19 Conceptual Clock Models

H. SALECKER opened the discussion by reporting on some conceptual clock models which he has analyzed in collaboration with E. P. Wigner. He first made some general comments on the fundamental nature of space-time distance measurements: In ordinary quantum mechanics the space-time point is specified by its four coordinates, but no prescription is given as to how these coordinates are to be measured. This, however, is in conflict with the principles of the general theory of relativity, according to which coordinates have no meaning independent of observation. A coordinate system can be defined only if space-time distance measurements can be carried out in principle without restrictions. SALECKER then went on to point out that the use of clocks alone is sufficient to measure both space-time and time-like distances. This is important since measuring rods, in contrast to clocks, are essentially macro-physical objects which will strongly influence other objects during the measuring procedure through their gravitational fields. Moreover, the measurement of distances between space-time points is considerably more complicated with measuring rods than clocks because of the Lorentz contraction of rods.

The method by which a clock can be used to determine a space-like distance between two events A and B is as follows: Let the geodesic world line of the clock pass through the event A at time , and suppose that the clock emits a continuously modulated light signal, e.g., monotonically changing color.¹ Let this signal be reflected at event B by a briefly exposed mirror. The time of emission of the reflected portion of the signal will be determined by inspection of the color of this portion when it eventually returns to the clock at time . The invariant distance S between A and B is then given by

provided is small compared to the radius of curvature of the space-time neighborhood of the two events.

A time-like distance can of course be measured by the obvious method of letting the clock pass along a geodesic between the two events.

SALECKER then described the results of considering the clock itself, and the process of reading it, in greater detail. Although his actual exposition suffered from lack of time in which to present the material, he very kindly gave the editors access to the manuscript of a paper which he will shortly publish on the subject. The following is a resume of portions of his paper. Certain simplifications have been made, leading to slight changes in numerical coefficients, for which the editors are entirely responsible.

One must first ask the question: What is the accuracy with which a clock can be read, independently of inherent inaccuracies of the clock itself? At the beginning of a time interval an observer may emit a light signal to read the clock. If the length of the signal’s train is and if the pulse is properly shaped, the root mean square uncertainty in the initial time measurement will be

The clock may simultaneously be made to emit a photon to compensate for recoil. However, an inevitable uncertainty in the clock’s momentum will remain, of amount

corresponding to an energy uncertainty in the light signal of amount

The clock will generally have also an initial position uncertainty giving rise to a momentum uncertainty

The total momentum uncertainty after the first reading is therefore

corresponding to a velocity uncertainty of amount

where the mass of the clock. (We neglect here certain relativistic corrections which SALECKER considered.) If is the mean distance from the observer to the clock then the time at which the observer receives the initial reading from the clock will be

The time at which the observer receives a second reading at the end of a (clock’s) time interval will be

giving for the total inaccuracy in the reading of the time interval

The minimum uncertainty is achieved by choosing , which yields

Evidently the accuracy of reading is greater the larger the mass of the clock. On the other hand, the gravitational field of the clock will be disturbing if its mass is made too large. One has therefore to consider the problem of how to construct a clock which shall be as light and as accurate as possible. One must at this point take into account the inherent inaccuracies in the clock itself, by considering its atomic structure.

A single atom by itself represents to a very high degree of accuracy an oscillator, but in spite of this it is not possible to take a single atom emitting radiation as a clock. Before an oscillator can be considered to be a clock it must be possible to register its information; i.e., the atom must be coupled with a device to count the number of times the emitted electromagnetic field strength reaches a certain value. Such a counting device would be in contradiction with the principles of quantum mechanics.

As his first model, therefore, SALECKER considered a statistical clock, composed of a certain number of elementary systems initially in an excited state. The systems were assumed to go over directly to a ground state and to be sufficiently well separated so as not to reexcite one another. If the decay rate is , then the probability of finding systems remaining in the excited state after a time is

where

The average value of and the root mean square deviation at time are given respectively by

The registering device in the clock has only to count the number of atoms remaining in the excited state (or, alternatively, the number of systems which have decayed) in order to record a statistical time given by

which has an uncertainty of amount

The minimum uncertainty is achieved by choosing , which yields

A final uncertainty arises from the fact that the registering device must distinguish between excited and unexcited systems, and for this purpose a time at least as great as is required, where and are respectively the elementary excited and ground state energy levels. In the most favorable case the decaying systems would undergo complete disintegration with, for example, the emission of two photons in opposite directions so as to eliminate recoil. The registering device might then be a counter to detect the photons, but however constructed it would have an absolute minimum time inaccuracy of amount , corresponding to photon energies of order where is the mass of a decaying elementary component of the clock.

The total inaccuracy in the measurement of a time interval by means of a statistical clock is therefore at least

where is the mass of the registering device plus the framework of the clock, and is the total clock mass. If one is clever enough in the construction of the clock the number may be kept from being excessively large. The minimum total inaccuracy for a given total mass is achieved by choosing

which yields

SALECKER considered also another type of statistical model, with essentially the same results. It should be pointed out immediately that for a given and (e.g., suitable for a certain experiment) it may be utterly impossible to choose the optimum value of , because of the very limited range of elementary-system masses occurring in nature. SALECKER therefore considered, as a third model, a very hypothetical type of clock, consisting of a single elementary harmonic oscillator, but operating at sufficiently high quantum numbers for it to be “read” like a classical device. Permitting himself to imagine an elementary oscillator which could function even at relativistic energies, he found for the minimum total uncertainty

where , being the rest mass of the oscillator and is its excitation energy.

The quantity is seen generally to depend on the time interval itself. When becomes of the order of then the time interval can no longer be measured. In every case the smallest time interval which can be measured is roughly

That is, the time associated with the mass of one of the elementary systems out of which the clock is constructed represents an absolute minimum for measurable time intervals. This minimum arises, of course, from the coupling between the registering device and the elementary systems, and would exist even if the registering device were removed from the immediate vicinity of the clock proper. (For example, the registering device might be the observer himself, but then, in the case of the statistical clocks, the observer would have to send out a large number of photons so as to “read” each elementary system separately, thus raising the reading error from to .) Since, for existing elementary systems, g, the corresponding time may, presently at least, be regarded as an absolute lower limit to measurable time intervals.

SALECKER finished his discussion by reporting on the results of applying such clocks to the measurement of gravitational fields. This part of his work has not yet been written up, so we can only record the final results which he quoted. Three typical measurements were considered by him:

1The direct measurement of .

2The measurement of through the observation of geodesics.

3The measurement of the scalar curvature of a three-dimensional space-like cross section of a quasi-static region of space-time. This is defined by

where is the area of a small triangle (or prism, in space-time) around which a test body is passed, and the are its angles. (He also proposed that a direct measurement of the Riemann tensor should be carried out, by the method suggested in a previous session by Pirani. He had not been aware of this possibility prior to the conference.)

Carrying out these conceptual measurements on the gravitational field of a particle of protonic mass, using a particle of electronic mass as a test particle, and choosing the most favorable clock model to serve as measuring apparatus, SALECKER found for the uncertainty in the resultant measurement of the mass producing the field

That is, the uncertainty is of the same order as the mass itself. From this, he concluded that the gravitational mass of a single proton is not strictly an observable quantity.

FEYNMAN asked SALECKER if he could write down a formula for in terms of fundamental constants, omitting dimensionless factors. SALECKER replied that this would be quite difficult. The difficulty seems to be that one has to account for the perturbing effect of the clock and, in effect, solve a two-body problem. The uncertainty in the measurement of time increases proportionally to some fractional power ( or ) of the interval being measured, and decreases with increasing clock mass. On the other hand the bigger the clock the greater its perturbing (or masking) effect. One has to carry out a minimizing procedure in which the interval , the protonic mass under observation, the clock mass , and, in the case of statistical clocks, the number are all involved. In the resulting numerical computations the fundamental constants get mixed up in a very complicated way. SALECKER did say, however, that is proportional to the mass being measured, the factor of proportionality depending on the ratio of the gravitational radius of this mass to that of the clock.

(The editors would like to suggest that, in view of the long range character of the gravitational force, discussed previously, the gravitational constant may, in the last analysis, not enter into the factor of proportionality. Furthermore, the value actually found for may in some way reflect the fundamental limitation in the clock, viz. that it cannot be constructed out of elementary systems having masses smaller than those found in nature, i.e., protonic.)

FEYNMAN remained unconvinced of Salecker’s result. He suggested the use of hypothetical atoms in which only two forces are operative: (1) the gravitational force and (2) some force which does not have an inverse square character. The relative phases of different time dependent states will then differ for atoms of different masses, and by waiting a sufficiently long time (e.g., 100 times the age of the universe!) one could measure the phase differences and hence infer the gravitational masses involved to as high a degree of precision as may be desired.

SALECKER replied that Feynman would still have to take into account the apparatus which measured this very long interval of time. Moreover, he pointed out that his own result did not take into account the possibility of pair production arising from necessity of measuring space-like intervals having dimensions less than the electronic Compton wavelength.

ROSENFELD asked how the expression for depends on .

SALECKER replied that he was unable to say, since his computations were started from the beginning with numerical values (e.g. g for the proton mass, etc.).

ROSENFELD then took the floor to make some remarks about the difficulties of trying to extend the theory of measurement to the gravitational field in the manner in which he and Bohr had done for the electromagnetic field. He first said that he had tried to see how far one can get, following Salecker, in determining the gravitational potentials by metrical measurements rather than by dynamical measurements. (By dynamical measurements one can get only the derivatives of the potentials, metrical measurements give the potentials themselves, apart from a gauge factor.)

Roughly speaking, the inaccuracy in the measurement of a gravitational potential - say - will be proportional to the inaccuracy in the determination of lengths. Therefore, if you can determine lengths with any accuracy, then you can also determine the potential with any accuracy. However, quantum considerations tell you that if the position of the measuring rod or clock is known to an accuracy then its momentum is uncertain by an amount . This gives rise to an uncertainty in the value of the gravitational field produced by the measuring instrument.

The factors which saved Bohr and Rosenfeld in the electromagnetic case were:

1Because of the existence of both negative and positive charge the perturbing field of the measuring instrument could be reduced to a dipole field.

2The charge to mass ratio of the measuring instrument could be controlled.

Therefore, Bohr and Rosenfeld came to the conclusion that the measurement of any component of the electromagnetic field could be carried out with arbitrarily high precision in spite of the quantum restrictions.

These saving features are not present in the gravitational case. Wheth-er one takes the measuring instrument heavy or light its perturbing effect will be roughly the same (proportional to ). Therefore ROSENFELD agreed with Salecker that there will be a fundamental limitation on the accuracy with which one can measure the gravitational field, although he could give no numerical estimate of this limitation.

WHEELER suggested that perhaps one should simply forget about the measurement problem and proceed with other aspects of theory. The history of electrodynamics shows that it is always a ticklish business to conclude too early that there are certain limitations on a measurement. He would propose rather to emphasize the organic unity of nature, to develop the theory (i.e., quantum gravidynamics) first and then to return later to the measurement problem. He suggested that this was particularly appropriate when we don’t even understand too much about the classical measurement process! We don’t know yet exactly what it is that one should measure on two space-like surfaces, i.e. the specification of the initial value problem.

He then went on to imagine what sort of ideas scientists might come up with if they were “put under torture” to develop a theory that would explain all the elementary particles and their interactions solely in terms of gravitation and electromagnetism alone! He first took a look a magnitudes and dimensions. In the Feynman quantization method one must “sum over histories” an amplitude which, in the combined electromagnetic-gravitational case has roughly the form

Therefore if one is making measurements in a space-time region of volume , contributions to this sum will be more or less in phase until variations in the electromagnetic and gravitational field amplitudes from their classical values become as large as

These represent the quantum fluctuations of the electromagnetic and gravitational fields. In the gravitational case, owing to the nonlinearity of the field equations truly new effects come into play at distances as small as cm where becomes of the order of unity. WHEELER envisages a “foam-like structure” for the vacuum, arising from these fluctuations of the metric. He compared our observation of the vacuum with the view of an aviator flying over the ocean. At high altitudes the ocean looks smooth, but begins to show roughness as the aviator descends. In the ease of the vacuum, WHEELER believes that if we look at it on a sufficiently small scale it may even change its topological connectedness, thus:

In this way he has been led to the concept of “wormhole” in space. A two-dimensional analog of a wormhole would look like this:

WHEELER pointed out another way in which a dimension of the order of cm could be arrived at, by considering only the fluctuations in the electromagnetic field. A fluctuation of amount would correspond to an energy fluctuation of amount in a region of dimension . The gravitational energy produced in this region by this fluctuation would be of the order of . The two energies become comparable when cm. In WHEELER’s “dream of the tortured scientists” the wormholes may serve as sources or sinks of electric lines of force, no charge being actually involved since the lines of force may pass continuously through the wormholes. A surface integral of the lines of force over a region of dimension would yield a value of the order of which would represent a rough average of the apparent charge associated with each wormhole. No quantization of charge is implied here. In fact the wormholes themselves have nothing directly to do with elementary particles. WHEELER envisaged an elementary particle as a vast structure ( ) compared to a wormhole. However, he left open the possibility that elementary particles might somehow be constructed out of wormholes. He compared the wormholes to “undressed particles,” their continual formation corresponding to pair production which is going on in the vacuum at all times. The electromagnetic mass associated with each wormhole is of the order of g, but this huge mass is almost entirely compensated by an equivalent amount of negative gravitational energy.

(Editor’s Note: No one at the conference thought to ask Wheeler why wormholes corresponding to magnetic poles would not be just as likely to occur as those corresponding to charges.)

WHEELER outlined five new concepts which have either already arisen or may yet arise as a result of pushing the ideas of general relativity to the limit:

• (1) Electromagnetism without electromagnetism.: By this, WHEELER was referring to MISNER’s work which shows that the electromagnetic field can be completely specified by its stress tensor alone. But the stress tensor is in turn specified by certain derivatives of the metric. Hence, merely by looking at the metric one can tell all about the electromagnetic field. One might call this the “unified imprint theory,” since the electromagnetic field leaves its characteristic imprint on the structure of space-time.

• (2) Mass without mass.: This concept is illustrated by geons, of which several varieties are currently under consideration:

• a) Electromagnetic geons.:

• b) Geons built out of neutrinos.:

• c) Geons built out of pure gravitational radiation alone.:

• (3) Charge without charge.: Wormholes.

• (4) Spin without spin.:

• (5) Elementary particles without elementary particles.:

As for the last two ideas, WHEELER stated that so far the “tortured scientists” have been unable to come up with anything plausible, except for some vague suggestions that may temporarily stave off the punishment which awaits them if they fail to produce an answer. He thought that spin might arise in a Feynman quantization procedure owing to a possible double-valuedness in the sign of which occurs in the action.

FEYNMAN objected at this point, saying that as long as is positive, is always of the same sign. A much more serious problem, he felt, would be to determine what happens if changes sign.

BELINFANTE pointed out that the square root does not occur in the Lagrangian if the metric field is described in terms of “vierbeine”.

WHEELER emphasized that he was not trying to give answers but merely pointing out what one might try to do if motivated by the overriding idea that “physics is geometry” and that everything can be derived from a master metric field.

WEBER wondered how one could get some quantities possessing spin and others which are spinless out of such a picture.

As for “elementary particles without elementary particles,” WHEELER suggested that one must first study the vacuum. The vacuum is in such turmoil, according to his picture, that it would be foolish to study elementary particles without first trying to understand the vacuum. He drew the following schematic dispersion curve for a wave disturbance (e.g., a photon or a graviton) propagating through the vacuum:

Over a tremendous range the wave satisfies the simple relation . When the wavelength becomes of the order of the size of the universe ( cm), however, the phase velocity is greater than , as has been shown long ago by Schrödinger (Papal Acad.). On the other hand, when the wavelength becomes of the order of cm the disturbance will be slowed down by the foam-like structure of the vacuum (and also by having always to climb over the metric bump which it is itself continuously creating due to its large energy concentration - Ed.). WHEELER suggested wedding the ultramacroscopic to the ultramicroscopic by finding the point of inflection on the dispersion curve. This would be the point at which disturbances would tend to hold themselves together. He also drew an analogy with superconductivity by suggesting that elementary particles might be the product of long range collective motions of the vacuum “foam,” the scale of elementary particles being much larger than the scale of the wormholes, just as the scale of superconducting regions is much larger than that of the elementary structures which give rise to it. In view of possibilities like these, WHEELER concluded finally that one could not absolutely deny the possibility of explaining all things in terms of metric.

GOLD suggested that this was an “answer without an answer.”

WEBER spoke for many by saying that he didn’t understand how such a large number of wormholes ( ) could give rise to such a well-defined quantity as the observed charge on an electron, except possibly by some statistical means.

FEYNMAN suggested that perhaps there was a unique stable ground state for a wormhole and that this would represent a sharply quantized “bare charge”.

WHEELER pointed out that the discrepancy between the charge associated with an undressed particle and the observed charge on an electron could be accounted for by vacuum polarization.

ROSENFELD then took the floor to express some second thoughts which he had had on the question of measurability. It seemed to him that, in principle, one could determine the mass of an arbitrarily small body to an arbitrarily high degree of precision by putting it on an ordinary spring balance and waiting long enough. Therefore he thought there was perhaps some doubt after all about Salecker’s limitations on the measurability of the gravitational mass of elementary particles.

FEYNMAN pointed out however that Salecker was trying to measure the gravitational field produced by a small mass whereas Rosenfeld was here considering the response of that mass to a given gravitational field.

ROSENFELD said, nevertheless, that perhaps his original pessimism in regard to the measurability of gravitational fields, as compared to the electromagnetic case, might be unjustified. For example, it might happen that the uncertainties and in the measuring instrument affect different components of the gravitational potential, leading to reciprocal relations of the form

which would not, however, prevent the precise determination of the value of a given component.

ANDERSON raised the question of the propriety of “measuring” quantities such as which have neither invariance nor tensor properties.

BARGMANN replied that the insistence on invariance has often been overdone. An invariant can always be defined provided one simply introduces the measuring apparatus into the mathematical description.

ANDERSON said that “gadgets” then evidently introduce “preferred” situations into the scheme of things and permit one to measure non-invariant quantities directly.

BERGMANN objected to this point of view.

PIRANI remarked that, in any case, if you are clever enough you should be able to construct true invariants or tensors (which don’t depend on your gadgets) from the results of reading your gadgets.

This concluded the discussion of the problems of measurement. Attention next turned to the technical aspects of the purely formal problems which arise when the attempt is made to apply mathematical quantization procedures to the gravitational field.

Footnotes