1        ABSTRACT

The purpose of this paper is to develop the kinematics for a non-local medium.  The long term goal of this paper is to use this as a basis for a non-local medium theory with application for a unified field theory. 

With the advent of EPR, Bell’s inequality theorem, and experimental evidence, any strict causal theory for a unified field theory must be non-local in nature.  The non-local nature of this medium theory gives it a foundation which has all the elements needed for a true causal unified field theory. 

2        Overview of a Non-Local Medium

A non-local medium is one that can have non-local flow.  The flow of the medium may go from one point in space directly to a distant point in space without traversing any path connecting the two.

The first paper in this series at http://www.nmtuft.com/nmtconcept/ gives an introduction into what a non-local medium is, why it has good possibilities for a unified field theory (UFT), and how it concept it might function in ways required by a UFT.

2.1       Sensible Non-Local Medium

A goal of this paper is to give a mathematical sense to the non-local flow. The local continuity laws will give some insight as to how the local three dimensions are so observed in the macro level in the universe while yet containing a non-local essence to its ontology.

3        Non-Local Medium Kinematics Terms

Kinematics is the study and description of all possible motions for a material.  Dynamics is the study of laws that describe how things interact and determines which of all the possible motions will be followed.  Kinematics is therefore the starting point for a description of the physics of motion of systems. 

3.1       Absolute Time - Needed by a Non-local Causal Theory

In a non-local medium every point can have an influence on every other point. There is instantaneous contact between every point in space.  A single time for the entire universe allows for a simple interaction perspective.  The first variable for this theory then is an absolute time, t, applicable to all points.  The state of the universe may be given by knowing the value of all kinematics terms at a single instance of time.

·       t – one absolute time for all space

3.2       Fundamental Local Kinematics Terms

The medium proposed is a compressible one perhaps like a fluidic or a gaseous medium.  A compressible medium will typically have the following terms:

·       S – a multidimensional spatial coordinate

·       t - a time coordinate

·       (S,t) a density scalar

·       (S,t) a local velocity vector

The multidimensional spatial coordinate, S, locates a point in space.  The simplest case of interest for a unified field theory is the three dimensional medium.  The density scalar, ρ(S,t) or simply ρ, is the density of the medium at point S and time coordinate t.  The local velocity (S,t) describes the local motion of the medium at point S and time t.

3.2.1     Additional Local Kinematics Terms

For reasons indicated by the conceptual investigation this medium with have a two dimensional density

·      

 Here the terms  and  are used to indicate to different dimensional terms.

3.3       Non-Local Kinematics Terms

The previous terms are all somewhat familiar to local flow medium theories. Non-local density flow is the main new concept being introduced by this paper.  This flow introduces some new terms not found in local flow mediums.

Experiments involving Bell’s Theorem have shown that any strict causal theory must have a non-local nature to it.  Therefore there must be a non-local nature to any strict causal theory.

It is instructive to ask a dynamics question at this point: “Why would material flow from point A to a distant point B without the traversal of any point in between?”  Another way to ask this question would be to ask: “Why if there is no non-local motion initially would there be a change in the non-local motion?”

For the case of the non-local medium, different points at separated points in space must have the ability for a causal relation.  There must be a causal relation between the two points that will allow for and give mathematical form to how medium at one point would flow into another point.

3.3.1     The Interconnection Bond (I-bond)

This non-local medium theory uses the concept of an interconnection bond (I-bond) to create the causal relation between two points in space and will be what allows for flow from one point to the other.  This is the term that will give mathematical sensibility to non-local flow.

The relationship between point A and point B is different from the relationship between point A and point C.  The kinematics terms must allow for a variable casual relationship between points in space over all space.

The strength of the I-bond indicates the strength of the connection. A stronger the I-bond will have a stronger relation.  Medium at one point will have a stronger force for flowing into the connected point. If the I-bond strength is zero between two points then there is no connection between the points.  If the strength is infinity, then the points will be inextricably connected.

It is proposed that the I-bond has a type of substantial existence.  It is a relation that exists and changes in response to the medium’s state.  As such it is considered to be an additional property of the non-local medium.  One way to view this is as a separate but interactive substance.  This substance principle will give it properties that allow for a mathematical description of its motions and interactions with the medium.

3.3.2     Non-Local Kinematics I-Bond Term

The term used for the I-bond strength is given by I(S1,S2,t) where I is the I-bond strength between the two points S1 and S2 at time t.

·       I(S1,S2,t) – Interconnection bond strength

For this medium there is a single interconnection bond which will apply uniformly to both medium densities.

3.3.3     Non-Local Flow Variable

Non-local flow is medium flow per unit time and describes the flow of the medium between distant points.  Some equivalent terms to designate this are:

·       f(Sin,Sout,t)

·        f(S1,S2,t)

·       fin®out

·       f1®2. 

Here Sin or S1 is the sink location from which the medium is flowing while Sout or S2 is the source location into which it is flowing.  The phrases “from which” and “into which” are only intended to indicate the direction of positive value flow.  A negative flow will be medium flow in the reverse direction.

There will be one flow term for each of the different densities.

3.3.4     Interconnection Bond Motion

The interconnection bond may have motions in the end points.  The motion will be described by the motions at each end point of the I-bond.  There must be a term for the velocity at one end point and another term for the velocity at the other end point.  Some equivalent terms to designate this are:

·       I(S1,S2,t)

·       I(Svel pt, Sconnection  pt, t)

The velocity term I(S1,S2,t) describes the velocity of the I-bond I(S1,S2,t) at point S1.  The term in the form of I(Svel pt, Sconnection  pt, t). is more descriptively subtitled to indicate the end point at which the velocity is being described.

3.4       Pressure Term

For a compressible medium pressure is an important term.  The term used for pressure is:

·       P(S,t)

This medium has two dimensions of density.  There will therefore be two dimensions of pressure, one for each of the density dimensions.

·       P1(S,t) – pressure in dimension 1

·       P2(S,t) - pressure in dimension 2

·      

These terms are related to the dynamics of the medium.  Any relationship between the two pressure dimensions is worked out in the dynamics of the medium.

3.5       Metric of Space

The metric of space will used to relate density of the medium to pressure.  The term for the metric of space is:

·       M(S).

For this medium there will be a single dimension to the metric of space. The one term will be used for determination of pressure from the density values in both dimensions.

·       P(S,t) = Function ( M(S,t), 1(S,t), 2(S,t) )

This indicates that the multidimensional pressure will be a function of the metric and the two densities.

Space may simply be a void or non-existent.  The medium may be something that “sits on nothing” and is the only thing that exists for relation.  In this case where space is void then M(S) would have the property of not having a direction and being the same everywhere. This would mean that the relation between pressure and density would be constant in the universe. 

However, this metric may have a connection to the gravitational field. It will be assumed initially that this metric is not constant.  That it may change in value.  The further development of is for the dynamics investigation.

4        Kinematics Equations

The kinematics equations use the previously described variables and bring out some other properties for this non-local medium theory.

4.1       Conservation of Medium

Conservation of the quantity of the medium will be considered a fundamental law. The sum total of the medium will remain constant.  Using C as a constant this law can be expressed as:

                                                                                                                           (1)

4.1.1     Conservation in Two Dimensional Medium

For purposes of this theory, the two different medium densities may have some type of influence on each other, but they will not flow into each other. Therefore where the term  is used the will be two equations, one for 1 and for 2.  This concept will be applied and used in the following equations also.

4.1.2     Conservation of Medium with Non-Local Flow

The quantity of medium flowing from a point S1 into a second point S2 must be the negative of the amount of medium flowing from S2 into S1.  This is expressed by:

                                                                                           (2)

This equation intuitively yields the following conservation integral equation:

                                                                                             (3)

This states that the sum total of non-local flow in the medium is zero.  Basically, this means that there are no overall sources or sinks.  For any source in one area there must be a corresponding sink in another area.

There are two sets of questions here, one for 1 and one for 2.

4.2       Euler's Equation of Continuity Modified

The conservation of medium brings about Euler's equation of motion for a compressible medium.  This is called the equation of continuity.  The equation of continuity for a local medium with conservation of mass is:

                                                                                          (4)

The divergence term here gives the change in density over time.  It is a general mathematical principle that the divergence of the product of a quantity and its velocity will yield the amount of change in a quantity over time.

For the case of a non-local medium we must add a source/sink term Q shown as follows:

                                                                                 (5)

Future more the quantity Q is directly related to the sum of all non-local flow:

                                                                                             (6)

This integral indicates that the change in quantity at a point S due to non-local flow is equal to the sum of all non-local flow into that point.  Q is the total volume integral over the variable S1.  This equation gives f some mathematical meaning to how non-local flow interacts with local flow.  The final full form of Euler's equation of continuity modified for non-local velocity is:

                                                             (7)           

There are two sets of questions here using 1 or 2 where the term  is used.

4.3       Euler's Equation of Continuity Applied to the I-Bond

The substance principle for the I-bond means that it has a quantity that does not appear and vanish.  When a material medium flows from one point into another the total quantity of medium remains constant.  The I-bond strength will be considered to be a conserved quantity.  Changes in strength will be the sum of changes in strength at end point.

              (8)

The term for the I-bond I(S1,S2,t) indicates the two points related at a point in time by the I-bond strength indicator.  The two points, S1 and S2, of the I-bond move independently.  The velocity of the I-bond I(S1,S2,t) at point S1 is described by I(S1, S2, t).  Applying Euler’s equation of continuity for the motion of the I-bond at point S1 yields:

                                 (9)

This implies the divergence of the product of the I-bond quantity term and its velocity at a point S1 will give the change in I-bond strength due to the motion of the I-bond at S1.

Similarly, the equation for the change in I-bond strength due to the motion of the I-bond at point S2 is given by:

                                 (10)

Note the difference between equations is the order of the S1 and S2 in the velocity term.

The change in the total I bond strength will be the sum of the change happening at point S1 and the change happening at point S2. Putting these two terms together yields the following equation of continuity for the I-bond:

             (11)

The zero on the right hand side indicates there will be no sources or sinks in the I-bond strength.  However, this theory uses infinity I-Bond strength values for certain purposes.  An infinity I-bond may be a source or sink to the I-bond.  Therefore certain considerations must be added when the nature of the infinite I-Bond is further explored.  There will be constraints to the infinite I-Bond resulting from dynamics. 

5        Laws of Local Continuity

A few laws of local continuity are introduced.  These laws should be provable by dynamics.  These laws help to constrain the medium parameters to allow for a sensible mathematical form.

5.1       Local I-Bond

The I-Bond as defined here is the causal bond between two points.  When its value is 0 between two points then there is no causal relation between them  When the value is infinity then the two points are directly connected.

A method to keep strong locality and the medium continuous is to let the I-bond strength go to infinity as two end points approach each other.

            lim S1àS2 I(S1,S2,t) = ∞

This law will indicate that local points are directly connected.

5.2       Law of Local Continuity in the Density

A method to keep strong locality and the medium continuous is to let the medium density become locally continuous.

lim S1àS2 r(S1) = r(S2)

This law should be a derivative of the local I-Bond.

5.3       Law of Local Continuity in the Metric

One more addition of continuity will be in the spatial metric.  The spatial metric should become locally continuous.

lim S1àS2 M (S1) = M (S2)

This law should be derivable from the previous local laws.

6        References

 [1] A. Cermal Eringen: Non-local Continuum Field Theories

This is the only real textbook on Non-local Continuum Field Theories that I have found. Although I derived my expression of kinematics are separate from this, I wish I knew of it before hand.  This gets much more in-depth into Non-local Field theory mathematics than I have attempted in this paper.

[2] D. Gilbertson: On the use of A Non-local Medium Theory for a Unified Field Theory

For a conceptual non-mathematical investigation into how this theory can account for the known phenomenon of physics see http://www.nmtuft.com/nmtconcept/.  This writing gives conceptual reasons for many initial questions about how the medium would have to act in order to be a unified field theory.

 

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