Causality

Causation is a form of event generation. To present an explicit definition of causation requires introducing some ontological concepts to formally characterize what is understood by ‘event’.

The concept of individual is the basic primitive concept of any ontological theory. Individuals associate themselves with other individuals to yield new individuals. It follows that they satisfy a calculus, and that they are rigorously characterized only through the laws of such a calculus. These laws are set with the aim of reproducing the way real things associate. Specifically, it is postulated that every individual is an element of a set s in such a way that the structure S = ⟨s, ◦, ◻⟩ is a commutative monoid of idempotents. This is a simple additive semi-group with neutral element.

In the structure S, s is the set of all individuals, the element ◻ ∈ s is a fiction called the null individual, and the binary operation ◦ is the association of individuals. Although S is a mathematical entity, the elements of s are not, with the only exception of ◻, which is a fiction introduced to form a calculus. The association of any element of s with ◻ yields the same element. The following definitions characterize the composition of individuals.

1. x ∈ s is composed ⇔ (∃ y, z) s (x = y ◦ z)
2. x ∈ s is simple ⇔ ∼ (∃ y, z) s (x = y ◦ z)
3. x ⊂ y ⇔ x ◦ y = y (x is part of y ⇔ x ◦ y = y)
4. Comp(x) ≡ {y ∈ s|y ⊂ x} is the composition of x.

Real things are distinguished from abstract individuals because they have a number of properties in addition to their capability of association. These properties can be intrinsic (P_i) or relational (P_r). The intrinsic properties are inherent and they are represented by predicates or unary applications, whereas relational properties depend upon more than a single thing and are represented by n-ary predicates, with n ≥ 1. Examples of intrinsic properties are electric charge and rest mass, whereas velocity of macroscopic bodies and volume are relational properties.

An individual with its properties make up a thing X : X =< x, P(x) >

Here P(x) is the collection of properties of the individual x. A material thing is an individual with concrete properties, i.e. properties that can change in some respect.

The state of a thing X is a set of functions S(X) from a domain of reference M (a set that can be enumerable or nondenumerable) to the set of properties P_X. Every function in S(X) represents a property in P_X. The set of the physically accessible states of a thing X is the lawful state space of X : S_L(X). The state of a thing is represented by a point in S_L(X). A change of a thing is an ordered pair of states. Only changing things can be material. Abstract things cannot change since they have only one state (their properties are fixed by definition).

A legal statement is a restriction upon the state functions of a given class of things. A natural law is a property of a class of material things represented by an empirically corroborated legal statement.

The ontological history h(X) of a thing X is a subset of S_L(X) defined by h(X) = {⟨t, F(t)⟩|t ∈ M}

where t is an element of some auxiliary set M, and F are the functions that represent the properties of X.

If a thing is affected by other things we can introduce the following definition:

h(Y/X ) : “history of the thing Y in presence of the thing X”.

Let h(X) and h(Y) be the histories of the things X and Y, respectively. Then

h(Y/X) = {⟨t,H(t)⟩|t ∈ M},

where H≠ F is the total state function of Y as affected by the existence of X, and F is the total state function of X in the absence of Y. The history of Y in presence of X is different from the history of Y without X .

We can now introduce the notion of action:

X ▷ Y : “X acts on Y”

X ▷ Y =^def h(Y/X) ≠ h(Y)

An event is a change of a thing X, i.e. an ordered pair of states:

(s₁, s₂) ∈ E_L(X) = S_L(X) × S_L(X)

The space E_L(X) is called the event space of X.

Causality is a relation between events, i.e. a relation between changes of states of concrete things. It is not a relation between things. Only the related concept of ‘action’ is a relation between things. Specifically,

C'(x): “an event in a thing x is caused by some unspecified event e^x_xi“.

C'(x) =^def (∃ e^x_xi) [e^x_xi ∈ E_L(X) ⇔ x_i ▷ x.

C(x, y): “an event in a thing x is caused by an event in a thing y”.

C(x, y) =^def (∃ e^x_y) [e^x_y ∈ E_L(x) ⇔ y ▷ x

In the above definitions, the notation e^x_y indicates in the superscript the thing x to whose event space belongs the event e, whereas the subscript denotes the thing that acted triggering the event. The implicit arguments of both C’ and C are events, not things. Causation is a form of event generation. The crucial point is that a given event in the lawful event space E_L(x) is caused by an action of a thing y iff the event happens only conditionally to the action, i.e., it would not be the case of e^x_y without an action of y upon x. Time does not appear in this definition, allowing causal relations in space-time without a global time orientability or even instantaneous and non-local causation. If causation is non-local under some circumstances, e.g. when a quantum system is prepared in a specific state of polarization or spin, quantum entanglement poses no problem to realism and determinism. The quantum theory describes an aspect of a reality that is ontologically determined and with non-local relations. Under any circumstances the postulates of Special Relativity are violated, since no physical system ever crosses the barrier of the speed of light.

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Left’s Alt-Right Hitmen…..

Enter ye in at the strait gate: for wide is the gate, and broad is the way, that leadeth to destruction, and many there be which go in thereat: Because strait is the gate, and narrow is the way, which leadeth unto life, and few there be that find it. Beware of false prophets, which come to you in sheep’s clothing, but inwardly they are ravening wolves. Matthew 7:13-21

Social Pathology………..As much as NRx is an intellectual movement it does need people. Not just in terms of thinking, but in terms of disseminating ideas, organising people and so on. It doesn’t need to be a “mass movement” but it does need to be a real movement, otherwise it’s simply pie in the sky ideas and will be without any historical relevance whatsoever. The problem is for NRx is not just diagnosis, the problem is how to move the world. The apparent current strategy, of being the last man standing, is false.  A collapsing society produces power vacuums and that vacuum is going to be filled either by NRx grasping power or someone else doing it. Power tempered by ideas is the ultimate imperative of NRx.

NRx seemed to provide a space where intelligent ideas could be discussed freely and a rallying point for those intelligent but dissatisfied people of the right. However, with the infusion of the alt-Right, thought policing–admittedly of different kind–has returned with methods of the Social Justice Warriors, driving away the intelligent people.

…For the Left, this state of affairs is particularly fortuitous and sometimes you have to wonder if they bring out their alt-Right hitmen every now and then to discredit intelligent Rightists through guilt by association. 

Through

Beginning of Matter, Start to Existence Itself

When the inequality

μ+3p/c2 >0 ⇔ w > −1/3

is satisfied, one obtains directly from the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ +3p/c²) + Λ

the Friedmann-Lemaître (FL) Universe Singularity Theorem, which states that:

In a FL universe with Λ ≤ 0 and μ + 3p/c² > 0 at all times, at any instant t₀ when H₀ ≡ (S ̇/S)₀ > 0 there is a finite time t_∗: t₀ − (1/H₀) < t_∗ < t₀, such that S(t) → 0 as t → t_∗; the universe starts at a space-time singularity there, with μ → ∞ and T → ∞ if μ + p/c² > 0.

This is not merely a start to matter – it is a start to space, to time, to physics itself. It is the most dramatic event in the history of the universe: it is the start of existence of everything. The underlying physical feature is the non-linear nature of the Einstein’s Field Equations (EFE): going back into the past, the more the universe contracts, the higher the active gravitational density, causing it to contract even more. The pressure p that one might have hoped would help stave off the collapse makes it even worse because (consequent on the form of the EFE) p enters algebraically into the Raychaudhuri equation with the same sign as the energy density μ. Note that the Hubble constant gives an estimate of the age of the universe: the time τ₀ = t₀ − t_∗ since the start of the universe is less than 1/H₀.

This conclusion can in principle be avoided by a cosmological constant, but in practice this cannot work because we know the universe has expanded by at least a ratio of 11, as we have seen objects at a redshift 6 of 10, the cosmological constant would have to have an effective magnitude at least 11³ = 1331 times the present matter density to dominate and cause a turn-around then or at any earlier time, and so would be much bigger than its observed present upper limit (of the same order as the present matter density). Accordingly, no turnaround is possible while classical physics holds. However energy-violating matter components such as a scalar field can avoid this conclusion, if they dominate at early enough times; but this can only be when quantum fields are significant, when the universe was at least 10¹² smaller than at present.

Because T_rad ∝ S⁻¹, a major conclusion is that a Hot Big Bang must have occurred; densities and temperatures must have risen at least to high enough energies that quantum fields were significant, at something like the GUT energy. The universe must have reached those extreme temperatures and energies at which classical theory breaks down.

Black Holes. Thought of the Day 23.0

The formation of black holes can be understood, at least partially, within the context of general relativity. According to general relativity the gravitational collapse leads to a spacetime singularity. But this spacetime singularity can not be adequately described within general relativity, because the equivalence principle of general relativity is not valid for spacetime singularities; therefore, general relativity does not give a complete description of black holes. The same problem exists with regard to the postulated initial singularity of the expanding cosmos. In these cases, quantum mechanics and quantum field theory also reach their limit; they are not applicable for highly curved spacetimes. For a certain curving parameter (the famous Planck scale), gravity has the same strength as the other interactions; then it is not possible to ignore gravity in the context of a quantum field theoretical description. So, there exists no theory which would be able to describe gravitational collapses or which could explain, why (although they are predicted by general relativity) they don’t happen, or why there is no spacetime singularity. And the real problems start, if one brings general relativity and quantum field theory together to describe black holes. Then it comes to rather strange forms of contradictions, and the mutual conceptual incompatibility of general relativity and quantum field theory becomes very clear:

Black holes are according to general relativity surrounded by an event horizon. Material objects and radiation can enter the black hole, but nothing inside its event horizon can leave this region, because the gravitational pull is strong enough to hold back even radiation; the escape velocity is greater than the speed of light. Not even photons can leave a black hole. Black holes have a mass; in the case of the Schwarzschild metrics, they have exclusively a mass. In the case of the Reissner-Nordström metrics, they have a mass and an electric charge; in case of the Kerr metrics, they have a mass and an angular momentum; and in case of the Kerr-Newman metrics, they have mass, electric charge and angular momentum. These are, according to the no-hair theorem, all the characteristics a black hole has at its disposal. Let’s restrict the argument in the following to the Reissner-Nordström metrics in which a black hole has only mass and electric charge. In the classical picture, the electric charge of a black hole becomes noticeable in form of a force exerted on an electrically charged probe outside its event horizon. In the quantum field theoretical picture, interactions are the result of the exchange of virtual interaction bosons, in case of an electric charge: virtual photons. But how can photons be exchanged between an electrically charged black hole and an electrically charged probe outside its event horizon, if no photon can leave a black hole – which can be considered a definition of a black hole? One could think, that virtual photons, mediating electrical interaction, are possibly able (in contrast to real photons, representing radiation) to leave the black hole. But why? There is no good reason and no good answer for that within our present theoretical framework. The same problem exists for the gravitational interaction, for the gravitational pull of the black hole exerted on massive objects outside its event horizon, if the gravitational force is understood as an exchange of gravitons between massive objects, as the quantum field theoretical picture in its extrapolation to gravity suggests. How could (virtual) gravitons leave a black hole at all?

There are three possible scenarios resulting from the incompatibility of our assumptions about the characteristics of a black hole, based on general relativity, and on the picture quantum field theory draws with regard to interactions:

(i) Black holes don’t exist in nature. They are a theoretical artifact, demonstrating the asymptotic inadequacy of Einstein’s general theory of relativity. Only a quantum theory of gravity will explain where the general relativistic predictions fail, and why.

(ii) Black holes exist, as predicted by general relativity, and they have a mass and, in some cases, an electric charge, both leading to physical effects outside the event horizon. Then, we would have to explain, how these effects are realized physically. The quantum field theoretical picture of interactions is either fundamentally wrong, or we would have to explain, why virtual photons behave completely different, with regard to black holes, from real radiation photons. Or the features of a black hole – mass, electric charge and angular momentum – would be features imprinted during its formation onto the spacetime surrounding the black hole or onto its event horizon. Then, interactions between a black hole and its environment would rather be interactions between the environment and the event horizon or even interactions within the environmental spacetime.

(iii) Black holes exist as the product of gravitational collapses, but they do not exert any effects on their environment. This is the craziest of all scenarios. For this scenario, general relativity would have to be fundamentally wrong. In contrast to the picture given by general relativity, black holes would have no physically effective features at all: no mass, no electric charge, no angular momentum, nothing. And after the formation of a black hole, there would be no spacetime curvature, because there remains no mass. (Or, the spacetime curvature has to result from other effects.) The mass and the electric charge of objects falling (casually) into a black hole would be irretrievably lost. They would simply disappear from the universe, when they pass the event horizon. Black holes would not exert any forces on massive or electrically charged objects in their environment. They would not pull any massive objects into their event horizon and increase thereby their mass. Moreover, their event horizon would mark a region causally disconnected with our universe: a region outside of our universe. Everything falling casually into the black hole, or thrown intentionally into this region, would disappear from the universe.