Original by Scott I. Chase.

Revisions by Jim Mitroy (January 1996), and Phil Gibbs (April 1997).

Current version (March 1999) by Momo Jeng, Department of Physics,

University of California at Santa Barbara, momo@sbphy.ucsb.edu

Additionally, we can make a very simple argument that antimatter should fall down, by assuming only that energy is conserved and that the values of fundamental constants do not vary with height. With these assumptions and some well-tested experimental results, we can show that

1/2 (*g*_{matter} + *g*_{antimatter}) =
*g*_{photon}

where *g*_{matter} is the acceleration of falling matter,
*g*_{antimatter} is the acceleration of falling antimatter, and
*g*_{photon} is a term that gives the gravitational redshift of photons
(this equation is explained and proven in the Appendix
below). One of the first predictions of general relativity was that
*g*_{matter} and g_{photon} should be exactly the same.

Probably no experiment can ever show that *g*_{matter} and
*g*_{photon} are exactly the same, but experiments have shown that the two
differ by less than 0.02%. See [3] for some background
information. If *g*_{matter} and *g*_{photon} differ
by less than 0.02%, then the equation above implies that *g*_{matter} and
*g*_{antimatter} cannot differ by more than 0.04%. So with only the
above assumptions, theory and experiment show that antimatter should fall down, and with
an acceleration fairly close to that of normal matter. Also, note that if matter
attracts antimatter, then in theory antimatter should attract matter. This follows
from Newton's third law.

While these theoretical arguments are convincing, experimental results are essential in physics, and we now turn our attention to them.

However, this does not necessarily tell us that matter and antimatter will fall similarly to the same level of precision when dropped on the Earth, because they could be affected differently be some new, unknown force that has too short a range to be seen on galactic length scales.

So what we would really like to have is a laboratory experiment where we simply drop some antimatter in a lab, and see how fast it falls. This has not yet been done. It is often thought that Fairbank and Witteborn at Stanford measured the fall of positrons and electrons and found them to fall at the same rate. In fact, they only measured the fall of electrons, and it's unclear to this day if, when measuring the fall of electrons, they were able to overcome the difficulties in properly isolating the electrons from stray electromagnetic fields. And they never reported on the fall of positrons at all.

In order to reduce the effect of stray electromagnetic fields, it would be nice to use objects with the same magnitude of electric charge as electrons and positrons, but with much more mass, to increase the relative effect of gravity on the motion of the particle. Antiprotons are 1836 times as massive than positrons, so give you three orders of magnitude more sensitivity. Gerald Gabrielse and his coworkers have been cooling and trapping antiprotons at CERN in order to attempt this and other experiments. The results are still inconclusive.

The key to testing the different theories of gravity will be the creation of the antihydrogen atom. Because they are electrically neutral, antihydrogen atoms are not as sensitive to stray electric fields as antiprotons. Unlike many composite objects containing antiparticles, antihydrogen atoms are also absolutely stable, making them the preferred objects to study.

CERN has recently announced the creation of about 10 antihydrogen atoms, but as these were formed in a beam travelling at close to the speed of light, they really can't be used for studying the properties of antihydrogen. At the moment, a number of different groups around the world are gearing up to form antihydrogen at low energies by introducing positrons into an ion trap that will contain lots of antiprotons. References [6], [7], [8] and [9] provide further reading on this.

Most people expect that antiatoms will fall down. But it is important to keep an open mind, since we have never directly observed the effect of gravity on antiparticles on Earth. This experiment, if successful, will definitely be "one for the textbooks."

1/2 (g_{matter} + g_{antimatter}) = g_{photon}

Our argument, which is an adaptation of the "Morrison argument" described in [2], will only make two basic assumptions. It will assume that
energy is conserved. And it will assume that "fundamental constants," such as the
inertial mass of the proton or electron, or the speed of light, do not vary with height
above the Earth. It will also use well-tested experimental results (such as the fact
that *E=m c*^{2} has been observed to hold to high accuracy). As
explained above, when this equation is coupled with gravitational redshift experiments, it
shows that antimatter must fall down with an acceleration within 0.04% of that of ordinary
matter.

Start with a chunk of matter and a chunk of antimatter, each of mass *m* (mass is
always understood here to mean inertial mass), at the top of a tower of height
*L*. These "chunks" could, for example, be a proton and antiproton, which
have been experimentally observed to have the same inertial mass to within one part in 100
thousand million. If we combine these two chunks, they form a photon (actually a
bunch of photons). If we measure the energy of these photons locally (by, for
example, looking at their frequency), special relativity tells us that we will see

*E* = *hf* = 2*mc*^{2}

This relation between energy and mass has been well-tested experimentally in many labs around the world (and thus at different heights above sea level).

Suppose that we now take these photons and send them to someone at the bottom of the tower. If that person measures the energy of the photons, they will measure a different energy that we did at the top of the tower, because the photons will be blueshifted. The photons will gain energy as they fall in a gravitational field. The equivalence principle implies that the energy the observer at the bottom measures is

*E = hf* = 2*mc*^{2} (1 + *g*_{photon}*L* /
*c*^{2}) = 2*mc*^{2} + 2*m g*_{photon}
*L*

We'll let this equation define *g*_{photon}. General relativity
predicts that *g*_{photon} should be exactly the same as
*g*_{matter} = 9.8 m/s^{2}, the rate at which normal matter
falls. But we don't want to assume this, so we'll keep *g*_{photon}
and *g*_{matter} distinct.

Now let's have the person at the bottom of the tower take these photons and turn them back
into chunks of matter and antimatter, each of mass *m* (for example, a proton and
antiproton). By special relativity, we know that the energy 2*mc*^{2}
is just enough energy to create chunks of matter and antimatter, each of mass
*m*. But the photons have some extra energy, 2*m g*_{photon}
*L*. This gives the matter and antimatter some extra energy (manifested as
kinetic energy). We want to use this extra energy to move the matter and antimatter
back to the top of the tower. This extra energy must be just enough to move them
back to the top of the tower, or else energy would not be conserved. In other words,
this cycle takes us back to the exact same conditions that we started with, so we had
better not have lost or gained energy in carrying it out.

So how much energy does it take to move these guys back to the top of the tower?
Well, the matter has an inertial mass *m *, and "feels" an acceleration
*g*_{matter}. So it feels a force *m g*_{matter, and
to move it a distance L requires energy m gmatter L.
Similarly, it takes an energy m gantimatter L to move the
antimatter to the top of the tower. To conserve energy, these two energies must add
up to be the same as the extra photon energy, so we need}

*m g*_{matter} *L* + *m g*_{antimatter} *L* =
2*m g*_{photon} *L*

Cancelling common factors, we get

1/2 (*g*_{matter} + *g*_{antimatter}) = *g*_{photon}

On rearranging:

*g*_{antimatter} = 2 *g*_{photon} - *g*_{matter}

or

*g*_{antimatter} = *g*_{matter} (2 ± 0.04) -
*g*_{matter} = *g*_{matter} (1 ± 0.04)

[1] "Gravity and Antimatter", Goldman, Hughes and Nieto, Scientific American, March 1988, pg 48. An easy-to-read explanation of gravity and antimatter.

[3] "New Tests of the Gravitational Redshift Effect", T. Krisher, Modern Physics Letters A, vol. 5, no. 23 pg 1809 (1990). Recent gravitational redshift experiments.

[5] "New Precision Tests of the Einstein Equivalence Principle from SN1987A," M. Longo, Physical Review Letters, vol. 60, no. 3 pg 173 (1988). Supernova 1987A.

[6] "Antihydrogen Physics", M. Charlton et al., Phys. Rep., vol. 241 pg 65 (1994)

[7] "The race to create an antiatom", P. Campbell, New Scientist, May 13, pg 32 (1995)

[8] "The production and study of cold antihydrogen: Letter of intent" By Antihydrogen Trap Collaboration (Gerald Gabrielse et al.). CERN-SPSLC-96-23, March 1996.

[9] "Antimatter Gravity and Antihydrogen Production", Holzscheiter, Goldman and Nieto, http://xxx.lanl.gov/abs/hep-ph/9509336.