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Space
Time and Entropy 25th March 2019
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Imagine that a colony of
amoebae is commuting between Ottawa and
Montreal, in a self-driving car named
Bubble, which has no windows, no brakes, no
accelerator, no steering-wheel and a
fuel tank with an unlimited capacity.
Furthermore in Bubble there is no
odometer, no clock, nor any GPS, to indicate
the exact location or time in a conventional
sense. Only information available to
the passengers or observers, is an
instrument similar to the fuel-gauge, called
action-gauge, which tells amoebae how
efficiently the fuel or the resources have
been consumed during their journey. The catch is that
the observers have to design the action-gauge
themselves.
Now let us assume that among the observers is
an enlightened amoeba named Aku, who is aware
of its own mortality. With help from his
fellow observers 133Cs and
HeNe, Aku is been able to develop a
system, not necessarily the precise
action-gauge, to get some sense of the
commute he is stuck in making.
If we think about it, Aku has not been given
any information about Bubble whatsoever except
for what Aku can measure. Aku is not
aware of the increasing entropy S, resulting
in increasing resource consumption for the
same destination, due to the lack of precision
in measurements. An
assumption can be made that entropy S does not
affect a sub-system inside the vehicle, and
the conditions such as Inertial Frames,
Unitarity and Locality etc. can be used to
describe the results measured by Aku inside
Bubble. The important thing to note, is
that the observers have no awareness of
"outside" Bubble. The states:
"inside" and "outside", are Aku's perceptions
localized within Bubble.
As it happens, the amoebae traveling to
greater distances begin to notice something
very odd going on. The precision
observers 133Cs
and HeNe both tell Aku, that per their
measurements, it is becoming increasingly
difficult to commute, as if Ottawa and
Montreal are growing distant from each
other. When different observers
commuting at different locations compare
notes they are forced to agree that
various cities, towns, and villages are
growing apart in similar manner, because
the fundamental standards of measurements which are
assumed to be absolute, are telling
observers as such. 1
An inevitable conclusion based on the measurements with
continuously deteriorating precision,
can be drawn that the blue marble is expanding
like a gas balloon, which actually is a
thermodynamic picture.2
But as outside observers we know that it is
not true. In particular the
constant which is directly linked to
the action-gauge itself, h^,
and the constant which expresses the
limitation of the Bubble due to inherent
entropy, c^,
both are bound to show the effect of entropy
if they are measured precisely enough.
(Another constant Aku has to worry about is G^,
which keeps Bubble on an affine
path, unless some measurable quantity
based on h^ and c^, can be maneuvered locally
inside the bubble.)
This required precision is impossible as the
observers 133Cs
and HeNe themselves are (i) inside the bubble
and (ii) the best they can do is to
approximate the actual action-gauge, but never
precise enough.3
They are affected by the same entropy as
Bubble. So all observers are in the same
boat essentially, with no outside observer (
||r to Obsi) to
validate their measurements.
Every measurement Aku can make, will have to
derive from the readings the action-gauge is
providing. It is equivalent to saying
that the actual
action-gauge providing the precise
state of the available resources, is the
topological space, Bubble is the manifold and
the measurements made by the Aku, 133Cs,
and HeNe are made using a measurement metric
based on the action-gauge of their
design, in this manifold. This implies
that the space (HeNe) and the time (133Cs),
both are linked to the action-gauge, and hence
they can not be measured independent of each
other.4
We can take the argument further, and assume
that there are different types of amoebae and
each of them riding their Bubbles of similar
nature, albeit with similar inefficient
action-gauges. This situation is like
different observers using different manifolds
and measurement metrics, while the underlying
topological space remains
unchanged. These observers can be
anything or anyone, from an atom to an amoeba,
each with their own unique characteristics and
hence unique action-gauges. The
bottom line remains that for each observer,
the action-gauge and hence the underlying
topology, does not change and the second law
of thermodynamics, ΔS ≥ 0,
is always applicable. Each Bubble will
be affected by the same entropy, resulting in
continuously increasing resource consumption
during the same quantum of journey.
Inside Bubble, Aku must complete a circuit
before it can start counting. The
sequence of events to accomplish this, is as
follows: (i)
Assume a VT-Symmetry, i.e. define
the origin based on precision measurements
of 133Cs
and HeNe. (ii)
Complete a circuit based on VT-Symmetry.
The circuit will be the curve of least energy
or the curve of least disorder for Aku. (iii)
Apply Stokes' theorem and Gauss'
theorem to form the Surface of Least
Disorder (SLD). (iv)
Aku can start counting using SLD inside
Bubble. We note that Aku is merely
counting, not measuring either length or time.
So how does the concept of time comes in to
the picture inside Bubble? Let us
assume the simplest case that
the circuit to be completed inside Bubble, is
a circle. And the observers measuring
the circle are: Aku, 133Cs
& HeNe inside Bubble, and an observer Obsi from
outside Bubble. For Aku, it is virtually
impossible to complete the required
measurements before the inevitable binary
fission takes place. In fact it may well
be an impossible task, for next few thousand
generations of Aku to complete the
measurements. On the other hand, for an
outside observer it is a single measurement to
determine the completed circuit as a
circle. The situation is as shown
below: Therefore in essence, the infinite time axis for Aku is nothing more than an instant for Obsi. The entropy affecting Aku's measurements, has no implication for Obsi measurements. Thus time is a relative concept which has significance only to the observer making measurements. Inside Bubble for Aku's measurements, time is absolute as Aku can not shake off the entropy inherent in its measurements. All the resources inside Bubble must therefore be calibrated against time-axis. Aku is in j-space, which is a dynamic measurement space, hence Aku must write his/her Lagrangian and minimize the action.5 It is not that difficult to imagine that inside Bubble there are billions and billions of amoebae and all of them are part of the same commute as Aku. Hence the time axis determined by Aku's measurements, will remain the same for all of them. We will continue to discuss Aku the amoeba, in forthcoming blogs. The concept of time and its correlation to the entropy ingrained in j-space measurements of an observer, has profound effect on how we should formulate problems in higher information space. An example is shown below:6 We
will discuss this example later on,
with respect to measurements in
j-space. Also now that we are able to
count, we should figure out how to build our
LEGO blocks.
___________________ 1.
The term "comparing notes by Akus" here,
requires adhering to Lorentz Invariance and Möbius
Transformations. Similarly fuel signifies
the resource utilization in j-space of
Akus. In essence we are discussing impossible-problem-1,
which is the resource
optimization in a closed system represented by
Bubble. 2. A
thermodynamic description implies that the
internal mechanism and symmetries of the
System, for e.g. spin, can not be precisely
determined. This is how Aku
approximates the interior of Bubble.
The interior of the System and the interior
of the Bubble are two different
descriptions. Bubble must follow the
topology which is set by the System.
Bubble is at the far edge of the System, yet
the interior of Bubble is all
Aku has for measurements. Entropy
represents the internal state of Bubble,
thus the value of entropy would remain the
same for any reference frame Aku is in,
inertial or non-inertial. The values
of dW, dQ, and T would change inside Bubble
for relativistic coordinate frames. 3.
Planck domain measurements are not
possible. 4.
Invariance of space-time
interval in General Theory of
Relativity. Entropy ensures that
the information gathered during each
successive space-time interval will be
continuously diminishing. 5. Aku
would have to assume action to be time
independent inside Bubble. We will discuss
the criterion behind this important assumption. 6. With grateful thanks from the notes of Dr. S. of Nikhef. ||r ≡ Similar *** |
Previous Blogs: Chiral Symmetry
Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis Majorana ZFC Axioms Set Theory Nutshell-2014 Knots in j-Space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-Field & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well Visual Complex Analysis The Einstein Theory of Relativity *** |
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