Baryosynthesis and Antimatter Generation 147Section 5.3 occur at the same rate in both directions of the arrow. To show this, one
still has to compensate for differences in phase space, i.e. the bookkeeping of energy
in endothermic and exothermic reactions.
Although time reversal is not very important in itself, for instance particles do not
carry a conserved quantum number related to T, it is one factor in the very important
combined symmetry CPT. According to our most basic notions in theoretical physics,
CPT symmetry must be absolute. It then follows from the fact that CP is not an abso-
lute symmetry, but slightly violated, that T must be violated by an equal and opposite
amount.
In a particle reaction, CPT symmetry implies that aleft-handed particle entering
the interaction region from the푥-direction is equivalent to aright-handed antiparticle
leavingthe region in the푥-direction. One consequence of this is that particles and
antiparticles must have exactly the same mass and, if they are unstable, they must
also have exactly the same mean life.
Needless to say, many ingenious experiments have been and still are carried out to
test CP violation and T violation to ever higher precision. CPT symmetry will probably
be tested when sufficient numbers of anti-hydrogen atoms have been produced, and
their properties will be compared with those of hydrogen.
Let us assume that general relativity is CPT invariant. It certainly is so in flat
space-time even if it has not been demonstrated in curved space-time. Although C
transforms particles into antiparticles, invariance under CPT implies that an addi-
tional PT transformation is needed, too. This part changes the sign of each component
of a four-vector and odd-rank tensor, not affecting even-rank tensors. C is ineffective
because, contrary to electrodynamics, there is no charge in the gravitational field.
When CPT is applied even-rank (odd-rank) tensors which are PT-even (PT-odd)
become CPT-odd (CPT-even). The change of sign of any d푥휇implies that the d푡of
antimatter will be reversed with respect to that of matter. Consequently the Lorentz
factor훾=d푡∕d휏will be negative for antimatter while a matter particle has훾=1in
its rest frame. Thus antiparticles are just particles travelling backwards in time. In
Einsteins equation there are only even-rank tensors and scalars, so it is even under
PT and CPT.
CPT invariance then implies that antimatter is attracted by antimatter in exactly
the same way as matter is attracted by matter, but it is not obvious what the interaction
between matter and antimatter is. In comparison with electrodynamics, the gravita-
tional field possesses no charge, but the four-momentum for a particle of mass푚
푝휇=푚
d푥휇
d휏, (6.118)
takes the role of a charge. Recall the expression for the four-acceleration in Equa-
tion (3.14)
d^2 푥휇
d휏^2=−훤휎휈휇d푥휎
d휏d푥휈
d휏= 0. (6.119)
Under (C)PT this is even because the product of the four-momenta of two particles is
even, the product of the four-momenta of two antiparticles is even, the affine connec-
tion is odd, and there is a minus-sign on the right.