).

=
(2.06) This can alternatively be expressed such that (Young & Freedman, 2004;Walker, 2016): .07 implies that the Lorentz factor is equal to 1 and implies that relative velocity or the velocity of an observer is zero ( = 0) for the purpose of this derivation (Young & Freedman, 2004;Walker, 2016).
Conclusively, inertial mass does not vary according to the Lorentz factor (Walker, 2016). In continuing the derivation of the VBGAP metric, a gravitational field between variable mass and a test particle of mass separated by distance is expressed such that (Young & Freedman, 2004;Walker, 2016): This can alternatively be expressed such that (Walker, 2016): Gravitational force varies according to variable mass which varies according to acceleration number (Walker, 2016). In the interest of applying this mathematical concept to the application of real world particle accelerators, the voltage required to accelerate the particles to produce a gravitational field has been formulated using the Lorentz force equation in the "Gravitational space-time curve generation via accelerated particles" paper (Walker, 2016) as shown below (see appendix A for the formulation of this equation) (Walker, 2016).
Where is the acceleration number, is the mass of the particles being accelerated, is the speed of light, is the charge of the particle(s), is the value of the associated magnetic field, and ∆ is the interval of time (Walker, 2016). apr.ccsenet.org Applied Physics Research Vol. 8, No. 5; Gravitational potential energy is given by the integral of gravitational force of equation 2.10 in respect to distance (Young & Freedman, 2004;Walker, 2016).
Distance is substituted for the Schwarzchild radius ( = ), therefore equation 2.12 can be re-expressed such that (Young & Freedman, 2004;Walker, 2016): Potential energy is set equal to the maximum value of kinetic energy of mass at the speed of light denoted (Young & Freedman, 2004;Walker, 2016).
This equivalence can be stated such that (Young & Freedman, 2004;Walker, 2016): Solving equation 2.15 for variable gives the value of the Schwarzchild radius such that (Wald, 1984;Young & Freedman, 2004;Walker, 2016): It is of great importance in formulating the VBGAP metric to state that the entire system being described has a sub-light velocity value of ( < Therefore, the compression factor denoted ( ) has a value and condition such that (Wald, 1984): ; 1 > ( ) (2.20) Compression factor ( ) is the compression of the geometry of space due to gravitational force produced by the particle accelerator. Therefore, it can be stated that the compression factor ( ) contracts space-time. A priori is that the compression or contraction of space-time is measured relative to radius . A displacement ц is expressed such that: (2.21) Thus displacement ц is defined at system velocity at a variable of time . It is important to cite that the variable of time is different from the constant value of time of equation 2.17. Time value varies while time value is a constant parameter set at a value of one second ( = 1 ). Hence, in regards to the operations of differentiation and integration in respect to time value , time value is treated as a constant. Time is simply a temporal parameter of measurement for the volume of space-time being influenced by the gravitational force of the particle accelerator. Displacement ц is the product between the displacement ц and the inverse of compression factor ( ) and its condition expressed below.
This can alternatively be expressed as: The most obvious question is "why is the displacement ц multiplied to the inverse of compression factor ( ) (expressed as ( ( )) )?". The displacement ц of equation 2.23 measures the movement of velocity at time relative to a stationary observer outside of the generated gravitational field which is also beyond the effects (in flat space) of the spatial compression of compression factor ( ). Therefore, the Lorentz factor for the measurement of displacement ц ( ц ) will be equal to 1 ( = 1) for the external stationary observer for having a relative velocity of zero( = 0) (Young & Freedman, 2004). The external stationary observer would perceive the system to be traveling a greater distance over time as the system transits over compressed space-time and thus spatial distance. Mathematically speaking, as measured by the external stationary observer, the displacement ц will be greater than displacement ц ( ц > ц ) in a time value of explaining the inverse coefficient of ( ( )) of the compression factor. An important fact to point out is that as the warped space caused by the gravitational fields generated by the particle accelerator passes through a region space, space flattens or reverts back to its linear form once the system has passed.
The basis of measurement in a four dimensional Minkowski space is defined by which is a collection of four 4-space basis vectors (Wald, 1984).
The four 4-space basis vectors of constitute the axis of measurement to an arbitrary external stationary observer geometrically measuring the system which also describes the coordinate axis used by the stationary external observer. Thus, the basis vectors are represented by a 4 by 4 matrix of equation 2.24. The expression of displacement ц in the new coordinate basis is the product of matrix and displacement ц as shown below.
Where column indices of matrix is set equal to index ц ( = ц), and where indices ц takes on values of {0,1,2,3} of Minkowski space (Penrose, 2004). As stated in section 1, the motion of the system is defined on a geodesic, this fact requires that the displacement ц ц as measured by an external stationary observer lie on the tangent plane defined on a differential space-time manifold (as dictated by General relativity) (Penrose, 2004). Moreover, in defining a metric in terms of general relativity or Einstein's field equations, the displacement ц ц must be associated with the curved surface of a space-time manifold. The field function ( ц ( )) is a smooth and continuous function of the Minkowski coordinates of ц = ц ( ), ( , , , ) describing a differential space-time manifold (Penrose, 2004), where ц is re-expressed such that ц = ц ( ) = ц . The displacement ц ц is stated to be a displacement on the tangent plane of the differential manifold of field function ( ц ( )) (Penrose, 2004), thus the displacement ц ц is expressed as a tangent vector such that: A very important mathematical clarification to equation 2.26 is that the tangent vector is the partial derivative ( ц ) ц ⁄ which has a value at Minkowski coordinate ц which takes on a value ц for the three spatial coordinates ( , , ). Therefore, in reference to the partial derivative ( ц ) ц ⁄ (or tangent vector), there is no need for the use of the chain rule ( ( ( ))( ( ))) expressing the variation in ц which vary in respect to parameter time (velocity is a constant ). The time coordinate is the exception as is shown in equation 2.29.
In giving an algebraic description of the displacement on the tangent plane, consider smooth functions ( ц ( )) and ( ц ( )) at Minkowski coordinates ц ( ) which describe a curved surface of a smooth space-time manifold. The tangent vectors to field functions ( ц ( )) and ( ц ( )) are denoted ( ц ( )) and ( ц ( )) . The tangent vector function ( ц ( )) relates to tangent vector function ( ц ( )) and compression factor ( ) such that: Research Vol. 8, No. 5; Equation 2.27 above conveys the form of equation 2.26. This implies that tangent vector function ( ц ( )) has a value such that ( ц ( )) = ( ) ( ц ( )) .The tangent vector function of ( ц ( )) represents variations on the space-time curve of field function ( ц ( )) compressed by compression factor ( ). However and as previously stated, the motion of the system traveling at velocity (and therefore displacement ц ц ) is measured by the amount of compressed spatial distance crossed as compared to an equal amount of uncompressed distance crossed as if it were in flat or linear space (as measured by a stationary observer outside of the generated gravitational field). Hence showing the use of the inverse compression factor ( ( )) as measured on the tangent plane.
At this juncture, the displacement ц ц has been sufficiently related to the tangent plane embedded on the surface of a differential manifold of field function ( ц ( )). Tangent vector ( ц ( )) ц ⁄ is an element of and represents an instantaneous rate of change in time and distance on the tangent plane (Wald, 1984). The scalar quantity of the differential element of displacement in time is denoted ( ∈ ), this is multiplied to the tangent vector such that (Wald, 1984) : (2.28) Therefore, equation 2.28 is a vector quantity in giving the distance on the tangent plane in respect to time . It is pertinent to recognize that the tangent vector component at the coordinate (or ( )) in (the 4-space Minkowski coordinates) takes on a value such that (Penrose, 2004): Where the chain rule is incorporated into equation 2.29, this implies that the basis vector at has a value of one ( = 1) for component ( ). Complex time component ( = ( )) is the product of the speed of light and time ( = ) in accordance to the Minkowski coordinates (Penrose, 2004) and the speed of light is set to unity ( = 1). Equation 2.29 is the tangent vector to the field ( ц ( )) (Penrose, 2004) at component ( ( ( ))). The tangent vectors for field function ( ц ( )) at components ( )-( ) (or the space-like components) are of the form of displacement ц ц of equation 2.28. Observe the symmetric tangent vector components below. (2.32) The Kronecker delta is expressed such that (Wald, 1984): Where the product of symmetric superposition terms is expressed such that: Applying the Kronecker delta to the product of 2.34 above gives (Wald, 1984):  Vol. 8, No. 5; One obtains the Pythagorean relation on the tangent plane to the space-time manifold which is a metric denoted . Metric is expressed such that (Wald, 1984): The component of metric at the time coordinate has a value of − ц ( = ) in accordance to equation 2.29, thus metric can be expressed such that: Recall that the basis vector at has a value of one ( = 1) and the speed of light is set to unity ( = 1). Substituting the value of equation 2.25 into equation 2.37, one obtains (Wald, 1984): The metric of equation 2.38 shown above is the "Velocity Based Gravitation by Accelerated Particles metric" (VBGAP) due to the fact that the displacement through space-time is measured at the system velocity (where velocity is less than the speed of light < ) and the compression or contraction of space-time ( ) caused by the gravitational fields generated by the Particle accelerators. Equation 2.37 can be stated as the alternative to the Alcubierre metric. A priori is that the VBGAP metric can be related to the length of a curve over the surface of field function ( ц ( )) for initial and final time values and as shown below.
The simplest form of the smooth and differentiable field function ( ц ( )) describing the gravitational space-time curve surface generated by the particle accelerator is expressed such that: Where the coordinate ц is of the form ц ( ц = ц ( ) = ц ), function ц ( ) Lastly, the spatial velocity as measured by a stationary observer outside of the volume of curved space corresponding to the gravitational field generated by the particle accelerators is given such that: Where velocity is greater than or equal to system velocity ( ≥ ). An important note is that gravitational time dilation will hinder the calculations of equations 2.38 and 2.41, the solution to this issue is the arrangement of two gravitational fields of equal magnitude producing an area of flat space-time between the two fields as will be shown in section 4.

The Incorporation of the VBGAP Metric to Einstein's Field Equations
Both the Alcubierre metric and the VBGAP metric describe warped space-time in terms of General relativity, for instance, the hypothetical vehicles described in both theoretical descriptions ride geodesic world lines on a space-time manifold. Thus, both theoretical descriptions incorporate aspects of Einstein's field equations which require that both theoretical descriptions be compatible to the field equations of General relativity. The most prudent approach to showing that the VBGAP metric is compatible to General relativity and hence Einstein's field equations is to derive the entire equation beginning with and based on the VBGAP metric. Hence, we now begin the heuristic derivation of Einstein's field equations with the VBGAP metric shown below. The VBGAP metric can alternatively be re-expressed as the product (or dot product) of vector valued symmetric differential terms such that: This can be re-arranged such that: The VBGAP metric is set equal to the metric tensor ( = ), this can be expressed such that (Wald, 1984): We must acknowledge the fact that when a coordinate is parameterized by time , many authors denote the metric tensor as , however to avoid a cluttered appearance in the mathematical exposition of the content, we simply use (Wald, 1984). The geodesic rule is given such that = 0 (Wald, 1984). Therefore, we look to obtain the straightest possible curve on the field function ( ц ( )) of the space-time manifold, the derivative is given such that (Wald, 1984): The chain rule must be applied due to the fact that the value of is parameterized by time ( = ( ) = ц ≡ ц ). The derivative is applied to the value of equation 3.04, resultantly one obtains equation 3.06 below (Wald, 1984). Thus applying the product rule or leibnitz rule to equation 3.10 above gives (Wald, 1984): This (equation 3.11) is equivalently expressed in short hand notation such that (Wald, 1984): = + ≡ 0 (3.12) Keep in mind that the terms are all symmetric ( = = ) (Wald, 1984). Equation 3.11 can be algebraically arranged such that (Wald, 1984 (Wald, 1984). An equivalence to equation 3.13 is given such that (Wald, 1984): (3.14) Observe the term which has a value = in equation 3.14 above (Wald, 1984). Applying the product rule or liebnitz rule to term , the term expand such that (Wald, 1984): ( 3 . 1 5 ) This can be expressed such that (Wald, 1984): = + (3.16) Substituting this into the original equation (3.14) gives (Wald, 1984): 17) The symmetric property is again applied ( = = ), therefore the left side of equation 3.17 above can be reduced such that (Wald, 1984): 18) This implies that (Wald, 1984): The inverse tensor (such that • = , where is the 4 by 4 identity matrix) is applied to the equation above such that (Wald, 1984): The values of the metric tensors are symmetric ( = = = ⋯ ..) permiting the distribution of the inverse tensor over equation 3.20 (Wald, 1984). Implementing the distributive property, one obtains an expression in terms of partial derivatives, where is the derivative of the straightest possible world line on the curved surface of the space-time manifold (Wald, 1984).
Setting this product equal to the Christoffel symbol gives (Wald, 1984) : = ≡ (3.22) Therefore this can be expressed such that (Wald, 1984): Where = = = are symmetric partial derivatives, subtracting the value of 2 from the derivative of permits the expression to equal zero as shown below (Wald, 1984).

25) Hence one obtains:
= − 2 ≡ 0 (3.26) Thus we have derived the geodesic equation (Wald, 1984). To verify the values of the VBGAP metric to the geodesic equation, simply substitute the VBGAP metric values of partial derivatives of equations 3.08 and 3.09 into equation 3.26. The values of equations 3.08 and 3.09 will satisfy equation 3.26 above. The values of partial derivatives and are of the form of the geodesic equation as shown below (Wald, 1984).
= − 2 ≡ 0 = − 2 ≡ 0 (3.27) As stated by Wald (Wald, 1984), the commutator for computing curvature is composed of derivatives and as shown below. apr.ccsenet.org Applied Physics Research Vol. 8, No. 5; [ , ] = − ≡ ( − 2 )( − 2 ) − ( − 2 )( − 2 ) (3.28) Therefore as stated by Wald (Wald, 1984), the Ricci tensor is the product of the commutator for computing curvature [ , ]and the metric tensor given such that: (3.29) Thus the Einstein tensor is expressed such that (Wald, 1984): Where is the scalar curvature computed by the coordinate component method or the orthonormal basis (Tetrad) method (Wald, 1984). Equation 3.31 can alternatively be expressed such that: Substituting the value of VBGAP metric (where = ) into the Einstein tensor or equation 3.31 gives the expression (Wald, 1984): Conclusively, the Einstein tensor can be expressed in terms of the values of the VBGAP metric such that: Setting the Einstein tensor equal to the stress energy tensor 8 ( = 8 ) gives (Wald, 1984): The formulation of the stress-energy tensor 8 in terms of the VBGAP metric is left as an exercise. Equations 3.31, 3.32, and 3.33 accomplish the goal of showing that the VBGAP metric has been formulated in accordance to Einstein's field equations

The Solution to the Gravitational Time Dilation Problem with Generated Gravitation Fields
As dictated and proven by general relativity, the gravitational fields generated by the particle accelerators will cause time to "slowdown" as compared to flat space located beyond the effects of the gravitational field. Resultantly, as the system moves spatially with a velocity over a proper time , time will be dilated by the gravitational field generated by the particle accelerators. The system will be traveling at a slower temporal or time-like rate as compared to linear or flat space which will diminish the system's space-like motion which is spatial displacement ц ( ц = ). Therefore, despite the system's compression of space-time in the direction of displacement ц , an object traveling in the same direction and velocity ( )in the same inertial frame but outside of (or beyond) the effects of the generated gravitational fields in flat or linear space will be traveling relatively faster. Section 4 gives the solution to the gravitational time dilation issue with presenting the notion of an equilibrium point between two gravitational fields produced by two particles accelerators that will compensate for the time slippage inherently caused by gravity.
Consider two gravitational force fields and (in the form of equation 2.10) produced by two particle accelerators. The acceleration numbers and correspond to gravitational forces and . Gravitational forces and at distance values and take on values and conditions such that: Where is the mass of a test particle, and masses and (where = ) are the mass values of the accelerated particles in each particle accelerator, the Gravitational forces and are equal in magnitude ( = ), this equivalence is expressed such that: The gravitational forces and have an equilibrium point between them at spatial coordinated , where the forces are equal. Mass value is located at equilibrium coordinate . Equilibrium coordinate is a distance from gravitational force (or the particle accelerator). Hence, equilibrium coordinate is a distance from gravitational force (or the second particle accelerator). Force value ( , ) is a function of distance values and and is the difference between force values and . Thus the combined force ( , ) is the total value of gravitational force acting or pulling on mass at the equilibrium coordinate located between both particle accelerators which implies that force value ( , ) is equal to zero at equilibrium coordinate . Force value ( , ) is then expressed such that: ( , ) = − = 0 (4.03) This can be expressed such that: Gravitational potential energy is the integral of gravitational force in respect to radius such that (Young & Freedman, 2004):

=
(4.05) Therefore, potential energy ( , ) is the integral of gravitational force ( , ) in respect to radius such that: is the initial point at the first accelerator and equilibrium point is the location of mass pertaining to distance . Coordinate is the initial point at the second accelerator and equilibrium point is the location of mass pertaining to distance . With aforementioned values being the limits of integration, potential energy ( , ) is expressed as the difference between two integrals in respect to and such that: Potential energy ( , ) takes on a value of zero ( ( , ) = 0 ). Potential energy ( , ) is the value of energy at the equilibrium coordinate between the two particle accelerators. A free falling particle of mass with an initial velocity of zero located at the equilibrium coordinate between gravitational forces and would have no work done on it and hence the resultant velocity induced by the two gravitational fields would be zero. Gravitational time dilation is expressed such that (Young & Freedman, 2004;Bench, 2016): Time is dilated time and time is the proper time for the purposes of the explanation (Young & Freedman, 2004). Kinetic energy is the energy of a particle of mass at the velocity which is the escape velocity required for the gravitational fields as shown below (Young & Freedman, 2004).  Vol. 8, No. 5; Therefore, = (4.12) Conclusively, at the equilibrium coordinate between the two gravitational fields and , the time value is the same (at proper time ) for an observer in flat or linear space. Every position ц within the two gravitational fields and that is not at equilibrium coordinate ( ≠ ц ) will experience time dilation or temporal slippage ( < ) produced by the gravitational fields.

The Two Gravity-Source Warp Field
This section gives the mathematical description of a hypothetical vehicle traveling at a sub-light velocity (velocity ) hoisting two particle accelerators that generate two gravitational fields of equal magnitude which produce an area of flat space-time at an equilibrium point (coordinate ) in between them as shown in the previous section. Hence an area of flat space-time is produced within the structure of the vehicle at equilibrium coordinate . As the vehicle transits space-time, the combined gravitational field produced by both particle accelerators compress space-time (and therefore spatial distance) toward the equilibrium coordinate in the vehicle's direction of travel. Equilibrium coordinate is the center of motion for the hypothetical vehicle. Therefore VBGAP metric is defined in terms of two gravitational fields whose space-time compression is measured from the equilibrium coordinate . This warping of space-time around the hypothetical vehicle generated by two particle accelerators of equal magnitude is referred to as the two gravity source warp field.  .The vehicle velocity and the compression or contraction of space-time are measured from the equilibrium coordinate which is also located at the center of mass due to the gravitational fields and having equal magnitudes ( = ). Center of mass between the two gravitational fields is presented such that (Young & Freedman, 2004): The distances and (this implies that = ) are the distances from the equilibrium coordinate between the two gravitational fields and of equal magnitude. The two gravitational force fields and are in close proximity to one another spatially. Therefore the two masses and combine to gravitationally attract a test particle of mass value which is located at a distance from equilibrium coordinate in the direction of vehicle velocity which is the vehicles direction of travel . The magnitude of the combined gravitational force of gravitational force fields and are measured from the equilibrium coordinate (or center of mass ) which correspond to the reduced mass equation. Observe the reduced mass equation (Young & Freedman, 2004): To avoid incorrect use of the reduced mass equation, we must point out the fact that the reduced mass in reference to gravitational interactions between masses and are not reduced. Therefore, in order to describe a gravitational interaction between masses and (or ) incorporating reduced mass, the reduced mass must be expressed such that (Nipoti, 2013): However, the gravitational force described is not the gravitational force between variable masses and . As previously stated, the combined gravitational force corresponding to the two variable masses and act on a test particle of mass value as measured from the equilibrium coordinate (or center of mass ). Hence, the combined gravitational influence of variable masses and on mass as measured from the equilibrium coordinate (or center of mass ) can be described using the reduced mass. Keep in mind that test particle mass is a distance from the equilibrium coordinate in the direction of travel. Thus, masses and in the reduced mass equation ( The combined gravitational field enacted on mass by mass values and at a distance as measured from the equilibrium coordinate (or center of mass ) between gravitational fields and of equal magnitude is expressed by substituting reduced mass for variable mass ( → ) into the gravitational force equation of equation 2.09 shown below (Young & Freedman, 2004). The corresponding value of gravitational potential energy is given such that (Young & Freedman, 2004): Where the distance from the equilibrium coordinate (or center of mass )of radius is set equal to the Schwarzchild radius ( = ), potential energy is expressed such that: In repeating the process of equations 2.14-2.16, the corresponding value of the Schwarzchild radius incorporating two-source gravity fields generated by the particle accelerators as measured from the equilibrium coordinate is expressed such that: Thus the distance and therefore Schwarzchild radius are measured from the center of mass (equilibrium coordinate ) between variable masses and corresponding to gravitational fields and generated by the two particle accelerators. The compression factor denoted ( ) for a two-source gravitational field has a value and condition as measured from the equilibrium coordinate such that:  Research Vol. 8, No. 5; Compression factor ( ) is the compression of the geometry of space-time due to gravitational force fields and produced by the two particle accelerators as measured from the equilibrium coordinate . Thus substituting the value of the two-source gravitational field compression factor ( ) into equation 2.22 shown below gives, ц = ц ( ( )) = ( ) ; ц > ц (5.17) And thus the displacement in terms of the two gravity source field is such that: Where the tangent vector component of metric at the time coordinate has a differential element of − ( ) (equation 2.29), the step by step process of section 2 is again applied accordingly. Applying equation 5.18 to equation 2.28 gives: Applying equation 5.19 to equation 2.38 gives the VBGAP metric in terms a two gravity source field as measured from the equilibrium coordinate which is the vehicle's center of motion.
More specifically, equation 5.18 describes the warping of space-time around the two particle accelerators generating gravitational forces and at a sub-light velocity as measured from the equilibrium coordinate (or center of mass ) in the direction of travel. Therefore equation 5.18 describes the two gravity source warp field. In reference to Einstein's field equations, the motion at equilibrium coordinate is the movement along a geodesic to the curved surface of field function ( ц ( )). The spatial velocity as measured by a stationary observer outside of the curved space of the gravitational (or warp) fields generated by the two particle accelerators is given such that: The voltage values and (in the form of equation 2.11) required by the particle accelerators to accelerate particle mass values and to the verge of the speed of light thereby generating gravitational forces and and hence the two gravity source warp field is given such that: , therefore the total required to generate the warp field is expressed such that: = + (5.24) Lastly, as shown in the previous section, proper time for a particle of mass is only experienced at the equilibrium point between the two gravitational forces and generated by the particle accelerators, hence where gravitational time dilation is zero as elucidated below. However, the gravitational potential energy ( , ) acting on the particle of mass at position (such that: ≠ ) within the generated gravitational fields and will be equal to the sum (or net value) of integrals in respect to distances and as shown below (Young & Freedman, 2004). Therefore, the limits of integration for both integrals of equations 5.26 and 5.27 range from initial points and at each particle accelerators to arbitrary a point within the warp field. Conclusively, a particle of mass within the warp field at any position that is not at the equilibrium point ( ≠ ) between the two gravitational forces and are described by the inequality below. > (5.29) An important consideration is that the two gravity source warp field also compresses space-time toward the vehicle in a direction opposite the direction of travel (which is opposite the direction of the velocity ) which would seem to compensate for the distance traveled through compressed space-time. The vehicle's velocity is measured from the equilibrium point between the particle accelerators generating gravitational forces and , therefore the compression of space-time toward the vehicle in the direction opposite of travel will experience a slower rate of time or the time dilation of inequalities 5.28 and 5.29. Conclusively, the vehicle at equilibrium point between the particle accelerators generating gravitational forces and will travel faster through time (at a time-like displacement) in the direction of velocity as compared to the compressed space-time behind the vehicle "squeezing" space-time towards the vehicle in the direction opposite of travel in a space-like sense. As stated, space-time will return to its flat form once the warp field passes through the region of space-time.

Conclusion
The quantitative description of a hypothetical vehicle incorporating two particle accelerators which generate a two gravity source warp field while transiting through space-time is obtained by inserting values into key equations describing the warp field. Table 1 below gives the value of the gravitational constant , the the speed of light , the acceleration numbers and corresponding to both accelerators, the total mass values and of the charged particles (or electrons) in each particle accelerator, the value of the magnetic field in each accelerator, the sub-light velocity of the vehicle , the charge of an individual electron , the interval of time ∆ to which motion is measured, and the constant parameter of time . To expediently obtain a realistic value of a magnetic field for calculations, the value of magnetic field in table 1 is obtained using the magnetic force equation for accelerating an electron of charge as stated by Young and Freedman (2004)  (or 9.109×10 ) (Young & Freedman, 2004), however mass values and are massive clouds of electrons in two hypothetical giant particle accelerators. Hence, the forces of and relate to pressure exerted on the cloud of particles and cross sectional chamber area of the particle accelerators such that (Young & Freedman, 2004): Therefore electromagnetic force is acting on a massive density of electrons (up to 600g in this example).
The values of table 1 are now applied to equations 5.14, 5.16, and 5.21 giving values of : shows that the warping of space-time of the two gravity source warp field produce an increased relative velocity which is equal to 55,937.4 / . Therefore, the increase in velocity from system velocity to relative | | = | + | = 4.4939×10 (6.7) One can conclude that the voltage required to utilize the two gravity source warp field for practical use is obscenely substantial and impractical to real world application as with the Alcubierre warp field (let along faster than light travel). The significance of the results of this paper is the elucidation of the theoretical possibility of generating a warp field of any magnitude with the use of available technology. Moreover, although the possibility of faster than light travel is seemingly distant, the results of this paper will hopefully represent continued research and experimentation to warp fields, which will inspire future advancements in the field. The task is to obtain the required voltage at a given acceleration number , this will require one to solve equation A.10 for electric field as shown below (Walker, 2016).
Recall that velocity is the particles' approximate velocity at 99% of the speed of light (Walker, 2016). Thus, velocity is simply the product of the speed of light and the value .99 ( = (.99× )) (Walker, 2016). The speed of light can then be distributed out of equation A.11, giving the value of equation A.12 such that (Walker, 2016 The value of electrical field is equal to the negative partial derivative of voltage in respect to length (Young & Freedman, 2004;Walker, 2016). Voltage is the product of electrical current and resistance ( = ) (Young & Freedman, 2004;Walker, 2016 sufficient to produce an acceleration that will generate gravity can be shown to exist in the real world with the condition of the inequality below (Walker, 2016).

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