Here, I make an attempt to
address these issues with as little algebra as possible, you might say, this
the Blahblah-version of my paper. These answers apply in the framework which I
proposed, and might not hold for whatever someone else thinks anti-gravitation
is. For details, please look up my paper.

I admit that the title
'Anti-Gravitation' might not have been the wisest choice, because it sounds a
bit Sci-Fi. However, the reason for this title was that I did not want it to be
confused with related approaches coming under the name 'Energy Parity' or
'ghost-condensate'.

**One can use anti-gravitating matter to build a perpetuum mobile!**

No,
of course one can not. Let me first explain how I think this brain-twist comes
along. Suppose you introduce negative charges into gravity by just switching
the sign in front of the energy-momentum tensor. Solving Einstein's field
equations, you then get a background field with either positive or negative
mass. If you now examine the geodesic equation in this background field, and
assume it to hold for all particles, then you come to the confusing conclusion
that a positive gravitational source attracts ALL matter, whereas a negative
source repels ALL matter.

Now
suppose you have a gravitating test-particle in an anti-gravitating field and
vice verse. Then forget about the larger mass being a background field and
shrink both systems down to two test-particles. You find that on the one hand,
the anti-gravitating particle repels the gravitating one (because the
anti-gravitating background field had repelled all matter), and on the other
hand the gravitating particle attracts the anti-gravitating one (because the
gravitating background field had attracted all matter).

Having
come to this point, it is easy to construct a perpetuum mobile. You pair both
to a gravitational dipole and the system will accelerate itself arbitrarily.

Though this
would definitely improve my financial situation, I am afraid such a
construction is not possible. The reason is easy to see: having introduced the
anti-gravitating particle from a gauge principle, it moves on a world-line
which one derives from the transformation properties under the gauge-group.
Take as an example Electrodynamics. The anti-particle comes along with the
appropriate gauge-covariant derivative, which is *not* identical to that
of the particle.

It
is the same reasoning with the anti-gravitation. The anti-gravitating particle
does not move on the same world-line as the gravitating one, because its
covariant derivative is modified (to what I call the anti-covariant derivate).
As a consequence, it does not move on a geodesic, which is the point where the
above construction of the perpetuum mobile goes wrong.

In
my approach, a gravitating source attracts gravitating test-particles, but
anti-gravitating test-particles are repelled. An anti-gravitating source
attracts anti-gravitating test-particles, whereas gravitating test-particles
are repelled. To summarize: like charges attract, unlike charges repel, as you
would have expected from the fact that gravity is a spin-2 field. (see e.g. A.Zee,
'Quantum Field Theory in a Nutshell' ,
Princeton University Press (2003)).

One
should in addition notice that the the above brain-twister explicitly
breaks the symmetry between negative and positive gravitational charges (switch
both and the dynamics should stay the same), which was my motivation to
write the paper in the first place.

**The vacuum is unstable and will decay.**

The
vacuum is unstable if you allow negative kinetic energies. Then, it would
become possible to produce a pair of particles basically out of nothing through
a quantum fluctuation.

This
is not possible when one introduces the gravitational mass as a charge. The
kinetic energy (or the kinetic energy momentum tensor, respectively) remains
unmodified bust must no longer be identical to the gravitational energy (or
gravitational energy momentum tensor, respectively.). The kinetic energy is
always positive and there is no vacuum decay possible.

Distinguishing
between the two types of energy is crucial. The kinetic energy momentum tensor
is derived from a variational principle by variation with respect to the field,
whereas the gravitational one is derived by variation with respect to the
metric tensor. For a gravitating field, both are identical. For the
anti-gravitating field, both differ by a relative sign.

In the Newtonian approximation, this is similar to saying that the inertial mass is
identical to the gravitational mass only up to a sign. The inertial mass of the
particle is what governs its kinetics and remains positive, whereas the gravitational
mass can be negativeo.

**A particle just can not 'fall up' because the equation of motion in General
Relativity is independent of the mass of the particle.**

It
is true that the equation of motion in General Relativity is independent on the
(inertial) mass of the particle. It does however, depend on the transformation behavior
under the Lorentz-group. When one introduces the anti-gravitating particle from
a gauge principle, it moves on a world-line which one derives from these
transformation properties. Since the covariant derivative is modified, the
geodesic equation is also modified to what I call the 'anti-geodesic' equation.

This
can be understood in a more intuitive way by recalling that the equations of
motion are indeed equivalent to the conservation of energy. A gravitating
test-particle in a gravitational field falls down by which it converts
potential energy into kinetic energy. In contrast to this, the anti-gravitating
particle moves up in the field to convert potential energy into-kinetic energy.

**There can be no symmetry between positive and negative charges in gravity,
because the Schwarzschild solution does not have this symmetry. The
Schwarzschild solution with positive mass has a horizon, whereas the solution
with negative mass has no horizon.**

Let
me rephrase 'horizon' as a 'trapped surface from which photons can not escape' and
you see the answer to this question. The Schwarzschild metric with positive
mass has a horizon for photons, but no horizon for anti-gravitating photons
(since these are repelled, how should they be trapped?) . On
the other hand, the Schwarzschild metric with negative mass has no horizon for
photons but it has a
horizon for anti-gravitating particles.

You
find the symmetry between both types of gravitational charges only if you
self-consistently replace the one with the other. That means, switching the
mass in the source from gravitating to anti-gravitating, you can only expect it to act in the
same way on anti-gravitating photons, which indeed it does.

**Is the equivalence principle still valid?**

The equivalence principle is relaxed: it is valid up to a sign. The gravitational
mass of a particle is either equal to its inertial mass or equal to the negative of the inertial mass.

**Is the theory still generally covariant? If so, how can it be that
the anti-geodesic equation is different from the geodesic equation? How can they both be covariant?**

The theory has general diffeomorphism invariance. However, not all quantities transform as usual
vectors and tensors. The newly introduced quantities obey a different
transformation
law. Nevertheless, they *have* a well defined transformation behavior and
all equations are independent of the choice of the coordinate system.
This is similar to the case when you look at a spinor field. It transforms
different
from a vector, but the transformation is well defined and everything is
independent
on the choice of coordinates.

The geodesic is defined as a curve which parallel transports its own tangential vector. In contrast to this, the anti-geodesic
curve anti-parallel transports its tangential vector. It is not the tangential vector which is modified, it is the prescription how it behaves
in a gravitational field - which is encoded in its behavior under local Lorentz-transformations.
Under an inifitesimal change of coordinates, both behave differently. Consequently, conserving the angle between curve
and tangential vector results in a different curve.

Both curves are invariant under a change of coordinate systems if you use the appropriate transformation
behavior for the quantities.

**This is just a messed up version of Dirac's see of anti-particles.**

Well.
This is not really a question, is it? I actually have not said anything about a
quantized version of the theory, and I admittedly have no idea how to do it.
Anyway, a particle of negative gravitational charge can not be replaced by a
particle with opposite charge and positive energy, because in this case the
energy is the charge.

**If there is anti-gravitating matter, why don't we see it?**

First,
recall that both types of matter repel. Thus, if there is anti-gravitating
matter, it would not stay here. It would move away as far as possible. Then the
question reduces to why we do not produce anti-gravitating matter in accelerators
or in ultra-high energetic cosmic rays.

The reason
is that the interaction between gravitating and anti-gravitating matter is
mediated by gravity only and is therefore naturally extremely weak. In
particular, it should become important only at Planckian energies. It follows
from the transformation properties of the particles that it just is not
possible to construct an interaction term which is both invariant under gauge-
and Lorentz-transformations.

Sadly,
this means even if the Planck scale is lowered to some TeV (as it is possible
in theories with large extra dimensions) and could be reached at the LHC in
2008, we could maybe produce these particles but we could not capture
them. They would basically add to the missing energy.

**If we don't see it, why should we care whether it exists?**

Even
though the existence of anti-gravitating matter might not influence our daily
life on earth, it will become important when the interaction between both types
of matter can no longer be neglected. That is, in the early universe or in
strong curvature regimes. It might have important implications for cosmology
and astrophysics. Not to mention that the asymptotic behavior of the theory
might be modified and it might shed new light on the singularity problem as well
as on the cosmological constant problem.