Modern Infinitesimal Analysis Applied to the Physical Metric dS and a Theoretical Verification of a Time-dilation Conjecture
aa r X i v : . [ phy s i c s . g e n - ph ] F e b Modern Infinitesimal Analysis Applied to the Physical Metric dS and aTheoretical Verification of a Time-dilation Conjecture.
Robert A. Herrmann9 FEB 2008
Abstract
In this paper, the modern theory of infinitesimals is ap-plied to the General Relativity metric dS and its geometric andphysical meanings are rigorously investigated. Employing resultsobtained via the time-dependent Schr¨odinger equation, gravita-tional time-dilation expressions are obtained and are shown to becaused by gravitationally altered photon interactions with atomicstructures.
1. Introduction.
In books that deal with General Relativity, one often sees the expression for the“metric” or line-element dS (= ds ) . This symbolism usually appears on the left-handside of an equation and the symbols dx j or dx η , where j, η vary from 1 to 4, appearon the right-hand side. So, the dS is related to the set of dx j . This, of course, doesnot indicate how this dS should be interpreted in a physical sense. The meaningsfor these symbols were thought to be well understood when they were first displayedwithin mathematics. However, there was a certain level of confusion and even errorin the physical interpretations that have developed since such interpretations wereintroduced.Since the time of Newton through 1855, the entire period when basic “calculus”notions were introduced, the application of these symbols to geometry and physicalscience was based upon the “intuitive” behavior of what was conceived of as the“infinitely small” and how ordinary Euclidean geometry and physical measures areperceived. For example, in calculus textbooks written from 1700 - 1850s a spacecurve is defined as an “infinite collection of infinitely small line segments.” Thisis distinct from how we actually go about physically measuring the “length” of acurve segment. If you take a segment of a two dimensional curve, draw it on a pieceof paper and lay a string on the entire path, then to measure the length you wouldlay out the string in a “right line” as they would call a “straight line” and place iton a ruler. Then simply read off the length of the curve in terms of a Euclideanunit.Students learned, by example, how to handle the mathematical symbols usedto identify infinitely small measures - the “little” o. There were no exact rules fortheir behavior. However, in 1826, it was shown that the known rules used to argue1n terms of the language of the infinitely small were mathematically inconsistent.This did not stop these inconsistent rules from being applied. The term infinitelysmall or the modern term infinitesimal is still used today as are the “old” rules forhow such symbols are to be handled. For example, in [7] on page 134, McConnellwrites, “Let P be a point whose coordinates are x r and let Q be a neighboring pointwith coordinate x r + dx r . If we denote the infinitesimal distance
P Q by ds , we call ds the element of length etc.” This can only mean that the dx r are infinitesimals,whatever they are. There is even a section entitled “Infinitesimal deformations,”where he only states that these are “ small homogenous deformations.” He statesthat these deformations are “infinitesimals of the first order.” He neither gives adefinition for the term “infinitesimal” or “small” nor any of the algebra for the“small” or the “infinitesimals.” The only way one could learn how to work withthe small is to replicate his “proofs” and nothing more could be known about thenotion of the small. Since this is not rigorous mathematics, it can only be hopedthat what properties of the small that are gleaned from these and other statementsmade by those who write in differential geometry and physics as to how the small behaves will not lead to one of the known inconsistencies.From these examples, it should be clear that to retain logical rigor within thecalculus one needs to find the proper algebraic properties for the infinitesimals sothat they can be properly manipulate and applied to the physical world.
2. The Modern Infinitesimals.
To rigorously understand dS, a “new” set of numbers - the actual infinitesimals -is adjoined to the real numbers IR (or even to the complex numbers) and a newrigorous infinitesimal arithmetic is needed and the order < needs to be properlyextended. These new infinitesimal numbers are often denoted by lower case Greekletters and the collection of these new infinitesimals as adjoined to IR is denotedby µ (0) . (Other new numbers are also adjoined to IR , and they are introduced asneeded.) The basic arithmetic is not too difficult to comprehend. For this article,here are the necessary “rules.”The “addition” and “multiplication” operations for the real numbers are ex-tended to members of µ (0) as follows: Any nonempty finite sum or multiplica-tion of members from µ (0) is a member of µ (0) . Members of µ (0) satisfy allof the basic “grouping” rules for IR . For example, if α, β, γ ∈ µ (0) , then α ( β + γ ) = αβ + αγ ∈ µ (0) . Let a = 0 , ( − a, a ) = { x | ( x ∈ IR ) ∧ ( − a < x < a ) } , and let f be any continuous function defined on ( − a, a ) such that f (0) = 0. Thenapplying the same “function” but extended to µ (0) will always yield members of µ (0) . µ (0), animportant application of this continuous function idea is when f ( x ) = gx, where g ∈ IR . Now substitute into this expression an infinitesimal ǫ and obtain g ǫ thatis also a member of µ (0) . This fact is a very important requirement when membersof µ (0) are used. On the other hand, if 0 = r ∈ IR , then, as intuitively expected, r + ǫ / ∈ µ (0), although it is a new number that behaves like a real number. The set µ (0) is shown to exist mathematically by various means, one of which is by pureabstract algebra as well as ideas from modern mathematical logic. Also, there isone important operation that is done with members of µ (0) and gives interestingnew numbers that are not in µ (0) . Let ǫ = 0 Then, although 1 /ǫ is one of these newnumbers that is adjoined to IR , /ǫ is not a member of µ (0) . This is a significantfact. An additional significant fact is that, with respect to the extended order < , if r is any real number and ǫ > , then r < /ǫ. If ǫ < , then 1 /ǫ < r. The major property for members of µ (0) is how the extended real numberorder behaves. Let 0 < r ∈ IR and ǫ ∈ µ (0) , then ǫ < r. If r < , then r < ǫ. Intuitively, the set µ (0) only contains the one real number 0 and all other membersof µ (0) “crowd around” 0 “closer than” any other real number. If f is a real valuedfunction defined on all the real numbers [ a, b ] = { x | ( x ∈ IR ) ∧ ( a ≤ x ≤ b ) } andthere is a real number B such that | f ( x ) | ≤ B for each x ∈ [ a, b ], then, for any ǫ ∈ µ (0) and any x ∈ [ a, b ] , ǫf ( x ) ∈ µ (0) . Any continuous f defined on [ a, b ] hasthis property. The behavior of the members of µ (0) appears to follow all of thenotions associated with the “small” notion and the original definitions used in thecalculus. Further, they should eliminate all of the known inconsistencies in the useof the terms “infinitely small” and “infinitesimal” throughout differential geometryand all other applications of the calculus. Also note that all ordinary real numberarithmetic holds for the ǫ and 1 /ǫ numbers.Similar to Cantor’s definition of the real numbers as equivalence classes ofrational number Cauchy sequences, certain sequences of real numbers that convergeto zero can be used to represent an infinitesimal. For example, consider the sequencewith values { G n = 1 /n } , where n is a nonzero member of natural numbers IN andwhere G = 0 . Then the sequence { G n } can “represent” an infinitesimal. There isan equivalence class of such sequences that are related in a special way and the G is one member. Mathematically, there are actual formal mathematical objects thatyield infinitesimals. But, what is the proper way to apply the calculus, using thesenow rigorously defined numbers, to real world applications?3 . The Geometric and Physical Meaning of the dS. ( In all that follows, all lower case Greek letters represent infinitesimals,where 0 is an infinitesimal. ) What is presented next could be classified asan overly-long explanation. However, if individuals want exposure to all of thecorrect steps that are usually hand-waved over in the ordinary course in calculusand physics, then what follows is necessary. To start, the most general form isnot discussed, but the 4-dimensional “form” for a physical metric is considered.Further, the Einstein summation convention is not used. One form is( dS ) = dS = h g ( dx ) + h g ( dx ) + h g ( dx ) + h g ( dx ) , (1)where the x , x , x , x denote distinct variables that are used to denote a “point”name ( x , x , x , x ) in spacetime and, when evaluated at point names, the functions g j ≥ , ≤ j ≤ . The h j = ± . Note that the dS is only consider, at the moment,as an abbreviation for this form. In this form, what do the dx i mean?It is often stated that they represent a “small” (infinitesimal) change. But,physically or geometrically a change in what? Consider an interval of real numbers[0 , . Take u as a parameter that varies over [0 , . Consider the four linear equations x = a + ub , x = a + ub , x = a + ub , x = a + ub . The collection of allfour-tuples generated by these equations is termed as a “linear path” from the pointwith “name” ( a , a , a , a ) to the point with name ( a + b , a + b , a + b , a + b ) . Suppose that [0 ,
1] contains all of the required real numbers as well as all of thesenew numbers; the non-negative infinitesimals (infinitely close but ≥
0) and all theones that look like r + ǫ, where 0 ≤ r + ǫ ≤ . The notation for this new view of[0 ,
1] is ∗ [0 , . Let ǫ > . Consider a “micro”-linear path from ( a , a , a , a ) to ( a + ǫb , a + ǫb , a + ǫb , a + ǫb ) . These are the end-points of a micro-line segment obtainedby varying infinitesimal γ , where 0 ≤ γ ≤ ǫ, and letting x j = a j + γb j , j =1 , , , . Consider the usual coordinate vector algebra and obtain the components( ǫb , ǫb , ǫb , ǫb ). Using the arithmetic of the infinitesimals, it follows that each ofthese components is an infinitesimal.By definition, each dx j means, for these equations, differences of this type.Hence, dx j = ǫb j , where j will always vary from 1 to 4, represents an infinitesimal(i.e. small) “change” in the point name. Now, letting each g j = 1, then dS = ( ǫb ) + ( ǫb ) + ( ǫb ) + ( ǫb ) = X j =1 ( ǫb j ) (2)and using the arithmetic for these new numbers dS = p ( b ) + ( b ) + ( b ) + ( b ) du, (3)4here du is the positive infinitesimal ǫ and dS is a non-negative infinitesimal.IF these point names are considered as names for points usinga Cartesian coordinate system, which is not easily drawn unlesscoordinates are suppressed, and physically the equations representa geometric linear path for such points, then this result might beclassified as the infinitesimal Euclidean micro-path length for themicro-linear path from ( a , a , a , a ) to ( a + ǫb , a + ǫb , a + ǫb , a + ǫb ) . This seems to be a “transfer” of the linear pathlength notion from the non-infinitesimal to the infinitesimal world.The next step is to generalize this to non-linear paths that start at u = 0 andend at u = 1 . Suppose that x = f ( u ) , x = f ( u ) , x = f ( u ) , x = f ( u ) , u ∈ [0 , , where paths between different points can usually be written so that only theinterval [0 ,
1] is employed. Can a collection of infinitesimal micro-linear paths beused and intuitively be joined together, so to speak, and the length of the entirepath between the two points ( f (0) , f (0) , f (0) , f (0) and ( f (1) , f (1) , f (1) , f (1))be obtained? Maybe, but as this demonstration progresses something else might benecessary.Consider the definite integral applied to equations such as (3). As viewed fromthe infinitesimal world, all the usual definite integrals are independent from theinfinitesimal used for the du. Intuitively, from the infinitesimal world, all ordinary integrals,relative to [0 , , are but extensions of ordinary finite sums P ni =0 k ( u i )∆ u , where the integrand function k is defined on [0 , , has specific properties and the u i must be located in restrictedpositions. The major difficulty is in obtaining the required “form” k ( u i ) du, where du ∈ µ (0) , and whether the integral yields the phys-ical measure being considered.The usual modeling approach requires the interval [0 ,
1] to be divided, in theinfinitesimal world, into infinitesimally “long” pieces. So as to conform to the usualnotion, the symbol du is used and is a positive infinitesimal. There are “numbers,”Γ , that behave in many ways like members of IN and if x ∈ IR , then x < Γ and1 / Γ ∈ µ (0) . These objects satisfy the Newton and Leibniz notion of the “infinitelylarge.” Consider − = du as the distance between division points. Consider all ofthe ∗ [0 ,
1] division points { u , u = u + du, u = u + du, u = u + du, . . . , u Γ = u Γ − + du = 1 } , or { u , du = u , du = u , du = u , . . . , (Γ − du, Γ du = 1 } . (4)5his gives a collection of (infinitesimal) du long subintervals { [0 = u , u ] , [ u , u ] , . . . , [ u Γ − , u Γ = 1] } . (5) . For each of these subintervals, one has the corresponding micro-linear pathsfrom ( k p g ( u i ) f ( u i ) , k p g ( u i ) f ( u i ) , k p g ( u i ) f ( u i ) , k p g ( u i ) f ( u i ) to( k p g ( u i + du ) f ( u i + du )) , k p g ( u i + du ) f ( u i + du ) , k p g ( u i + du ) mf ( u i + du ) , k p g ( u i + du ) f ( u i + du )) , where i = 0 , , , , . . . , Γ − k j = 1 or k j = i = √− . The idea of using “complex” coordinate names and, hence, complex geometryis not new. It was introduced into differential geometry in 1822 [9]. Further, inrelativity theory, it was introduced by Minkowski in about 1906.Next, the method used to arrive at (2) is applied to each of these subintervals,where the primary square root is used. This gives dS i = vuut X j =1 h j ( q g j ( u i + du ) f j ( u i + du ) − q g j ( u i ) f j ( u i )) . (6) . Assume that g j and f j are each continuous on [0 , . In this case, a major propertyfor continuous functions is that each of terms in the right-hand side of (6) is aninfinitesimal. Thus dS i ∈ µ (0) for each i = 0 , , . . . Γ − . Notice that the du doesnot appear outside the radical. However, intuitively, one might claim that the lengthof the original micro-line segments is being altered.All of these dS i are added in order to obtain the form Γ − X i =0 dS i = Γ − X i =0 vuut X j =1 h j ( q g j ( u i + du ) f j ( u i + du ) − q g j ( u i ) f j ( u i )) . (7) . Once again if h j g j ( u ) = 1 , then (8) represents the Euclidean length of a micro-polygonal path. This is a collection of attached micro-line segments. In general,if some 0 = h j g j = 1 , then this can be viewed as a shift to a different (possiblycomplex coordinate) polygonal path. (Such “paths” were first investigated in 1822[9].) However, there are two difficulties. First, can the left-hand side of (7) beput into the proper form so that an integral can represent the values and does the S actually measure the modified path length notion represented by the function ℓ ( x, y )?The most basic property a definite integral displays is that it is “additive.” For afunction like ℓ ( x, y ), this means that for three parameter intervals [ a, b ] , [ b, c ] , [ a, c ] ,ℓ ( a, b ) + ℓ ( b, c ) = ℓ ( a, c ) . If you assume this property, then there is another property6hat the function S must satisfy before it can represent ℓ . This property requiresthat each infinitesimal ℓ i ( u i , u i + du ) be “closer” to its approximation dS i than just“infinitely close.” This concept states that for S and any non-zero infinitesimal du there must exist an infinitesimal β i such that ℓ ( u i , u i + du ) = ( S ( u i + du ) − S ( u i )) + β i du. (8)In general for other measures, the S can be replaced with other appropriatefunctions h such as ( h ( u i + du ) − h ( u i )) du. In this case, the statement is written inthe form ℓ ( u i , u i + du ) du = ( h ( u i + du ) − h ( u i )) dudu + β i =( h ( u i + du ) − h ( u i )) + β i . (9)Note that the wrong notation for these statements is being employed. In all theabove cases, each of the functions must be “extended” to the entire set ∗ [0 ,
1] anda new symbol is used for these extended functions. This notation has not beenused in order to minimize notation. If expression (8) holds, then ℓ and S are saidto be indistinguishable of order 1 or infinitely close of order 1. [Note: Ifadditional requirements are imposed upon ℓ and S , then ℓ and S automaticallysatisfy this requirement.]Thus, it is assumed that ℓ and S are infinitely close of order one and this is ap-plied to the left-hand side (7). It is known that if S can be obtained by an integral,then it is differentiable on [0 , . Hence, assuming this, from the Fundamental The-orem of Differential Calculus in infinitesimal form there are α i and u ′ i ∈ [ u i , u i +1 ]such that S ( u i + du ) − S ( u i ) = S ′ ( u ′ i ) du + α i du = dS i + α i du. Under the assumptionthat ℓ and S are infinitesimally close of order 1, then ℓ ( u i , u i +1 ) = S ( u i + du ) − S ( u i ) + β i du = S ′ ( u ′ i ) du + ( α i + β i ) du. (10) Γ − X i =0 ℓ ( u i , u i +1 ) = Γ − X i =0 (( S ′ ( u ′ i ) du + ( α i + β i ) du ) , (11)where we note that ( α i + β i ) = ν i ∈ µ (0) . Now comes the interesting fact about this type of “summation.”
It behavesin many ways just like finite summation.
Since every nonempty finite set ofreal numbers contains a maximum number, the set of { ν i } as i varies from 0 to Γ − ν . In what follows, the triangle inequality holds for a *-sumfrom 0 to Γ − | P Γ − ν du | = | ν | | P Γ − du | = | ν | (1) = | ν | since there are Γ constant du terms in this series and Γ du = 1 . Thus, using the *-triangle inequality −| ν | ≤ Γ − X i =0 ℓ ( u i , u i +1 ) − Γ − X i =0 S ′ ( u ′ i ) du ≤ | ν | , (12)and, hence, P Γ − i =0 ℓ ( u i , u i +1 ) is infinitely close to P Γ − i =0 ( S ′ ( u ′ i ) du. The *-series P Γ − i =0 S ′ ( u ′ i ) du = P Γ − i =0 dS i is in the exact form needed to replaceit with R dS = R S ′ ( u ) du = S (1) − S (0) if S ′ ( u ) is Riemann integrable. [Theintegral is actually infinitely close to such sums and to obtain S (1) − S (0) thestandard part operator is employed.] The function S ′ is integable if it is continuous.This will be the case here. Hence, under this assumption R S ′ ( u ) du = S (1) − S (0).But, P Γ − i =0 ℓ ( u i , u i +1 ) = ℓ (0 ,
1) by additivity. Hence, S (1) − S (0) and ℓ (0 ,
1) are“infinitely close” as Newton might say. But they are real numbers and cannot be“infinitely close” unless they are equal. Hence, ℓ (0 ,
1) = S (1) − S (0) . For the right-hand side, additional properties will lead to the integral form.Consider the right-hand side of Γ − X i =0 dS i = Γ − X i =0 vuut X j =1 h j ( q g j ( u i + du ) f j ( u i + du ) − q g j ( u i ) f j ( u i )) . (13) . One of the properties for a continuous function defined on [0 ,
1] is that it is “uni-formally continuous.” From the infinitesimal view point, this yields that for any v ij ∈ [ u i , u i +1 ] there is an α ij such that p g j ( v ij ) = p g j ( u i ) + α ij and a β ij suchthat p g j ( v ij ) = p g j ( u i + du ) + β ij . Substituting and using the arithmetic of in-finitesimal numbers yields Γ − X i =0 dS i = Γ − X i =0 vuut X j =1 h j g j ( v ij )( f j ( u i + du ) − f j ( u i ) + γ ij ) = Γ − X i =0 vuut X j =1 h j g j ( v ij )( f j ( u i + du ) − f j ( u i ) + ν ij du ) . (14)Assuming that each f j is continuously differentiable on [0 ,
1] (i.e. the path issmooth), apply the (extended) Mean Value Theorem for Derivatives. Thus, thereis for each du a u ′ ij ∈ [ u , u i +1 ] such that f j ( u i + du ) − f i ( u i ) = f ′ j ( u ′ ij ) du + ǫ ij du. Substituting yields that Γ − X i =0 dS i = Γ − X i =0 vuut X j =1 h j g j ( v ij )( f ′ j ( u ′ ij ) + ǫ ′ ij ) du. (15)8quaring the terms inside the radical and using infinitesimal number arithmetic,(15) is re-written as Γ − X i =0 dS i = Γ − X i =0 vuut X j =1 h j g j ( u ′ ij )( f ′ j ( u ′ ij )) + ǫ i du (16)where each v ij = u ′ ij .Taking a closer look at the expression under the radical, it follows that (16)can be re-written as Γ − X i =0 dS i = Γ − X i =0 vuut X j =1 h j g j ( u ′ ij )( f ′ j ( u ′ ij )) du + δ i du, (17)The expression under the radical is now in the proper form. Using the exact samemethod as used to obtain equation (12) yields −| ν | ≤ Γ − X i =0 dS i ! − Γ − X i =0 vuut X j =1 h j g j ( u ′ ij ) f ′ j ( u ′ ij ) du ≤ | ν | . (18)This implies, since the integrals exist, that ℓ (0 ,
1) = Z vuut X j =1 h j g j ( u ) f ′ j ( u ) du. (19)Hence under the stated requirements, the modified path measure is what themetric is trying to display in differential form. Intuitively, the ℓ (0 ,
1) is obtainedby considering the h j g j as modifying the micro-line segments for a *-polygonalpath without a gravitational field present and uses standard Euclidean-styled lengthmeasures for this new curve. This is accomplished by simply shifting to a different*-polygonal path. This is equivalent, but not intuitively, to considering the lengthof each micro-line segment as being modified from the Euclidean length. In gen-eral, the calculus uses Euclidean-styled notions in the infinitesimal world to achievebehavioral results. But, here, there are a few problems in interpretation. First, oneof the terms is a “time” statement, and this needs to be interpreted as such in allcases within General Relativity. Then some of the h j may be = − .
4. Two Basic General Relativity Interpretations.
Today, in physical science, it is often necessary that a specific method to measurequantities such a “length” and “time” be included in an interpretation. Einstein9ntroduced special modes of measurement (i.e. Einstein or apparent time, and Ein-stein distances [2, p. 368, 417]) into his Special and General Theories of Relativityand the General Theory reduces, in a local infinitesimal sense, to the Special Theory.
Such measurements incorporate the basic properties for light propagation.
When thisis done, the Special Theory “chronotopic interval” statement dS = c dt − dr isobtained. This form is actually obtained by consider the behavior of “light proroga-tion,” where “relative velocity” is measured by a special method. The method is the“radar method,” the transmission of a light-pulse that is reflected from an objectand returned to the sender. Such modes of measurement must be used to measurethe t and the r in this expression. (The expressions for the Einstein measures aregiven in section 5.) Hence, in any of the General Relativity metrics this mode or anequivalent mode of measurement must be done. Although it was usually mentionedin the past [2, p. 368], today this fact seems never to be mentioned. Einstein timeand distance use the constancy of the measured to-and-fro speed of light. Further,within the physical world, the wave property that this speed does not change rela-tive to the speed of a source is used. Electromagnetic radiation paths, as modeledby photons, appear to be the only physical world entities that can be infinitesi-malized and that yield easily observed effects. The light-propagation device thatcan be used for Einstein measurements within the infinitesimal world, at least asan analogue model for such physical behavior, is the “infinitesimal light-clock” [5].The “geometry” used for Relativity is called chronogeometry [4].It has been shown in [5], with respect to gravitational fields and relative ve-locities, that certain physical changes in behavior take place when behavior is com-pared to behavior where there are no gravitational or relative velocity effects. Thesechanges are all produced by changes in the (analogue) infinitesimal light-clock mea-suring devices. This is further related to how simple behavior within a substratum(the “medium”) yields behavior within our universe as modeled by infinitesimallight-clock behavior. It is interesting that the notion of a substratum absolute timeand distance is still a valid notion and, indeed, the original Einstein derivation thatyields the Special Theory uses such an absolute time notion.From the viewpoint of infinitesimal analysis, a path within a gravitationalfield is equivalent to a *-polygonal (i.e. micro-linear) path within the substratummedium, that is being modified within the physical world by a gravitational field.The micro-length of each micro-line segment is altered by a constant multiplicationfactor. But, the same effect can be obtained by simply altering the *-polygonal micro-linear path itself. This yields the alteration in the micro-length via a Euclidean-styledmeasure.
The effect may not be any more mysterious than that if using complex10oordinates is not considered as mysterious.However, General Relativity also applies to certain behavior that is not statedin terms of “paths.” In all of these cases, there are gravitational “potentials” in-volved. There are two approaches. One method is to simply drop the notion of a*-polygonal path and replace it with a “*-sequence of infinitesimal *-linear effects”(i.e. micro-linear effects). Note that the “time” coordinate is necessary due tohow “velocity” or “*-sequence of infinitesimal *-linear effects” are measured or ap-plied, respectively. This approach would apply to “very small” real time intervals.The second approach is to assume that the spatial point is fixed. This yields onlythe “time” term in the metric. The expression obtained is then related to othermeasures.
5. “Time” Measurable Aspects of dS.
The numbers being expressed by (1) need to be associated with physical measures.As shown [5] and noted in [2, p. 368], the t and r must be Einstein (radar) measuresusing light prorogation properties. [Note: In what follows, the “times” t , t , t are all being viewed from the medium. Such “times,” that have been termed as“epochs,” should not be considered as physical world measures, as yet. In section7, it is shown how all of these times are directly related to physical world timemeasurements. However, their meanings with respect light propagation still remainvalid. ] The points P and R originally coincide where the “clock” values coincide.The expression “clock” means an infinitesimal light-clock or its counter values.A light-pulse leaves position P at “clock” time t , arrives at position R at R “clock” time t . A type of “reflected” light-pulse arrives back at P at time t . TheEinstein time t E is obtained by considering the “flight-time” that results using thewave property that all to-and-fro standard measures for c are not altered by thevelocity of the source, where the sources are considered at standard locations (i.e.not infinitely close). This Einstein approach assumes that the light-pulse path-length from P to R equals that from R back to P. Thus, the Einstein flight-timeused for the distance r E from P to R is ( t − t ) /
2. The Einstein time t E , used forthe “clock” time t at R, satisfies t − t E = t E − t . The “times” t , t are localtime values observed within the infinitesimal part of the medium. This yields theEinstein measures t E and r E as follows: t = t E = t + t , r E = c t − t , . (20)For comprehension, let , x = t. (Note that the h j g j “time” position is often denotedas g .) Hence, consider (1) in the form dS = h g ( dt E ) + h g ( dx E ) + h g ( dx E ) + h g ( dx E ) , (21)11here dS ≥ . When the integral expression for S is considered, under any “regular” coordi-nate transformation where the integral expression is correctly altered by the trans-formation, the path length is of a fixed numerical value relative to standard units.This does not hold for every conceivable coordinate transformation but only forthose that carry the additional property of “regularity.” The formal regularityrequirement is not examined in this article.To investigate a further physical meaning for (21), the idea of a “point particle”is introduced. Such a particle may have physical measures attached to it, but itis a mere “point location” in spacetime as far as the particle coordinate name isconcerned. Assume that there is a “clock,” attached to the particle. Further, theusual modeling method of assuming physically simple behavior within the mediumis used. From the particle P , observations are made of another particle Q at adifferent spacetime location as it “moves” relative to P within spacetime. For theseobservations, particle Q is assumed to “move” in the medium along a *-polygonalmicro-linear path to the spacetime point Q ′ . For the beginning of each piece of the micro-line segment, the P “clock” reads t as the “clock” time for the “to” part of the to-and-fro light-pulse model for Q. The t is an unknown Q “clock” reading and t is the P reading for the “fro” part.At the other end of the micro-line segment, the readings are t , t . If the functions f j that describe the spacetime location of Q relative to P are defined in terms of t E , then the inverse function theorem implies that t E − t E ∈ µ (0) . Notice that,for infinitesimal regions, simple dynamics is assumed and this would imply that t E = t + β , t E = t + β for the micro-line segments. Since this is all relative tothe behavior of light-paths, then the proper (or local) Einstein time invariant for P observations of the two spacetime positions Q, Q ′ , is the right-hand side of (19)divided by c, where u = t E . But, what would this all become for two spacetimeevents occuring only to P and P ′ ?Let Q be located such that t − t = ǫ . Hence, t E = t + β. This yields thatfor the Einstein times dt E = t E − t E = t − t + γ = dt + γ. The time t is thetime, as originally viewed from the medium. It represents, where each f j is definedon standard t ∈ [ a, a ] , the “beginning” for an event at P composed of infinitesimalmicro-linear changes occuring over a “time” interval. (Some authors actually use,in this context, the phrase “infinitesimal observers” for observations at P .)The metric (21) needs to be expressed entirely in terms of t . To do this, let x = cf ( t ) , u = t and let T be an additive function measuring the “local elapsedtime” experienced by this specific type of ”clock” located at P . As before, it is12ecessary to assume, in order to obtain the integral result, that the measure T isinfinitely close of order 1. Now using the previous infinitesimal arithmetic methodsalong with the continuity of f ′ j for the path functions f j , where f j ( t E ) = f j ( t ) + α j and dt E = dt + β, the local elapsed time interval measured from the medium forthe two events P, P ′ is, where a = t , ( S ( a ) − S ( a ))(1 /c ) = T ( a ) − T ( a ) = (1 /c ) Z aa vuut h g c f ′ ( t ) + X j =2 h j g j ( t ) f ′ j ( t ) dt. (22)What is needed where the point aspects are not relative to paths? In such acase, there is no change in the spatial location. Hence, in this case, (22) reduces tothe “particle’s medium view” of the cumulated micro-linear time changes. T ( a ) − T ( a ) = (1 /c ) Z aa q h g c f ′ ( t ) dt = Z aa q h g f ′ ( t ) dt, (23)where h g ≥ . The problem is that this is the medium view. What is needed is that (23) berelated to other viewpoints. To do this, the medium view is compared with themedium view where the gravitational field reduces to the Special Theory that is“infinitely close” to the General Theory at a point in spacetime [2, p. 416]. This isbut the Special Theory chronotopic interval expression. For an event that occurs at P , f ( t ) = t, and note that t = t = t E = t, from the medium viewpoint. Hence,(23) reduces to T ( t ) − T ( t ) = Z tt p h g dt. (24)From the chronotopic interval, letting h = 1 and dr = 0 , it follows that dt s = (1 /c ) dS = dT = p g ( t ) dt m . (25)(Letting h = 1 is based upon the requirement that the spatial point is fixed and“time” is varying (a timelike metric) and that the metric reduces to the mediumchronotopic interval metric and that there is no change in h . ) A basic principleis that for infinitesimal regions the gravitational potentials are infinitely close toconstants. The “s” means a medium (substratum) “clock” located at P, wheregravity (or acceleration) affects the “clock” located at P from the medium view-point. The chronotopic interval is used since locally the gravitational alterationsproduce measurements that satisfy this interval statement. The “m” indicates thecoordinate “clock” measured time interval viewed from the medium where there are13o gravitational field effects. In general, g may be time dependent, but the spatialcoordinate names are fixed. Obviously, g is a unitless number.Suppose that g is not time depended (the field is “static”) and, hence, itbehaves like a constant, relative to “time,” at the point. Then this gives for P ∆ t s = √ g ∆ t mP . (26)
6. Medium Time-dilation Effects.
How should equation (26) be interpreted? Consider another spatial point R withinthe gravitational field, where for the two points P, R the expression g is written as g ( P ) , g ( R ) , respectively. Considering the point effect at each pont and applyingthe relativity principle, this gives, in medium t time, that∆ t sP p g ( P ) = ∆ t m = ∆ t sR p g ( R ) (27)Equations like (27) are comparative statements. This means that identical lab-oratories are at P and R and they employ identical instrumentation, definitions,and methods that lead to the values of any physical constants. Since infinitesimallight-clocks are being used, standard “clock” values can take on any non-negativereal number value. The ∆ t sP , ∆ t sR represent the comparative view of the gravi-tationally affected “clock” behavior as observed from the medium where there areno gravitational effects. (The 1 / √ g removes the effects.) Assume a case like theSchwarzschild metric where real √ g <
1. Consider two different locations
P, R along the radius from the “center of mass.” Then there is a constant r s such that r − r s r P ∆ t sR (in R − digits) = r − r s r R ∆ t sP (in P − digits) . (28)where r s ≤ r P , r R . [The cosmological “constant” Λ modification (Λ is not assumedconstant) is s − r s r P − (1 / r P c ∆ t sR = s − r s r R − (1 / r R c ∆ t sP . ] (28) ′ Of course, these (28) [(28)’] are comparisons that must be done with the sametype of “clocks.” As an example for (28), suppose that r s /r P = 0 . r R =100 , r P . Then r s /r R = . . t sR = 0 . t sP .Hence, ∆ t sR = 316 . t sP . Thus, depending upon which “change” is known, thispredicts that “a change in the number of R-digits” equals “316.2262 times a change14n the number of P-digits.” Suppose that at P undistorted information is received.Observations of both the “P-clock” and the “R-clock” digit changes are made.(The fact that it takes “time” for the information to be transmitted is not relevantsince our interest is in how the digits on the “clocks” are changing.) Hence, ifthe “clock” at P changes by 1-digit (the “clock” tick), then the change in theR-digits is 316.2262. The careful interpretation of such equations and how their“units” are related is an important aspect of such equations since (27) representsa transformation. Using a special “ clock” property, if the “R-clock” changes itsreading by 1, then at P the “P-clock” shows that only 0.003162 “P-clock” time haspassed. If you let R = ∞ , then ∆ t sR = 316 . t sP and, in a change in the readingof 1 at P , the R-reading at ∞ is 316.2278. Is this an incomprehensible mysteriousresults? No, since it is shown in [5] that the gravitational field is equivalent to atype of change in the infinitesimal light-clock itself that leads to this result. But, forour direct physical world, thus far, the answer is yes if there is no physical reasonwhy our clocks would change in such a manner. The equation (28) [(28) ′ ] must berelated to physical clocks within the physical universe in which we dwell.
7. The Behavior of Physical Clocks.
Einstein did not accept general time-dilation for the gravitation redshift but con-jectured that such behavior, like the gravitational redshift, is caused by changeswithin atomic structures rather than changes in photon behavior during propaga-tion. This was empirically verified via atomic-clocks. To verify Einstein’s conjecturetheoretically and to locate the origin of this atomic-clock behavior, the compara-tive statement that dt s = √ g dt m is employed. Using special techniques and thetime-dependent Schr¨odinger equation, it is shown in [5] that certain significant en-ergy changes within atomic structures are altered by gravitational potentials. Onceagain, consider identical laboratories, with identical physical definitions, physicallaws, construction methods etc. at two points P and R and within the medium.When devices such as atomic-clocks are used in an attempt to verify a statementsuch as (28), the observational methods to “read” the clocks are chosen in such amanner that any known gravitational effects that might influence the observationalmethods and give method-altered readings is eliminated. For point P , let E sP , de-note measured energy. (The “s” always means gravitationally affected behavior andthe “m” always means the medium view where there are no gravitational effects.)In all that follows, comparisons are made. Using the principle of relativity, thefollowing equation (29) (A) holds, in general, and if g is not time dependent, then(B) holds. 15A) p g ( P ) dE sP = dE m = p g ( R ) dE sR , (B) p g ( P )∆ E sP =∆ E m = p g ( R )∆ E sR . (29)This is certainly what one would intuitively expect. It is not strange behavior.Hence, in the case that g is not time dependent, then p g ( P )∆ E sP = p g ( R )∆ E sR , (30)Quantum mechanics states, at least for an atomic structure, that the totalenergy is controlled by the time-dependent Schr¨odinger equation. For this appli-cation, equation (29) corresponds to the transition between energy levels relativeto the ground state for the specific atoms used in atomic-clocks. But, for this im-mediate approach, the atomic structures must closely approximate spatial points. Further, at the moment that such radiation is emitted the electron is consideredat rest in the medium and, hence, relative to both P and R . The actual aspect ofthe time-dependent Schr¨odinger equation that leads to this energy relation is notthe spatial “wave-function” part of a solution, but rather is developed from the“time-function part.” The phrase “measurably-local” means, that for the measuring laboratory thegravitational potentials are considered as constants. ) Diving each side of (B) in (29)by Planck’s (measurably-local) constant in terms of the appropriate units, yieldsfor two observed spatial point locations
P, R that p g ( P ) ν sP = p g ( R ) ν sR , (31)Equation (31) is one of the expressions found in the literature for the gravitationalredshift [6, p. 154] but (31) is relative to medium “clocks.” Originally (31) wasverified for the case where p g ( R ) ≪ p g ( P ) using a physical clock. Note thatsince the P and R laboratories are identical, then the numerical values for ν sP and ν sR as measured using the altered medium “clocks” and, under the measurably-localrequirement, are identical. Moreover, (31) is an identity that is based upon photonbehavior as “clock” measured.What is necessary is that a comparison be made as to how equation (31) affectsthe measures take at R compared to P, or at P compared to R . Suppose that | ν sA | indicates the numerical value for ν sA at any point A . To compare the alterationsthat occur at P with those at R , | ν sR | P is symbolically substituted for the ν sP andthe expression p g ( P ) | ν sR | P = p g ( R ) ν sR now determines the frequency alterationsexpressed in R “clock” units. As will be shown for specific devices, this is a real effect16ot just some type of illusion. This substitution method is the general method usedfor the forthcoming “general rate of change” equation.
As an example, supposethat for the Schwarzschild metric R = ∞ and let | ν sR | = ν . Then ν s ∞ = ν = p − r s /r P ν . This result is the exact one that appears in [1, p. 222]. However,these results are all in terms of the behavior of the “clocks” and how their behavior“forces” a corresponding alteration in physical world behavior and not the clocksused in our physical world. These results need to be related to physical clocks.Consider atomic-clocks. At P , the unit of time used is related to an emissionfrequency f of a specific atom. Note that one atomic-clock can be on the first floor ofan office building and the second clock on the second-floor or even closer than that.Suppose that the identically constructed atomic-clocks use the emission frequency f and the same decimal approximations are used for all measures and f satisfiesthe measurably-local requirement. The notion of the “cycle” is equivalent to “onecomplete rotation.” For point-like particles, the rotational effects are not equivalentto gravitational effects [8, p. 419] and, hence, gravitational potentials do not alterthe “cycle” unit C. Using the notation “sec.” to indicate a defined atomic-clocksecond of time, the behavior of the f frequency relative to the “clocks” requires,using equation (31), that p g ( P ) 1CP − sec . = p g ( R ) 1CR − sec . . (32) p g ( P ) 1P − sec . = p g ( R ) 1R − sec . . (32) ′ For measurably-local behavior, this unit relation yields that p g ( P )( t R − t R )(R − sec . ) = p g ( R )( t P − t P )(P − sec . ) . (33)Hence, in terms of the atomic-clock seconds of measure p g ( P )∆ t R = p g ( R )∆ t P . (34)Equation (34) is identical with (28), for the specific g , and yields a neededcorrespondence between the “clock” measures and the atomic-clock unit of time.Corresponding “small” atomic structures to spatial points, if the gravitational fieldis not static, then, assuming that the clocks decimal notion is but a consistentapproximation, (34) is replaced by a (28) styled expression p g ( P, t P ) dt sR = p g ( R, t R ) dt sP (35)17nd when solved for a specific interval correlates directly to atomic-clock measure-ments. Also, the Mean Value Theorem for Integrals yields q g ( R, t ′ P )( t R − t R ) = q g ( R, t ′ R )( t P − t P ) , (36)for some t ′ P ∈ [ t P , t P ] , t ′ R ∈ [ t R , t R ] . Equations (34), (35) and (36) replicate, viaatomic-clock behavior, the exact “clock” variations obtained using the mediumtime, but they do this by requiring, relative to the medium, an actual alterationin physical world photon behavior. The major interpretative confusion for suchequations is that the “time unit,” as defined by a specific machine, needs to beconsidered in order for them to have any true meaning. As mentioned, the “unit”notion is often couched in terms of “clock or observer” language. The section 6illustration now applies to the actual atomic-clocks used at each location.For quantum physically behavior, how any such alteration in photon behavioris possible depends upon which theory for electron behavior one choices and someaccepted process(es) by which gravitational fields interaction with photons. Are thealterations discrete or continuous in character? From a quantum gravity viewpoint,within the physical world, they would be discrete if one accepts that viewpoint.This theoretically establishes the view that such changes are real and are due to“the spacings of energy levels, both atomic and nuclear, [that] will be differentproportionally to their total energy” [3, pp. 163-164]. Further, “[W]e can rule outthe possibility of a simple frequency loss during propagation of the light wave. . . .”[8, p. 184]. This gravitational photon frequency redshift is not the only redshift thatoccurs in the behavior of electromagnetic radiation. For example, for the behaviorof photons, there is the derivable Special Theory alterations as well as the cosmic“redshift.”Although the time-dependent Schr¨odinger equation applies to macroscopic andlarge scale structures via the de Broglie “guiding-wave” notion, the equation hasnot been directly applied, in this same manner, to such structures since they arenot spatial points. However, it does apply to all such point-approximating atomicstructures since it is the total energy that is being altered. One might concludethat for macroscopic and large scale structures there would be a cumulative effectfor a collection of point locations. Clearly, depending upon the objects structure,the total effect for such objects, under this assumption, might differ somewhat atdifferent spatial points. However, the above derivation that leads to (34) is forthe emission of a photon “from” an electron and to simply extend this result toall other clock mechanisms would be an example of the model theoretic error ingeneralization unless some physical reason leads to this conclusion.18quation (34) is based upon emission of photons. Throughout all of the atomicand subatomic physical world the use of photon behavior is a major requirementin predicting physical behavior, where the behavior is not simply emission of thetype used above. This tends to give more credence to accepting that, under themeasurably-local requirement, each material time rate of change, where a physicallydefined unit U that measures a Q quality has not been affected by the gravitationalfield, satisfies p g ( P )∆ Q P in a P − sec . = p g ( R )∆ Q R in an R − sec ., ∆ Q R in an R − sec . = p g ( P ) p g ( R ) | ∆ Q R | P , (37)Equations (37) give a comparative statement as to how gravity alters such atomic-clock time rates of change including rates for other types of clocks.Prior to 1900, it was assumed that a time unit could be defined by machinesthat are not altered by the earth’s gravitational field. However, this is now knownnot to be fact and as shown in [5], such alterations in machine behavior is probablydue to an alteration in photon behavior associated with a substratum stationarysource that undergoes two types of physical motion, uniform or accelerative. Thereis a non-reversible substratum process that occurs and that alters photon behavioras it relates to the physical world. These alterations in how photons physicallyinteract with atomic structures and gravitational fields is modeled (mimicked) bythe defining machines that represent the physical unit of time, when the mathe-matical expressions are interpreted. The observed accelerative and relative velocitybehavior is a direct consequence of this non-reversible process. As viewed from thesubstratum, every uniform velocity obtained from the stationary first requires accel-eration. This is why the General Theory and the Special Theory are infinitesimallyclose at a standard point.It is claimed by some authors that regular coordinate transformations for theSchwarzschild solution do not represent a new gravitational field but rather allowsone to investigate other properties of the same field using different modes of obser-vation. When such transformations are discussed in the literature another type ofinterpretation appears necessary [5, p. 155-159]. Indeed, what occurs is that theoriginal Schwarzschild solution is rejected based upon additional physical hypothe-ses for our specific universe that are adjoined to the General Theory. For example,it is required that certain regions not contain physical singularities under the hy-potheses that physical particles can only appear or disappear at chosen physical“singularities.” Indeed, if these transformations simply lead to a more refined view19f an actual gravitational field, then the conclusions could not be rejected. Theywould need to represent actual behavior. One author, at least, specifically statesthis relative to the Kruskal-Szekeres transformation. In [10, p. 164], Rindler rejectsthe refined behavior conclusions that would need to actually occur within “nature.”“Kruskal space would have to be created in toto : . . . . There is no evidence thatfull Kruskal spaces exist in nature.”One way to interpret the coordinate transformation that allows for a descriptionof “refined” behavior is to assume that such described behavior is but a “possibility”for a specific gravitational field and that such behavior need not actually occur. Thisis what Rindler appears to be stating. But, since such properly applied coordinatetransformations also satisfy the Einstein-Hilbert gravitational field equations, then,from the medium view, using collections of such “possibilities” is equivalent toconsidering different gravitational fields. For the medium view of time-dilation, thisleads to different alterations in the atomic-clocks for each of these “possibilities.”These results, as generalized to the behavior exhibited by appropriate physicaldevices, imply that no measures using these devices can directly determine theexistence of the medium. Although Newton believed that infinitesimal values didapply to “real” entities and, hence, such measures exist without direct evidence,there is a vast amount of indirect evidence for existence of such a medium. References [1] Bergmann, G., Introduction to the Theory of Relativity, Dover, New York, 1976.[2] Craig, H. V., Vector and Tensor Analysis, McGraw-Hill, New York, 1943.[3] Cranshaw, T. E., J. P. Schiffer and P. A. Egelstaff, Measurement of the red shiftusing the M¨ossbauer effect in Fe ,,