Gravitational Time-Dilation

Robert A. Herrmann
Revised 26 JAN 2008 and 6 JUN 2012.

In modern physics, "time" is considered as an actual physical primitive. As such its properties are only operationally represented. That is, the properties of (physical or observer) time are relative to its measure and the instrumentation used to make such measurements. In this article, the notion of time-dilation is generally restricted to "local" time-dilation. From this operational approach, the notion of time-dilation needs to be restricted to the specific physical objects used to measure time, general known as clocks, and, in particular, the constituents of specific clocks. A major question that needs to be investigated relative to clock constituents is "how" gravitational fields might affect these constituents and yield the actual experimental evidence that indirectly verifies time-dilation and, whether all constituent physical behavior is affected, in this time-dilation manner, by a gravitational field. That is, does it actually apply to all physical objects that can be considered as clocks? In this paper, I use my own analysis as well as a considerable amount of analysis that appears in the referenced literature. In section 3, experimental evidence is discussed that indirectly verifies a predicted physical mechanism for certain modes of time-dilation.

1. The Equivalence Principle.

An Equivalence Principle appears to be necessary in every General Theory of Relativity (GR) derivation for expressions that predict gravitational effects attributed to "(local) time-dilation." The question is which Equivalence Principle? Einstein first stated his equivalence principle as follows:

[Let K' be a system of reference such that] relative to K' a mass sufficiently distant from other masses has an accelerated motion such that its acceleration and direction of acceleration are independent of its material composition and physical state.

Does this permit an observer at rest relative to K' to draw the conclusion that he is on a "really" accelerated system of reference? The answer is negative; for the above mentioned behavior of freely moving masses relative to K' may be interpreted equally well in the following way. The system of reference K' is unaccelerated, but the space region being considered is under the sway of a gravitational field, which generates the accelerated motion of the bodies relative to K'. (Ohanian and Ruffini, 1994, p. 53)

However, Fock writes:

As was mentioned, Einstein considered that from the point of view of the Principle of Equivalence it is impossible to speak of absolute acceleration just as it is impossible to speak of absolute velocity. We consider this conclusion of Einstein's to be erroneous . . . (1959, p. 208)

Associated with this Einstein description is the famous "elevator" illustration. However, Fock gives an example that uses a rotating non-infinitesimal (i.e. non-local) physical structure and this example explicitly contradicts the above Einstein statement. It seems that under most conditions experienced within our universe such effects, for macroscopic entities, can be differentiate one from another. This led Fock to modify, at the least, Einstein's original hypothesis so as not to forget "that the nature of equivalence of fields of acceleration and of gravitation is strictly local." (Fock, 1959, p. 369) [Also see pages 206-210 of Fock's text for an extensive and excellent analysis of this concept.]

This is not the last word on this principle, however, for what would be needed is an actual physical experiment that would also demonstrate Einstein's error. The mathematical model chosen to model the Einstein theory of gravity uses the infinitesimal calculus. The methods of corresponding physical behavior to the mathematical structure were specifically ignored. One needs to approximate physical infinitesimal measures before such a structure can be considered as a meaningful mathematical model. Gravitational tidal effects are the force-like effects that a gravitational field has upon physical objects. Because of the existence of a special instrument called a gravity gradiometer, an instrument that can measure the local differences in the tidal effects or what is termed the tidal fields, the above Einstein equivalence principle can be experimental falsified.

Further, the gravity gradiometer instrument can be reduced to a comparatively small size and this reduction in size represents an approximate physical "infinitesimalizing process." Apparently, as this instrument is reduced in size it is less likely to measure differences between gravitational tidal forces and the tidal forces associated with pure acceleration.

One may be able to determine by further analysis a better way to state such an equivalence principle. For this theory, the term point particles intuitively refers to physical entities that are small in size or massless. Further, such entities are, usually, restricted to behavior within "small" spacetime intervals (i.e the notion of local) and are assumed not to have a significant gravitational field themselves so that they do not measurably influence the "stronger" gravitational field being investigated. It is the mathematical "differential" model that, by a special type of summation of these effects, allows one to describe behavior of point particles over macroscopic or large-scale spacetime. The following is a general description relative to all possible gravitational fields within our universe.

Local experiments can distinguish between a reference frame in free fall in a gravitational field and a truly inertial reference frame placed far away from all gravitational fields. Local experiments can distinguish between a reference frame at rest in a gravitational field and an accelerated reference frame far away from all gravitational fields. Gravitational effects are not equivalent to the effects arising from an observers acceleration . . . . Gravitation and acceleration are only equivalent as far as the [local] translational motion of point particles is concerned (this amounts to what we call the Galileo principle of equivalence, sometimes also called the weak principle of equivalence.) If rotational degrees of freedom of the motion of masses is taken into consideration, then the equivalence fails.(Ohanian and Ruffini, 1994, p. 53)

This last statement assumes that one is not considering a "perfectly homogeneous gravitational field" (Ohanian and Ruffini, 1994, p .53), which, using the usual secular methods,is not known to exist since the formation our basic "material" universe.

With respect to rotation, Ohanian and Ruffini show specifically that it follows from GR that even ". . . Galileo's principle of equivalence fails for spinning particles." (Ohanian and Ruffini, 1994, 419.) Actually, it might be better to state that the principle "approximately" fails, in this case, since one can state a certain degree of failure with respect to "free-fall" motion. Hence, from what is stated above, to apply an Equivalence Principle in an exact manner locally, the principle would be the Galileo principle applied to the non-rotational behavior of single objects that approximate point-like particles. Such objects exist in the molecular, atomic and subatomic regions, or when applicable they can be the so-called massless "particles" such as photons. Obviously, this does not mean that specific time-dilation effects do not occur for collections of such constituents and, indeed, this does appear to be the case for experiments involving such gravity-produced effects, when such local effects are properly extended to non-local regions. This includes non-point-like rotations since such behavior is modeled by considering it as "directed" local (infinitesimal) linear behavior.

Relative to derivations for the gravitational effects attributed to time-dilation, we are told that ". . . it [gravitational field] exerts an indirect effect on the relative rates of clocks at different positions." (Ohanian and Ruffini, 1994, p. 166) But after the clock rate derivation given from pages 167 through 171, we have "Note the crucial role played by the equivalence principle in several stages of the above arguments." (Ohanian and Ruffini, 1994, p. 171) "And note that the radar-ranging [macroscopic and large-scale] procedure implicitly relies upon the equivalence principle: in the local inertial reference frame of the freely falling clock, light propagates as in the absence of the gravitational field, at its standard speed." (Ohanian and Ruffini, 1994, p. 169) After this derivation, there is the Einstein styled energy derivation relative to photons on page 186 of Ohanian and Ruffini (1994) and, yet, another derivation based upon the Galileo equivalence principle on page 187 in the terms of "clocks." Note that the equivalence of what is called inertial mass and gravitational mass, which is sometimes called Newton's equivalence principle, leads, using Newtonian approximation, only to the Galileo equivalence principle (Ohanian and Ruffini, 1994, 21-24) assuming the objects have the same initial velocity.

2. The Model Theoretic Error of Generalization

One of the basic logical errors within mathematical modeling is called the model theoretic error of generalization. This has been known to be an error in scientific discourse for many years. For example, in 1850, John Stuart Mill mentions this error from a purely philosophical viewpoint (Mill, 1963-1992, p. 785). The error is very simple to state. A derivation that a specific physical event will occur under certain circumstances uses a fixed language and, usually, scientific logic. From the viewpoint of pure formal logic, this language corresponds to predicates of one form or another that are restricted to certain domains. A model theoretic error of generalization occurs when one claims that the predicted events now hold for other domains, where no derivation is given with respect to these other domains.

The above Einstein description, his Equivalence Principle, is an explicit example of such an error being made. There are various "derivations" for gravitational effects attributed to dilation of "clock" rates using the Einstein concept of equivalence. These so-called derivations do not even define specifically the term "clock" except to say that it uses some type of process to "measure" this quantity. However, as discussed above, the only correct derivations using such a general term as "clock" appear to require an equivalence principle restricted to the Galileo principle. Hence, unless one can give a derivation for gravitational effects attributed to time-dilation that does not involve the Galileo equivalence principle, which applies to approximating point particles, then claiming that such gravitational effects are attributable to time-dilation and hold for "all" clocks, macroscopic, large-scale and atomic would appear to be a specific example of the model theoretic error of generalization within a physical theory.

There are, of course, significant examples within mathematical modeling. Suppose that you model a specific measure only using the natural numbers greater than 0. Then, assuming that your model does predict physical behavior, you could truly state that any set of such measures would contain a smallest measure. [Note: that if your model uses decimals with a fixed number of significant digits, these measures can be considered as modeled by the natural numbers.] If you then state that these measures are but approximations for measures that are actually being modeled by the real numbers, you cannot say that any such set of measures has a smallest member. You cannot extend this simple property of the natural number "ordering" to the ordering of the real numbers, where it does not hold.

To further establish that such gravitational effects attributed to time-dilation should first be restricted to atomic, subatomic or massless objects, there is in Herrmann (1994, p. 73-75) and Herrmann (1995) a derivation using the time dependent Schrödinger equation. The differential is often used to justify a type of summation of physical effects through the use of the integral. As discussed later in this article, this process must be carefully considered for the effects of gravitational time-dilation. If one conceives of a macroscopic device as composed of coupled atomic entities that are influenced by gravitational time-dilation, one might assume that as a coupled group the entire macroscopic entity viewed as a single object would be so influenced. However, such a macroscopic device considered as composed of such a coupled group would not satisfy the Galileo equivalence principle except very approximately. The only possible and exact Galileo equivalence principle effect would be "derivable effects" associated with each individual point-like particle or the totality of these effects as produced by each individual particle.

3. Location of Some Gravitational Time-Dilation Effects

Using special techniques, it is possible to view our universe from a Euclidean-type "medium." As viewed from that medium, our universe seems to behave as if it "likes" local linear paths and any deviation from such paths requires a "price to be paid," so to speak. Additionally, within the medium, "light" paths are only linear in a very local sense. How they divert from such linear paths in a global sense is what leads to the Special Theory of Relativity. For the General Theory, where the notion of "mass" and a possible non-zero cosmological constant are adjoined, the General Theory deviations physically require types of "forces" that are measured in terms of "accelerations." Locally, a gravitational field and an acceleration field yield identical results. The only difference appears to be causal. The following discussion is related to a gravitational cause, but applies equally well to the notion of an acceleration field. The time-dilation notions consider here are couched in terms of gravitational acceleration. For these gravitational effects, it is sometimes difficult to determine which gravitational effects are caused by what has been termed as time-dilation.

One problem for certain physical objects such as photons is whether the dilation is a local or global effect. From the material in section 1, local time-dilation effects, the basic type being considered here, are associated with the Galileo equivalence principle and appear to be associated with entities of a small spacetime "size." [Most often the "time" is actually "fixed" and the interval only relates to a spatial concept.] However, it is the derivation that specific gravitational effects should apply to a particular physical scenario that tends to indicate which effects are associated with measurable alterations that are customarily termed as "time-dilation." Once one has made a determination relative to such gravitationally produced alterations, there is an additional question that may be difficult to answer for all scenarios. Is the gravitational field producing such effects independent from the machines used to observe the effect, or is the gravitational field simply altering the machines that observe a specific physical scenario and the field is not actually altering the physical scenario as predicted by a time-dilation derivation? The derivations for such time-dilation as they appear in the references imply that time-dilation is but alterations in measuring instruments.

As discussed in section 5, the use of the infinitesimal light-clock concept, that in the non-infinitesimal state was first introduced by Einstein, and what follows in this section will aid in determining what are (local) time-dilation effects and exactly "where" the effects take place in spacetime. As discussed in Herrmann (1994) and to avoid the model theoretic error of generation, such a device must be one that approximates an infinitesimal light-clock or an equivalent device. It is alterations in the behavior of such light-clocks that yields actual physical causes, distinct from the gravitation field itself, for local alterations in other measuring devices as well as alterations in other physical behavior.

Using the statement in Bergmann (1976, p. 222), one assumes that "emission" from an atomic structure occurs when a particle is momentarily "at rest" in the gravitational field. The method illustrated in Herrmann (1994, p. 73-75) and Herrmann (1995) using the time dependent Schrödinger equation can be applied trivially to the Schrödinger equations that model more complex atomic structures assuming that if one has an explicit solution, then it can be expressed in a factored time-function form. This form has always been the case for all verified applications of the separated operator method. For such systems, the derivation yields that the changes in the total energy at a "specific gravitational potential," when viewed from the medium, are altered when compared with energy changes in our physical world where the gravitational potential is ignored. This "dilation" is expressed by the equation

where is the usual GR factor for an electrically neutral non-rotating centrally symmetric and homogenous spherical object (a Schwarzschild configuration). Dividing this equation by Planck's constant, this result can be written in terms of any associated change in frequency (the Greek nu),

which for emissions of electromagnetic radiation yields the exact GR prediction for the observed "gravitational redshift." In this case, the frequency with the "s" superscript is the laboratory measured frequency where gravity is ignored. The Special Theory chronotopic interval is used as an important requirement in the derivation since locally the General Theory is "infinitely close" to the Special Theory. (The similar expression can be obtained for a second location and a quotient compares the two frequencies.) The frequency with the "m" superscript is the observed or "created," or altered frequency that occurs a radial position "r" units from the center, but exterior to, a Schwarzschild configuration and as viewed from a medium known as the NSPPM-field. In this case, this is equivalent to viewing the effects of the frequency alteration, where there is no measurable gravitational field at a great distance from the gravitational center. The usual approach is to consider the "s" or standard frequency as measured in a very weak gravitational field and the observed altered frequency "m" occurs in a very strong field. In this case, this expression is the same as the one that appears in Bergmann, (1976, p. 222).

The Bergmann derivation is obtained via time-dilation. The Schrödinger equation approach verifies that an alteration takes place within the atomic structure. This is, at present, a statement for a total energy change that depends upon the values of M and r. This would lead to the appropriate adjustment in the energy levels for the constituents. Depending upon the how a gravitational field actually interacts with a quantum physical structure, this could be but an acceptable approximation. On the other hand, one can suppose that the gravitational field is really an example of a pure primitive continuous field that appeared at the same moment that all of the other physical aspects of the universe appeared and that any law of gravity that incorporates such parameters simply displays relations that would exist between the measured numbers M and r, relations that allow the effects of the field to be measured and behavior predicted. From the quantum-physical viewpoint, a particle "free in space" would require a type of "continuous" alteration. This would make a field particle such as the proposed graviton but a convenient fiction used to comprehend somewhat the interaction of this field with atomic structures. Personally, I don't believe that there is a "correct" humanly comprehensible solution to this "interaction" problem with the exception of the purely operational approach used in properton theory, where no such interaction notions are required.

To obtain the more exact frequency variation statement for comparison purposes, relative to Schwarzschild gravitational potentials, one would simply consider the ratio expression derived form two of the above frequency statements. The result, in this case, is then the exact same statement as appears in Lawden (1982, p. 154). For other physical configurations, such as one where we drop the non-rotating or non-electrically neutral character of the Schwarzschild configuration and others, the only difference in the above frequency equation is how the γ is expressed. My derivation of this gravitational redshift expression is obtained using changes in the behavior of infinitesimal light-clocks as further mentioned in the last section of this article.

Of significance is that the Schrödinger equation derivation in Herrmann (1995) was for alterations in the total energy Es not just the total energy changes. The reason it is restricted in the derivation to the emission of electromagnet radiation follows from the notion of "momentary at rest." As mentioned in Herrmann (1994), the same derivation applies to other atomic structures, among others, relative to the total energy. Then depending upon the structure, one needs to investigate whether only one or more aspects of the total energy is being affected by the gravitational potential.

One of the first laboratory experiments relative to a test of this frequency relation was done in 1960 by T. E. Cranshaw, J. P. Schiffer, and A. B. Whitehead. Although the Schrödinger equation derivation was not available at that time and the light-clock methods used for such derivations demonstrate that the results are more likely attributable to electromagnetic properties of atomic structures, a possible atomic structure correspondence was stated by these researchers.

From the point of view of a single coordinate system two atomic systems at different gravitational potentials will have different total energies. The spacings of their energy levels, both atomic and nuclear, will be different in proportion to their total energies.(Cranshaw et al., 1960, p. 163-164)

These researchers, using "nuclear clocks" and gamma ray emissions, showed that this variation in frequency did occur to within an accuracy of about 2%. A Schrödinger equation derivation predicts such a change and verifies the Cranshaw et al. speculation as well as a less specific Einstein conjecture.

The most famous and direct laboratory measurement of this dilation effect on atomic structures was that of Pound and Rebka, Jr. in 1959-60 and by Pound and Snider (1965, p. B788-B803) and showed that the above predicted alteration in gamma ray frequency is correct within an experimental error of 1%. Ohanian and Ruffini state relative to these experiments:

The latter experiment does not give quite as direct an indication of whether the frequency shift between the absorbed gamma rays and the natural nuclear oscillations is due to slower oscillation rate of the emitter or a loss of frequency suffered by the gamma ray as it climbs upward in the Earth's field. However, we can rule out the possibility of a simple frequency loss during propagation of the light wave . . . . Experiments with flying [atomic] clocks, in aircraft and rockets, have a crucial conceptual advantage over the gamma-ray experiment, in that the former experiments show in the most direct way that clocks in a gravitational potential run slower (Ohanian and Ruffini, 1994, p. 184)

Hydrogen-maser clocks and other types of atomic clocks have confirmed to a great degree this same "frequency variation" in any atomic structure where variations in total energy lead to frequency measurements. Astronomical measurements also confirm the predicted frequency alteration discussed previously that produces a particlar form of the "gravitational redshift" for light emitted from atoms on "surfaces" of stars.

If the correct methods for combining infinitesimal changes are applied to specific macroscopic objects, then depending upon how the total energy variations are distributed, other measures that characterize physical behavior are altered, when compared to a standard. Although such alterations might not be classified as variations due to time-dilation, they actually are related to, at least, one time-measuring device. The alterations may be obtained from other derivations, but since for the derivations considered here the "time" depended Schrödinger equation is used to obtain the alterations, then they are associated with alterations in, at the least, a type of light-clock. From the viewpoint of some cosmologies, this might exacerbate what are the known energy problems associated with the cosmology and gravitational fields. No such problems exist from the properton and medium view point.

A "fourth test of GR" was purposed in 1964 (Shapiro, 1964). This is another test using photons to measure a predicted alteration in their "global" measured speed. This alteration is postulated to be caused by a gravitational field as a type of retarding medium. Of course, this uses the speed and wavelength model rather than the probabilistic (Feynman, 1985) model. This is a "time delay" test associated with a photon as it passes through a gravitational field. This delay would be cumulative and the analysis of this delay can be done with respect to an infinitesimal time change of tm as it is being measured by an infinitesimal light-clock within the gravitational field as viewed from the medium. When infinitesimal light-clocks are used for the derivation, the expression obtained is the exact same one as [50] in Ohanian and Ruffini (1994, p. 198) as it is derived from expression [52] (Ohanian and Ruffini, 1994, p. 202). How infinitesimal light-clocks enter into these calculations will be discussed more fully in the last part of this article.

The derivation as given in Ohanian and Ruffini (1994, p. 202) for this delay shows that two infinitesimal light-clocks within the gravitational field are being considered as analogue models and as such are being used to measure this "global" photon speed. [Infinitesimal light-clocks do not predict such alterations through any alteration of the photon speed within the light-clock. Alterations in the light-clock counts is produced by considering different infinitesimal light-clocks where there is a "bounded" alteration in their construction.]

4. How the Differential is Used as a Model for Physical Behavior

Suppose that a physical measure  F  is represented by a real or complex number, a vector or the like. The values for  F  depend upon or are influenced by a set of independent variables. Suppose that for this discussion there is but one such variable  x. To directly employ the differential in any of its usual forms,  x , as a physical measure, must behave in a certain manner. It must satisfy the Leibniz' principle. For the infinitesimal calculus, this is also called the infinitesimalizing process. This principle states

1. First, assume or establish that an observed change in physical behavior being measured by  F  corresponds to a change in physical qualities being measured by values of the variable  x. If it is reasonable to assume that the physical qualities being measured by the values of  x  can be altered in such a manner that a change in the value of  x  can be made extremely small in "value" (and I mean just that a number or vector "length," or the like that is intuitively near to zero), then the differential infinitesimalizing process holds and differentials may be an appropriate model that will predict the value of the  F. How accurate this model will be in making such a prediction depends upon how "close" this nonzero infinitesimalizing process approximates zero. [Note that infinitesimal type objects and "differentials" mathematically exist under more general conditions than illustrated here. Further, nothing in these statements should preclude the indirect use of differentials as a mere modeling artifact, where one simply "reasonably smooths out" discrete behavior.]

2. If an overall change in behavior of a particular physical entity is a type of summation of the infinitesimal changes in  F  that occur when the  x  changes by such an infinitesimal amount, then differential models may be an appropriate model for such changes in physical behavior.

Our modern knowledge of the mathematical structure refines the usual intuitive concept as previously used within theoretical physics as it is explained by Max Planck.

[A] finite change in Nature always occurs in finite time, and hence resolves into a series of infinitely small changes which occur in successive infinitely small intervals of time. (Planck, 1932, p. 2.)
It took more than 300 years to discover the correct refinements to this Planck statement (Herrmann, 1990). There is a much closer relationship between the physical world and the measures considered than expressed by this Planck statement and the terms such as "infinitely small" and "successive" could not actually be formally modeled until after 1961. A most important aspect of the second requirement above is the "summation" concept. There are two possible "summations" allowed.

3. The differential can be applied to a specific entity influenced by these infinitesimal changes in the behavior of  x  if such changes affect the behavior of the entity as a whole. Then a change in the behavior of the entire entity is a special summation of these infinitesimal changes.

4. It has been shown specifically that if an infinitesimal change in  x  "approximates" a non-infinitesimal change in the behavior of a particular physical entity, denoted by  A , and the change in the behavior of another physical entity  B  depends completely upon the combined effects of each of an enormous but finite (not infinite in the usual sense) number of the  A  entities, then the differential can be used as an ideal approximating model that predicts the behavior of the  B  physical entity.

The facts are that (3) and (4) are stating different requirements. Statements (3) and (4) express how one must modify the Planck statement, a statement that is experientially obtained and that is not derivable. Indeed, there are examples where if these rules are not followed, then the approach using the differentials fails to provide an adequate model for physical behavior. In Herrmann (1990,) an example is given that shows that, in general, one cannot use in (4) an infinite series of objects that are not measured by infinitesimals to model the cumulative behavior of a set of approximating physical objects.

5. Infinitesimal Light-clocks

In 1905, relative to the Special Theory of Relativity, Einstein introduced into scientific modeling a specific approach where he described devices that would provide the needed measurements for the quantities being measured. In particular, the well-known light-clock constructed from the "radar-ranging" concept (i.e. photons reflecting back-and-forth between mirrors.) Further, he introduced the operational approach. Since his introduction of this approach, it has become required practice to associate physical measures with measuring devices, when a mathematical model is being constructed. This process uses physical terms and associates the entities within an abstract mathematical structure with specifically named physical entities within an assumed or perceived physical world.

However, this light-clock notion considered as a physical model could not be extend, at that time, to the differential calculus. One reason for this is that there did not exist mathematical entities that modeled the intuitive concept of the infinitesimal as introduced by Newton. It was not until 1961, that Abraham Robinson made one of the most significant mathematical discoveries of the twentieth century. It was at that time that the mathematical properties of the infinitesimals first appeared as a portion of his discovery (Robinson, 1961).

The line element methods used to derive the time-dilation effects of gravitational fields use specific infinitesimal changes in "time" to measure the predicted and observed changes taking place with respect to physical objects. There is no doubt that the model uses these infinitesimal changes in "time" to predict how specific timing devices will behavior when located within a gravitational field and to predict cumulative effects that occur with respect to certain individual point-like objects. It is the infinitesimal changes in "time" that models each of these pure time-dilation effects. Using the modern Einstein approach and our present day knowledge as to the correct rules for infinitesimal modeling, then these infinitesimal changes in "time" must be modeled, approximately, by an actual device that satisfies all of the requirements.

The modern approach to mathematical modeling does not allow for the philosophical concept of "time" to be considered at all but forces upon the scientific world the requirement that an actual "infinitesimalizable device" be described. Assuming that the Riemannian geometry is an appropriate analogue model for gravitational field investigations, such an infinitesimal-type clock is constructed in the paper by Marzke and Wheeler (1964). However, this construction is after the fact rather than before. Although the mathematics may seem formidable, using the modern theory of infinitesimal and infinite numbers, various infinitesimal light-clocks are described in Herrmann (1994, Article 2, section 6) using only two observed properties of electromagnetic radiation and nothing more. These infinitesimal light-clocks are utilized to investigate various aspects of the Special and General Theories of Relativity in the search for a specific physical cause for such behavior as time-dilation. Of course, one should not expect the answer to be easily obtained when one considers that it took over 300 years to discover the formal mathematical properties for the infinitesimals.

These infinitesimal light-clocks, since they display pure Robinson infinite numbers with their infinitesimal inverses, are appropriate for any theory consistent with their construction that uses differentials as models for physical changes. The most significant aspect of this more basic infinitesimal light-clock interpretation is that it is the alteration in the behavior of this specific "clock" that models the alteration in behavior displayed by other physical entities, regardless of how these other physical entities are employed for the purpose of physical measurement. The infinitesimal light-clocks are analogue models that are considered to undergo the physical alterations due to gravitational fields for their application to GR. For time-dilation, only the timing infinitesimal light-clock is used to measure gravitational alterations in appropriate entities that satisfy the requirements of the Galileo equivalence principle over a local spacetime interval. Although their use is analogue in character, there is no doubt that infinitesimal light-clocks imply that such gravitational alterations attributed to time-dilation are actually alterations in behavior associated with an interaction with the gravitational field.

It is an interesting exercise to investigate relative to gravitational effects which of the two types of "summation," 3 or 4, applies to a specific problem. There have arisen various theories as to exactly what "gravity" may be from a more fundamental viewpoint and some of these theories present distinctly different causes for such time-dilation. Although it is of no significance to the material presented within this article, infinitesimal light-clocks can be used to develop such a theory and this theory is discussed in more detail in Herrmann (1994).

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Math. Dept., U. S. Naval Academy, 572C Holloway Rd., Annapolis, MD 21402-5002

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