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.
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
the mass of the clock. (We neglect here certain relativistic corrections which SALECKER
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
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
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.
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
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
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
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.
(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.)
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
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
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
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
becomes of the order of unity. WHEELER
In this way he has been led to the concept of “wormhole” in space.
(Editor’s Note: No one at the conference thought to ask Wheeler
• (1) Electromagnetism without electromagnetism.: By this, WHEELER
• (2) Mass without mass.: This concept is illustrated by geons,
• 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
As for “elementary particles without elementary particles,” WHEELER
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.).
which would not, however, prevent the precise determination of the value of a given component.
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.
SALECKER let the position of event B be simply determined by the time of emission of a pulsed signal. The editors have inserted the idea of color modulation in order to avoid objections based on the principle of causality.