Absolute Maximum Proper Time to an Initial Event, the Curvature of Its Gradient Along Conflict Strings and Matter

Home , Proper time

Journal of Modern Physics, 2013, 4, 791-806 http://dx.doi.org/10.4236/jmp.2013.46109 Published Online June 2013 (http://www.scirp.org/journal/jmp)

Absolute Maximum Proper Time to an Initial Event, the Curvature of Its Gradient along Conflict Strings and Matter

Eytan H. Suchard R&D Algorithms Department, ANB/ANT—Applied Neural Biometrics/Technology, Netanya, Israel Email: [email protected], [email protected]

Received January 22, 2013; revised February 28, 2013; accepted April 1, 2013

Copyright © 2013 Eytan H. Suchard. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT Einstein equation of gravity has on one side the momentum energy density tensor and on the other, Einstein tensor which is derived from Ricci curvature tensor. A better theory of gravity will have both sides geometric. One way to achieve this goal is to develop a new measure of time that will be independent of the choice of coordinates. One natural nominee for such time is the upper limit of measurable time form an event back to the big bang singularity. This limit should exist despite the singularity, otherwise the cosmos age would be unbounded. By this, the author constructs a scalar field of time. Time, however, is measured by material clocks. What is the maximal time that can be measured by a small microscopic clock when our curve starts at near the “big bang” event and ends at an event within the nucleus of an atom? Will our tiny clock move along geodesic curves or will it move in a non geodesic curve within matter? It is almost paradoxical that a test particle in General Relativity will always move along geodesic curves but the motion of matter within the particle may not be geodesic at all. For example, the ground of the Earth does not move at geodesic velocity. Where there is no matter, we choose a curve from near “big bang” to an event such that the time measured is maximal. Without assuming force fields, the gravitational field which causes that two or more such curves intersect at events, would cause discontinuity of the gradient of the upper limit of measurable time scalar field. The discontinuity can be avoided only if we give up on measurement along geodesic curves where there is matter. In other words, our tiny test particle clock will experience force when it travels within matter or near matter.

Keywords: Foliation; Field Curvature; General Relativity; Accelerated Cosmic Expansion; Quantum Gravity; Dark Matter; Chameleon Scalar Field

1. Introduction: Square Curvature in atomic force deviates from any non-Newtonian Positive Definite Metric Spaces model in that the force is trajectory and speed dependent because it means that a particle that moves A) Ideas along a trajectory and indeed measures the upper  If spacetime is homotopic to a single starting event, limit can’t be geodesic near matter. say “big bang” then maximal proper time curves can  The laws of Nature will never directly involve any be drawn back from any event, to a limit that con- absolute time. Instead, we will be coerced to define verges at the big bang singularity. Along these curves them by using the gradient of such time, which is in- we can imagine a tiny clock that travels and measures deed local. This point is crucial to the understanding time. This time is considered as a scalar field. Clock of this paper! Also crucial is to understand the time tick is different under local space location due to discussed in this paper is well defined as an upper gravity. The scalar field therefore, has a significant limit on measurable proper time. gradient by space. Where there is matter, however, a  A nice, though less important issue, is that decon- tiny particle-like clock will never move along geo- structing space time into 3 + 1 dimensions require lo- desic curves because as we shall see, it would lead to cal time orthogonality unlike in Kerr solution. This discontinuities in the gradient of our scalar field of idea will also be addressed though it is a bit specula- upper limit of measurable time. The expected sub- tive.

2. Classical Matter: The Gradient Need Not force that prevents any microscopic real world clock to Be Parallel to any Geodesic Curve, from moving along geodesic curves. Our field of time Resolving Singularities then sets an upper limit to measurable time by any such tiny clock particle. The conflict is apparent also on the Our strategy now will be to see what would happen to the edges of the ball because matter is granular, that is to say gradient of the upper limit of measurable time from any that the mass is not evenly distributed. Particles measur- event backwards, if no forces act on our test-clock-parti- ing absolute maximum proper time from near “big bang” cles. Then to show that forces are indeed required for along curves that enter the ball, must pass through the avoiding singularities at curve intersections. The direc- walls of the ball or hyper-cylinder in 4 dimensions. There- tion in space time of the particles measuring the maxi- fore, gravitational lenses are formed and events in the mum proper time forms a geodesic curve but near matter, hollowed part of the ball are accessible by more than one without the existence of forces, not necessarily the gra- curve as depicted by Figure 1(b). These singularities can dient of the field would be parallel to a geodesic curve be resolved too if any real world particle-like clock will because: 1) more than one curve could reach the same not move along geodesic curves in the microscopic vi- event; 2) at that event near matter, even known forces cinity of matter, e.g. due to Casimir/Casimir Lifshitz [3]. would cause any test particle clock to move along non Again, the newly presented type of gradient conflicts geodesic curves; and 3) one idea, even without explicitly (also caused by absolute maximum proper time inter- assuming forces near matter, is that in a quantum model sections) events can be avoided by either providing that the intersection of such curves does manifest singularity the matter’s time and location are uncertain or by force of the gradient but by coupling by multiplication of the exerted on the test-clock-particles. If we say that matter time field and a wave function of the entire system of is measured by such conflicts/intersections, then the fact particles, the 4-loaction of the singularity can be blurred we also have a microscopic, though negligible geodesic by uncertainty. Whichever model we choose 2 or 3, a conflict in the center of the ball, attests to the existence of real world clock will not move along geodesic curves a new type of mass, we can call “Secondary Dark Mat- within matter, otherwise its measurement will result in ter”, that mass is not additional to the mass of the ball but discontinuities or singularities of the gradient of our up- simply means that some force field which relates to mat- per limit on measurable time. The idea of such particle ter can weakly extend beyond the microscopic scale. In clocks is not quite new [1] and is important in order to have physical meaning. A good example of discontinuity Hollow ball of mass is the center and edge of a hollowed ball of mass, see Figure 1(a). Due to General Relativity, the clock ticks in Does conflict have measure 0 where there is no mass? the gravitational field of the ball are slower than far from the ball. As a result, max proper time geodesic curves from say “big bang” event, must come from outside the Max proper time curves from big bang. ball into the ball. The time at the center of the ball is also a geodesic curve but it is in the time direction in (a) Schwarzschild coordinates due to symmetry. The vector field of the lines is therefore discontinuous and we have a non-zero Euler number [2] of the gradient field. As was already mentioned, one way to resolve such singularities is that our particle clock will experience force. Space- time in a hollow ball of mass is conic i.e. the metric co- efficient grr of the radius difference dr is greater than 1, and is thus flat but with a zero measure singularity of the Gaussian curvature at the center which is the tip of 4 dimensional cone. In any case such force has to be negligible though it should exist and should disallow (b) geodesic movement in the sub atomic scale that will otherwise manifest the gradient singularity. Figure 1. (a) The line of the max proper time field from “big bang” is discontinuous in the middle of a hollowed ball of If we consider the ideas of Quantum Gravity, “con- mass and therefore a real world clock will not move along flict” events where curves intersect do not have zero geodesic curves at such points, no matter how negligible is measure. But as was already mentioned, even without such an effect; (b) On the edges, gravitational lenses due to quantum gravity it is not difficult to imagine that such granularity cause geodesic conflicts. The particles form an singularities can be avoided in the sub-atomic level by obstacle that is bypassed by the entering curves.

Copyright © 2013 SciRes. JMP E. H. SUCHARD 793 big words, the theory that is behind this paper says that ture or Field Curvature of a vector field V in Rn , with indeed, laws of physics are local but the entire geometric positive definite Euclidean geometry, is. The same for- context have influence on a new effect we have just malism is easily extended to Riemannian geometry. We named, “Secondary Dark Matter”. Contrary to the abso- also define Bending Energy BE as the Square Curvature lute maximal proper time from “big bang” measured by a multiplied by the square norm of the gradient of a scalar microscopic clock, most geodesic curves—though they field. We would like the field V to reduce or increase its also use travelling clocks—usually measure only local differential in directions that are perpendicular to the maximum proper time. The curves of the global/absolute direction of the field. This requirement is also compre- upper limit on measurable time from the past to an event hensible when the metric tensor of a manifold with coor- usually have tangents that point to a 4-direction in space dinates in Rn has only positive eignevalues in local time along which the time changes. In free of matter orthogonal coordinates and we shall see that the operator space, in perpendicular (Lorentzian) to the direction of that describes Field Curvature has quite the same formal- the gradient, the differential should be zero. Therefore, in ism in Riemannian manifolds. We will start with an in- geodesic coordinates such that the time is parallel to one tuitive description of the operator and later give a proof it of the absolute maximum proper time curves, the mixed is the square curvature of the vector field. Given two terms of the metric tensor vanish. Locally, the separation infinitesimally close points in Rn , q1 and between space and time works also in metrics such as the qqhV21 for some infinitesimal h , we would like Kerr metrics and time appears perpendicular to 3D space that Vq 21  Vq  will be as parallel as possible to the manifolds along the maximum proper time curves, e.g. in a set of rotating reference frames. Separation of space field Vq 1 . and time is important and can be achieve at least locally By Pythagoras, that can be written as the following even along closed time-like curves, however, this paper problem locally minimize has a much higher priority motivation, to get an equation Vq 21  Vq  Vq 21 Vq  that depends only on geometry. To show that time is an   emergent dimension is secondary in this paper. hh B) Open questions—emergent time unsolved issue 2 (1) The question is: Can inverted logic work? By mini- Vq21 Vq Vq 1     2 mizing an action operator on three dimensional mani- h h Vq1  folds, can a degree of freedom yield multiple solutions  for the metric tensor, such that: When  is the inner product in Rn . 1) The action can serve as a homotopy [4] parameter. Vq1 2) The action will be invariant under Lorentz—like ro- Here Vq21 Vq  represents the pro- tations in the resulting four dimensional manifold. Vq1 These questions, to the author’s opinion justify further jection of the derivative matrix of the vector field Vq   research. multiplied by the field direction in space. We would like to describe the curvature of the gradient In other words, since h2 is arbitrarily small, our ob- of the absolute maximum proper time from near “big jective is to minimize, bang” as a scalar field that is measured by our microscopic particle-like clocks and show its possible relation t 2 VVVV to Ricci curvature and to Einstein’s tensor. The idea is VV    V VVVV that the gradient of the scalar time field of absolute  (2) maximum proper time from “Big Bang”, forms curves 2 VV  VV VVVt   that have non vanishing curvature where there is matter.  Again, it is important to say these gradients are local and VV VV that our time is an upper limit on all possible tiny test particles. Vi Here V means the matrix aij  . Intuitive discussion about the second power of curva- X j ture of a conserving vector field and about “bending en- Figure 2 shows the geometric idea of how to measure ergy” perpendicular changes in the gradients of conserving We now need tools to assess the curvature of a tra- vector fields. jectory of a test-clock-particle as it interacts with mate- The following next Figure 3 shows us two curves one rial force fields. on the left for which BE is zero and one on the right The reader can find the origin of this work in [5]. So for which BE is positive by comparison to parallel let us begin. We will now define what the Square Curva- curves.

V1 = V = V(q1)   m PP ,m 1 PP ,m P Vm  and  . PPi t i 32 i PPi V V VV 22VhVq Vh V dP i V and P  for coordinates x . If P is the upper i dxi limit of measureable time from near big bang to any event then (3) is the second power of the curvature of the dP t gradient P  . VV i i Vh dx  VV Don’t confuse the scalar function t with maximum proper time. We choose a simpler expression for our upper limit from near “big bang”, namely  , and where

2 there is no matter in space-time we expect one of the t VVVV following to be true. hV  V V VVVV  PP  , is real (4)

2 Figure 2. “Bending Energy” which is the Field Curvature PP**, P  is complex. (5) multiplied by the squared norm of the gradient and its (4) is offered for a deterministic theory and (5) for quan- Euclidean geometric meaning—how much the field changes in direction perpendicular to itself. tum. Please note that Square Curvature (P) = Square Cur- vature (kP) for constant k . Please note that in the model presented in (5), the time  is coupled with a wave function  and there is only a need for P   to have 3rd order derivatives but not for  alone. The formalism of (5) possibly needs mul- 8πK tiplication by  for each differentiation, such c4 that K is the constant of gravity and  is the Planck constant divided by 2π . Figure 3. Parallel deviation on the right. dP Repeating our question: Can P  be non-tan- 3. Tensor Formalism of the Square dx Curvature gent to a geodesic curve: 1) If there is more than one geodesic curve that con- As will be discussed, in tensor formalism, derivatives are nects the “Big Bang” event to say event “e”, then obvi- replaced by covariant derivatives and are denoted by ously P need not be geodesic in a small neighborhood semi colon and derivatives by comma. Upper and lower of “e”. indices represent the covariant and contra-variant proper- 2) Intersection of different curves if coordinates are ties and upper and lower indices sum according to Ein- uncertain in quantum sense. stein convention so (2) can be written as a tensor density Development of (3) can be from the following: with local coordinates in Rn . We will also divide (2) by 2 VV in order to achieve a fully geometric action op-  smkPP , Pm 1 PP ,,msk PP g  m  or. Regarding the square root of the determinant of L  erat 23 the metric tensor g , following are tensor densities 4 PPii PP ii [6], that yield tensor equations [7], 1 j Square Curvature   UUj (6) 4 2  smk m PP,, PP P PP ,,msk PP g PP ,m P (3) m  1   UP 23g or mmi 2 4 ii PP i PP PP i PPi ii Obviously UPm  0 . The vector U describes the 2 m m 1m 1 that can be written as VV g for direction and intensity of the curvature of the field P m  m 4 t which is a change perpendicular to P .

From minimum action of dinary General Relativity matter-geometry equation.

k There is always a way to solve the following equation  ZZPPk 1 k Z PP and U 2 and LUUk 11  k   Z Z 4 UU UUgk T for an ordinary dust en- 42 R Ricci curvature.  ergy momentum tensor and therefore (9) is consistent Action Min RL gd   with existing theories and is an important link to well established work on General Relativity. See Appendix A, In other words, curvature of the gradient of the upper 1 LUUZPPik and limit of measurable time from near “Big Bang” to an 4 ik event is equivalent to Ricci curvature. If that is true then   m we can have an equation that is based solely on geometry. 1 PP ,m P  2;PPPk  In any case we have a nice action (without spinors [10] 4  Z 3 k   and other advanced mathematical technology, though 2 4-force means rotation) of the form: s  m ZPs PP Z 1 1 22;PP VV  such that V is a vector field and is a 32m  2  ZZZ  t t 2 scalar field. If the definition is in 3 dimensions, it hints at   PZ k ZZ PP 11  ZZ  2  ggk 4 dimensional Lorentzian metric geometry. U is in ZZ23Z 22Z2 1 2 units of . For the complex square (second power PZ  PP , Pm  Length ZZ PP m  23 2 3PZ of) curvature operator we set the curvature vector ZZZ Z   iik PP;*ik PPP ;**i P   m U . 1 PP ,m P  kL   k kLPP** P P 2;3 P PPkL** P P kL 4  Z     2 Obviously UP *0 . Bearing in mind that in our s  m ZPs PP Z PP22; kkk1 k 32 case PPkkk*** PP P P we calculate ZZZ  2   ZZ PP   1 k  1 kk mkPP2 UU UUg  UUkk* U * U g Square Curvature 2 4 ZZ 2  (7)  1 j LjLik 1  PPPPPPPPg;* *; *; ;* RRg 1  2 ij kL ij kL 2   4 PPPP**i  as we shall see later, a simple solution to the Euler La-  i   grange equations by P and it’s derivatives yields  2 ij  m  ms PPP;** P *; PP  Z PP ,ms P ZZ ZP ij   k    g 23;;mkP 34 3 ZZZZi   PPPP**2  i   and (10)

m  m  PP ,m P Z  kk 4. Quantum Gravity in a Nutshell ;mPU;k;k0 (8) Z Z 2  The observable in the offered theory is a scalar field of 2 and therefore (7) becomes time. Time can be either P   , or PP**   where the theoretical formalism is the complex formalism 11 k 1 (of Square Curvature) although (10) is incomplete without UU UUgk R Rg (9) 422 8πK d multiplication of each derivative by  4 k . The and we have LR, such that R is the Ricci cur- cxd vature tensor [8,9]. If we ignore (6), and the coming change in the direction of the gradient of the time field is (18)-(24) and (34)-(36), it is a bit disappointing that after due to the need to avoid discontinuity of gradient meas- all the efforts we simply get [8,9] which look like an or- urement by particle clocks in max proper time curve in-

tersections. Discontinuity of the gradient is avoided by If V is a conserving field then VV ,,rr  and thus uncertainty of the intersection events/strings. Then 1 VV,,r  Norm2  * could be the probability of the 4-location of such  r 2  avoided geodesic conflict in the middle of a constellation   2 of particles. The coupling of  and  has one impor- 2 VV VV Curv 22 tant meaning which is that quantum uncertainty resolves Norm Norm the discontinuities of the gradient of  and prevents its 1 Norm22,, Norm g k measurement. In the classical model of gravity in this  k  4 (15) theory,  is “smoothened out” by the Equation (7) or (9) 4 Norm 2 which is an approximation or a limit of a Quantum ef- 1 Norm2 , V g sr  sr fect. The classical model is sufficient for a giving a new 4 Norm3 description of matter, however,  is required for re-  solving gradient singularities of  that do not exist in Writing the last term in Riemannian geometry is the the classical model. same field curvature operator that we chose.

5. Proof That Square Curvature Is the 6. Byproduct—Separable Coordinates Square (to the Second Power of) Field Where the Time Coordinate Is Parallel to Curvature at Least One of the Global/Absolute Maximum Proper Time Curves We restrict the proof to the Euclidean case. The square curvature is defined as The following is a bit speculative but may be important. We see that 3 dimensions hint at 4 dimensional action. ddV V Curv2    g  (11) This is done by looking at the action (3) in three dimen- ddttkk sions and observing the following way to write it, VVkkVV  such that g is a diagonal unit matrix. For conven- m PP,,,,m P PP012 PP PP k d  ience we will write Norm V V and VV  . 3 PPiii PP PP k   PPi 2 iii dt i For some parameter t . Let W denote:  PP , Pm  m d VV V 3 WVV gk (12) i 2 3 k PPi dtNork mNorm  VVk  1000  PP ,0 00 01 02 Obviously i 0 ggg PP  i VVg ks VVg 0 ggg10 11 12 WVgkkksVVg   kk3 20 21 22 PP ,1 Norm Norm 0 ggg (13) i kk PPi VVgkk VVg  0  Norm Norm PP ,2 i Thus PPi    s 1000 2  VVg VVgs  k Curv W W  V V g 00 01 02 24k 0 ggg Norm Norm   (14) g  101112   2 0 g gg VV VV  20 21 22 22 0 ggg Norm Norm  ggg00 01 02 V Since is the derivative of the normalized ij 10 11 12 (16) Norm and qggg  20 21 22 curve or normalized “speed”, ggg ddddxVrrVrwhere g  is the metric tensor in 4 dimensions and qij VVrr V V,r ij ddtddxx t Norm Norm is in 3 dimensions. q implicitly refers to a local submersion [11] where time is locally held constant. such that xr denotes the local coordinates. Can we do the opposite, look at 4 dimensions and re-

Copyright © 2013 SciRes. JMP E. H. SUCHARD 797 duce the problem to 3 without violating the principle of 8. History of the Paper’S Concept of Time covariance? The idea of an absolute time, such as maximum proper First, our upper limit proper time curves are intrinsic time from a common event, i.e. “big bang” is not new and do not depend on the coordinates unless more than [12,13] and it appears in Hebrew writing such as the one curve intersect with the same event. Book of Principles by Rabbi Josef Albo 1380-1444. We can therefore agree that the absolute maximum Rabbi Josef Albo wrote about time that can be measured proper time curves are different than ordinary geodesic by devices and another aspect of time which he termed curves on which only local maxima of proper time can be immeasurable. The maximum proper time can’t be meas- measured. ured by devices on Earth because due to General Relativ- We choose to describe (3) on our space-time in our ity, clock ticks are slowed down by the gravitational field. special coordinates. Under correct local choice of coordi- It is an upper limit. nates, the direction in space time of the maximum proper time is an eigenvector of the metric tensor with the big- 9. Conservation of Known Matter from the gest eigenvalue, our metric tensor is of the form pre- Euler Lagrange Equations sented in (16) for which the mixed space time terms are zero. Also, Finally we get the following zero divergence:

 PPPP 11 PP,1,2,3 d Ld L k  0  UU g W ; dxdxPP, k and  g 0 PPPP01,,,, 02 03 10P20,,0P 30  Please note: We can assume as possible where W  is obtained from the subtraction of (35) from (36), PP1230, 0,P 0 especially if multiple maximum proper time curves to the same event “e” exist. Instead of see Appendix A. (3) we reduce the action to become three dimensional, Variation by P and its derivatives is a special case Tweaked Square Curvature LLd  UUk  g 0 .  k 2 PPdx ,  smkPP , Pm  1 PP ,,msk PP g  m g 23 4 ii  m s  11PPiiPP Z ZZ ZPs  P 4;4m P  4 ; 23 3 ZZ Z or  (18) (17) 2 Tweaked BE m m  ZP ZPZm m 2 PP26  0 smkm  34 1 PP,,msk PP g PP ,mP ZZ g 4 ii P 11PPiiPP  Multiplication by and contraction yields, 4 i 1PPi  Square Curvature  s  m Z ZPs P Z Z   m This means that on our three dimensional sub-mani- 23;;Z 2 Z ZZ  folds (“Leaves of a foliation”), there is a corresponding  (19) action operator that is free of derivative dependence on m 2 ZPm time. 0 Solving the Euler Lagrange equations for the Tweaked Z 3 Square Curvature and receiving a plurality of solutions is  s  Z ZPs P indeed a promising direction of research!   23;; ZZ 7. Unsynchronizability  2 (20) m ZPm Since P is not constant on the 3 dimensional sub- 1 ZZm m  230 manifolds perpendicular to the global/absolute maximal Z ZZ  proper time curves, these manifolds are not synchroni- zable and are therefore not the ideal inflating S(3) i.e. and as a result of (20) the following terms from (7) van- Friedmann-Robertson-Walker. ish,

2  ms 10. Chameleon Fields or Pressure? PP ,ms P ZP PP 2;PPPk  2 Z3k Z3Z As we can see, the more general case is,  2 ZPsm P Z P m   sm ZP ZZ P 4;4PP 2;PP  2 34 22m ZZ ZZZ   m 2 Z ZZ  ms  W ;4;P4P m  ; PP ,ms P ZP P P 23  k   ZZ (25) 2;33PPPk 2  ZZZ 2  ZPm m  m  ZPZm m   2243P  ZP ZZ P ZZ 2;22m PP 2  ZZZ   0 2  ms  PP ,ms P ZP P P 2;PPPk 2 ddLL 33k  UUk  g ZZZ k  ddxxPP, (26) m    ZP ZZ P Wg;0 2;m PP 2 ZZ22Z  instead of

 m  m Z PP ,m P LLd k  k  UU g 0 (27) 2;23m P ;  k ZZ PPdx ,  

2 An effect which is contrary to gravity will add a posi-  s  (21) 1 ZZ ZPs tive delta to the Ricci curvature and therefore from (7),   PP0 k 23 multiplication by the metric tensor g and contraction Z ZZ   yields, Which yields a simpler Equation (9). Recall that 2 PP , PmsZP m    msk Z s  2;22;PZ   Z PP 332k   Z  s  ZZZ U  2 ,  (28) Z Z  ZZ Z m Z 20 And that  UUU   Z 2 Z  An effect which adds gravity, will add negative delta 2  s m mto Ricci curvature and therefore, Z ZPs P 1 ZZ ZPm m ;; 2323 2 ZZZ ZZ  msm  PP ,ms P ZP Z   k  2;22;332PZk  U 11 ZZZ ;;UUm  U  (29)  m  (22) ZZ Z  ZZ m Z 202 11 m Z 2 UZ UUm Z Z Known matter will be simpler 1  U ;0 2 Z  msm PP ,ms P ZP Z   k   2;22;PZk  Which proves (8) ZZZ332   (30) U ;0 (23, see 8)  ZZ Z 20 And proves the simple representation of the field equa- m Z 2 tions It is possible that either (28) or (29) is mathematically That we saw in (7) not valid. Additional terms can’t violate the vanishing of

11 k 1 the divergence of Einstein tensor. UU UUgk R Rg (24, see 9) 422 If one of them is correct then there is a particle which

Copyright © 2013 SciRes. JMP E. H. SUCHARD 799 its force fields decay very fast with distance, in other the scope of Friedman-Scarr representation of accelera-   words, discovery of weakly interacting particles may im- tion AUa , speed UU  1 and acceleration ply either (28) or (29). matrix A A . High order derivatives of the metric tensor and pressure were studied by Deser and Tekin [14]. For applica- 12.2. Unconventional Conclusions tion of (28) and/or (29) to space-time warp drive please There is experimental evidence regarding high gradient refer to [15]. of an electric field in which force that acts on metal balls 11. Acknowledgements is represented as the ordinary force on the induced dipole as in ordinary dielectrophoresis plus an unexpected I would like to thank my boss, Mr. Yossi Avni for ap- “force” that depends on mass [17]. [17] can attest to the proving my idea of using Riemannian geometry and a existence of true force field, which is not gravity, that curvature operator in minimum cost functions in the depends on mass, as also seems to be an outcome of this company’s research of hand signature recognition. His theory. This is one way to achieve a unique trajectory of approval lead not only to new technology but also to this the maximally measured proper time by any massive test paper. I also would like to thank him for helping me to particle including zero mass Chronons [1]. In this case, prepare a poster for an Israeli physics conference IPS 9, [17] can be a residual force of the original force field. December, 2012. Another effect is that (28) and (29) allow gravity to be Thanks also to supporters of this theory, Dr. Sam generated from non linear force fields. If the 4-force is Vaknin who presented to me the idea of Chronons as not linear then (28) or (29) are possible which allow clock particles and to Professor Larry (Laurence) Hor- warps or gravitational dipoles [15]. In order for such an witz whose advices were very useful. My special thanks effect to be consistent with the vanishing divergence of to emeritus professor David Lovelock for his help in Einstein tensor and with conservation laws, (28) and (29) corrections of the Euler Lagrange equations several years mean the creation of gravitational dipoles. The overall ago. momentum—energy must be preserved. In this case, the simple Equation (7) can’t be reduced 12. Conclusions—Test to the Theory to (9). 12.1. General REFERENCES Maximal time is measured by particles. This time sets an [1] S. Vaknin, “Time Asymmetry Revisited [microform],” upper limit on measurable time quite similar to the way Ph.D. Dissertation, Library of Congress, Microfilm, 1982. the speed of light sets an upper limit on speed. Since time [2] J. W. Milnor, “Topology from the Differentiable View- is measured by material clocks, these material clocks are point,” Princeton Landmarks in Mathematics, Princeton influenced by forces. The particle that can measure the University Press, Princeton, pp. 32-41. maximal possible time from the “big bang” to an event [3] J. N. Munday, F. Capasso and V. A. Parsegian, Nature, within matter will therefore be influenced by such forces. Vol. 457, 2009, pp. 170-173. doi:10.1038/nature07610 For the trajectory to be meaningful, either there exists a [4] V. Guillemin and A. Pollack, “Homotopy and Stability,” force which can’t be gravity since gravity is not a force, Differential Topology, Princeton Landmarks in Mathe- such that the force will depend on mass or that only a matics, Princeton University Press, Princeton, pp. 33-35. unique particle can enter the force field. Evidence of [5] Y. Avni and E. Suchard, “Apparatus for and Method of mass dependent force will be shortly discussed. Such an Pattern Recognition and Image Analysis,” US Patent No: effect can also attest to gravity and it is therefore difficult 7424462. to distinguish between the two. [6] D. Lovelock and H. Rund, “The Numerical Relative Although the upper limit of measurable time is meas- Tensors,” Tensors, Differential Forms and Variational ured along curves, the gradient of that time is local, Principles, Dover Publications Inc., Mineola, pp. 113- 114. which makes the use of such scalar field feasible to theoretical physics. [7] D. Lovelock and H. Rund, “Tensors, Differential Forms and Variational Principles,” Dover Publications Inc., Mi- Non geodesic motion as uniform acceleration is well neola, pp. 323-325. defined by Friedman-Scarr representation and has a lin- [8] D. Lovelock and H. Rund, “Tensors, Differential Forms ear interpretation by an anti-symmetric tensor [16] which and Variational Principles,” Dover Publications Inc., Mi- also indirectly describes a more intuitive alternative to neola, p. 262. spinors (well at least if we can blissfully afford to ignore [9] D. Lovelock and H. Rund, “Tensors, Differential Forms wave functions, and group representation), however, the and Variational Principles,” Dover Publications Inc., Mi- most interesting effects that this theory offers, are beyond neola, p. 261.

[10] E. Cartan, “The Theory of Spinors,” Dover Publications 1946. Inc. Mineola, p. 149. [14] S. Deser and B. Tekin, Physical Review Letters D, Vol. [11] V. Guillemin, “Submersions, Local Submersion Theo- 67, 2003, Article ID: 084009. rem,” Differential Topology, Princeton Landmarks in doi:10.1103/PhysRevD.67.084009 Mathematics, Princeton University Press, Princeton, p. [15] M. Alcubierre, Classical and Quantum Gravity, Vol. 11, 20. 1994, pp. L73-L77. [12] Rambam, “The Guide For The Perplexed By Moses Mai- [16] Y. Friedman and T. Scarr, Physica Scripta, Vol. 86, 2012, monides (Moreh Nevuchim), Part. II, Chapter 13,” Rout- Article ID: 065008. ledge and Kegan Paul Ltd., London, 1904. doi:10.1088/0031-8949/86/06/065008 [13] J. Albo, “Immeasurable Time-Maamar 18, measurable, [17] T. Datta, M. Yin, A. Dimofte, M. C. Bleiweiss and Z. Cai, Time by Movement,” Book of Principles (Sefer Hai- “Experimental Indications of Electro-Gravity,” 2005. karim), Chapter 2, Chapter 13, (Circa 1380-1444, un- arXiv:physics/0509068 [physics.gen-ph] known), The Jewish Publication Society of America

Appendix A: The Euler Lagrange Equations of the Square Curvature Action We will not solve the entire system

ZZPPk 1 Z PP and U k and LUUk Z Z 2 4 k R Ricci curvature. (31)

1 k  RUUgk d0  4

But rather focus on  UUk gd    k 

2  PZ LZPP s.t.  Z 3    s Lgd  Lg PP ,s P    mmi   mim m 2;;3 PPPPPPmmim PPPimPPP ggxZ,d     m   2 2  s  s  m PP,, P PP, P PP P   s  s 1  m  23PZ PP  g 3432 ZZPPi 2   i    s PP ,s P 2PPP mim;  PPP  PPP  im  3 mimim Pi P i PP ,, Ps  s   sm PP ,s P 26PPP mmr PPP PP ,  g 34 rm PPii PP  ii  2  ms  s PP ,,ms P ZPPP PP s P  k     2;22PPPk   PZ ii333i PP PP PPi  Z ii  22 PZ PZ  1  PP (32) 33ggi  2 ZZPP i 

ZZ LZPP s.t.  Z 2 

Lg  Lg  mmi   m im  d  PPZPPZ;;mmimi PPZm PPZ Z Z m 2 22  ii gg dx  ,m  PP PP  ii PP,, PPsmk g PP ,, PPsmk g msk  1 msk  2 32PP g (33) ii2 PPiiPP

PPZ mim;  PPZ  PPZ  im  PPZ mrPP ,  mim imrm 242 3 g PPi PPi  i i  ZPPZZm ZZ  1 ZZk  2;PP  2 k g   g 2222m i ZZPPZZi 2

ZZPPk ZPP and Uk and LUU Z Z 2 k 2 Lg Lg  ms  d  PP ,ms P ZP PP 2;PPPk 2 m 33k g dxZgZ ,m Z  22 (34) s  1 PZPP,s P PZ PP gP 2Z   2 ZZ33 Z 3Z  ZPm ZZ P1 ZZk ZZ  2;PP 2 k gg   22m 22 ZZZ 2 ZZ

2 ZPs  s   LZPPZPP3 s.t.  and mm, Z Lg  Lg PP;  PPik d  s  ik  m 4Zs Pg3  ggdxZ  ,m PP;   PPi   si sPP 4ZPs3g4 ZPs, 3g (35) ZZ 2 PP m  ZPm s  ZPZm m  12Zs PZgg43 2 6 4Pg ZZZ 2 s  m  m ZPs P ZPZ ZPm   m  4;233 P  6 4 Pg ZZZ 

s ZZs  LZPPZPP2 s.t.  and mm , Z Lg  Lg PZ;   PZik m d  ik ZZm  m 4423 g Pg gg dxZ ,m Z (36) PZ;  PZii     PZ  PZ ii  i 4823gPPgi ,  Z i PPi Z ZZm 4;4 m Pg  23 ZZ

Appendix B: The Scalar Time Field of the but in the middle of a hollowed ball of mass the gradient Schwarzschild Solution of the absolute maximum proper time from “Big Bang” derivatives by space must be zero due to symmetry We would like to calculate which means the curves come from different directions

2 to the same event at the center. Close to the edges, gravi-  smkPP , Pm  PP ,,mskPP g  m tational lenses due to granularity of matter become cru-  23cial. PPiiPP  ii The speed U of a falling particle as measured by an observer in the gravitational field is in Schwarzschild coordinates. This theory predicts that URGM2 2 where there is no matter, the result must be zero. V 2  (37) The result also must be zero along any geodesic curve CrC22r

Copyright © 2013 SciRes. JMP E. H. SUCHARD 803 where R is the Schwarzschild radius. If speed V is So we can write U normalized in relation to the speed of light then V  . 22 R 1 2 C NPP 1 Prt P r R For a far observer, the deltas are denoted by d,dtr  1 r and, 2 Rr rR 2drR2   11 2 r V1  (38) rR Rr dtr  rR because ddrr 1Rr and ddtt 1Rr. N 2  2 (41) Rr 2 t R ddrt dN 2  N 2  And we can calculate Pt1d    r R dx 0 1 r 22 RR1 2 1 RR NN22 2 1  rRr t  22 (42) R rr 2 rR 1d t N 2  r R  0 1 Rr r

2 We continue to calculate ttRR 1dtt1d 2  2 RRR1 00rr NPtt1 2 rRr r Which results in, dPR and Pt 1 (39) dtr 2  NPtt RRR1 1  (43) Please note, here t is not a tensor index and it de- R rRr 2 r 1 notes derivative by t !!! r On the other hand Please note, here t is not a tensor index and it de- r R 11 notes derivative by t !!! Pr1d  2  r r R 1 RRRr2 1 r 11NPrr  (44) rrRR r 2 Rr r 1 rR 1 Please note, here r is not a tensor index and it de-  dr  2 notes derivative by r !!!  R R 1 1 r r RRrR1  NP2  1    2  rR rRr  R r rrr R ddrr and rR R  22 2 r 2  RR1 NP 1 2 rRr Which results in  (45) dPr rR P  (40) 2 r drR Rr Please note, here r is not a tensor index and it de- So notes derivative by r !!! 22 2 RR1 For the square norms of derivatives we use the inverse 2  1 NP 2 of the metric tensor, So we have  rRr 32 (46) N 2 rR R 1 1 R  2 1 and 1 Rr r R R r  1 1 r r And finally, from (42) and (46) we have,

2 is inevitable.  smkPP , Pm PP ,,msk PP g  m   Center analysis if there is an atom of movement length ii23Without even negligible forces acting on a test particle PPii PP  and without quantum center location uncertainty, in the 2 22 2  middle of a hollowed ball of mass the gradient of abso- NN NP  lute maximal proper time is discontinuous due to sym- 23 (47) NN22 metry. Suppose that the difference between the gradient 22 22 at the center where r  0 and where, rr  , such that RR11 RR   r is small, results in  N 2 0 . We want to measure 1122  rRrrRr  the second power of the curvature of the gradient of ab- 220 rR rR solute maximum proper time due to that difference. 22 Suppose that the change happens smoothly within a Rr Rr small radius from the center, measured around r  0 . Which shows that indeed the gradient of time meas- We assume that such curvature measures how much ured, by a falling particle until it hits an event in the the gradient is not geodesic due to curve intersections. gravitational field, has zero curvature as expected. 3,3 R 0,0 R 2 rR Consider g 1 and g 11 The term N  2 is slightly disturbing be- r0 r0 Rr r cause at very far distances, becomes significant. R PP0, 0 1 ,0 (49) R tr   r0 Moreover, if R has a lower atomic limit, then for such r  R the term is a whole number!!! R r0 R PrPrtr,1, (50) rR We now return to the discussion about a hollowed ball 0 of mass. 22 It is clear that the maximum proper time from “Big RR 11 Bang” curves entering the ball are symmetrical in rela- 2 rr00r0 R tion to the center and therefore  N 01  RRRr0 11 (51) dP r rr 0 00 Pr 00  drR r 0 1 where r0 is the radius in the far coordinate system of R dPR r the hollowed ball of mass. However, Pt 1. 0 dtr 22 1 0 NN003,3 3,3 R R gg1 Writing the gradient in two dimensions in t, r, ignoring  rrRR 0 the gravitational lenses due to mass granularity, and ig- rr11(52) rr00 noring quantum uncertainties of coordinates and of energy momentum, we have r0 R 11  Rr0 Rr rr1, 0  R rR  N 2 01  N 2 0 r R r gg0,0 0 3,3 PP,0r r 1, 0 (48) tr 0  tt' rR0  R   N 2 01 R r r 01 ,0 0 3,3   g (53) r0 r  r 0 R dP r0 The last result Pr 00  is an inevitable r0 drR 2 1  N 0 R outcome of the symmetry in the center of the ball. The  RRrr gradient by the space coordinates must be zero and the rr1100 change of direction in the gradient means that curvature rR00 rR

22 2 Given the radius  r that is seen within the gravita- rr00r0 111 tional field, the surface of a small ball around the center 22 RR RR NN 221(54) is smaller than expected in flat space-time, 2 r r  r 0 rr0 R 4π 3 R Vol r 1 (63) 2 3 r r R 0 0 11 22  NN Rr0 We now calculate the curvature and then multiply it by 2 2 (55) 2 the volume of the ball in which the direction of the gra- N 2 r0 R  r 2 dient changes towards the center as seen in (49),(50), Rr0  2 2 NP2  1 NN  r0 RR SquareCurvature  11 2223 2 Rrr 4 NN  N 0 0,0 00   gPt  (56) tr 2 22 2  1 NN NP rrR   00 23 Vol 2 11 4 22  N 0 RRr NN gP3,3  0 (57)  rr r 4π 3 rR1 2 r  00   r  2 (64) r0 R 34r  rrR  11 0 0   2  Rr0 R r0 2π NP  (58) r      rr0 R 3 2 (64) is very interesting because it depends only on  r r0 R 11 and not on the mass of the gravitational source. 2 2  Rr0 r0 R NP 2 2 (59)  r Rr0 Appendix B2: Approximated Validity Test 2 at the Planck Scale r0 R 11 (64) imposes some strict limits on the offered theory. For Rr0 r0 R 2 2  2 2 the following we assume that  rr 1 Rr is big NP  r Rr0 enough in comparison to the Schwarzschild radius R 3 3 2 otherwise none of the following calculation will be valid, N r0 R 2 Suppose that all the matter we have is due to force field Rr0  (60) acting along the distance  r . Then by (9) and the fol- 2  r0 R lowing conclusion that LR and by Einstein equa- 11 Rr 8πK  0 tion of Gravity TG , and from (64) we have 2 c4  2 r0 R  r 2 the following, Rr0 2π 8πK 2  r R  rE 4  (65) 2 0 11 3 c 22 NP2   NN  Rr0 232 2 (61) where  E is achieved via integration of energy on 22 NN 2 r0 R space. K is the known Gravity constant  r 2 Rr 0 K  6.67384 80 1011 m 3 kg 1 s 2 2 rR00rR and 222 2  NN() NP Rr 81 1  0 c  2.99792458 10 m s . 232 2 22 2 NN 2 rR  r 0 So we can divide the equation by  r rR (62) 0 So we have 4 2r0 2 cE 2 Force (66)  rrR  2 R 12Kr  0  r 1 r0 That is quite a strong force, about

1.0086146094754842869840358898085*1043 Newtons. Recall the definition of work as Force multiplied by On the other hand if our energy is within a ball of ra- length on which the force acted, we have from (69) and dius  r and  r is also the uncertainty of the space from (66) coordinate of the center then we have by the law of un- cK456  c certainty of Quantum Mechanics Force r  12KKc3 24  Pr  (67)  3.9929699575019974555997778404945 (70) 2 108 Joules.  1.05457172647 1034 J s and in the inequality ex- tremity of equality, This value is quite close to the Reduced Planck Energy 5  c Pr  (68) which is defined as . and since it is smaller than 2 8πK Now consider (66) which is a very strong force, acting c5 Planck Energy , our calculations imply that a sin- on a small enough particle so virtually we can say that K the speed of the particle is approximated by an average gle-photon black hole is highly unlikely to exist if there speed which the speed of light. So is a Planck length limit. Also pairs of photons will 12K probably not create a microscopic black hole. cPr  E r  P c3 (69) 66KK  r  cr33  c