

Space Time and Entropy
25^{th} March 2019

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Imagine that a colony of amoebae is commuting between Ottawa and Montreal, in a selfdriving car named Bubble, which has no windows, no brakes, no accelerator, no steeringwheel 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 fuelgauge, called actiongauge, 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 actiongauge 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 ^{133}Cs and HeNe, Aku is been able to develop a system, not necessarily the precise actiongauge, 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 subsystem 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 ^{133}Cs 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 actiongauge 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 ^{133}Cs and HeNe themselves are (i) inside the bubble and (ii) the best they can do is to approximate the actual actiongauge, 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 Obs_{i}) to validate their measurements. Every measurement Aku can make, will have to derive from the readings the actiongauge is providing. It is equivalent to saying that the actual actiongauge providing the precise state of the available resources, is the topological space, Bubble is the manifold and the measurements made by the Aku, ^{133}Cs, and HeNe are made using a measurement metric based on the actiongauge of their design, in this manifold. This implies that the space (HeNe) and the time (^{133}Cs), both are linked to the actiongauge, 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 actiongauges. 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 actiongauges. The bottom line remains that for each observer, the actiongauge 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 VTSymmetry, i.e. define the origin based on precision measurements of ^{133}Cs and HeNe. (ii) Complete a circuit based on VTSymmetry. 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, ^{133}Cs & HeNe inside Bubble, and an observer Obs_{i }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 Obs_{i}. The entropy affecting Aku's measurements, has no implication for Obs_{i} 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 timeaxis. Aku is in jspace, 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 jspace 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 jspace. 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 jspace of Akus. In essence we are discussing impossibleproblem1, 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 noninertial. 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 spacetime interval in General Theory of Relativity. Entropy ensures that the information gathered during each successive spacetime 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
Sigmaz and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous zAxis Majorana ZFC Axioms Set Theory Nutshell2014 Knots in jSpace Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, bField & 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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