We Christians like to get into debates that there really is

no answer to. And we get into a huge huff about it too. One old debate was the

one about Angels and Pin points (not pin heads as is often cited although it

does result in a certain amount of pin-headishness). I found a few articles:

one that confirmed that there was some wacky discussions during the medieval

period (though this one may not have been one of them), one that offers a

methodology to answer the question and one (here’s the link and it’s also in

the read more) that actually mathematically counted how many Angels can dance

on the point of a pin.

## Quantum Gravity Treatment of the Angel Density Problem

*[EDITOR’S NOTE: we apologize for the lack of clear formatting,in this
web version, of the mathematical formulae.]*

#### Abstract

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.[5]

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.[2]

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 (RÂ¼1 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 Ã…ngstrÃ¶m 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.[1]

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 Ã…ngstrÃ¶m (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[3] 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.[7] 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 Â¼3.8807*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/*c*exp2+kD),

very slightly below mcrit. This corresponds to the maximal angel density

of Nmax=8.6766*10exp49 angels (see figure).

#### Dance Dynamics

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).[4]

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 Ãx?R we find that the uncertainty in momentum must be

Ãp?hbar/R, and this leads to a velocity uncertainty Ãv>hbar/Rm. If m=

mcrit we get Ãv>> 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.

#### Discussion

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 Â¼3.8807*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.

#### Bibliography

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>.

of Yamada Conference, Kyoto, Japan, April 1998

7.

*The Physics of Immortality*, Frank Tipler,. Macmillan, London,

1994.

Â© Copyright 2001 Annals of Improbable

Research (AIR)