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二国
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Is ours only one of three universes?
by Dietrick E. Thomsen
The usual big-bang cosmology con-
nected with Einsteinian general rela-
tivity has the universe starting from a
point of space-time that is called the
singularity. "Singularity" is a mathema-
tician's euphemism for something dif-
ficult to deal with, a point at which
physically the universe has no dimen-
sions and infinite density. From this
point the universe expands as time pro-
ceeds, extending its dimensions and
lowering its density.
Such is the usual picture of the ex-
panding universe. But this universe oc-
upies only one region of the space-
ime that physicists are used to dealing
with, the region that lies to the future
of the singularity. The question arises:
What happens in the other regions of
pace-time that physicists are able to
magine? Does anything happen in the
ingularity's past? Can anything happen
eside it, so to speak in the regions of
pace-time called spacelike?"
The answer, says J. Richard Gott III
f California Institute of Technology,
yes. Writing in the latest ASTROPHYS-
CAL JOURNAL (Vol. 187 No. 1), he
hows that if we look for the most gen-
tal solutions of Einstein's equations,
flat space-time, we come up with
ree universes. One is our own, which
e have just described, lying in the
ngularity's future and dominated by
dinary matter. Let us call it Universe
Universe II lies in the singularity's
ist and is dominated by antimatter.
niverse III lies in the spacelike region
space-time and is inhabited by ta-
lyons, particles that travel faster than
ht.
To understand the geometry of this
ther mind-boggling concept, it is nec-
sary to spend a few words on a gen-
al description of space-time. In true
ace-time there are three spacclike
mensions and one timelike dimension.
r graphic purposes two of the space
nensions are suppressed, and a two-
mensional graph is drawn in which
vertical axis is time and the hori-
tal space.
Every point in this two dimensional
ce-time represents an event: It speci-
both the location and the time at
ich something happens. The start of
articular particle's flight may be one
ot; its finish, another. The slope of
line that joins them represents the
ocity of the flight.
Calculation shows that the lines run-
g at 45 degrees to the time and
Space-time
diagram of Gott's
proposed three-
universe cos-
mological model.
Gott/Astrophysical
Journal
space axes are of particular importance.
They represent objects moving at the
speed of light (they define what is
called the light cone), and in ordinary
physics one cannot cross them in going
from event to event. The light lines
(or the light cone in more than two
dimensions) divide space-time into two
regions, the timelike (in the upper
and lower quadrants) and the spacelike
in the right and left quadrants.
For two events in the timelike region
(where we live) it is possible to find
an observer moving in such a way
that the two events seem separated in
time only. If observer A sees a particle
moving from x to y while the time
goes from t1 to t2, observer B, who
happens to be going along with the
particle, will see the time change only.
If the particle was in his hand at the
start of the flight it will be in his hand
at the end. In the spacelike region, in a
similar way one can find an observer
for whom two events are simultaneous
but appear to represent an instantane-
ous translation in space. Thus in the
spacelike region our usual perceptions
of space and time and cause and effect
are overthrown, but we need not worry
about it since we can never get there.
When observer B moves with respect
to observer A, from A's point of view
the motion represents a skewing of his
time axis in the direction of the light
line. It can also be shown that his space
axis will skew and also in the direction
of the light line. The faster B goes, the
narrower becomes the angle between
his space and time axes. When he
reaches the speed of light his space
and time axes meet in a grand flash
of-well that's the singularity, as Gott
considers it.
There's no crossing it. Gott puts our
universe in the upper quadrant to the
future of the singularity. His time-re-
versed antimatter universe lies in the
lower quadrant to its past. And his
tachyon universe lies in the spacelike
region, which is not two regions but
one. This can be seen if we add a
third dimension and imagine the dia-
gram rotated around the time axis: Re-
gions I and II become cones; region
III becomes a wedge-shaped ring.
There is no communication across
the singularity. Antimatter and tachyons
can exist in our universe occasionally
and ephemerally-they are not visitors
from the other universes. They are pro-
duced here. There are differences in
perception: Our view of Universe II,
if we could see it, would be that it is
dominated by matter and contracting.
To its own inhabitants it looks as if
antimatter dominates and it is expand-
ing. Finally the principal of causality,
which says that neither information nor
energy can be transmitted faster than
light, is not violated in the tachyon
universe. Though the tachyons them-
selves go faster than light, their radia-
tion, which is the only way they can
transmit energy or information, does
not.
Gott concludes: "The model we have
presented is a unified, time-symmetric
model treating matter, antimatter and
tachyons in a natural and equal fashion.
The model is consistent with our pres-
ent observations of the universe and
could gain support from an experimen-
tal discovery of tachyons. . .
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...

369
(11.58) is in qualitative agreement with the observed spin-orbit splittings
in nuclei.
11.6 Proper Time and the Light Cone
:
In the previous sections we have explored some of the physical con-
sequences of the special theory of relativity and Lorentz transformations.
In the next two sections we want now to discuss some of the more formal
aspects and to introduce some notation and concepts which are very useful
in a systematic discussion of physical theories within the framework of
special relativity.
In Galilean relativity space and time coordinates are unconnected.
Consequently under Galilean transformations the infinitesimal elements
of distance and time are separately invariant. Thus
ds2 == dx2 + dy2 + d22 = ds'2
dt2 == dt'2
(11.59)
For Lorentz transformations, on the other hand, the time and space
coordinates are interrelated. From (11.21) it is easy to show that the
invariant "length" element is
ds2 = dx2 + dy2 + dz2 - c2 dt2
(11.60)
This leads immediately to the concept of a Lorentz invariant proper time.
Consider a system, which for definiteness we will think of as a particle,
moving with an instantaneous velocity v(t) relative to some coordinate
system K. In the coordinate system K' where the particle is instantaneously
at rest the space-time increments are dx' = dy' = dz' = 0, dt' = dr. Then
the invariant length (11.60) is
-c2 d+2 = dx2 + dy2 + dz2 - c2 dt2
(11.61)
In terms of the particle velocity v(t) this can be written
2
৳2
dT = dt
1ª
(11.62)
Equation (11.62) shows the time-dilatation effect already discussed. But
much more important, by the manner of its derivation (11,62) shows that
the time T, called the proper time of the particle, is a Lorentz invariant
quantity. This is of considerable importance later on when we wish to
discuss various quantities nd their time derivatives. If a quantity behaves
in a certain way und
`ntz transformations, then its proper time
CLASSICAL ELECTRODYNAMICS
JOHN DAVID JACKSON
JOHN WILEY & SONS 1962
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1
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s rest frame precess
v xa
(11.54)
valid if v <c. We
precession by noting
acceleration. If a
ien there is a Thomas
ssion of the magnetic
ed by the screened
locity is
dV
dr
(11.55)
-
. contribution to the
: spin-orbit coupling
), yielding
AV
(11.56)
dr
mic electron.
:elerations due to the
es are comparatively
nucleons as moving
attractive, potential
addition a spin-orbit
ctromagnetic contri-
(11.57)
. The form of wy is
he nuclear spin-orbit
(11.58)
we note that both V
so that the signs of
t in nuclei the single
jonable form for Vy,
:
[Sect. 11.6]
Special Theory of Relativity
on /t = 817
:

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CE -1 CSOKOR KSET
370
Classical Electrodynamics
BA Ict
realantin xxct
Future
0
x
Past
=
Fig. 11.10 World line of a system and the
light cone. The unshaded interior of the cone
represents the past and the future, while the
shaded region outside the cone is called
"elsewhere." A point inside (outside) the
light cone is said to have a time-like (space-
like) separation from the origin.
derivative will behave in the same way because of the invariance of dr.
But its ordinary time derivative will not have the same transformation
properties. From (11.62) we see that a certain proper time interval
(72 - 71) will be seen in the system K as a time interval
Ist' = st
N
dr
1 -
v2(T)
02
(11.63)
where t1 and t2 are the corresponding times in K.
Another fruitful concept in special relativity is the idea of the light cone
and "space-like" and "time-like" separations between two events. Con-
sider Fig. 11.10, in which the time axis (actually ct) is vertical and the space
axes are perpendicular to it. For simplicity only one space dimension is
shown. At t = 0 a physical system, say a particle, is at the origin. Because
the velocity of light is an upper bound on all velocities, the space-time
domain can be divided into three regions by a "cone," called the light cone,
whose surface is specified by x2 + y2 + 22 = c212. Light signals emitted
at t = 0 from the origin would travel out the 45° lines in the figure. But
any material system has a velocity less than c. Consequently as time goes
on it would trace out a path, called its world line, inside the upper half-
cone, e.g., the curve OB. Since the path of the system lies inside the upper
half-cone for times t > 0, that region is called the future. Similarly the
lower half-cone is called the past. The system may have reached O by a
path such as AO lying inside the lower half-cone. The shaded region
outside the light cone is called elsewhere. A system at O can never reach
or come from a point in space-time in elsewhere.
The division of space-time into the past-future region and the elsewhere
region can be emphasized by considering the invariant separation between
two events P1(x1, 3/1, 21, 11) and P2(x2, 12, 22, 12) in space-time:
$122 = (x1 - x2)2 + (3/1 - 3/2)2 + (21 - 2.))2 -- c2(t1 - 12)2 (11.64)
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[Sect. 11.7]
For any :
(2) $122 -: :
because i.
coordinat.
That is, t .:
time. In: .
lies in cls.
ation. T !:
= = 2, 3
In the co.
but are se.
other is i-
The di-
classes ----
invariant
system ha
that two
actions p
than that
related.
only by t
11.7 1
.
The L.
are lines
(x', y', z',
This con
rotation .
duce the
then the .

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371
[Sect, 11.7]
Special Theory of Relativity
For any two events P1 and P2 there are two possibilities: (1) $122 > 0,
(2) $122 < 0. If $122 > 0, the events are said to have a space-like separation,
because it is always possible to find a Lorentz transformation to a new
coordinate system K' where (41' - 12) = 0 and
$122 = (21' - x2')2 + (y1' - y2')2 + (z1' - z2')" > 0 (11.65)
That is, the two events are at different space points at the same instant of
time. In terms of Fig. 11.10, one of the events is at the origin and the other
lies in elsewhere. If $122 < 0, the events are said to have a time-like separ-
ation. Then a Lorentz transformation can be found which will make x1'
= x2', y]' = y2', 21' = z2', and
5122 = - c2(t1' -t2')2 < 0
(11.66)
In the coordinate system K' the two events are at the same space point,
but are separated in time. In Fig. 11.10, one point is at the origin and the
other is in the past or future.
The division of the separation of two events in space-time into two
classes -- space-like separations or time-like separations-is a Lorentz
invariant one. Two events with a space-like separation in one coordinate
system have a space-like separation in all coordinate systems. This means
that two such events cannot be causally connected. Since physical inter-
actions propagate from one point to another with velocities no greater
than that of light, only events with time-like separations can be causally
related. An event at the origin in Fig. 11.10 can be influenced causally
only by the events which occur in the past region of the light cone.
. 11.7 Lorentz Transformations as Orthogonal Transformations in
Four Dimensions
The Lorentz transformation (11.19) and the more general form (11.21)
are linear relations between the space-time coordinates (x, y, z, t) and
(x', y', z', '), subject to the constraint,
" x2 + y2 + 2 - c2+2 = x2. + y"2 + 2"2 - c212
(11.67)
This constraint is very reminiscent of the constraint involved in the
rotation of coordinate axes in three space dimensions. In fact, if we intro-
duce the four space-time coordinates,
x1 = x, x2 = y, x3 =2, x4 = ict
(11.68)
then the constraint becomes
R2 = x,2 + x,2 + x32 + x12
(11.69)
2
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of a system and the
ed interior of the cone
the future, while the
: the cone is called
inside (outside) the
ve a time-like (space-
From the origin.
invariance of dr.
me transformation
oper time interval
11
(11.63)
ea of the light cone
two events. Con-
rtical and the space
space dimension is
the origin. Because
ies, the space-time
Called the light cone,
ght signals emitted
¡in the figure. But
uently as time goes
ide the upper half-
lies inside the upper
ture. Similarly the
Ive reached O by a
The shaded region
. O can never reach
n and the elsewhere
separation between
-time:
c2(11 - 12)2 (11.64)