HADRONIC JOURNAL 17, 257--284 (1994)

    :::  Copyright 1994 by The Institute for Basic Research,
P.O.Box 1577, Palm Harbor, FL 34682, USA  :::


                  ANTIGRAVITY

             Ruggero Maria Santilli
           The Institute for Basic Research
       P. O. Box 1577, Palm Harbor FL 34682, USA
      Fax 1-813-934-9275, E-mail ibrrms@pinet.aip.org

      Received December 9, 1993; Revised April 20, 1994


       Abstract:

   Recent advances in the integral, axiom-preserving isotopies and
isodualities of the Minkowskian and Riemannian geometries permit
a quantitative formulation of antigravity conceived as the reversal
of the gravitational attraction, which is here submitted as a
scientific curiosity for young minds. According to these advances,
the interior gravitational problem of matter is characterized by
the isoriemannian geometry and that of antimatter by a novel
antiautomorphic map known as isoduality. The corresponding exterior
gravitational problems of matter and antimatter in vacuum are
characterized by the conventional Riemannian geometry and its
isodual. The covering isogeometries permit quantitative studies on
the origin (rather than the description) of the gravitalion itself,
which yield the identification of the gravitational and
electromagnetic interactions in the structure of matter (i.e.. Ihe
electromagnetic origin of mass plus short range contributions). The
understanding of the origin of Gravitation then yields the
capability to reverse the gravitaional force exactly as it occurs
for the Coulomb force. In particular, antigravity emerges as tbe
projection of the isodual gravitational field of antimatter in the
gravitational field of matter (or viceversa). While antigravity is
prohibited by conventional geometries, the covering isogeometries
and related isoduals predict that antiparticles experience a
repulsive force in the gravitational field of matter. The proposed
antigravity is fully testable with current technology, e.g.; via
the comparison of suitable interferometric measures on thermal
beams of neutrons and antineutrons in the gravitational field of
Earth.

:::  Copyright 1994 by The Institute for Basic Research, P.O.Box
1577, Palm Harbor, FL 34682, USA  :::



(Page 2)

1. lNTRODUCTION

   The search for antigravity dates back to the beginning of
physics (see, e.g., the comprehensive review by Nieto and Goldman
[1a]) and includes attempts which have been even palented (see
[1b,c] and quoted literature). All existing studies deal either
with a reduction of the gravitational attraction or with mechanical
means to bypass gravity. None of them deals with the conception of
antigravity as the reversal of the attractive character of the
gravitational field. This is due to the fact that the Riemannian
geometry (see, e.g.. [2]) permits no known possibility of reversing
tne attractive charactcr of gravitation.

   To put it clearly, by its very nature antigravity is "beyond
Einsteinian theories" that is, it requires theories of gravitation
structurally more general than Einstein's gravitatton. The
understanding is that if antigravity is experimentally established
via the tests proposed in this note or other approaches, Einstein's
gravitation is not "violated", but "inapplicable", simply because
Einstein did not formulate his theory to study antigravity.

   Antigravity therefore requires the identification of the arena
of conception and applicability of Einstein gravitation, with the
understanding that its assumption as being universally valid under
whatever conditions exist in the universe has no scientific value.


   As clearly identified in his limpid writings, Einstein conceived
his gravitation for the exterior gravitational problem of matter
in vacuum. The first arena of inapplicability (and not "violation"
of Einstein's gravitation is therefore the exterior problem of
antimatter in vacuum. After all, antimatter did not exist at the
time of Einstein's conception of his gravitation.

   Our first condition for the study of antigravity is therefore
the assumption of the Riemannian geometry and Einstein's
gravitation as being valid for the exterior problem of matter in
vacuum and the search for a different geometry and gravitational
theory for the exterior problem of antimatter in vacuum.

   The known unresolved problematic aspects afflicting Einstein's
gravitation have no bearing for this note owing to the dominance
of the Riemannian geometric profile over the explicit form of the
field equations which can be defined in it. As such, these
problematic aspects will be ignored hereon.

   The second condition is that the reversal of the sign of gravity
is evidently linked to the problem of the origin (rather than
"description") of the gravitational field. As such, it requires
geometries which are first suitable to represent actual interior
conditions, and then capable of reducing the gravitational field
to primitive interactions originating mass itself.

(Page 3)
   Now, interior dynamical problems, such as missiles in
atmosphere, are arbitrarily nonlinear in the velocities,
nonlocal-integral in various variables (e.g., dependent on the
shape of the body) and non-(first-order)-Lagrangians (variationally
nonselfadjoint systems [3,4]). The assumption of the
local-differential-Lagrangian Riemannian geometry as being exactly
valid for interior problems also has no scientific value.

   As an example, interior problems such as gravitational collapse
are not given by a large yet finite set of isolated points, but in
the physical reality they are composed of extended
charge-distributions/wavepackets/wavelengths of hadrons in
conditions of total mutual penetration and compression in large
number into small regions of space. The consequential emergence of
the indicated nonlinear-nonlocal-nonlagrangian conditions is then
beyond credible doubts.

   Our second condition for the study of antigravity is therefore
the search for suitable covering geometries for the quantitative
representation of nonlinear-nonlocal-nonlagrangian interior
gravitational problems of matter and, separately, of antimatter,
as well as for the complete reduction of the gravitational field
to the primitive fields originating matter itself, the
electromagnetic interactions plus short range contributions. It is
evident that the latter interior generalizations must be able to
reproduce the preceding representations of the exterior
gravitational problems of matter and antimatter in vacuum.

   Our third condition for the study of antigravity is that,
whatever geometric description of the gravitational field of
antimatter is assumed, that description must recover all other
experimentally established behaviour of antimatter, e.g., under
electromagnetic interactions.

   In different terms, the gravitational characteristics of
antiparticles are theoretically and experimentally unsettled at
this writing. However, their behaviour under electromagnetic
interactions is fully established. Any theory of antigravity based
on antimatter must therefore recover the conventional
electromagnetic behaviour of antiparticles in its entirety.

   The only known mathematical formulations which meet all the
above requirements are given by the so-called isotopies and
isodualities of the Riemannian geometry. The general lines of
isotopies were first introduced by this author in 1978 [3] and then
studies in Its various aspects in [4-15]. The first specific
application to gravitation was done in 1988 [7] and then studied
in detail in [8,1O,11]. The most recent isotopies and isodualities
of gravitation have been studied in [14] and those of the tangent
Minkowskian geometry in [15]. Independent reviews can be found in
[16-20]. Topological aspects of isomanifolds and isotensors defined
on them are studied in [21].

   The above studies have identified tne following four primary
geometries at the foundations of this note [10,11]:

  1. Riemannian geometry, for the exterior gravitational problem
of matter in vacuum;
  2. Isodual Riemannian geometry, for the exterior gravitational
problem of antimatter in vacuum;
  3. Isoriemannian geometry, for the interior gravitational problem
of matter; and
  4. Isodual isoriemannian geometry, for the interior gravitational
problem of antimatter.

   In this note we shall study only the exterior problem of
antigravity, that in vacuum, and therefore limit our analysis to
the Riemannian geometry and its isodual (a detailed
nonlinear-nonlocal-nonlagrangian treatment of the interior problem
is available in Vol. II of ref.s [4]). The local tangent geometries
of our study are given by the conventional Minkowskian geometry for
the case of matter and the isodual Minkowskian geometry for the
case of antimatter [6,15].

   Intriguingly, the theory of antigravity submitted in this paper
is reducible to the interplay belween two primitive symmetries, the
conventional Poincare symmetry P(3.1) for matter in the tangent
plane and its isodual image P^d(3.1) for antimatter also in the
tangent plane, with isotopic generalizations P^(3.1) which have
resulted to be directly universal for all possible exterior and
interior gravitational problems of matter and antimatter,
respectively (see [5] for the original proposal, [6,10,11] for
detailed studled, [13] for recent advances and [20] for an
independent review).

   More particularly, the lack of antigravity in contemporary
gravitation appears to be due to its lack of a universal symmetry,
such as the Poincare symmetry of the special relativity. The
identification of the universal isopoincare symmetry for the
conventional gravitational field of matter then leads naturally and
uniquely to the isodual isopoincare symmetry for antimatter.
Antigravity then follows uniquely and unambiguously.


  [The rest of this article is in graphic form, as the files
ANTIGR04.GIF through ANTIGR28.GIF]