Databases Reference
In-Depth Information
•
Space-time
•
Godel's theorem
•
Arithmetic indeterminism
•
Turing halting problem
Additionally, some well-described physics as well as neuro-biological events
appear to be truly random [2]; as well as the mathematical descriptions that
are randomly uncertain. If they are, there is no way of avoiding causal impre-
cision.
It may well be that a precise and complete knowledge of causal events is
not possible or at least uncertain. On the other hand, we have a commonsense
belief that causal effects exist in the real world. If we can develop models
tolerant of imprecision, it would be useful. Also, to some degree, the degree
of importance that some of these items have decreases as grain size increases.
4.3 Granular Space-Time
One of the key principles of space-time is that of
background independence
.
This principle says that the geometry of space-time is not fixed. Instead,
the geometry is an evolving, dynamical quantity. A closely related principle is
diffeomorphism invariance
. This principle implies that unlike theories prior to
general relativity, one is free to choose any set of coordinates to map space-
time and express the equations. A point in space-time is defined only by what
physically happens at it, not by its location according to some special set of
coordinates (no coordinates are special).
Modern physics has developed a theory that entails that space and time
are granular [31]. This is an extension of quantum theory. Quantum mechanics
require that certain quantities, such as the energy of an atom, can only come
in specific, discrete units. Over the last few years, theory has evolved con-
cerning quantum gravity and quantum space-time. This area of endeavor is
sometimes called
loop quantum gravity.
(The term
loop
arises from how some
computations in the theory involve small loops marked out in space-time.)
The work is concerned with quantum theory of the structure of space-time at
the smallest size scales.
What concerns us in this chapter is that there are apparently limits on
fine grain size. These limits apply to areas, volumes, and time [31]. There is
a non-zero minimum area (about one square Planck length, or 10
−
66
cm
2
)
and a discrete series of allowed quantum areas. Similarly, there is a non-zero
absolute minimum volume (about one cubic Planck length, or 10
−
99
cm
2
)and
it restricts the set of larger volumes to a discrete series of numbers. Time is
also discrete; it comes in “clicks” of approximately the Planck time. Time
does not exist between the clicks; there is no “in between,” in the same way
that there is no water between adjacent molecules of water.
This information should influence how we think about causality. If the
universe is fundamentally granular, causal descriptions need to somehow deal
