Abstract

In physics the application of Riemann manifolds and the corresponding analytical apparatus is confined to namely the General Theory of Relativity and the lightspeed preserving space-time coordinate systems which place restrictions on the form of the metric, i.e. only transformations which preserve the speed of light are considered, and the spaces considered are 4D. 3 space and one time. In Non-relativistic theories i.e., classical or Newtonian mechanics, Riemann manifolds are confined to 3 space dimensions and the time is considered separate. The cited reason for this is that in general an arbitrary space time transformation from one coordinate system to another will not preserve the speed of light, maybe so, but this should not prevent a non light speed preserving metric from expressing an equation of motion, for what is a geodesic in one coordinate system is also a geodesic in any other coordinate system, even those which do not preserve the speed of light.  Realizing this key point allows the full power of Riemann metrics to be exploited. In relativity theory the metric is restricted so that the speed of light is always the same, and only transformations between such metrics are permissible. In this paper no such restrictions are placed on the metric, an arbitrary transformation from one reference system to another gives rise to a new metric in which perfectly reasonable equations are obtainable.  For example consider two Cartesian coordinate systems on the plane. Let one be inertial and the other be rotating. By definition in the inertial frame free particles follow straight lines, and in the rotating frame the same particles follow curved lines. If it is assumed that the particles follow geodesics then it is easy to find a metric in the inertial system which give rise to the geodesics, i.e. chooseand it does not matter which constants are chosen, just so long as they are constant.

  Let denote the metric in the rotating system. can easily be determined from if the transformation which takes the inertial points to the rotating frame is known. Once the  are determined, the paths can be calculated using the geodesic equations:
  
  =
  =        
  
  
  
  
  Calculating the geodesics in the rotating system we get we get;
  
  
  
  Now because which in the inertial frame is constant.
  
  For the remaining we have
  
  
  
  
   This example serves to illustrate the point that the metric space apparatus and space-time transformations can be employed to do Newtonian
physics.

Space- time having four dimensions is an illusion because it lacks concrete definition, all that exists are measurements and the relations between them, which are what physical laws, are. Each measurement or observable is a dimension in its own right, so for example if an experiment is being performed in which ten observables are being recorded then the space is ten dimensional or eleven dimensional if the experimenter also is using a clock as an extra observable. Also if the experimenter does intend to use a clock then using more than one would add redundancy as well as present calibration problems. Late that night the experimenter looks at all the data, row upon consecutive row of ten or eleven columns each and tries to form a summary of it all which can also be used to predict the out come of future experiments, i.e. he tries to find a physical law. Suppose also, that day, a second experimenter was collecting the same data but was using completely different Instrumentation, but was still recording snapshots of ten observables. He also went home that night to try and make a summary of his data but was unable to do so. Later he discovers that the first experimenter did manage to find a physical law or summary of his data and that it was successful in predicting the outcome of future experiments. But then he realizes that this law is no good to him because his experimental apparatus use’s different instrumentation to observe the same phenomenon. He ponders the problem and realizes all is not lost because all he needs to know is the relationship between the two sets of instrumentation, i.e. he needs to know the transformation from the first set of observables to the second set. Since a summary has already been found for the first set then it should be easy to find a summary for the second set.
    
  y1(x1......x10)
  y2(x1......x10)
  .....
  y10(x1.....x10)
  
  The x's are the readings on the first experimenter's instruments who found the physical law and the y's are the readings on the second experimenter’s set of instruments. Also note that once a summary is found it alone is not enough to describe the phenomenon. What is also required is a description of the instrumentation. This way if the same phenomenon is being observed using different instrumentation, inferences can be made by transforming the readings from one set of instruments to the other.  As an example Suppose an experiment is being performed to measure the interaction between two particles.  Seven instruments it is felt should supply enough information to determine a summary of the interaction. Six transducers to measure the x, y and z coordinates of the particles, if a rectangular reference system is chosen and one clock. Note, an entire array of clocks placed at different points in the coordinate system, it is felt, is not required, as this would supply to much information as well as lead to calibration problems. So the system can be thought of as existing in a seven dimensional space, it may be also be possible to obtain a predictive summary by only having two instruments, one, a transducer to measure the particle separation and a clock, in which case the problem is only two dimensional.  In any case each configuration is regarded as a distinct point in the 7D or 2D space, which ever is being attempted, and as the measurements evolve they trace out a path in the space. One approach would be to assume that the permitted paths follow geodesics in the configuration space. To do this requires imposing a metric on the configuration space to form a Riemann manifold where neighboring points are related by
  
    
  
  If the metric can be determined then it is straightforward to calculate the path by using
the geodesic equations. Moreover if the same phenomenon is observed using different instrumentation, i.e. in a different configuration space, and the transformation is known, i.e. , then the can be easily determined and hence geodesics calculated. Imposing a metric on the configuration space does not imply that the background physical space  (PBS)  is curved. The PBS does not really exist, only measurements exist and the relationships between them. In fact we are not placing any restriction on the kinds of experiments being performed and the observables may not even have anything to do with the so called PBS, for example each of out observables could be commodity prices just to make the point, but for the most part the spirit of this investigation shall be directed at physical phenomenon. In what follows the observables is assumed to have a lagrangian description. By considering a certain class of lagrangians it will be strongly suggested that geodesic motion in configuration spaces is the dominant one for a wide range of natural phenomenon, not just gravity.
  The concept of a configuration space is not new, and in fact treating all observables as a single point in a multi dimensional space is implicit in the lagrangian formulation of mechanics. However the idea of embedding a Riemann metric on such a multi dimensional space would appear to be a novel one and is the focal point of the analysis presented in this paper. Traditionally when Riemann manifolds have been employed in mechanics, as in General Relativity (GR), the approach has been to model the background 4 space-time as a Riemann manifold on which particles move and from which observations are taken. In GR the properties of the manifold are determined from the distribution of the particles, or more generally the distribution of matter and energy, and each particle is assumed to move along geodesic paths in the 4D space-time manifold. If a particle does not follow a geodesic, as for example an electron in an electric field, then the use of a 4D space-time manifold fails to provide a complete description, hence the need for a unified field theory. Approaches to finding such a theory have centered on extending the number of dimensions that make up the the physical background space and reexpressing Einstein's equations of general relativity in these higher dimensional spaces. Such approaches have met with varying degrees of success and skepticism alike and have their current incarnation in string theory and M theory. The extra 'physical' dimensions required to bring about unification are curled up in little hyper spheres and are therefore hidden from everyday observation and participation in experiments.
   By regarding all experimental observations as a single point in a multi dimensional Riemann manifold, the interactions and couplings between each of the observables arises naturally by virtue of the metric coefficients.  In the analysis to follow it will be shown that the most natural form  (and it is conjectured but not proven to be the only form) the system's lagrangian can take is that of the distance element in the multidimensional metric space, and therefore the system evolves so as to trace out the shortest distance in the Riemann configuration space. The metric coefficients are determined by requiring the lagrangian reproduce Einstein's mass-energy relationship for a single particle and that the classical lagrangian (T-V)  is recovered when all velocities are small when compared to the speed of light. The 'distance element' Lagrangian, in essence, represents what has been traditionally termed, the Relativistic correction, and it will be shown how it consistently and naturally extends beyond the single particle case.

Analysis

  Consider an experiment with n observables, and a clock.
  Also form a metric space  by imposing  the metric, , on the configuration space. (,=t,i=1,n)
  
  [] , i=1,n  
  
  Satisfies i=1,n. (1)
  
   define Δφ=[]  ,=,
   
   =>, i=1,n
   
   define []=[], i=1,n, j=0,n  ,  
   
     always exists since []= i=1,n
   
   => =++==Δ
   
   Now since , then
   
   +[]+where i=1,n  and k=1,n.
   
  Now in general  , so equating the coefficients of the 's and 's  is NOT IMPLIED here. But in what follows the classes of considered will be such that
  
    k=1,n     or    with k=0,n         (2)
  
  => , i=1,n                                                              (3)
  
and , i=0,n                                                                   (4)

Combining equations (1) ,(3) and (4) yields

i=1,n                                                        (5)      

where L=

for  i=0  and using equations (2) ,(3) and (4)



using equation (1)

Therefore

i=0,n. The case i=0 describes the energy of the system, and the other case describe the momentum.   (6)

   A  Hamiltonian for the system (6) can be defined  as
   
H=-L.

It has to be noted that this is not the same for the case when i=0  in equation (6). H can be thought of as a 'Hyper Energy' of the system and at this point its physical significance is not known.

From equation (2) it can be easily seen that

H= -L=0                                                                           (7)

  H is a scalar and must be identically zero in arbitrary frames of  reference, since L is a scaler and and transform as contravariant and covariant vectors respectively.
  
  Equation (7) places a restriction on the form of L
  
  L=A+B where A and B are constants, is a rank two tensor and is a vector.       (8)
  
  An example of L could be
  
L=A+B

where    is the metric tensor and Ψ is a scalar potential.
  

Determination of Δφ

for  i=0,n

  L is a scalar function, therefore
  
  However it is not clear that
  
=0.

In what follows this will be shown to be in fact the case.

suppose that ,form a metric space, M, so that .

Define the basis vectors, , such that =, =, where is the inverse of .

It can be shown that

() transforms as a vector .                                  (9)

Also

ΔL=is a scalar .                                                    (10)

when , (9) => transforms as a vector which => from (10) that

must transform as a vector also, which => that

()  must  transform  as a scalar for non zero     (11)

subtracting (11) and (10) , ie

(11)-(10) =>   that ( )will transform as a scalar

which =>  that () will  transform as a vector.

Define P=as the momentum vector and grad(L)=() .

then -grad(L) is the same in all coordinate systems and further more

-grad(L)==0                                                      (12)

which => =0  is true in all coordinate systems since the are linearly independent and =

Now since L and dφ are scalers,  the action quantity A=s a scalar across all coordinate systems, i.e. Let us restrict ourselves to the  case when the action is stationary, i.e.  when  equation (6) is true.

Let denote the primed coordinate system and R the unprimed coodinated sytem.

Let represent the transformations from , i.e.  :

where the are independent parameters which can be varied at to generate different coordinates systems  R.

suppose for , and   is given by T[0],  i.e. (13)

and that

=0, =0

Now let the transformation be perturbed by shifting the by so that and are given by .

Under the new transformation the action , since all we are doing is changing the reference system R and scalar  quantities don't care about the details of the reference system. Likewise

All taken together => that

                          (14)

Now the approach taken here is that which is typically employed to prove Noether's theorem to show conservation laws. However unlike  Noether's theorem  we are not restricting oursleves to any specific reference system, i.e., cartesian, or a specific class of transformations,  i.e. translational or rotational. No restrictions are being placed  on the reference system, R, or the transformations.   

Consequently since the are arbitrary  the only way for  (14) to be true is for =>=0 from (6)

which  => L=K , a constant             (15)

How can (15) be reconciled with  (8) . The solution is in the choice of Δφ.

If  L=A+B choose  K Δφ=A+B

=> L==K   Q.E.D.

This gives the form of to be

=K =A+B where we have defined =1.                             (16)

Going  back to our earlier example, but in this case for , gives

=A + B

Conservation laws in arbitrary coordinate systems .

Conservation laws can be determined from an inpection of the Lagragian alone and the approach employed in Noether's theroem is not required. For example suppose that in a particular reference system the lagrangian, L, is found to have the  property that

L[]=L[]

then this => that

, which from (6) gives .

what's more , conservation laws are dependent on the frame of reference in which the lagrangian is inspected. Conservation of energy arises if the L in the above example is independent  of time or if there is a translational invariance with  respect to , i.e. if

L[]=L[], which again from (6) gives

or simply where .

Again let L=A+B.  

Since is identically true for any ,  then setting  L= should lead to the same equations of motion.  In this case E becomes

                   (18)

and similarly

                                          (19)

Classical correspondence

Allow a system consisting of n observables to have a total potential energy, V(,,.......,t), and a total Newtonian kinetic energy, T(,,......). The classical lagrangian, , has the form
=T-V. Also V can be written as, V=M+ Φ, where Φ(,,.......,t)  is the total  interaction potential between all the observables, eg if we are dealing with a system of charges then  Φ would be the columb potential energy, and M is the total rest mass of the system. For the 'distance element' Langrangian , , to be consistent with when the system's velocities are small compare to c, the following form of   is chosen:

=. => =, ==0, for i=1,2.....n.               (20)

when  T << V, can be, using a Taylor series expansion to the first order,  approaximated as;

~~-. The minus sign is just a constant of proportionality and has no effect on the equations of motion, ie - will produce the same equations of motion as .

Using   in equation (18) we obtain the energy, E, of the system  to be;

E==.                                 (21)

For a single particle moving with velocity, u, in the absence of a potential, ie with Φ=0, we  obtain

  E==,  where T=      (22)
  which is in agreement with Einstein's energy-velocity relationship.

Correction to the classical equations of motion

Define  = as the Lagrangian  operator.

                    (23)
  
  defines the classical equations of motion. However because of the form   in equation (20), equation (23)  should receive a modification of the form
  
           (24)
  
  where should  be small under normal conditions. In order to quantify , the following approach is taken.
  
  Observe that
  
      (25)
  
  Since is a linear operator then,
  
              (26)
  
  Also for any functional ,
  
         (27)
  
  Therefore  since
  
        (28)
  
  Using the fact that and  ,  (28) simplifies to,
  
     (29)
  
  Also since , , and   is a constant of the motion,   (29) simplifies again to,
  
       (30)
  
  Giving
  
         (31)
  
  Under everyday conditions   and   since it would be expected that,   <<    in the expression  , which => that,
  
  0

    

Implications

Non conservation of Newtonian energy and momentum

Even though translational invariance of the Lagrangian  with respect to space and time coordinates can lead to conservation of momentum and energy laws, such as equations (18) and (19), the
expressions for energy and momentum generated by , as has been already stated, only approximate their Newtonian counterparts when velocities are small compared to c. This means that in general the Newtonian energy, T+V, and the total Newtonian momentum along a particular axis, , will not be conserved. This can be seen most clearly from the term in equation (31) which essentially, from the Newtonian viewpoint, represents a non conservative force, and which effectively behaves like an external force in a classical system, altering its conserved quantities. Newtonian energy may be only globally conserved if a system returns to its original potential energy, , at some point during its evolution since according to equation (21), the systems Newtonian kinetic energy will also be its original, .  The variation of Newtonian energy is explained in the theory of Special Relativity by requiring that the mass of accelerated particles increase to compensate for their reduced Newtonian energy if it was assumed that the mass otherwise stayed the same. However an alternative explanation would be to have the mass stay the same and require that an accelerated particle radiate away some of its received energy in order to compensate for the Newtonian mechanical energy it would otherwise have had. It therefore should not be suprizing that this is exactly what charged particles do when accelerated by electric fields. For example according to equation (21), if a particle, initially with infinite potential energy, is accelerated from rest by the same potential  then it will ultimately attain a speed of c when the potential is zero, with a value of   for the Newtonian kinetic energy. This seems to make more sense than  requiring the particle receive an infinite mass. In this view, throughout the evolution of a system, it is continuously radiating and reabsorbing energy. Under suitable conditions it may be possible to configure a system so as to absorb and radiate energy in a desired away, or to put it another way, construct an energy pump to move energy from an incoherent to a coherent form.

Galilean invariance

Suppose we have a system of particles each of mass , under the influence of a potential, and  confined to move in one dimension along a straight line.  

= and equation (30) becomes,

      (32)

suppose we define a Galilean reference frame as a particle in the system, ,  for  which and which has speed . Then

                                  (33)


(32) and (33) together imply that,

    (34)

where

=.

So from the point of view of an observer moving along with the partical , the equations of motion are the same. More generally a Galilean reference system can be defined as a group of particles
all moving with the same speed . In any case the result (34) will be the same.

Gravity and Inertialess Propulsion

From the previous example particles which do not experience the potential , satisfy

                                  (35)

and form what we have defined as a Galilean frame of reference if they all start of with the same velocity. An observer in such a system would not experience any forces of acceleration despite the
fact that the system as a whole would be accelerating. In this sense the experience would be the same as that of an observer in freefall in a gravitational field. For this reason it is conjectured that gravity has its origin in equation (35) and this conjecture will be explored in a future paper. Moreover equation  (35) hints at how an inertialess propulsion system might be designed by somehow
generating the large potentials required to make the term significant. Equation (35) has the exact solution,

                              (36)

where the particles accelerate from velocity to velocity when the overall potential energy changes from to . In order to achieve a change in velocity the particals must be moving to begin with. Particles which are stationary will remain stationary. Moreover the direction of motion is preserved, particles moving in a particular direction will maintain their course.  Because of the Galilean  invariance of  (35) an accelerated observer will see the entire universe accelerate in the opposite direction but without experiencing any forces of acceleration, again the freefall experience.

Weber's Electrodynamics

In order to incorporate Electrodynamics into this formulation of mechanics it is felt that the Weber force law extended to a multiple particle system will provide a more natural route as opposed to any attempt to incorporate Maxwell's formulation. In any case the Weber lagrangian tends to a lagrangian density as the number of observables or particles tend to infinity.  Weber's force law for a pair of interacting charges can be described by the Weber classical lagrangian ,

                      (37)

where is the Coulomb potential.

For a system of particles in one dimension it is here conjectured that (37) generalizes to,

                                    (38)

If then  (38) conserves linear momentum as it is easy to see that,




Also as the number of particles tends to infinity and assuming the total mass, , and remain bounded as then

                 (39)


The function describes the path of a continuum of particles that form a fluid. The parameter can be thought of as a particle identifier in a fluid and is the continuous version of the discrete particle identifier, i.e.  

       Path of a finite number of particles.

               Path of an infinite number of particles or path of a fluid.

In three dimensions, in order to describe the path of a fluid, a single parameter is not sufficient even though a single discrete parameter is sufficient to identify a discrete number of particles regardless of the number of spatial dimensions they are permitted to move in. This is because a finite set is always countable, but in general infinite sets are not. in particular the set    is not countable. Therefore in order to map all the particles in a fluid in 3 spatial dimensions we need three parameters. So for example the path of a 3D fluid could be described as,

  ([],   [,t],    [])  ,  since any particle in the fluid can be uniquely identified by the tuple   . Getting back to the 1D case above, the complete equations of motion are yielded by equations (38) and (30) to give,
  
     (40)
  
  where all the particles have the equal mass .
  
  As     equation (40) becomes,
  
  

  


     

Summary

Like the original formulation of Lagrangian mechanics a set of experimental observations is treated as a single point in a multidimensional configuration space where each dimension in the configuration space represents each of the experiment's observables. By requiring that a universal time variable, φ, be defined in terms of the experimental observations and that φ be independent of the details of the instrumentation used to obtain the observables it was concluded that the most general lagrangian had to be of the form of a distance element in a multidimensional Riemann manifold embedded in the configuration space of observations. By further requiring that in the limit of everyday observations the classical lagrangian be recovered from the distance element lagrangian, it was possible to infer the Einstein energy-velocity relationship in the absence of a potential to within the rest mass constant and obtain a correction to the classical equations of motion. The correction has the characteristics of a gravitational field and hence if the potential in our lagrangian is the coulomb potential, a possible connection has been suggested between electromagnetism and gravity. No attempt was made to define a physical background space or any of its properties nor was it felt necessary to do so in this formulation of mechanics. It was noted that only when the number of observables tends towards infinity, does the notion of a 3D-background space become apparent.