The clock problem (clock paradox) in relativity

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The clock problem (clock paradox) in relativity  (1959) 
by Mildred Catherine Benton


Theories, Both Pro and Con, Recorded in the Literature
An Annotated Bibliography

Compiled by Mildred Benton
Library Branch, Technical Information Division

May 1959



Washington, D.C.


Current interest in the possibilities of space flight and the prediction that atomic clocks in earth satellites may be utilized to check Einstein's theory have focused attention on the clock problem in relativity, or the so-called time-traveler paradox and its implication. General theory holds that a precise clock will run more slowly at extreme altitudes than an earthbound clock because of differences in gravitational fields. Special theory holds that rates of two clocks will vary because of relative motion between the two. Implication is that, of two observers who part company, travel with a relative speed, and rejoin one another, one will record a shorter lapse of time than the other. Thus, a time advantage will exist for the traveler on an extended voyage. The question naturally arises, Will space travel lengthen life? Some experimenters vouch that it will. Others debunk such a theory.

Einstein first introduced the clock paradox in 1905, although Michelson touched upon the subject in a report of his experiments in 1882. Since that time many thoughts on the subject have been expressed and the question posed, discussed and interpreted by many scientists and writers of popular articles.

The purpose of this bibliography is to serve, through the medium of recorded references, as a sort of debating platform for what has been said and done in an effort to prove a point one way or the other.

Periodical articles predominate, most of them in English, but some in foreign languages. A list of periodicals cited together with the abbreviations used may be found on page iii. There are a few books and three research reports. Arrangement of the references is alphabetical by author. For the benefit of those who may wish to investigate further, or for those who have only a cursory interest, annotations indicative of the contents of each item are included. Whenever abstracting publications are quoted, the source is stated. It is believed that the majority of items are available for consultation in the larger public, or research libraries.

Sources Searched


U. S. Naval Research Laboratory

Indexes and Abstract Journals

Applied Science and Technology, 1913 - March 1959.

Battelle Technical Review, 1956 - March 1959.

British Interplanetary Society. Journal, 1947 - 1958.

Dissertation Abstracts, 1956 - 1958.

Engineering Index, 1919 - 1957.

Science Abstracts. Section A. Physics, 1898 - February 1959.

Abbreviations Used in Citations to Periodicals

The following is a list of abbreviations used in citing references to periodicals, followed by the complete title of the periodical.

Acad.​Sci.​Paris.​Compt.​Rend. Academie des Sciences, Paris. Comptes Rendus Hébdomadaires des Séances.
Accad.​Naz.​Lincei.​Atti. Accademia Reale Nazionale dei Lincei Atti.
Acta Phys. Acta Physica.
Aero.​Dig. Aero Digest.
Akad.​Nauk.​SSSR.​Dok. Akademiya Nauk,SSSR Doklady.
Am.​J.​Phys. American Journal of Physics.
Am.​J.​Sci. American Journal of Science.
Am.​Phil.​Soc.​Proc. American Philosophical Society. Proceedings.
Am.​Phys.​Teacher. American Physics Teacher.
Am.​Scientist. American Scientist.
An.​Real.​Soc.​Espan.​Fís.​Quím. Anales de la Real Sociedad Espanola de Físicay Química.
Ann.​Math. Annals of Mathematics.
Ann.​Physik. Annalen der Physik.
Atlantic Mon. Atlantic Monthly.
Austral.​J.​Phys. Australian Journal of Physics.
Bell Lab.​Rec. Bell Laboratory Record.
Brit.​Interplan.​Soc.​J. British Interplanetary Society. Journal.
Camb.​Phil.​Soc.​Proc. Cambridge Philosophical Society. Proceedings.
Can.​Aero.​J. Canadian Aeronautical Journal.
Centro.​Aero.​Atom.​Ital. Centro Aeronautico e Atomico Italiano.
Commercial and Finan.​Chron. Commercial and Financial Chronicle.
Dansk.​Videns.​Selskab.​Math.​Fys.​Medd. K. Danske Videnskabernes Solskab, Copenhagen. Mathematisk-fysiske Meddelelser.
Engr. Engineer (London).
Fortschr.​Physik. Fortschritte der Physik.
Frank.​Inst.​J. Franklin Institute. Journal.
Ges.​Wiss.​Göttingen.​Nachr. Gesellschaft der Wissenschaften zu Göttingen.Nachrichten.
Helv.​Phys.​Acta. Helvetica Physica Acta.
Hibbert J. Hibbert Journal.
Inst.​Radio Engrs.​Proc. Institute of Radio Engineers. Proceedings.
J.​Astronautics. Journal of Astronautics.
J.​Phys.​et Radium. Journal de Physique et le Radium.
J.​Space Flight. Journal of Space Flight.
Jet Propul. Jet Propulsion.
Junior Inst.​Engrs.​J. Junior Institute of Engineers. Journal (London).
Mech.​Eng. Mechanical Engineering.
Metropolitan Vickers Gaz. Metropolitan Vickers Gazette.
Natl.​Geog.​Mag. National Geographic Magazine.
Naturw. Naturwissenschaften.
Ned.​Akad.​Wetens.​Proc. Koninklijke Nederlandse Akademie van Wetenschappen.
Opt.​Soc.​Am.​J. Optical Society of America. Journal.
Phil.​Mag. Philosophical Magazine.
Phys.​Rev. Physical Review.
Phys.​Soc.​Proc. Physical Society (London). Proceedings.
Phys.​Today. Physics Today.
Physik.​Z. Physikalische Zeitschrift.
Prog.​Theor.​Phys. Progress of Theoretical Physics.
Redstone Arsenal Ordnance Missile Labs.​Quart.​Res.​Rev. Redstone Arsenal Ordnance Missile Laboratories Quarterly Research Review.
Rev.​Mexicana Fis. Revista Mexicana de Fisica.
Rev.​Mod.​Phys. Reviews of Modern Physics.
Rocket News Ltr. (Chic.​Rocket Soc.​) Rocket News Letter (Chicago Rocket Society).
Roy.​Astron.​Soc.​Mon.​Not. Royal Astronomical Society. Monthly Notices.
Roy.​Dublin Soc.​Sci.​Proc. Royal Dublin Society. Scientific Proceedings.
Roy.​Soc.​Edinburgh.​Proc. Royal Society of Edinburgh. Proceedings.
Roy.​Soc.​London.​Proc. Royal Society of London. Proceedings.
Sci. Science.
Sci.​Am. Scientific American.
Sci.​News Ltr. Science News Letter
Scripta Math. Scripta Mathematica.
Space J. Space Journal.
Studia Filoz. Studia Filozoficzne.
Ver.​Deut.​Ing.​Z. Verein Deutsche Ingenieure Zeitschrift.
Z.​Physik. Zeitschrift für Physik.


A Bibliography


In French.

Translated title: Physical space- and space-time of an observer in relation to other observers.

Continuing an investigation of the structure of the space-time of a solitary observer, a discussion in general terms is now given of the restrictions imposed if this observer exchanges signals with other observers.


In Spanish. Not examined.

"The equations of motion in Birkoff's flat space-time theory of gravitation are solved to the first order of approximation for small velocities. A particle moving approximately in a circular orbit is found to have a small acceleration perpendicular to the plane of the orbit. The effect is too small for observation, but might have applications in cosmology." Sci.Abs. 56A:595, 1953.


A space-space representation is developed for nonrelativistic wave mechanics as was done for classical mechanics, by adding to the system a rotating body to serve as a clock, and applying the time-independent forms of the equations of motion. The usual Schrödinger equation is obtained in this way, however, only in the limit as the energy of the clock becomes very large. The reason for this restriction is found in an appendix, by comparison with the classical equations. Other appendixes deal with certain aspects of the Hamiltonian representation of the problem.

4. Bacon, R.H. ON THE PARADOX OF THE TWINS. Am.J.Phys. 26:502-503, Oct.1958.

Refers to the clock paradox controversy and suggests use of a clock whose behavior is totally unaffected by motion and as easily observable by the stay-at-home brother as by the traveling brother. "The well-known variable stars constitute such a clock."

5. Band, William. IS SPACE-TIME FLAT? Phys.Rev. 61:698-701, June 1, 1942.

It is not claimed that flat-space theory is false, but that in practice it would be an inconvenient method of description that can hardly be identified with that actually employed in astronomy and physics.

6. Band, William. ON IMAGINARY TIME IN SPECIAL RELATIVITY. Am.Phys.Teacher 6:323-324, Dec.1948.

A discussion of time measurement.

7. Bastin, E.W. THE CONCEPT OF ORDER. I. THE SPACE-TIME STRUCTURE. Camb.Phil.Soc.Proc. 50:278-286, Apr.1954.

Even if ideas of measurement did not exist, a great deal of the simple mechanical and electromagnetic structure of the physical world could be understood. To see how this can be so, the orthodox ideas of the relationship of mathematics to physics are inadequate. In this paper, a development of this fundamental nonmetrical physics is made to depend on a different view of this relationship.

8. Becquerel, Jean. REMARQUES SUR LE RATENTISSEMENT DU COURS DU TEMPS PAR L'EFFET D'UN CHAMP DE GRAVITATION. Acad.Sci.Paris.Compt.Rend. 232:1617-1620, Apr.30,1951. In French.

Remarks on the slowing down of time by the effect of a gravitational field.

9. Becquerel, Jean. SUR LA STRUCTURE DE L'ESPACE-TEMPS ET LA NOTION PHYSIQUE DU TEMPS DANS UN CHAMP DE GRAVITATION STATIQUE. Acad.Sci.Paris.Compt.Rend. 233:590-593, Sept.10, 1951. In French. Translated title: On the structure of space-time and the physical notion of time in a static gravitational field.

The space-time interval between neighboring events for an observer moving with a particle is expressed in terms of the coordinates of an observer at rest relative to the center of spherical symmetry of a gravitational field. Assuming Newtonian acceleration, and postulating the equality of inertial and gravitational mass, it is shown that the methods of special relativity indicate a metric of the Schwarzschild form.


Time, p.19-30.

11. Bergmann, O. COMMENTS ON THE CLOCK PARADOX. Acta Phys. Austriaca 11:377-389, 1957.

In German. Not examined.

12. Bergmann, P.G. FIFTY YEARS OF RELATIVITY. American Astronautical Society. Proceedings, 3rd, Dec.6-7, 1956, p.135-155, New York, The Society, 1957.

The special and general theories of relativity and the significance of these concepts are reviewed with brief mention of the clock paradox.

13. Bickerdike, C.H. RELATIVITY CLOCKS AND MEASURING RODS. Phil.Mag. 45:327-335, Feb.1923.

The author convicts, by quotations, even such experts as Einstein and Eddington of confusing the minds of readers of their popular writings on relativity by the use of language naturally interpretable as ascribing physical reality to the supposed behavior of the purely conventional contracting measuring rods and compensating clocks assumed for the purpose of popular explanation.

14. Blackett, P.M.S. FURTHER MEASUREMENTS OF THE COSMIC-RAY ENERGY SPECTRUM. Roy.Soc.London Proc.A. 159:1-18, Mar.1937.

Measurements on the momentum distribution of µ mesons at sea-level. F.S. Crawford (See Item 45) connects this with his clock paradox assumptions.


In German.

Translated title: The homogeneous gravitation field and the Lorenz transformation.

"The connection between the special and the general theory of relativity lies in that in any given gravitational field, local rigid systems of reference exist for every point at which the special theory of relativity is valid. For all freely moving bodies whose masses are so small that their effect on the field can be neglected, the gravitational acceleration is nil at a point with reference to an infinitesimal system of reference which is freely moving and therefore participates in the gravitational motion. In such a local inertial system

- 3 a gravitational field is no longer present. Freely moving bodies move according to the inertia law, clocks and measuring rods behave according to the laws of the special theory of relativity. The local inertial systems have a relative acceleration with respect to each other, and it is in general not possible for a finite region to make a transformation of the gravitational field. The author now deals with a homogeneous gravitational field which he regards as one whose field effect disappears throughout its whole extent when considered from a suitable rigid system. According to classical mechanics all things would have independent accelerations, which is impossible in the inertial system where the special theory of relativity holds good. The first part of the present paper discusses this homogeneous gravitational field but is restricted to processes in a plane. The second problem dealt with is the determination of the time surface for an observer in the gravitational field, for which the special theory of relativity supplies the necessary basis. The last section of the paper makes a comparison with the general theory. The paper is mathematical throughout." Sci.Abs. 25A:2216, 1922.


In German.

Translated title: Space and time in relativity theory.

"The author follows Kant in assuming that space and time are not derived from experience, but exist a priori in the mind, and also in the view that the inductive investigation of natural science is justified by the conformity to law of the relations between natural objects. Also in his conclusion that the minimum of congruence and uniformity that is necessarily conjoined with the concept of this conformity, therefore represents the a priori foundation of natural science. In order to make definite the concept of the physical continuum discussed at length in the paper, coordinates have to be introduced. Such a continuum can be constructed from a finite number of observations, provided that the distance between two adjacent points is represented by a definite quadratic differential form of the relative coordinates dx: in this case any infinitesimal rigid point system can move freely throughout the continuum. The only continuum throughout which any rigid point system can move freely is the Euclidean continuum. In the case of a four-fold world representation, not resolvable into space and time, it will always be possible to order the representation in accordance with the requisite functional conditions, since the sum total of the a priori foundations is equivalent to the two conditions: (1) that it is possible by observation to determine the space-time coincidences; (2) that from a finite series of such determinations all other world-points can be determined. Here is found the minimum of theoretical knowledge required to make the knowledge of nature a possibility. As its consequence it appears that for the continuum to be measureable, it must contain infinitesimal congruent systems of world-points, or in other words, equidistant event-pairs must be capable of existing in every part of it." Sci.Abs. 26A:1777, 1923.


In German.

Translated title: The Lorenz contraction and the clock paradox.

"The author continues the investigation, begun in a former paper, of a rod moving in an inertial field in such a manner as to remain either unstrained or without change of strain, and illustrates graphically some of the conclusions. He obtains a general solution of the case in which a body receives acceleration not with consequent strains, but by the application of an impulse, such as might be communicated to a train by a push from a locomotive, so that after the dissipation of the resulting strain waves, the motion becomes steady. As a consequence of this general solution, it is shown that a moving body, on becoming free from strain, immediately undergoes the exact Lorentz contraction, without the intervention of any forces of unknown origin, the necessity of which has been postulated by some relativists. The author next proceeds to deal with the so-called clock paradox. He takes the case of the observer A at rest in an inertial system, his world-line being OA, perpendicular to OX, the x-axis. Then OA is the time axis, and an observer B, moving along an arbitrary course in the space-time, is supposed to meet the observer A at O and A, and at each point they compare their clocks. The space-time interval OA is then common to the two tracks, so that it must be the same for each observer. The author verifies this by a rather lengthy analysis, using the formulae obtained in the paper and embodying the assumptions that A measures his time so that the light velocity in any inertial system is constant, while B measures his so that the backward and forward velocities are everywhere equal to each other. He then explains the rate of B's clock appears to A to be determined by: (1) the retardation due to the uniform motion of special relativity together with the parallel displacement of the time-plane; (2) the retardation of the rate wherever B's motion is retarded, both accompanied by corresponding rotations in the time-plane." Sci.Abs. 26A:824, 1923. 18. Bolton, L. INTRODUCTION TO THE THEORY OF RELATIVITY. 177p.,diags., London, Methuen, 1921.

Simultaneousness of events and rates of clocks, p.71-74.

19. Bondi, H. RELATIVITY AND INDETERMINACY. Nature 169:660, Apr.19,1952.

The flow of time has no significance in the logically fixed pattern of events demanded by deterministic theory, time being a mere coordinate. In a theory with indeterminacy, however, the passage of time transforms statistical expectation into real events.

20. Bondi, H. THE SPACE TRAVELLER'S YOUTH. Discovery 18:505-510, Dec.1957.

The writer contends that the problem posed by the clock paradox of relativity is essentially one of the comparison of different motions, and the part of physics that is principally concerned with this kind of question is the theory of relativity, the special and the general. Brief reference is made to the character of each part, and the extent which we have to make use of it. Various illustrations and tables are utilized to explain facts.

The article concludes with the statement that "there is no clock 'paradox' since it is not paradoxical for two persons with different experiences to find that the consequences of their experiences differ. There is simply the result that high-speed travel makes the route dependence of time reckoning evident, whereas low-speed travel does not."

21. Born, M. and Biem, W. ON THE CLOCK PARADOX. Ned.Akad.Wetens.Proc. 61B:110-120, 1958.

In German. Not examined.

22. Boskovic, R.J. O PROSTORU, VREMENU I RELATINOSTI. Predgovor, Izbor Tekstova i Provod Dusana Nedelkovica [Belgrad] Kultura, 1956. 64p.

In Russian. Not examined.

Translated title: On space, time and relativity.


The basic equations of a new space-time representation are derived in an heuristic fashion, and a simple application is presented to illustrate the point of view. 24. Brewster, W.R.,Jr. THE PARADOX OF MOTION. Atlantic Mon. 200:155-156, Oct.1957.

Discusses the clock paradox based on Einstein's principle that an observer cannot detect uniform motion of his own system, and shows how Einstein's other basic principle - the constancy of the speed of light - resolves the paradox.

25. Broad, C.D. EUCLID, NEWTON AND EINSTEIN. Hibbert J. 18:425-458, Apr.1920.

Clocks, time, motion, and relativity are discussed, in an attempt to "put into simple terms" the main points of Einstein's principle of relativity.

26. Builder, G. THE CLOCK-RETARDATION PROBLEM. Austral.J.Phys. 10:424-428, Sept.1957.

27. Builder, G. THE RESOLUTION OF THE CLOCK PARADOX. Austral.J.Phys. 10:246-262, June 1957.

It is shown that the paradox can be resolved completely in terms of the restricted theory. It is also shown that the general theory can add nothing of physical significance to an analysis correctly made in terms of the restricted theory.

28. Campbell, J.W. CLOCK PROBLEM IN RELATIVITY. Phil.Mag. 15:48-51, Jan.1933.

It is proved that if a clock X moves in a Galilean frame of reference (where gravitational fields are defined by their potentials) over any closed path and in any manner, then between any two passages through the same point with the same velocity, the gains, as seen by an observer moving with X, of all clocks at rest in the Galilean frame and not in the vicinity of matter are the same. These gains are precisely equal to the loss of the clock X as seen by observers stationed with the other clocks.

29. Campbell, J.W. CLOCK PROBLEM IN RELATIVITY. Phil.Mag. 16:529-533, Sept.1933.

Clock problem treated for case of two observers, A and B. A remains at rest in Galilean frame of reference, while B moves relative to A with varying speed along radial straight line; problem is treated by another method: viz, that of moving axes.

30. Campbell, J.W. CLOCK PROBLEM OF RELATIVITY. Phil.Mag. 19:715-720, Mar.1935. "In the particular case discussed by Eddington, a body B projected from the earth returns to find that he has lived a shorter time than a body A which has remained on the earth undisturbed. The author shows that B may have been disturbed in such a way as to leave the earth and return to find that he had lived a longer time than A who remained on earth. It is shown that time is increased over some disturbed paths and that the explicit effect of disturbances themselves on Time is zero. The special case is considered where B is disturbed so as to follow a cometary orbit with perihelion distance equal to the perihelion distance of the earth and with longitude of perihelion sufficiently different from that of the earth so that the earth would not appreciably perturb B's motion on subsequent near approaches, and at some later approach it is again projected to the earth at a perihelion passage. The effects of the following four factors are considered: (1) sun's gravitational field, (2) earth's gravitational field, (3) the projecting disturbances on B, (4) the disturbances (molecular bombardments by the earth) on A." Sci.Abs. 38A:1994, 1935.

31. Campbell, W.F. THE CLOCK PARADOX. Can.Aero.J. 4:316-319, Nov.1958.

The paradox depends on the theory of relativity and the strange phenomena that result when a body travels at a velocity near the velocity of light. An experimental verification one way or the other may be imminent.

32. Carr, L.H.A. NOTES ON SOME RECENT ADVANCES IN SCIENCE. Junior Inst.Engrs.J. 59:41-47, Nov.1948.

Explains briefly the theory of relativity, and then discusses the special or restricted theory, "which is that part of the theory which deals with velocities, propounds and asserts the only simple hypothesis on which the theory can be made to agree with observational facts."

The author contends that proof is too long to include, but "the result is that the dimension of length in the direction of motion in the case of a body moving relative to an observer appears to that observer to be shortened in the ratio $\sqrt{1 - v^2/c^2}$ where v is the relative velocity between the two bodies and c is the velocity of light. The same ratio obtains on the moving body, as determined by the moving observer and the stationary observer respectively. This is not to say that either observer will see anything different from what is normally to be expected. Each observer will use his own length and time scales to determine their relative movement. They will agree as to the speed at which they are approaching or receding from each other, but the time at which they record the happening of some outside occurrence will not necessarily be the same for both of them.

"This has led to some curious misconceptions (due to insufficient knowledge) which have crept into print. There is the story that a person travelling at great speed to Sirius and back will take less time than we think that it does and will come back younger in age than expected. This is just bunkum. The only difference is that we will see him dashing towards Sirius, while he will think that Sirius is dashing towards him, but the distance, speeds and times are all the same to both of us.

"The question whether the contraction in length is 'real' or not is meaningless. It depends on what is meant by 'real.' To one observer the length 'appears' unchanged, to another it 'appears' to be contracted, and this is all that can be said."

33. Carr, L.H.A. THEORY OF RELATIVITY. Metropolitan-Vickers Gaz. 22:378-383, Sept/Oct.1948.

Theory is divided into two parts, "special" or "restricted" theory, which deals with relative velocities, and "general" theory, which deals with relative accelerations.

34. Casci, C. and Bertotti, B. ON THE SLOWING DOWN OF TIME. Jet Propul. 27:665-666, June 1957.

Explains Einstein's reasoning as expressed in Die Naturwissenschaften 6:697, 1918. (See Item 67).

35. Cimino, M. EINSTEINIAN CORRECTION OF TIME IN A PLANETARY MOTION. Accad.Naz.Lincei.Atti. 16:223-227, Sept.1932.

In a previous paper the author worked out a law of areas for Einsteinian motion in a field considered as Euclidean. In the present one he extends it so as to find a law for cosmic time and the periods of revolution. This enables corrections to be applied to the data of observation; which corrections are extremely minute.

36. Cleugh, M.F. TIME AND ITS IMPORTANCE IN MODERN THOUGHT. 308p., London, Methuen, 1937.

Chapter II is devoted to the physical aspects of time including brief mention of space and time.

The time-traveler paradox is related on p.211-213.

- 9 37. Cochran, W. A SUGGESTED EXPERIMENT ON THE CLOCK PARADOX. Nature 179:977-978, May 11,1957.

Suggests that, in principle, the controversy between Dingle and McCrea can be settled by deflecting the meson beam in the experiments of Durbin, Loar and Havens by one or more electromagnets so that it returns to intercept its own track and then measuring the proportion of particles returning.

38. Cochran, W. A SUGGESTED EXPERIMENT ON THE CLOCK PARADOX. Camb.Phil.Soc.Proc. 53:646-650, Jly.1957.

A fuller account than the article of same title appearing in Nature 179:977-978, May 11, 1957. (See Item 37).

39. Contopoulos, G. TIME IN PHYSICS. Texnika Xponika 32:259-266, Sept/Oct. 1955.

In Greek. Not examined.

Changes of our notion of time by the theory of relativity.

40. Corben, H.C. TIME DILATION EFFECTS IN SPACE TRAVEL. Lecture for course Engineering X461 entitled "Space Technology." University of California, Berkeley, Calif., Lecture 5C, 6p., Feb. 17-19, 1958.

41. Costa de Beauregard, O. LE REALISME DE L'ESPACE TEMPS; SUR DEUX PROBLEMES D'INTERPRETATION EN MECANIQUE ONDULATOIRE. Acad.Sci.Paris.Compt.Rend. 243:1838-1840, Dec.5, 1956.

In French.

Translated title: The realism of space-time; on two problems of interpretation in wave mechanics.

Microscopic phenomena are regarded as rigourously symmetric in past and future, the retarded actions of macroscopic physics being statistical effects. This point of view is applied in a discussion of certain classical quantum mechanical paradoxes.


In French.

Translated title: Relation between the proper time of a terrestrial clock and the astronomical Schwarzschild time to the 10-12 approximation.

- 10 To the 10-12 approximation, a clock located on the earth's surface is slow when referred to a similar one at rest in the solar system and infinitely distant from the sun. There are four causes of the phenomenon: mass of the sun, circulation of the earth around the sun, mass, and rotation of the earth. The corresponding effects are calculated here without reference to the influence of the moon: qualitative arguments show that this influence is negligible to the aforesaid approximation; a rigorous theory of the moon's effects would involve a two-body relativistic theory.

43. Courvoisier, L. THE INFLUENCE OF THE "LORENZ CONTRACTION" OF THE EARTH ON THE RATE OF QUARTZ CLOCKS. Experientia 9:286-287, 1953; 13:234-235, 1957.

In German. Not examined.

"It is maintained that part of the observed fluctuations is caused by the Lorentz contraction of the earth. It is also stated that clock corrections should have opposite signs in the northern and southern hemispheres. Data from northern and southern observatories are quoted in support of the author's thesis." Sci.Abs. 57A:3055, 1954.

In Part II, further evidence is derived from observations of relative corrections at northern and southern stations for 1951-1954.

44. Crawford, F.S. THE 'CLOCK PARADOX' OF RELATIVITY. Nature 179:1071-1072, May 25, 1957.


Reveals use of accelerator µ-mesons, under Atomic Energy Commission auspices, to verify clock-paradox prediction and cites other experiments on which verification is based, i.e., work by Blackett (Item 14), Rasetti (Item 184) and Rossi et al (Item 189).

46. Crew, Henry. RISE OF MODERN PHYSICS. 2nd ed., 434p.,illus., Baltimore, Williams & Wilkins, 1935.

The Einstein dilatation of time, p.407-408.


The argument about the life of space travelers, in which one side finds cogent support in Einstein's principles and the other in

- 11 Einstein's equations, suggests that some basic inconsistency has been overlooked.

The section, p.70-76, entitled The Fallacy of the Clock Paradox, attempts to explain the misunderstanding.

48. Darwin, C.G. THE CLOCK PARADOX IN RELATIVITY. Nature 180:976-977, Nov.9,1957.

This writer contends that "there is no doubt whatever that the accepted theory of relativity is a complete and self-consistent theory and it quite definitely implies that a space-traveler will return from his journey younger than his stay-at-home twin brother."

He cites a numerical example which he thinks makes the matter easier to follow than would any mathematical formula.

49. Das, Anadijiban. THE ARTIFICIAL SATELLITE AND THE RELATIVISTIC RED SHIFT. Prog.Theor.Phys. 18:554-555, Nov.1957.

Refers to formula derived by Singer and extended by Hoffman which, according to the author, does not follow directly from the solutions of the field equation. This the author proceeds to do.

50. Dingle, Herbert. THE "CLOCK PARADOX OF RELATIVITY." Nature 179:1242-1243, June 15,1957.

A distinction is made between the "postulate of relativity" and the "postulate of constant light-velocity," the former being taken to imply that if two bodies separate and reunite, there is no observable phenomenon that will show in an absolute sense that one rather than the other has moved. The claim that the second postulate involved no asymmetry is reiterated.

51. Dingle, Herbert. THE CLOCK PARADOX IN RELATIVITY. Nature 180:1275-1276, Dec.7,1957.

The writer has repeatedly pointed out that symmetrical ageing is an inevitable requirement of the postulate of relativity. He considers the problem in more detail in order to bring out what he believes to be misconception underlying such arguments as that of Sir Charles Darwin. (See Item 48).

52. Dingle, Herbert. MODERN ARISTOTELIANISM. Nature 139:784-786, May 8, 1937.

Refers briefly to E.A. Milne's manner of reconciling clock readings.

- 12 53. Dingle, Herbert. A PROBLEM IN RELATIVITY THEORY. Phys.Soc.Proc. 69A:925-934, Dec.1956.

The paper analyzes a form of the familiar "clock paradox" in which two observers separate and reunite, but in which the accelerations normally necessary at starting, reversing and stopping are eliminated, so that the problem lies wholly within the scope of the special theory of relativity. The importance of the problem lies in the fact that it embodies all the essential principles of that theory, about which recent correspondence in 'Nature' has shown much disagreement to exist. The points specially brought out are the following: (1) The theory is not basically concerned with a comparison between different observers but between different coordinate systems which are available to a single observer. (2) It is not basically concerned with a comparison between stationary and moving systems but between descriptions of a single system with respect to different arbitrarily chosen standards of rest. (3) The familiar expression, "time retardation," does not relate to a physical change experienced by a clock but to a comparison of times of an event by two clocks, at least one of which is not present at the event. When both are present the discrepancy vanishes. A similar remark applied to the "Lorenz contraction" of moving rods.

54. Dingle, Herbert. THE RATE OF A MOVING CLOCK. Nature 146:391-393, Sept.21,1940.

The author replies to correspondence connected with a previous article in which it was maintained that there is no foundation for the statement that the rate of a moving clock is reduced by a certain factor compared with that of a similar stationary clock. It is stated that the result of comparison between the stationary and moving clocks is quite unpredictable without an exact specification of the principle of operation or detailed mechanism of the clock. The experimental discovery of the dependence of length on velocity affects the whole of physics, and its effect is to show that certain conceptions thought to be independent and necessary are actually different forms of others already adopted. These results in no way remove the dependence of laws of physics on experience.

55. Dingle, Herbert. RELATIVITY AND SPACE TRAVEL. Nature 177:782-784, Apr.28,1956.

Calls for a re-examination of assumptions. Refers to ideas expressed by Sir George Thomson in his book "The Foreseeable Future;" and by W.H. McCrea, P.S. Epstein, and Einstein.

- 13 56. Dingle, Herbert. RELATIVITY AND SPACE TRAVEL. Nature 177:785, Apr.28,1956.

A reply to McCrea's criticism, Nature 177:784-785, Apr.28, 1956.

57. Dingle, Herbert. RELATIVITY AND SPACE TRAVEL. Nature 178:680-681, Sept.29,1956.

A statement prepared in conclusion of clock paradox argument with W.H. McCrea.

58. Dingle, Herbert. THE RELATIVITY OF TIME. Nature 144:888-890, Nov.25,1939.

According to the restricted theory of relativity, moving measuring rods are shortened in the direction of motion, and moving clocks run slow in the rates (1-v2/c2)½: 1; this view is correct so far as measuring rods are concerned but is incorrect with respect to clocks. Circumstantial evidence for the contraction of a moving rod is given by the Michelson-Morley and other experiments; there is no evidence for the definite retardation of clocks. There is in physics, no explicit definition of a clock. By consideration of a clock of hour-glass type it is shown that the ratio of time intervals recorded by stationary and moving clocks is different according as the interval is measured by the number, by the total weight or by the total volume of similar particles falling into a receptacle. The corrections to be applied to the readings of clocks in motion in order to realize the readings of an ideal clock if it were set moving are almost as various as the constructions themselves. The transformation formula for t is a logical consequence of that for x; it is the necessary consequence if Newtonian mechanics is to hold good for uniformly moving systems despite the change in x. The Kennedy-Thorndike experiment does not afford a confirmation of the Lorenz transformation for time, but shows that light behaves as an ideal clock. The statement that relativity has caused a fusion of time and space is not true; the conformability of light to Newtonian mechanics, established by the Michelson-Morley and Kennedy-Thorndike experiments, makes it possible to define corresponding units of space and time in terms of light.

59. Dingle, Herbert. SPECIAL THEORY OF RELATIVITY. 3rd.ed., 94p., New York, Wiley, 1950.

Chapter IV: Time.

- 14 60. Dingle, Herbert. THE TIME CONCEPT IN RESTRICTED RELATIVITY. Am.J.Phys. 10:203-205, Aug.1942.

Takes issue with Epstein's article on "The time concept in restricted relativity."

61. Dingle, Herbert. THE TIME CONCEPT IN RESTRICTED RELATIVITY. Am.J.Phys. 11:228-230, 1943.

A criticism of Epstein's "misinterpretation" of his opinion relative to time and space.

62. Dingle, Herbert. THE UNDERSTANDING OF RELATIVITY. Nature 122:673-674, Nov.3,1928. .

Briefly mentions the clock paradox.

63. Donahue, T.M. and Leffert, C.B. THE CLOCK PARADOX AND THE PHYSICS OF DISCONTINUOUS GRAVITATIONAL FIELDS. 25p., figs., Pittsburgh, Pa., University of Pittsburgh, Oct. 1957. (Rpt. 1) (Contract Nonr-624(06))

The motion of the clock as represented in a solution of the clock paradox offered by C. Møller is studied in detail. It is found that in the non-inertial rest frame of the accelerated clock the free clock suffers discontinuities in velocity whenever the "gravitational field" in that frame abruptly changes. Although these discontinuities occur at points of discontinuity of the metric tensor, examination of a case in which the gravitational field is turned on smoothly reveals that the effect is a real one within the framework of the general theory of relativity. By the principle of equivalence it follows that even in a real gravitational field when that field changes in time velocity dependent terms in the acceleration of a particle exist. These become impulsive for abruptly changing fields and can even cause an acceleration in the direction "opposite" to the field.

64. Eddington, A.S. THE MATHEMATICAL THEORY OF RELATIVITY. 2nd.ed., 270p., Cambridge, University Press, 1952.

Rectangular coordinates and time, p.13-16.

65. Eddington, A.S. SPACE, TIME AND GRAVITATION: AN OUTLINE OF THE GENERAL RELATIVITY THEORY. 218p., Cambridge, University Press, 1920.

Effects of velocity on clocks, p.58, 74.

66. Eddington, A.S. THE THEORY OF RELATIVITY AND ITS INFLUENCE ON SCIENTIFIC THOUGHT. 32p., Oxford, Clarendon Press, 1922. (The Romanes Lecture, 1922).

- 15 67. Einstein, Albert. DIALOG UEBER EINWAENDE GEGEN DIE RELATIVITAETSTHEORIE. Naturw. 6:697-702, Nov.29,1918.

In German.

Translated title: Dialog on the arguments against the theory of relativity.

Clarification of the clock paradox. Einstein's answer to the question whether a moving clock is left behind a fixed clock is "yes."

For English interpretation of this article, see Casci, C. and Bertotti, B., "On the slowing down of time." Jet Propul. 27:665-666, June 1957.

68. Einstein, Albert and Infeld, Leopold. THE EVOLUTION OF PHYSICS. THE GROWTH OF IDEAS FROM EARLY CONCEPTS TO RELATIVITY AND QUANTA. 319p., New York, Simon and Schuster, 1938.

The clock problem, p.189-194.

69. Einstein, Albert, Infeld, Leopold and Hoffman, B. THE GRAVITATIONAL EQUATIONS AND THE PROBLEM OF MOTION, Ann.Math. 39:65-100, Jan.1938.

Investigation of the extent to which the relativistic equations of gravitation determine the motion of ponderable bodies.

70. Einstein, Albert. MEANING OF RELATIVITY. 135p., Princeton, N.J., University Press, 1945.

Subject of space and time is discussed.

71. Einstein, Albert. PHYSIK UND REALITAET. Frank.Inst.J. 221:313-347, Mar.1936.

In German.

Translated title: Physics and reality.

Translation by J. Piccard, p. 349-382.

Brief mention of concept of space and time.

72. Einstein, Albert. RELATIVITY, THE SPECIAL AND GENERAL THEORY . 168p., New York, Holt, 1921.

The fundamental idea of this book is to show how the views of relative motion employed by Newton and Galileo can be replaced by a more general principle which explains at once the impasse introduced by the Michelson-Morley experiment, the shortening proposed by Fitzgerald and Lorenz, and various other phenomena connected with the motion of bodies.


In German.

Translated title: On the influence of gravity on the speed of light.

One of the first mentions of the clock paradox.

74. Einstein, Albert. ZUR ELEKTRODYNAMIK BEWEGTER KORPER. Ann.Phys. 17:891-921, 1905.

In German.

Translated title: On the electrodynamics of moving bodies.

For translation, see Item 148.

The clock paradox is introduced.

75. Epstein, P.S. TIME CONCEPT IN RESTRICTED RELATIVITY. Am.J.Phys. 10:1-6, Feb.1942.

Criticisms of Dingle's ideas on special relativity. It is assumed that, when a clock is set in motion, it does in fact alter its rate of working, and not that special relativity merely gives the transformation rule from the time-coordinates of one observer to those used by another, relatively moving. A clock is defined as a succession of events which can be identified by different observers without recourse to coordinate systems. Such clocks always show the same changes of rate when set in motion. It is claimed that the Lorenz contraction is real. If the equation for the period of a simple pendulum is transformed according to the Lorenz rules for the transformations of mass, length and electromagnetic force, the period changes according to the rule for the "dilatation of time." It is argued that this is a dynamical explanation of the dilatation, which is therefore also real.

76. Epstein, P.S. THE TIME CONCEPT IN RESTRICTED RELATIVITY. A REJOINDER. Am.J.Phys. 10:205-208, Aug.1942.

Further testimony in the Dingle-Epstein controversy.

77. Essen, L. THE CLOCK PARADOX IN RELATIVITY. Nature 180:1061-1062, Nov.16,1957.

The writer contends that the clock paradox in relativity is due to an error in Einstein's paper (Ann.Phys. 17:891, 1905) concerning acceleration effects.

78. Fahy, E.F. THE CLOCK PARADOX IN RELATIVITY. Austral.J.Phys. 11:586-587, Dec.1958.

- 17 Contends that it is feasible to perform astronomical observations which could provide an experimental basis for choosing between the two points of view expressed in the various discussions of the subject.

79. Feenburg, Eugene. DOPPLER EFFECT AND TIME DILATATION. Am.J.Phys. 27:190, Mar.1959.

A brief comment on the clock paradox controversy in which the author suggests "a trivially simple derivation of the time relation" and claims that "since the observers are not equivalent, there is no paradox."


In Spanish. Not examined.

"The field is first obtained for a particle moving uniformly relative to an inertial frame, and then generalized on the assumption that it is independent of the acceleration of the particle. The potentials, expressed in terms of retarded position and velocity 4-vectors, have the form m(2vivj-?ij)/|p|, where ?ij is the metric tensor for flat space-time." Sci.Abs. 56A:594, 1953

81. Fiorio, Franco. HOW TO TRAVEL OUTSIDE OUR SOLAR SYSTEM. Missiles and Rockets 3:89-90,93,illus.,Apr.1958.

An article based on thoughts expressed by G. Crocco in the Italian magazine, Civilta Delle Macchine. He seeks an application of the relativistic theory to applied cosmonautics.

82. Fisher, Ronald and McCrea, W.H. SPACE TRAVEL AND AGEING. Discovery 18:56-58,diags.,Feb.1957.

Letters referring to correspondence between McCrea and Dingle appearing in Nature. There the latter named men disputed the truth of the statement which appeared in Sir George Thomson's book, "The Foreseeable Future." This said that a person who ventured out from the earth for a long journey in a space-ship would find, when he returned after several years, that he had aged less than the people who had stayed at home. In other words, space travel might be a way of putting off old age.


The empirical result of Michelson and Morley is symbolized in the behavior of a spherical light wave inside a rigid spherical

- 18 mirror. This affords the model of an idealized instrument which is clock and yardstick both in one. It is shown that the focusing of the spherical light wave is not disturbed by accelerating the rigid mirror.

84. Francis, Montgomery. NEW CONCEPTS OF TIME AND SPACE: WHAT EINSTEIN HAS DONE TO THESE FUNDAMENTAL NOTIONS. Sci.Am. 124:146-147,diags.,Feb.19,1921.

Einstein, for the first time, brings mechanical, electromagnetic and gravitational phenomena within one structure.

85. Frank, Phillip. EINSTEIN: HIS LIFE AND TIMES. 198p., New York, Knopf, 1947.

Includes on page 57-63 an explanation of the relativity of time which led to drastic changes in our use of the words "space" and "time."

86. Fremlin, J.H. RELATIVITY AND SPACE TRAVEL. Nature 180:499-500, Sept.7,1957.

Suggests an alternative approach to the clock paradox by making use of the Fitzgerald-Lorenz contraction.

Ascribes many of the false conclusions which have been reached in this matter to the unconscious introduction of an asymmetry into what is essentially a symmetrical situation.

See reply by Herbert Dingle, Nature 180:500, Sept.7,1957.

88. Frye, R.M. and Brigham, V.M. PARADOX OF THE TWINS. Am.J.Phys. 25:553-555, Nov.1957.

There is no ambiguity about the relationship between time and motion, either in terms of experiments with rapidly moving mesons or canal rays, or in terms of general relativity theory. But contradictory results are obtained when it is attempted to make computations using only the special relativity theory. A number of opinions to the contrary have recently found their way into print.

89. Gardner, J.P. REALITY, RELATIVITY AND COMMON SENSE. Space J. 1:14-22,illus., Mar/May 1959.

A re-examination of our ideas of space, time and motion with illustrations emphasizing the importance of the relative velocity concept. The clock paradox is mentioned.

90. Gerharz, Reinhold. ON THE UNIVERSAL STANDARD OF TIME AND THE VELOCITY OF LIGHT. Inst.Radio Engrs.Proc. 45:1549-1550, Nov.1957.

- 19 Contends that "the velocity of light should regain its character as the fundamental reference which it deserves...this constant demonstrates the most intimate union between space and time."


The purpose of this communication is to show that the controllable orbit of an artificial planet can be selected to yield a very large relativity advance of perihelion under certain conditions.


In German.

Translated title: Experimental proof of theory of special relativity.


Paper given at the International Geophysical Year, Rocket and Satellite Conference, Washington, D.C., Sept. 30-Oct. 5, 1957.

In a suggestion for measuring the gravitational displacement of frequency it is proposed that one measures not the frequency but the difference between the readings on clocks on earth and the readings of clocks on the satellite.

94. Golay, M.J.E. NOTE ON RELATIVISTIC CLOCK EXPERIMENT. Am.J.Phys. 25:494-495, Oct.1957.

A further note concerning the writer's query of atomic clock makers about the eventual feasibility of clocks possessing such precision as to permit the detection of a relativistic effect, by comparing the time signals of two identical synchronized clocks carried to two altitudes, A and B, differing by some 5,000 meters.

95. Gunn, J.A. THE PROBLEM OF TIME. AN HISTORICAL AND CRITICAL STUDY. 460p., London, George Allen & Unwin, 1929.

The clock problem, p.190-198.


- 20 The author assumes a fixed either and the Fitzgerald-Lorenz contraction; restricted relativity; that the elastic constants are infinite (eliminating the effects of centrifugal force) and Euclidean space.

97. Haskins, J.R. ASYMMETRICAL AGING IN SPACE TRAVEL. 15p., Huntsville, Ala., Redstone Arsenal, Ordnance Missile Laboratories, Feb. 15,1958. (Rpt. 2R26P)

The author concludes that it seems correct to say that the general theory of relativity of Einstein does predict asymmetrical aging but that the part of the theory enlisted in proving asymmetrical aging has not been proven (nor disproven) by experiment.

98. Haskins, J.R. NOTES ON ASYMMETRICAL AGING IN SPACE TRAVEL. Redstone Arsenal. Ordnance Missile Labs.Quart.Res.Rev. 15:22-30, Dec.1,1957.

99. Haskins, J.R. NOTES ON ASYMMETRICAL AGING IN SPACE TRAVEL. Redstone Arsenal. Ordnance Missile Labs.Quart.Res.Rev. 18:11-16, Sept.1,1958. (Rpt. 2A18)

100. Haskins, J.R. NOTES ON ASYMMETRICAL AGING IN SPACE TRAVEL. Army Rocket and Guided Missile Agency. Quart.Res.Rev. 20:81-83, Mar.1,1959. (Rpt. ARGMA Tech.Note 1C20N-1)

In an illustration, the traveler T is stationed on the earth, e, and a star, s, a distance 4c^2/g away begins to send out light pulses. All three are at rest. At the time shown the first pulse from the star has arrived at e and there are 16 units of pulses in the space between e and s. At this time the traveler begins to accelerate. This diagram considers the question from the point of view of T who will consider himself to be kept stationary in a gravitational field by his rocket motor, while e and s are falling freely in this field.

101. Herman, R. RELATIVITY AND SPACE TRAVEL. A REPLY TO PROFESSOR DINGLE. Brit.Interplan.Soc.J. 16:564-574, Nov/Dec.1958.

An answer is given to Professor Dingle's argument that if a spaceship were to travel away from the Earth at a velocity comparable with that of light and subsequently return, then the duration of the two-way journey, as measured by a clock in the spaceship would be substantially the same as that measured by a clock on the Earth.

It is shown in the paper that the correct statement corresponding to the postulate of general relativity is that all coordinate systems

- 21 are equivalent for the formulation of the general laws of nature, and that this does not imply complete equivalence between all frames of reference.

102. Hill, E.L. THE RELATIVISTIC CLOCK PROBLEM. Phys.Rev. 72:236-240, Aug.1,1947.

The theory of uniformly accelerated motions based on the conformal group of transformations in space-time is applied to the clock problem of relativity theory. Two solutions are found, both of which are at variance with the usual theory. The bearing of the problem on the relation between mechanics and electromagnetic theory is discussed.


An examination is made of the result of restricting translations and Lorentz transformations in space-time to those with rational coefficients. This removes the major defect of Schild's model of discrete space-time by elimination of the lower bound on the relative velocities of reference systems. The theory is formulated in two stages. In the first, the energy and momentum components of a particle are restricted to a countable set satisfying the relativistic energy-momentum space in which wave functions are almost periodic functions. In the second state, the space-time variables are restricted to rational values. This leads to the theory of discrete space-time.

104. HISTORIC FIRSTS: RETARDATION OF MOVING CLOCK. Bell Lab.Rec. 29:60-61, Feb.1951.

Note in respect to work of H.E. Ives in connection with experimental verification of one of consequences of Einstein's restricted theory of relativity.


Singer's formula for the general relativistic red-shift on an earth satellite is modified to take account of the diurnal rotation of the earth and the lack of spherical symmetry of its gravitational field. It is shown that the Singer rates of the earth and satellite clocks need slight modifications, but that these modifications tend to cancel each other except at large distances from the earth, so that when one uses a mean radius of the earth in Singer's formula, the formula is adequate for present purposes.

- 22 106. Holmes, E.C. THE MAIN PHILOSOPHICAL CONSIDERATIONS OF SPACE AND TIME. Am.J.Phys. 18:560-570, Dec.1950.

Includes a brief discussion of the simultaneity of events and the constancy of distance in connection with relativity.

107. Infeld, Leopold. CLOCKS, RIGID RODS AND RELATIVITY THEORY. Am.J.Phys. 11:219-222, Aug.1943.

Claims that a journey covering an exact number of hours will show that a moving clock has slowed down.

108. Ingraham, R.L. RELATION OF ISOTOPIC SPIN SPACE TO SPACE-TIME. Phys.Rev. 106:595-597, May 1,1957.

The customary form of physical space as the direct product of space-time and isotopic spin-space can be replaced by a certain fusion of these two spaces into one space if one widens the group from the Lorentz group to the conformal space-time group which endows particles with an intrinsic finite size.

109. INTERPLANETARY SPACE CLOCK. Mech.Eng. 76:442-443,illus.,May 1954.

Said to be the world's first interplanetary space clock, the timepiece, designated the Hamilton Space Clock, was created to demonstrate the difference between earth time and the time on other planets. The clock simultaneously records the hours, date, month, and year on Earth and the planet Mars.

110. Isaak, G.R. THE CLOCK PARADOX AND THE GENERAL THEORY OF RELATIVITY. Austral.J.Phys. 10:207-209, Mar.1957.

It is suggested that since the paradox is resolved by the general theory, an experiment (involving unstable charged particles) equivalent to the clock problem would provide a test of general relativity.

111. Ives, H.E. ABERRATION OF CLOCKS AND THE CLOCK PARADOX. Opt.Soc.Am.J. 27:305-309, Sept.1937.

Continuing the investigations of a previous paper, the problem of the change in setting of clocks transported over the same measured distance at different times when the velocity of the observing platform is different is investigated. The result is related to stellar aberration. The behavior of separated and reunited clocks is investigated, and the "clock paradox" which arises from the supposition that only relative velocities are of significance is resolved by allowing for their motion relative to the ether.


Considers measurements of lengths and times on relatively moving bodies, in which light signals are not involved.

113. Ives, H.E. THE CLOCK PARADOX IN RELATIVITY THEORY. Nature 168:246, Aug.11,1951.

A discussion of clocks which reverse their direction of motion is made in terms of an auxiliary moving clock to which the epochs of the original clocks are communicated on passing.

114. Ives, H.E. DERIVATION OF THE LORENTZ TRANSFORMATIONS. Phil.Mag. 36:392-403, 1945.

The derivation of the Lorentz transformations here given has for its central feature the confronting of the Maxwell electromagnetic picture of radiation with the laws of conservation of energy and momentum. The solution of apparent conflicts demands first the variation of mass with velocity, and then in turn the variation of linear dimensions and clock rate.


Considers an optical experiment which is competent to decide the value of n.

116. Ives, H.E. and Stilwell, G.R. EXPERIMENTAL STUDY OF THE RATE OF A MOVING ATOMIC CLOCK. Opt.Soc.Am.J. 28:215-226,illus.,Jly.1938.

Verifies the dependence of the passage of time upon motion. Shows by experiments on high-speed hydrogen canal rays that the frequency of a moving radiating source is altered by the factor (1 - v2/c2)1/2, where v is the velocity of the source, and c the velocity of light.

117. Ives, H.E. and Stilwell, G.R. EXPERIMENTAL STUDY OF THE RATE OF A MOVING ATOMIC CLOCK. II. Opt.Soc.Am.J. 31:369-374,illus.,May 1941.

Direct experimental verification of retardation using atomic clocks.

- 24 118. Ives, H.E. EXTRAPOLATION FROM THE MICHELSON-MORLEY EXPERIMENT. Opt.Soc.Am.J. 40:185-191, Apr.1950.

The invariance with motion of the measurement of two-way light signals in the Michelson-Morley experiment is extrapolated to one-way measurements. The time of arrival of signals at a distant point is accepted as indeterminate, in accordance with the before-and-after characterization of Robb; but it is shown that, in spite of this indeterminancy, definite expressions can be obtained for the operations involved in measuring one-way signals. Extension of the same reasoning leads to Lorentz-type transformations expressed in terms of observable rod and clock readings, which include the self-observed velocities of moved clocks used to set the epochs of distant clocks.

119. Ives, H.E. THE FITZGERALD CONTRACTION. Roy.Dublin Soc.Sci.Proc. 26:9-26,illus.,May 1,1952.

Considers Fitzgerald's contraction as applied to clocks.


A discussion of the theory in connection with a change in clock rate.

121. Ives, H.E. HISTORICAL NOTE ON THE RATE OF A MOVING CLOCK. Opt.Soc.Am.J. 37:810-813, illus.,Oct.1947.

The history of the idea of variation of frequency with velocity is followed through Voigt, Larmor, Lorentz and Einstein. The Michelson-Morley experiment is explained.


Considers the effects of variations in the properties of rods and clocks due to their motion through the ether.


Transformations of the Lorentz type are derived in terms of distance and time measurements made by rods and clocks subject to the Fitzgerald-Larmor-Lorentz contractions by their motion through the ether. These transformations contain terms involving the self-measured velocity of a transported setting clock which

- 25 must be used to establish times at distant points. When the setting clock velocity is small, the expressions approximate to the original Lorentz transformations.


A new approach to the measurement of the velocity of light is discussed, using modern technical advances to construct two accurate clocks to measure the velocity over a single path instead of an up-and-down path. Relativistic implications and equations of such a measurement are discussed.

125. Ives, H.E. MEASUREMENT OF VELOCITY WITH ATOMIC CLOCKS. Sci. 91:79-84, Jan.26,1940.

Discusses what happens to the measurement of velocity when the clocks used for the determination of time are atomic clocks, which vary in their rate, when moving, according to a relation for which experimental evidence has been obtained.

126. Ives, H.E. REVISIONS OF THE LORENTZ TRANSFORMATIONS. Am.Phil.Soc.Proc. 95:125-131,diags.,Apr.30,1951.

This study has been based on the use of three postulates: first, the physical fact of the independence of the velocity of light from its source or other matter; second, the principle of relativity of Poincaré; third, the operational principle of Bridgman, which requires that all symbols represent purely physical operations and observations. By the use of these, new expressions for the description of events on relatively moving bodies have been obtained. These expressions are similar to the Lorentz transformations as modified by Poincaré but contain additional terms which provide the operational definitions lacking in the Lorentz-Poincaré equations. From the philosophical standpoint the importance of these revised equations lies in the fact that they dissipate the logical impasse and the mysticism which have been long associated with this subject. Also of philosophical import is that with the abandonment of the "principle" of the constancy of the velocity of light, the geometries which have been based on it, with their fusion of space and time, must be denied their claim to be a true description of the physical world.

127. Jackson, J. SHORT CLOCKS AND THE EARTH'S ROTATION. Roy.Astron.Soc.Mon.Not. 89:239-250, Jan.1929.

Detailed account of performance and conditions affecting it.

- 26 128. Janossy, Lewis. COSMIC RAYS. 424p., Oxford, Clarendon Press, 1948.

Relativistic time contraction. p.22,177,201.

129. Janossy, Lewis. REMARK ON THE THREE OBSERVED EFFECTS OF GENERAL RELATIVITY. Acta Phys. 5:215-228, 1955.

The three observed effects supporting the general theory of relativity are interpreted in a purely phenomenological manner. It is shown what kind of ad hoc assumptions about the effects of the gravitational potential on physical quantities are suitable to account for the three effects (the perihelion motion of the planets; the deflection of light in a gravitational field; and the gravitational red shift.)

130. Jeffreys, H. THE CLOCK PARADOX IN SPECIAL RELATIVITY. Austral.J.Phys. 11:583-586, Dec.1958.

Refers to a new analysis produced by G. Builder and criticized by H. Dingle, then states that "both introduce concealed hypotheses, and that the methods of the special theory cannot produce a unique answer."

131. Johnson, Martin. THE MEANINGS OF TIME AND SPACE IN PHILOSOPHIES OF SCIENCE. Am.Scientist 39:412-421, Jly.1951.

A discussion of various notions of time, including brief mention of the clock paradox.


Bibliography, p.184-186.

133. Jones, H.J. EINSTEIN AND THE THEORY OF RELATIVITY. Engr. 132:529, Nov.18; 556-558, Nov.25; 588-589, Dec.2,1921.

Discusses distance between two points; time between two events; and apparent contraction.

134. Jones, R.C. ON THE RELATIVISTIC DOPPLER EFFECT. Opt.Soc.Am.J. 29:337-339, Aug.1939.

The theory of the recent experiment of Ives and Stilwell, concerning the changes to be expected in the rate of a moving clock, is developed on the basis of the special theory of relativity.

- 27 135. Kaplan, Joseph. HOW MAN-MADE SATELLITES CAN AFFECT OUR LIVES. Natl.Geog.Mag. 112:791-810,illus.,Dec.1957.

The satellite may check Einstein's theory by using two so-called atomic clocks, one of which would remain on earth while the other goes aloft.

136. Karapetoff, Vladimir. GENERAL OUTLINE OF RESTRICTED RELATIVITY. Scripta Math. 8:162-163, Sept.1941.

A discussion of space-time relationships, including slowing down of clocks.

137. Kennedy, R.J. and Thorndike, E.M. EXPERIMENTAL ESTABLISHMENT OF THE RELATIVITY OF TIME. Phys.Rev. 42:400-418, Nov.1,1932.

The experiment was devised to test directly whether time satisfies the requirements of relativity.

138. Kennedy, R.J. SIMPLIFIED THEORY OF THE MICHELSON-MORLEY EXPERIMENT. Phys.Rev. 47:965-968, June 15,1935.

It is shown that a correct application of Huygen's principle in the theory of the experiment leads to the same expression for the expected result as derived in the simple classical theory. The effect due to path difference is shown to be the same as the effect derivable from the relative rotation of the interfering beams. Critics of the classical theory have mistakenly regarded the latter as a compensating factor almost exactly offsetting the first.

139. Kooy, J.M.J. SPACE TRAVEL AND FUTURE RESEARCH INTO THE STRUCTURE OF THE UNIVERSE. Brit.Interplan.Soc.J. 15:248-259,diags.,Sept/Oct.1956.

According to the leading cosmological theories, intergalactic space is spherical, the radius of curvature varying with time. According to Eddington, there are only two ways to account for the large speeds of recession of the spiral nebulae, they are produced either by a scattering force, (called cosmical repulsion), predicted by the general theory of relativity, or the receding velocities now observed have existed from the beginning. For completeness, he advances as a third possibility an oscillating model of the universe. Today the question which of these two ways must be taken is still unsettled. As Eddington remarked, our first hope for further progress is some quantitative test. In this paper an extensive program for astronomical observation is indicated in order to test the alternate cosmological theories.

- 28 140. Kooy, J.M.J. SPHERICAL SPACE-TIME AND THE SPECIAL THEORY OF RELATIVITY. Brit.Interplan.Soc.J. 16:172-173,illus.,Jly/Sept.1957.

Further remarks on space and time, used as notes to the author's paper entitled "Space travel and future research into the structure of the universe."

141. Krause, H.G.L. RELATIVISTISCHE RAKETENMECHANIK. Astronautica Acta 2:30-47, 1956.

In German.

Translated title: Relativistic rocket mechanics.

In this paper the mechanics of the special theory of relativity are extended to systems with timely changeable rest masses (rockets). The theorem of impulse and energy is examined as well as the law of the decrease of the mass of a rocket with an optional acceleration in a free space without outward power, that is, in the system of the resting ground observer and in the system of the astronaut, moved with the rocket. Two special cases are treated, namely, the movement of a rocket with constant self-acceleration and the movement of a rocket with constant amount of mass flow rate (thrust).

142. Langevin, Paul. L'EVOLUTION DE L'ESPACE ET DU TEMPS. Scientia 10:31-54, 1911.

In French.

Translated title: The evolution of space and time.

Discussion, in detail, of the clock paradox.

143. Leffert, C.B. and Donahue, T.M. CLOCK PARADOX AND THE PHYSICS OF DISCONTINUOUS GRAVITATIONAL FIELDS. Am.J.Phys. 26:515-523,figs.,Nov.1958.

The motion of the clocks as represented in a solution offered by C. Møller is studied in detail. It is found that in the non-inertial rest frame of the accelerated clock the free clock suffers discontinuities in velocity whenever the "gravitational field" in that frame abruptly changes. Although these discontinuities occur at points of discontinuity of the metric tensor, examination of a case in which the gravitational field is turned on smoothly reveals that the effect is a real one within the framework of the general theory of relativity. By the principle of equivalence, it follows that even in a real gravitational field when that field changes in time velocity, dependent terms in the acceleration of a particle exist.

- 29 These become impulsive for abruptly changing fields and can even cause acceleration in the direction "opposite" to the field.

144. Lieber, Lillian. EINSTEIN THEORY OF RELATIVITY. 324p.,illus., New York, Rinehart, 1945.

A popular discussion, including the clock paradox.

145. LIFE SPAN SAME IN SPACE AS ON EARTH. Sci.News Ltr. 70:371,Dec.15,1956.

According to Dr. William Brewster, Jr., of Harvard Medical School, many popularly held theories that man will not age as fast in space as he does on earth are not so. Einstein's theory suggests to some that space travel may be the fountain of youth, but "slowing up life's processes works both ways," Dr. Brewster reported at an American Rocket Society meeting. "It depends on whether you say the space ship is moving. away from the earth or the earth is moving away from the space ship. No matter which twin brother you are, you will see the other aging faster. When the two of you are back home, one will not be any older than the other."

146. Loos, H.G. TIME DILATATION IN SPACE FLIGHT. Jet Propul. 27:665, June 1957.

Contends that since "time dilatation is a straightforward result of the hypothetical basis of the theory of relativity" there is no reason to doubt its existence.

147. Lopez, J.A. THE TIME CONCEPT IN SPECIAL RELATIVITY. 29p., diags., Cordoba, Edicion Scientia, 1956.

The clock paradox, p.15-19. A discussion of three definitions with reference to various writings on the subject.

148. Lorentz, H.A., Einstein, Albert, Minkowski, H. and Weyl, H. THE PRINCIPLE OF RELATIVITY. A COLLECTION OF ORIGINAL MEMOIRS ON THE SPECIAL AND GENERAL THEORY OF RELATIVITY, translated by W. Perrett and G.B. Jeffrey. 216p., New York, Dover, 1923.

On the electrodynamics of moving bodies, by Albert Einstein, p.37-56, (translated from Zur Elektrodynamik Bewegter Körper, Annalen der Physik 17, 1905); Space and time, by H. Minkowski, p.75-91, (a translation from an address delivered at the 80th Assembly of German Natural Scientists and Physicists, at Cologne, 21 September 1908).


Retardation of moving clocks is discussed on p. 91,94,98,103,253.

150. McCrea, W.H. THE CLOCK PARADOX IN RELATIVITY THEORY. Nature 167:680,Apr.28,1951.

The writer contends that it has never been made quite clear that there really is no paradox.

151. McCrea, W.H. THE FITZGERALD-LORENTZ CONTRACTION - SOME PARADOXES AND THEIR RESOLUTION. Roy.Dublin Soc.Sci.Proc. 26:27-44,illus.,May 1,1952.

152. McCrea, W.H. RELATIVISTIC AGEING. Nature 179:909-910, May 4,1957.

Discussion of relativistic versus asymmetric ageing in connection with space travel.

153. McCrea, W.H. RELATIVITY AND SPACE TRAVEL. Nature 177:784-785, Apr.28,1956.

Contends that Professor Dingle's exposition of relativity is wrong.

154. McCrea, W.H. RELATIVITY AND SPACE TRAVEL. Nature 178:681-682, Sept.29,1956.

Statement prepared in conclusion of argument with Herbert Dingle.


It is shown that a particle-clock which describes a circular orbit in a central gravitational field appears to go slow as compared with a similar clock at a greater orbital distance. The factor appearing in the comparison can be interpreted by a combination of the effects concerned in the gravitational red shift and in the clock paradox.

156. McMillan, E.M. THE "CLOCK PARADOX" AND SPACE TRAVEL. Sci. 126:381-384, Aug.30,1957.

- 31 This article has three sections. In the first, the "paradox" is stated and resolved, using only inertial coordinate systems; in the second, a treatment of accelerated coordinate systems based on the principles of special relativity is given; and in the third, the possibility of practical implications for space travel is examined.

157. McVittie, G.C. DISTANCE AND RELATIVITY. Sci. 127:501-505,Mar.7,1958.

Discusses large-scale distances: time and distance and special relativity.

158. McVittie, G.C. GEODESICS OF EXPANDING SPACE-TIME. Roy.Astron.Soc.Mon.Not. 94:476-483, Mar.1934.

The equations of the geodesics of expanding space-time, which pass through the origin, are worked out in terms of "cosmic" time and a distance-variable which corresponds with that used in observational astronomy. It is shown that certain moving observers can synchronise their clocks so that the clocks read cosmic time. It is also shown that particles possessing the properties of the "statistical" particles of Milne's kinematic theory can be found by the method of general relativity. The law for the acceleration of these particles differs from Milne's and the discrepancy is probably due to the new method for calculating acceleration proposed by Milne. Reasons are given for supposing that Milne's interpretation of the cosmical constant and the gravitational constant is not valid.

159. Majorana, Q. SPACE AND TIME. Accad.Naz.Lincei.Atti. 12:481-486, May 1952.

In Italian. Not examined.


Various definitions of uniform acceleration are possible in relativity theory. Two of these are considered with particular reference to their physical realizations.

161. Martinelli, E. and Panofsky, W.K.H. THE LIFETIME OF THE POSITIVE p-MESON. Phys.Rev. 77:465-468,Feb.15,1950,

Describes an experimental method to determine the lifetime of the p-meson which consists of measuring the attenuation, beyond geometrical losses, of a beam of mesons as a function of the number of cyclotron orbits it has traversed.

- 32 It is interesting to note that if the experiment were carried out to higher accuracy for both fast and slow mesons, it constitutes the experimental equivalent of the clock paradox. It would be observed that the fast meson returning to its starting point has actually lived longer in the laboratory frame as compared to a slow one. This experiment therefore in principle depends on the fact that the laboratory frame is a preferred one in relation to the masses in surrounding space.

162. Maugin, C. ASTRONAUTIQUE ET RELATIVITÉ. A L'ASSAUT DE L'ESPACE TEMPS. Acad.Sci.Paris.Compt.Rend. 234:1004-1007, Mar.3; 1329, Mar.24,1952.

In French.

Translated title: Astronautics and relativity. The test of space-time.

The relationship between the distance x (in light years) travelled by a hypothetical vehicle, in a system in which the vehicle is at rest at zero time and the time t (in years) measured in the vehicle is derived for the case in which the vehicle is moving with a constant acceleration of 950 cm.sec.-2 (measured with respect to a system instantaneously at rest relative to the vehicle at any instant). In this case, x = cosh t-1, the corresponding time which elapses in the fixed system is t = sinht.

163. Meksyn, D. ON THE METRIC OF FOUR-DIMENSIONAL SPACE-TIME. CONTINUUM. Phil.Mag. 50:1227-1244, Dec.1925.

Demonstrates that the sizes of bodies, intervals of time and mass are not affected by motion as such.

164. Meksyn, D. RELATIVITY OF ACCELERATED MOTION. Nature 160:834-835, Dec.13,1947.

The difficulty of reconciling experimental evidence with the general relativity theory which states that all accelerated motion is relative.

165. Michelson, A.A. INTERFERENCE PHENOMENA IN A NEW FORM OF REFRACTOMETER. Phil.Mag. 13:236-242, 1882.

Michelson devised "an exceedingly clever, beautiful and accurate instrument - his well-known ether-drift interferometer - for the particular purpose of discovering whether there is any difference in the time demanded for a beam of light to make a round trip over the same distance, along and across the direction of the earth's motion." The experiment furnishes evidence for thinking that the relative motion of the earth and the ether is almost zero.

- 33 166. Michelson, A.A. and Morley, E.W. INFLUENCE OF MOTION OF THE MEDIUM ON THE VELOCITY OF LIGHT. Am.J.Sci. 31:377-386,1886.

A verification of Fizeau's experiment in comparing the speed of light traveling along with a current of water with the speed of an identical beam traveling against the same current of water.

167. Mikhail, F.I. THE RELATIVISTIC CLOCK PROBLEM. Camb.Phil.Soc.Proc. 48:608-615,1952.

The problem is treated purely by general relativity by considering a particular example in which the two observers are attached to two test-particles moving freely in the field of gravitating mass; one of these makes complete revolutions in a circular orbit while the other moves radially outwards and inwards. The time-interval between two successive encounters is shorter in the reckoning of the former than in that of the latter. The difference is found to agree qualitatively with a naive application of special relativity.


A repeat of the Michelson-Morley experiment, under a variety of conditions, with distinctly improved apparatus.

169. Milne, E.A. and Whitrow, G.J. ON THE SO-CALLED "CLOCK PARADOX" OF SPECIAL RELATIVITY. Phil.Mag. 40:1244-1249,Dec.1949.

From the viewpoint of kinematical relativity, there are no clock paradoxes. References to previous discussion are given. The ideas developed are applied briefly to photons.


Offers a solution to the clock paradox.


Appendix B. Satellite problems.

The idea which suggests itself is an attempt at using atomic clocks to verify the relativistic formula for the rate of clocks placed at different potentials in a gravitational field.

- 34 172. Møller, C. THE THEORY OF RELATIVITY. 386p., London, Oxford University Press, 1952.

The clock paradox, p.48-51; 258-263.

173. Munitz, M.K. SPACE, TIME AND CREATION. PHILOSOPHICAL ASPECTS OF SCIENTIFIC COSMOLOGY. 182p., Glencoe, Ill., The Free Press and The Falcon's Wing Press, 1957.

Time and the use of clocks in physics, p.137-142.

174. Newton, R.H.C. INTERSTELLAR FLIGHT. Brit.Interplan.Soc.J. 11:297-298,Nov.1952.

Draws attention to clock paradox, by commenting on article by L.R. Shepherd.

Dr. Shepherd's reply appears on p.298-299,Nov.1952.

175. Nordmann, Charles. THE TYRANNY OF TIME: EINSTEIN OR BERGSON. 217p., London, Unwin, 1925.

Endeavors to bring some clearness and precision into the question raised by Einstein's conception of time.


More about time, p.48-50.

177. Palacios, J. SHOULD RELATIVITY THEORY BE REVISED? An.Real.Soc.Espan.Fís.Quím. 53(A):31-42, Jan/Feb.1957.

In Spanish. Not examined.

"Several explanations of the 'clock paradox' are criticized and, as they do not seem satisfactory, the author suggests to replace the equations of Lorentz by other equations, which, being in agreement with the principle of Galileo and with the invariance of the velocity of light, do not lead to the mentioned paradox. The new equations have the same physical consequences as the theory of Einstein, but the value of the constant of Planck is no more invariant. Instead, it changes, when measured in a moving frame, in such a way that the red shift of the canal rays and the anomalous half-life of rapid muons are at once explained." Sci.Abs. 61A:7668, Nov.1958.

178. Partel, Glauco. CONGRESO INTERNAZIONALE ASTRONAUTICO DI BARCELLONA - 6-12 OTTOBRE 1957. Centro.Aero.Atom.Ital. 20:12-26,Nov.1957.

- 35 In Italian.

Translated title: The International Astronautical Congress at Barcelona - 6-12 October 1957.

Survey of developments and discussion of the behavior of clocks during travel in the curved space-time of the general theory of relativity, and analysis of the effects observable in journeys in the solar system or on earth satellites.

179. Peschka, W. UEBER DIE UEBERBRUECHUNG INTERSTELLARER ENTFERNUNGEN. Astronautica Acta 2:191-200,1956.

In German.

Translated title: On the bridging of interstellar space.

Shows that photon rockets traveling close to the speed of light may eventually make possible return journeys to points 25 light years away. Time dilatation effects will become appreciable, and enormous mass ratios will be required.

180. Preston-Thomas, H. TWO ASPECTS OF THE TIME ELEMENT IN INTERPLANETARY FLIGHT. In International Astronautical Congress, Innsbruck, 1954. Proceedings 5th:150-156, Wien, Springer Verlag, 1955.

Discusses power requirements and transit times for Earth-Mars transits.

181. Proell, Wayne. RELATIVITY AND SPACE TRAVEL: A COMMENTARY. J.Space Flight 1:8-11,Aug.1949.

A consideration of the criticism by John Spaulding of the author's former paper on the subject, with the conclusion that further work appears merited both as a test of certain aspects of relativity and as a means of predicting the future of interstellar travel.

182. Proell, Wayne. THE SPEED OF LIGHT AND SPACE SHIP NAVIGATION. Rocket News Ltr. (Chic.Rocket Soc.) 2:9-12,Nov.1948.

Presents data to show that both astronomically and terrestrially, relative velocities greater than c are in existence.

183. Quan, Pham Mau. MOUVEMENTS PERMANENTS D'UN FLUIDE PARFAIT THERMODYNAMIQUE. Acad.Sci.Paris.Compt.Rend. 238:324-325,Jan.18,1954.

In French.

Translated title: Permanent motions of a perfect thermodynamic fluid.

- 36 The author's theory in which space-time formalism is extended to include thermodynamic terms is applied to the case of stationary Riemannian space-time.

184. Rasetti, Franco. DISINTEGRATION OF SLOW MESOTRONS. Phys.Rev. 60:198-204,diags.,Aug.1,1941.

Reports measurement of the mean life of p-mesons at rest. F.S. Crawford (See Item 45) relates this finding to his assumption no. 2, namely "the acceleration of an ideal clock relative to an inertial system has no influence on the rate of the clock, and the increase in the proper time of the clock at any time is the same as that of the standard clocks in the system in which the clock is momentarily at rest."

185. Robb, A.A. ABSOLUTE RELATION OF TIME AND SPACE. 80p., Cambridge, University Press, 1921.

Asserts that although all events have temporal relations to some events, no events have temporal relations to all.

186. Robb, A.A. GEOMETRY OF TIME AND SPACE. 2nd.ed., 408p., New York, Macmillan, 1936.

A new view of the theory of relativity involving optical geometry of motion. Introduces the "before and after" characterization.

187. Robinson, J.D. and Feenberg, Eugene. TIME DILATATION AND DOPPLER EFFECT. Am.J.Phys. 25:490-491,Oct.1957.

A discussion of the well-known twin problem based on the relativistic formula for the longitudinal Doppler effect and the recent innovation of educational television.

188. Romer, R.H. TWIN PARADOX IN SPECIAL RELATIVITY. Am.J.Phys. 27:131-135,Mar.1959.

The familiar "traveling twin paradox" is discussed in a particularly simple manner, using special relativity only. The asymmetrical aging of the twins is predicted using only the relativistic definition of simultaneity and the relativistic time dilatation. Possible objections to such a treatment are discussed.


- 37 F.S. Crawford (See Item 45) states that the first quantitative check of the assumption that the time dilation of special relativity holds for uniform motion is contained in the combined experiments of Rossi, Hilberry and Hoag; Rasetti; and Blackett (See also Items 184 and 14).

190. Rowland, E.N. A NOTE ON SPACE TRAVEL IN A GRAVITATIONAL FIELD. Brit.Interplan.Soc.J. 16:216-221, Oct/Dec. 1957.

The behavior of clocks during travel in the curved space-time of the general theory of relativity is discussed, and effects observable in journeys in the solar system or on earth satellites are examined.

191. Rülf, B. DIE RELATIVITAETSTHEORIE VON EINSTEIN UND DIE GRUNDLAGEN DER MECHANIK. Ver.Deut.Ing.Z. 64:593-598,Jly.31; 623-625,Aug.7,1920.

In German.

Translated title: The Einstein theory of relativity and the basic principles of mechanics.

Includes mention of constancy of the expansion of light in all moving systems; the Einstein formula for speeds; and relativity of time.

192. Ruse, H.S. MEASUREMENT OF SPATIAL DISTANCE IN A CURVED SPACE-TIME. Roy.Soc.Edinburgh Proc. 53:79-80,1932.

A definition of spatial distance is given, which, it is claimed, corresponds to that determined by the use of rigid measuring rods. The formula derived for spatial distance corresponds with that given in the author's previous paper and with that derived from Whittaker's definition of spatial distance.

193. Russell, Bertrand. HUMAN KNOWLEDGE. 524p., New York, Simon & Schuster, 1948.

Chapter V. Time, public and private.

194. Sampson, R. A . CONCEPT OF TIME. Roy.Astron.Soc.Mon.Not. 94:603-630,May 1934.

Postulating Euclidean space-time and the energy of a particle E=c (?p2+ P1p12 + P2p22 + P3p32)1/2, where p, P1, P2, P5 are momenta, conjugate to displacement cdt, dq1, dq2, dq3, and the coefficients ?, P1, P2, P3 are functions of the coordinates and include the effect of a gravitational field, the case is discussed of two particles, related by the equation E + E' = H. It is shown that an

- 38 equation p1dq1 + … + p cdt + p1'dq1' + … +p'cdt' - Hdt = 0 obtains identically, also H = constant, and that p = mc if t does not occur explicitly. Further, the complete solution for two particles in space-time is given, and the results shown to be the same as those of general relativity. It is concluded that the curvature of space-time is not an essential.

195. Sänger, Eugen. THE ATTAINABILITY OF THE STARS. Santa Monica, Calif., Rand Corp. Translation. 25p.,Dec.26,1956.

Original article a report of the Furschungsinstitut für Physik der Stahlantriebe, Stuttgart-Flughaven.

Paper read at the Seventh International Astronautical Congress, Rome, Sept. 1956.

"The yet hypothetical quantum rockets have jet-velocities equal to the velocity of light, so that also their flight velocities may approach the optic velocity.

"From the laws of classical mechanics, there would follow that the limited human lifetime and the limited mass-ratio of the rocket would permit ranges of some tenths of light years, i.e., over a very limited space of our galaxy and to the very next fixed stars only.

"From the laws of relativistic mechanics, however, follows for those very neat optic-velocities a considerable dilatation of proper time on board of the vehicle relative to the terrestrial time, so that life of the crew and action of the rocket-motor occur slower, than would correspond to terrestrial time scale.

"From this follows that within the life span of the crew and with limited mass-ratios of the rocket, every thinkable distance in space, up to nebulae millions of light-years distant can be covered, so that, expressed in technical terms, and from the standpoint of the crew, the vehicle seems to be able to move with considerable super optic-velocity."

196. Sänger, Eugen. CONCERNING SOME LIMITS OF ASTRONAUTICS. Astronef 1:8-10,Sept.1950.

In French. Not examined.

A discussion of some of the relativity effects on space ships traveling near the speed of light.

197. Sänger, Eugen. TIME AND THE SPACE TRAVELER; WITH EDITORIAL COMMENT. Atlantic Mon. 200:153-155,Oct.1957.

Postulates a space journey of 11 years which would be the same as 1,000 years earth time. The apparent time difference is an observational phenomena on the part of the terrestrial observer.

- 39 198. Sänger, Eugen. ZUR FLUGMECHANIK DER PHOTONENRAKETEN. Astronautica Acta 3:89-99,diags.1957.

In German.

Translation, with title Flight mechanics of photon rockets, in Aero.Dig. 73:68-70,72-73,Jly.1956.

Consideration of relativity with reference to time and distance for interplanetary spacecraft; theoretical consideration under which time span of one human life would be long enough for travel around the total static universe.

199. Schlegel, R. SPACE AT MID-CENTURY. Am.J.Phys. 20:38-49,Jan.1952.

Effects of uniform motion on space and the simplest currently accepted relations between space and time are stated.


Is concerned with an explanation of the general theory of relativity.

201. Schouten, J.A. ZUR GENERALLEN FELDTHEORIE, RAUMZEIT UND SPINRAUM. Z.Phys. 81:405-417,Mar.22,1933.

In German.

Translated title: Space-time in the unitary field theory.

From the point of view of ordinary relativity, there are two invariant planes in spin space. We need, therefore, only admit transformations which leave these planes invariant. In projective relativity these two planes do not lose their invariance, but they are no longer constantly covariant, and we must therefore adopt general linear transformations in the spin space. In the present paper the author shows how the spin space is connected with the local space-time world, what invariant structures exist in spin space, and how an invariant calculus may be constructed.

202. Schrödinger, Erwin. SPACE-TIME STRUCTURE. 119p., Cambridge, University Press, 1950.

Intends to show the simplest access to possible geometrical models of space-time.

203. Schwinge, H.T. and Wilson, C.E.,Jr. A STUDY OF THE ACCURACY OF SPACE-TIME SYSTEMS. 24p.,illus., Holloman Air Force Base, N.Mex., June 1957. (Air Force, Holloman Air Development Center, Rpt. Tr-57-5).

- 40 A study is made to ascertain the limitations and inherent errors present in space-time measuring systems used on high-speed tracks for determining velocity and acceleration. (Space-time systems determine either the instants of time at which a vehicle passes given positions in space or the position of a vehicle for known time intervals). Equations are derived for the errors of velocity, acceleration and the rate of change of acceleration, when the space and time errors inherent in the measuring system are known.

204. Sheldon, H.H. SPACE, TIME AND RELATIVITY. THE EINSTEIN UNIVERSE. 104p.,illus., New York, University Society, 1932.

A popular explanation, with illuminating illustrations, directed toward helping the layman understand the apparent slowing up of time.

205. Shepherd, L.R. INTERSTELLAR FLIGHT. Brit.Interplan.Soc.J. 11:149-167,Jly.1952.

Illustrates the time-dilatation effect as it affects interstellar flight.


In view of the paucity of experimental tests for the general theory of relativity, it is desirable to consider the uses to which a satellite vehicle could be put. The advance of the perigee is calculated similarly to the perihelion advance of Mercury; it amounts to only 15 seconds of the arc per year. However, the effect on a satellite clock is large and could be measured. With respect to an earth clock it is calculated to be a "red shift" for low-altitude orbits, zero shift for an orbit of one-half the earth's radius, and a "violet shift" for higher altitudes,, where it approaches 7 x 10-10. Some experimental schemes for the measurement of the clock shift are discussed; a counting technique seems to be best suited since it is capable of higher ultimate accuracy and avoids signalling problems during intercomparison arising from the motion of the satellite.

207. Singer, S.F. MEASUREMENTS OF THE EARTH'S MAGNETIC FIELD. In Van Allen, J.A.,ed. Scientific Uses of Earth Satellites, p.215-233, Ann Arbor, University of Michigan Press, 1956.

- 41 Discusses the application of satellite observations to magnetic measurement on three different time scales.

208. Singer, S.F. RELATIVITY AND SPACE TRAVEL. Nature 179:977, May 11,1957.

Comments on the Dingle-McCrea interchange on the subject of relativity and space flight and suggests an experimental approach involving long accumulation times.

209. Singer, S.F. SATELLITES FOR PHYSICISTS. Phys.Today 9:21-23,Apr.1956.

Mentions possible aid in revealing nature of relativistic earthbound clock.

210. Singer, S.F. SPACE VEHICLES AS TOOLS FOR RESEARCH IN RELATIVITY. In American Astronautical Society Proceedings; 3rd Annual Meeting, December 6-7,1956, p.121-125, New York, The Society, 1957. Also in J.Astronautics 4:49-51, Autumn,1957.

The twin paradox from the special theory of relativity is discussed. Special attention is given to a presentation of the application of close orbit earth satellites, with the aid of atomic clocks, to perform an experimental test of relativity. The author discusses the long-standing controversy of the twin paradox and states conclusively that a time advantage will exist for the traveler on extended space voyage.

211. SLOW TIME. Sci.Am. 196:63-64,Mar.1957.

Refers to the "long-standing dispute about the behavior of time in the theory of relativity" which "appears to have been settled."

The question is: Would space travelers moving at close to the speed of light age more slowly than people on the earth? According to F.S. Crawford of the University of California, "the answer is definitely 'yes' at least if the travelers are mu-mesons. (See Item 45).

"Comparisons of the intensity of mu-mesons on mountaintops and at sea level show that their half-life in flight is about 30 millionths of a second against only two millionths of a second at rest. The difference seems to be evidence for a relativistic fountain of youth."

212. Snyder, H.S. QUANTIZED SPACE-TIME. Phys.Rev. 71:38-41, Jan.1,1947.

Although it is usually assumed that space-time is continuum, it is pointed out that this assumption is not required by Lorentz invariance.

- 42 213. SPACE PROGRAM OUTLINED. Sci.News Ltr. 75:99, Feb.14,1959.

Among the experiments included in U.S. space program for 1959 is that involving a further test of Einstein's general theory of relativity. This would be done by comparing the time kept by an atomic clock in an earth-circling satellite with a similar clock at a ground station during a month or more.

214. SPACE TRAVEL AND AGEING. Discovery 18:174-175, Apr.1957.

Letters to the editor from Halsbury; Herbert Dingle; Bennett Weston; E. Hogben; R.A. Fisher, relative to the controversy between Dingle and McCrea, and W.H. McCrea's reply to these letters.

215. Spaulding, John. THE SPECIAL THEORY OF RELATIVITY AND SPACE NAVIGATION. I. J.Space Flight 1:1-8,Aug.1949.

Demonstrates that two bodies cannot approach each other faster than the velocity of light.

216. Stehling, K.R. SPACE TRAVEL AND RELATIVITY OR HOW TO KEEP FROM GROWING OLD. Jet Propul. 26:1105-1106, Dec. 1956.

A review of opinions and arguments concerning time dilatation.

217. Steinmetz, C.P. FOUR LECTURES ON RELATIVITY AND SPACE. 126p.,diags., New York, McGraw-Hill, 1923.

Effect of relative motion on length of time, p.3-6; time effects, p.67.

218. Stephenson, G. and Kilmister, C.W. SPECIAL RELATIVITY FOR PHYSICISTS. 108p., New York, Longmans, Green, 1958.

The clock paradox, p.42-44.

219. Stoyko, N. L'HORLOGE ATOMIQUE ET L'IRREGULARITÉ DE LA ROTATION DE LA TERRE. Acad.Sci.Paris.Compt.Rend. 244:43-45, Jan.2,1957.

In French.

Translated title: The atomic clock and the irregularity of the rotation of the earth.

The variations in the rotation of the earth as determined from observations with the caesium frequency standard constructed by Essen at the NPL are compared with the extrapolated values used at the Bureau International de l'Heure to allow for the seasonal

- 43 irregularity of the earth's rotation. The agreement is very close but the caesium values show more detail.

220. Strömberg, G. SPACE-STRUCTURE AND MOTION. Sci. 76:477-481, Nov.25; 504-508, Dec.2,1932.

A general discussion of the meaning of motion is undertaken. A distinction is made between the kinematic description of motion, in which forces are distinct from the space-structure, and the dynamical, in which forces play no part, but alterations in space-structure control the motions of bodies.

221. Subotowicz, M. SATELLITES FOR CHECKING EINSTEIN'S RELATIVITY THEORY. Missiles and Rockets 2:57-59,illus.,Feb.1957.

According to time-lengthening theories in Einstein's general theory of relativity, a clock in an earth satellite would run more slowly than one at relative rest on earth. The writer of this article says the difference in a year could be measured with available equipment.

222. Synge, J.L. RELATIVITY: THE SPECIAL THEORY. 450p.,diags., New York, Interscience Publishers, 1956.

The clock problem is discussed on p.7,16,118,122 and 189.

Chapter I is entitled. The space-time continuum and the separation between events.

223. Thomson, G.P. THE FORESEEABLE FUTURE. 166p., Cambridge, University Press, 1955.

On page 89 there is a discussion of a journey to the nearest star and back in just over seventeen years by terrestrial clocks. The author states that owing to "the relativistic contraction of time: the travelers would have "aged" by only 14 1/2 years and adds the following footnote: "The bearing of the relativistic contraction of time on this problem has been questioned, since it leads to an apparent paradox, but the best opinion is that the contraction would occur and that the returning astronaut would, in fact, find that time has gone more rapidly on the earth than in his space ship."

224. TIME OUT. Sci.Am. 195:58,60,Dec.1956.

Quotes statements of Herbert Dingle and W.H. McCrea which have appeared in recent issues of Nature in connection with the controversy over motion causing space and time to contract.

- 44 "A method for settling the argument experimentally without waiting for interstellar space ships was suggested by H. Herman of the Standard Telecommunication Laboratories in England. He pointed out that any radioactive atom of known half-life is a clock. He proposed that a beam of such atoms be accelerated to relativistic speed for a measured time in a synchroton and their aging be measured afterward. If the decay of the atoms was less than would normally be expected, this would show that the clock has indeed slowed down."

225. Tolman, R.C. RELATIVITY, THERMODYNAMICS AND COSMOLOGY. 501p., Oxford, Clarendon Press, 1934.

Gives extension to general relativistic treatment of the clock paradox, p.192-198.

226. Truitt, R.W. ANALOGY OF SPECIAL THEORY OF RELATIVITY TO STUDY OF COMPRESSIBLE FLUID FLOW. 20p., Raleigh, N.C., 1949. (N.C. State Col.Dept.Eng.Res.Bull. 44)

Relativistic idea that speed of light is limiting speed between observers if replaced by restricted universe where speed of sound is imposed limit; Lorentz transformation used to relate time and space units between two frames of reference; one frame of reference is airfoil and the other surrounding air; contraction of body as it approaches speed of light is seen by analogy to be explanation for forces involved.


Relativity of space and time, p.146-156.

228. Voigt, W. UEBER DAS DOPPLERISCHE PRINCIP. Ges.Wiss.Göttingen Nachr. 10:41-51,Mar.10,1887.

In German.

Translated title: Concerning the Doppler principle.

The first suggestion that a "natural" clock would alter its rate on motion appears in this paper.

229. von Gleich, Gerold. ZUR DEFINITION DES ZEITBEGRIFFES. Z.Phys. 47:280-298, 1928.

In German.

Translated title: Definition of the notion of time.

Modern speculations concerning the nature of time and space are discussed, and it is pointed out that the space-time world of the relativity theory is merely an artificial image. The two

- 45 different ideas of time of the relativity theory are solely consequences of physically unfounded transformations of the coordinates of the image, or of unjustified applications of the theory of invariants.


In German.

Translated title: Two flaws in the theory of relativity and its proof.

Discussion of the clock paradox.

231. Walker, A.G. A THEORY OF REGRADUATION IN GENERAL RELATIVITY. Roy.Soc.Edinburgh Proc. 62A:164-174, 1946.

The regraduation of an observer's proper-time clock is discussed, starting from the local Galilean coordinate system at each point of a Riemannian space-time. In order to show that nontrivial regraduations exist, a new interpretation of Einstein's gravitational equations is proposed, viz., that regraduations preserve the form of the equations but alter the values of the gravitational constant and the cosmical constant. No experimental evidence is adduced for these assumptions. Particular cases are worked out in detail with special references to Lemaitre expanding universes. It is shown that certain regraduations are possible which transform the cosmical constant to zero.

232. Weyl, Hermann. SPACE-TIME-MATTER, TRANSLATED FROM THE GERMAN BY HENRY L. BROSE. 330p., New York, Dover, 1922.

233. Weyl, Hermann. ZUR ALLGEMEINEN RELATIVITAETSTHEORIE. Phys.Z. 24:230-232, June 1,1923.

In German.

Translated title: General relativity theory.

Formulates simple principle enabling an observer in an arbitrary gravitational field to estimate change in wavelength for any given motions of himself and a point source of light.

234. Whittaker, Edmund. FROM EUCLID TO EDDINGTON. A STUDY OF CONCEPTIONS OF THE EXTERNAL WORLD. 212p., Cambridge, University Press, 1949.

A history of the evolution of concepts and principles, especially such as have provoked long controversies. Time is discussed throughout the book.

- 46 235. Wigner, E.P. RELATIVISTIC INVARIANCE AND QUANTUM PHENOMENA. Rev.Mod.Phys. 29:255-268, Jly.1957.

Considers the quantum limitations on the accuracy of the conversion of time-like measurements into space-like measurements, illustrated in figure 4, p.260. Space-like distances are measured by means of a clock.

236. WILL SPACE TRAVEL LENGTHEN LIFE? Popular Sci. 171:103,Oct.1957.

Refers to an advertisement in Time magazine by the Martin Company titled " What is Time?" and asks why the Company should pose such an esoteric question.

"The reason is that the problem of speed vs time is no longer a matter for Einsteins. With rockets and satellites bristling all over, it becomes practical to know for sure whether time (hence life) is affected by speed."

According to a scientific experiment with a clock, or "a handy substitute - the meson" explained in the article, "a meson in flight lives about 15 times longer than a meson at rest. You can indeed put the brakes on time." (See Item 14 for scientific experiment.)

237. Windred, G. EVOLUTION OF THE NOTION OF TIME. Am.J.Sci. 30:383-391,Oct.1935.

A sketch is given of the ideas of time found in mathematical physics from the time of Newton to the present day.


In German. English summary.

Translated title: Relativistic time dilatation in an artificial satellite.

Applies the general theory of relativity to the problem of time-measurement in a satellite. Concludes that there should be a measurable apparent slowing of satellite-borne clocks, of the order of several thousandths of a second annually.

239. Witten, Louis. EXPERIMENTAL ASPECTS OF GENERAL RELATIVITY THEORY. J.Astronautics 4:46-48,illus.,Autumn,1957.

The general relativity theory is discussed, and an outline of the several experimental verifications of the theory is reviewed. Consideration is given to the use of atomic clocks

- 47 having accuracies of the order of 1 part in 1012 in discerning the time shift predicted by the theory of relativity.

Also in Proceedings of the 3rd American Astronautical Society, December 6-7, 1956, p.127-135.

240. WORLD'S FIRST SPACE CLOCK INTRODUCED BY HAMILTON WATCH CO. Commercial and Finan.Chron. 179:157, Jan.14,1954.

Brief description of the space clock conceived by I.M. Levitt of Fels Planetarium.


In Russian.

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This work is in the public domain in the United States because it is a work of the United States federal government (see 17 U.S.C. 105).