SANS/NADA, Royal Institute of Technology, Stockholm, Sweden
[EDITOR'S NOTE: we apologize for the lack of clear formatting,in this web version, of the mathematical formulae.]
We derive upper bounds for the density of angels dancing on the point of a pin. It is dependent on the assumed mass of the angels, with a maximum number of 8.6766*10exp49 angels at the critical angel mass (3.8807*10exp-34 kg).
Ancient Question, Modern Physics
"How many angels can dance on the head of a pin?" has been a major theological question since the Middle Ages.
According to Thomas Aquinas, it is impossible for two distinct causes to each be the immediate cause of one and the same thing. An angel is a good example of such a cause. Thus two angels cannot occupy the same space. This can be seen as an early statement of the Pauli exclusion principle. (The Pauli exclusion principle is a pillar of modern physics. It was first stated in the twentieth century, by Pauli.)
However, this does not place any upper bound on the density of angels in a small area, because the size r of angels remains undefined and could possibly be arbitrarily small. There have also been theological criticisms of any assumption of angels as complete causes.
Stating the Question Correctly
The basic issue is the maximal density of active angels in a small volume. It should be noted that the original formulation of the problem did not refer to the head of a pin (R?1 mm) but to the point of the pin. Therefore, the point, not the head, of the pin is the region that will be studied in this paper.
One of the first reported attempts at a quantum gravity treatment of the angel density problem that also included the correct end of the pin was made by Dr. Phil Schewe. He suggested that due to quantum gravity space is likely not infinitely divisible beyond the Planck length scale of 10exp-35 meters. Hence, assuming the point of the pin to be one Ångström across (the size of a scanning tunnelling microscope tip) this would produce a maximal number of angels on the order of 1050 since they would not have more places to fill.
While this approach does produce an upper bound on the possible density of angels, it is based on the Thomist assumption of non-overlap.
Since angels can be presumed to obey quantum rules when packed at quantum gravity densities, the uncertainty relation will cause their wave functions to overlap significantly even if there is a strong degeneracy pressure. If the non-overlap assumption is relaxed, this approach cannot derive an upper bound.
Quantum Gravitational Treatment
A stricter bound based on information physics can be derived that is not based on overlap assumptions, but merely the localisation of angelic information.
Assuming that each angel contains at least one bit of information (fallen / not fallen), and that the point of the pin is a sphere of diameter of an Ångström (R=10exp-10 m) and has a total mass of M=9.5*10exp-29 kilograms (equivalent to that of one iron atom), we can use the Bekenstein bound on information to calculate an upper bound on the angel density. In a system of diameter D and mass M, less than kDM distinguishable bits can exist, where k=2.57686*10exp43 bits/meter kg. This gives us a bound of just 2.448*10exp5 angels, far below the Schewe bound.
Note that this does not take the mass of angels into account. A finite angel mass-energy would increase the possible information density significantly. If each angel has a mass m, then the Bekenstein bound gives us N<kD(M+Nm). Beyond mcrit>1/kD ?3.8807*10exp-34 kg this produces an unbounded maximal angel density as each angel contributes enough mass-energy to allow the information of an extra angel to move in, and so on.
However, if angels have mass, then the point of the pin will collapse into a black hole if c2R/2G< Nm (here I ignore the mass of the iron atom at the tip).4 For angels of human weight (80 kg), we get a limit of 4.2089*10exp14 angels. The maximal mass of any angel amenable to dance on the pin is 3.3671*10exp16 kg; at this point there is only room for a single angel.
The picture that emerges is that, for low angel masses, the number is bounded by the Bekenstein bound, and increases hyperbolically as mcrit is approached. However, the black hole bound decreases and the two bounds cross at mmax=1/(4GkM/cexp2+kD), very slightly below mcrit. This corresponds to the maximal angel density of Nmax=8.6766*10exp49 angels (see figure).
Maximum number of angels for a given mass. The allowed region is bounded from above by the line c2R/2G=Nm (gravitational collapse) and the curve N=kD(M+Nm) (information density) which has an asymptote for mcrit, and from below by N=0. The maximal number of angels occurs at the intersection of the gravitational bound and the asymptote at mcrit
If the angels dance very quickly and in the same direction, then the angular momentum could lead to a situation like the extremal Kerr metric, where no event horizon forms (this could also be achieved by charging the angels). Hence the number of dancing angels that can crowd together is likely much higher than the number of stationary angels.
However, at these speeds the friction caused by their interaction with the pin is likely to vaporise it or at least break it apart. Even for a modest speed of 1 m/s the total kinetic energy of Nmax angels of mass mcrit would be 1.682*10exp16 J. In the case of charged angels at relativistic densities, pair-creation in their vicinity would likely cause the charge to dissipate over time,6 and charge transfer to the pin would also likely induce electromechanical forces beyond any material tolerances.
The uncertainty relation also imposes a limitation on the dance. Since the uncertainty in position of the angels by assumption is less than the size of the point ?x?R we find that the uncertainty in momentum must be ?p?hbar/R, and this leads to a velocity uncertainty ?v>hbar/Rm. If m= mcrit we get ?v>> 8.6766*10exp59 m/s (>> c), which shows that:
(1) the angels must dance with speeds near the velocity of light in order to obey quantum mechanics;
(2) a full relativistic treatment is necessary; and
(3) that the precision of the dance must break down due to quantum effects.
This can be used to rule out certain types of dance due to their high precision requirements.
We have derived quantum gravity bounds on the number of angels that can dance on the tip of a needle as a function of the mass of the angels. The maximal number of angels -- 8.6766*10exp49 -- is achieved near the critical mass mcrit>1/kD ?3.8807*10-34 kg, corresponding to the transition from the information-limited to the mass-limited regime. It is interesting to note that this is of the same order of magnitude as the Schewe bound.
Angel physics has until now mainly employed theological methods, but as this paper shows, modern information physics, quantum gravity and relativity theory provide powerful tools for exploring the dynamics and statics of angels.
These bounds are only upper bounds, and do not take into account the effects of a finite number of available angels, degeneracy pressures if angels obey the Pauli exclusion principle as suggested by Aquinas, or the theo-psychology of the angels themselves. The exact dance dynamics also clearly play a major role. A full relativistic treatment of the dance appears as a promising avenue for further tightening of the bounds.
1. Reported in the New York Times, November 11, 1997, "Science Q \& A: Dancing Angels," by C. Claiborne Ray <http://www.nytimes.com/learning/students/scienceqa/archive/971111.html>. Dr. Phil Schewe presented his idea at a meeting of the Society for Literature and Science in 1995. It should be noted that the given density in the New York Times is wrong, as it did not count the area, just the length.
2. T. Aquinas, Summa Theologiae, vol. 52, no. 3, 1266.
3. J. Bekenstein, Physical Review D, vol. 23, no. 287, 1981.
4. Gravitation, C. Misner, K. Thorne, and J. Wheeler, W.H. Freeman and Co., 1973.
5. Memoirs of the Extraordinary Life, Works and Discoveries of Martinus Scriblerus, 1741, A. Pope, J. Swift, J. Gay, T. Parnell, and J. Arbuthnot, ed. by C. Kerby-Miller, 1950; republished 1966.
6. "On the Dyadosphere of Black Holes," R. Ruffini, in Proceedings of Yamada Conference, Kyoto, Japan, April 1998. <http://xxx.lanl.gov/abs/astro-ph/9811232>.
7. The Physics of Immortality, Frank Tipler,. Macmillan, London, 1994.
_____________________This article is republished with permission from the May-June 2001 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!
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