IS OURS ONLY ONE OF THREE UNIVERSES?

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. ? ILT 
s ours only one of three universes? 
y Dietrick E. Thomsen 
The usual big-bang cosmology con-
ected with Einsteinian general rela-
vity has the universe starting from a 
oint of space-time that is called the 
ingularity. "Singularity" is a mathema-
cian's euphemism for something dif-
cult to deal with, a point at which 
hysically the universe has no dimen-
ions and infinite density. From this 
oint the universe expands as time pro-
eeds, extending its dimensions and 
mering its density. 
Such is the usual picture of the ex-
anding universe. But this universe oc, 
upies only one region of the space-
me that physicists are used to dealing 
ith, the region that lies to the future 
f the singularity. The question arises: 
/hat happens in the other regions of 
mce-time that physicists are able to 
nagine? Does anything happen in the 
ngularity's past? Can anything happen 
cside 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-
,/iL JOURNAL (Vol. 187 No. 1), he 
,iows that if we look for the most gen-
1-al solutions of Einstein's equations, 
t flat space-time, we come up with 
ree universes. One is our own, which 
e have just described, lying in the 
gularity's future and dominated by 
dinary matter. Let us call it Universe 
Universe II lies in the singularity's 
St and is dominated by antimatter. 
iverse III lies in the spacelike region 
space-time and is inhabited by ta-
yons, particles that travel faster than 
ht. 
To understand the geometry of this 
her mind-boggling concept, it is nec-
ary to spend a few words on a gen-
1 description of space-time. In true 
ce-time there are three spacelike 
ensions and one timelike dimension. 
r graphic purposes two of the space 
ensions are suppressed, and a two-
ensional graph is drawn in which 
vertical axis is time and the hori-
tal space. 
very point in this two dimensional 
ce-time represents an event: It speci-
both the location and the time at 
'eh something happens. The start of 
articular particle's flight may he one 
nt; its finish, another. The slope of 
line that joins them represents the 
city of the flight. 
.alculation shows that the lines run-
at 45 degrees to the time and 
1-1,-ceifiari in Part - 
Space-time 
diagram of Gott's 
proposed three-
universe cos-
mological model. 
Gott/Astrophysical 
Journal 
? 
f:=411 
F..-
174, 
frz. 
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 
reaions, the timelilce (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 mode! 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. . . ." 0 
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U C? 
rest frame precess 
x a (11.54) 
ialid if v < c. We 
)recession by noting 
acceleration. If a 
2n there is a Thomas 
sion of the magnetic 
xt by the screened 
ocity is 
iv 
Ir 
(11.55) 
contribution to the 
spin-orbit coupling 
), yielding 
/V 
(11.56) 
dr 
mnic electron. 
:elerations due to the 
es are comparatively 
nucleons as moving 
attractive, potential 
iddition a spin-orbit 
ctromagnetic contri-
(11.57) 
[Sect. 11.6] Special Theory of Relativity 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 _F dz2 = ds'2 
dt2 = de2 
(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, di' = dr. Then 
the invariant length (11.60) is 
? c2 dr2 = dx2 dy2 -1- dz2 c2 di2 (11.61) 
In terms of the particle velocity v(t) this can be written 
) ..., ----1 
1. The form of o32, is i /. z- i 11 /?--e-