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In article <1992Nov7.013240.1@stsci.edu> zellner@stsci.edu writes:
>In article <1992Nov06.211115.134895@watson.ibm.com>, lucas@watson.ibm.com (Bruce D. Lucas) writes:
>> How do we know space is three-dimensional, or space-time
>> four-dimensional?  Is there a simple experiment one can do to
>> determine the dimensionality of space or of space-time?
>
>Keplerian orbits of planets around the sun.  They are stable in 2 or 3
>dimensions, but I think not in most higher dimensionalities or in things
>like 3 + delta dimensions.  Can someone confirm that?
>
>Ben
>
	For the question of orbital stability, refer to list of essays below.


The dimensionality of spacetime has always been an interest of mine.
 
Below is a list of some papers and essays on the subject.
I haven't read and worked through all of these.  So, I can't vouch for
the validity of all of the various arguments.
 
        In my opinion, extrapolating physical laws into another dimension is
not trivial and not unique.  One has to define what one takes as the "law"
and its "natural" generalization, mathematically.  One should try to
capture "just the physics" in the mathematics, leaving out extraneous things
like a particular choice of coordinate-system or gauge. Then, do mathematical
operations, and then _try_ to read off the "physics".
        Consider Newtonian gravitation.  Is the "law" that mass points
attract each other with an inverse-square force law, independent of the
dimension of space?  Or as 1/r^(n-1)), where n is the dimension of
space?  Or is it that the n-dimensional Poisson equation (del^2 phi= -rho)
be satisfied?  These three are more-or-less equivalent in three space
dimensions, but not in two dimensions, for example.
        Just a thought: It might be nice to formulate the laws of physics
in a "dimension-free" way and observing which phenomena arise naturally,
independent of dimension, and which arise from picking out a particular
choice of dimension.
        It might be also good to remember that what we are doing is
playing with our_mathematical_formulations_of_physical_laws, not the
physical laws themselves.
 
        An interesting curiosity is described in Barrow's Dimensionality
paper.  Suppose one takes as fundamental the natural constants of quantum
mechanics and relativity, namely, the Planck constant h and the speed of
light c.  And suppose one varies the coupling constants G, for gravity,
and e, for electromagnetism, by insisting that the n-dimensional Poisson
equation be satisfied.  Then, the dimensionless combinations of these
constants are for n=1,2,3,4,5: G/hc^3, Ge^2/c^4, e^2/hc (of course),
e^6/Gh^4, ce^4/Gh^3.  One gets the general combination using that
[h]=ML^2/T, [c]=L/T, [G]=L^n/MT^2, [e^2]=ML^n/T^2, (M=mass, L=length,
T=time, n=dim of space), and solving an algebraic equation, which can
be written as [e^(2(n-1)) h^(-2(n-2)) G^-(n-3) c^(2(n-4))]=1.  This
shows that for n<=4, one member of the set {h,c,G,e} does not appear
in the dimensionless combination.  Is there any physics buried in here?
or is it just a mathematical curiosity?
 
 
ESSAYS:----- (some of these refer to the STABILITY OF ORBITS question)
Ehrenfest, Paul. "In what way does it become manifest in the fundamental
        laws of physics that space has three dimensions."
        Koninklijke Akademie Van Wetenschappen Te Amsterdam, vol XX,
        200-209 (1917). [Also in "Collected Works of Ehrenfest"(title?)]
Ehrenfest, P. and Uhlenbeck, G.E. "On the connection of different
        methods of solution of the wave equation in multidimensional spaces."
        Collected Works of Ehrenfest (title?), 526-531 (1926).
Barrow, J.D. "Dimensionality."
        Phil. Trans. R. Soc. Lond. A 310, 337-346 (1983)
Barrow, J.D. and Tipler, F.J. The Anthropic Cosmological Principle. Ch 4.8
Freeman, I.M. "Why is space three-dimensional?..."
        Am.J.Phy. 37, 1222 (1969).
 
GENERAL RELATIVITY:---
Lovelock, D. "The Uniqueness of the Einstein Field Equations in a
        Four-Dimensional Space." Archive for Rational Mechanics and Analysis
        vol 33, 54-70 (1969).
Lovelock, D. "The Four-Dimensionality of Space and the Einstein Tensor."
        J.Math.Phy. 13, 874-876.
Deser, S., Jackiw, R., and 't Hooft, G. "Three-Dimensional Einstein Gravity:
        Dynamics of Flat Space." Annals of Physics 152, 220-235 (1984).
Giddings, S. ,Abbott, J., and Kuchar, K. "Einstein's Theory in a
        Three-Dimensional Space-Time." Gen. Rel. and Grav. 16, 751-775 (1984).
Tangherlini, F.R. "Schwarzschild Field in n Dimensions and the
        Dimensionality of Space Problem."  Il Nuovo Cimento, XXVII,
        636-651 (1963).
Collas, P. "General relativity in two- and three-dimensional space times."
        Am.J.Phy. 45, 833-837 (Sep 1977).
 
GENERAL PHYSICS:------
Giddings, S. "Incoherent radiation in an n-dimensional space."
        Am.J.Phy. 52, 1125-1126 (Dec 1984).
Hochberg, D. and Wheeler, J.T. "Spacetime dimension from a variational
        principle." Phys.Rev.D. 43, 2617 (1991).
Lapidus, I.R. "Fundamental units and dimensionless constants in a universe
        with one, two, and four space dimensions."  Am.J.Phy. 49,
        890-891 (Sep 1981).
Lapidus, I.R. "Classical electodynamics in  a universe with two space
        dimensions."  Am.J.Phy. 50, 155-157 (Feb 1982).
Landsberg, P.T. and Alexis De Vos "The Stefan-Boltzmann constant in
        n-dimensional space." J.Phys.A:Math.Gen. 22, 1073-1084 (1989).
 
Rob Salgado
sal7@ellis.uchicago.edu
rbsalgad@rodan.acs.syr.edu
-- 
ROB SALGADO                                      There is no such thing as
sal7@ellis.uchicago.edu                          gravity....      
rbsalgad@rodan.acs.syr.edu                         ...the earth just sucks.