One of the biggest misconceptions deep-rooted in the human
mind is the one-dimensionality of time. When physicists discover a theory that
calls for multiple extra dimensions, such as the string theory, their first
reaction is to assign those extra as spatial dimensions. They do so because
they abhor the plurality of time a).
In order to give the reason for their invisibility, the
physicists hypothesize that the extra dimensions are curled up into tiny loops b).
However, as there are so many possibilities on how those extra-dimensions may
be curled up, the outcomes of such string theory can reach billions. It gives us a good reason to allow the Occam
razor getting rid of this dire hypothesis without delay.
Internal symmetry
The second reason for the invisibility of the
extra-dimensions is that they are time dimensions. A world with multiple space
and time dimensions certainly complies with the principle of the relativity. We
would have in our worlds no more bizarre things such as donut-like or
Calabi-Yau tiny manifolds. As such, the world may preserve its internal symmetry
in which changing [reversibly] the time dimension with the space dimension
leaves the physical laws identical.
It is evident that the physical laws' formulation in the
world with many time dimensions would be exceedingly complicated. However, we
may have a more natural way to solve the formulation by transforming all but
one time dimensions into space dimensions. Having done that, we get a
simple system of higher-dimensional space with only one temporal dimension
without altering the outcome of the result.
To give an illustration, let us take the example of our
space (or 3-brane as you wish c)) which we interpret as embedded in
a 10-ambient spacetime. Under multiple time dimensions framework, such a system
consists of three space dimensions and seven temporal dimensions. By
transforming all temporal dimensions
(except the highest dimensional one) into space dimensions, we get a
system having nine space dimensions and one temporal dimension (a 9-brane embedded in the same 10-ambient
spacetime). The physical laws' formulation in the latter system is much simpler
than that in the first one d).
Under such a system we can also transform only some of the
time dimensions into space dimensions and keep the remaining intact. As such,
we would have various dimensional branes ranging from 3-brane to 9-brane
embedding in the same 10-ambient spacetime without even changing the outcome of
the result.
These are the various dimensional branes which we encounter
in the superstring theory. However, in most of the cases, each of those branes
is assumed to have only one temporal dimension. As such, the physicists who
deal with those branes have not any simple way to solve the respective
formulation e).
Even in dealing with the 9-brane, we may have no good
solution. We may, in this case, extend the dimensions of the ambient spacetime
higher and higher until we get the solution f).
We may say that the higher the brane's dimensions are, the
flatter it is. This situation makes the
mathematical formulation of the physical laws simpler.
Figure-1 diagrammatically shows how events which are not
simultaneous at a particular time dimension t1 become
simultaneous at a higher time dimension t2. It is as though
space flatten out as the number of temporal dimensions becomes higher that
makes the physical laws formulation more straightforward g).
Notes:
a) Physicists
wrongly regard the 4-spacetime as having three space dimensions and one
temporal dimension which should be inherently multidimensional. The underlying
of what we know as 4-spacetime is 4-dimensional time having a 3-dimensional
cross-section (3-brane) in it.
b) Kaluza
and Klein first introduced Kaluzathe idea to unify the electromagnetic field
with that of gravity by adding the curly fifth dimension to the classical four.
Later on, the idea is extended for much higher dimensions in which the loop
size of the extra dimensions is at the order of Planck size (10-33
cm).
c) We
use the notations of space, hypersurface, hyper-interface or brane
interchangeably.
d) A
system consisting of 3-space embedded in 10-dimensional ambient spacetime can
be formulated as Hypercomplex function:
q = x1 +x2 + x3 + ic1t1
+ jc2t2 + kc3t3 +lc4t4
+ mc5t5 + nc6t6
+ oc7t7
under
a coordinate patch consisting of three real x1, x2, x3
and i, j, k, l, m, n,
and o as independent imaginary numbers as the basis coordinate
representing seven different time dimensions, ci is the speed of
light of the respective temporal dimensions ti .
It
is identical to Octonion :
q = x + ic1t1 + jc2t2
+ kc3t3 +lc4t4 +
mc5t5 + nc6t6 + oc7t7
where
x = x(x1 , x2 , x3)
The
physical laws prevailing in such a system would be extremely complicated. Under
the internal symmetry, we can make it much simpler by changing the extra time dimensions
into spatial ones transforming the system to get a system consisting of 9-space
embedded in 10-dimensional ambient spacetime.
Its
mathematical formulation then becomes a simple ordinary complex number:
q = x1 +x2 + x3 + x4
+x5 + x6 + x7 + x8 + x9+ oc7t7
Denoting
x = x (x1, … x9), we get a simple form:
q
= x + oc7t7
e) For
example, if we have a system consisting of 6-brane embedded in 10-ambient
spacetime (a system with 6 space dimensions and 4 time dimensions):
q = x1 +x2 + x3 + x4
+x5 + x6 +lc4t4 + mc5t5
+ nc6t6 + oc7t7
Usually
we consider such a system having only one time dimension and disregard the
other three:
q = x1 +x2 + x3 + x4
+x5 + x6 +lc4t4
the
solution of this formulation would be the only approximation of the former.
f) The
dimensions of macro-cosmos are assumed to be [quasi] infinite. It happened that
we need only an ambient spacetime having 11 dimensions embedding a
10-brane as in the case of supergravity theory.
g) In
such a diagram, simultaneous events would flatten their loci, an n-surface
(hyperinterface, brane) embedded in (n+1) ambient spacetime; otherwise, the
surface would not be flat. Figure-1B shows events which happen simultaneously
at a certain time dimension t2 while they do not happen
simultaneously at a lower time dimension t1 (Figure-1A).
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