# Matter Fields

In classical relativity theory, one generally takes for granted that all there is, and all that happens, can be described in terms of various “matter fields,” each of which is represented by one or more smooth tensor (or spinor) fields on the spacetime manifold M. The latter are assumed to satisfy particular “field equations” involving the spacetime metric gab.

Associated with each matter field F is a symmetric smooth tensor field Tab characterized by the property that, for all points p in M, and all future-directed, unit timelike vectors ξa at p, Tabξb is the four-momentum density of F at p as determined relative to ξa.

Tab is called the energy-momentum field associated with F. The four- momentum density vector Tabξb at a point can be further decomposed into its temporal and spatial components relative to ξa,

Tabξb = (Tmbξmξba + Tmbhmaξb

where the first term on the RHS is the energy density, while the second term is the three-momentum density. A number of assumptions about matter fields can be captured as constraints on the energy-momentum tensor fields with which they are associated.

Weak Energy Condition (WEC): Given any timelike vector ξa at any point in M, Tabξaξb ≥ 0.

Dominant Energy Condition (DEC): Given any timelike vector ξa at any point in M, Tabξaξb ≥ 0 and Tabξb is timelike or null.

Strengthened Dominant Energy Condition (SDEC): Given any timelike vector ξa at any point in M, Tabξaξb ≥ 0 and, if Tab ≠ 0 there, then Tabξb is timelike.

Conservation Condition (CC): ∇aTab = 0 at all points in M.

The WEC asserts that the energy density of F, as determined by any observer at any point, is non-negative. The DEC adds the requirement that the four-momentum density of F, as determined by any observer at any point, is a future-directed causal (i.e., timelike or null) vector. We can understand this second clause to assert that the energy of F does not propagate with superluminal velocity. The strengthened version of the DEC just changes “causal” to “timelike” in the second clause. It avoids reference to “point particles.” Each of the listed energy conditions is strictly stronger than the ones that precede it.

The CC, finally, asserts that the energy-momentum carried by F is locally conserved. If two or more matter fields are present in the same region of space-time, it need not be the case that each one individually satisfies the condition. Interaction may occur. But it is a fundamental assumption that the composite energy-momentum field formed by taking the sum of the individual ones satisfies it. Energy-momentum can be transferred from one matter field to another, but it cannot be created or destroyed. The stated conditions have a number of consequences that support the interpretations.

A subset S of M is said to be achronal if there do not exist points p and q in S such that p ≪ q. Let γ : I → M be a smooth curve. We say that a point p in M is a future-endpoint of γ if, for all open sets O containing p, there exists an s0 in I such that, ∀ s ∈ I, if s ≥ s0, then γ(s) ∈ O; i.e., γ eventually enters and remains in O. Now let S be an achronal subset of M. The domain of dependence D(S) of S is the set of all points p in M with this property: given any smooth causal curve without (past- or future-) endpoint, if its image contains p, then it intersects S. So, in particular, S ⊆ D(S).

Let S be an achronal subset of M. Further, let Tab be a smooth, symmetric field on M that satisfies both the dominant energy and conservation conditions. Finally, assume Tab = 0 on S. Then Tab = 0 on all of D(S).

The intended interpretation of the proposition is clear. If energy-momentum cannot propagate (locally) outside the null-cone, and if it is conserved, and if it vanishes on S, then it must vanish throughout D(S). After all, how could it “get to” any point in D(S)? According to interpretive principle free massive point particles traverse (images of) timelike geodesics. It turns out that if the energy-momentum content of each body in the sequence satisfies appropriate conditions, then the convergence point will necessarily traverse (the image of) a timelike geodesic.

Let γ: I → M be smooth curve. Suppose that, given any open subset O of M containing γ[I], ∃ a smooth symmetric field Tab on M such that the following conditions hold.

(1) Tab satisfies the SDEC.
(2) Tab satisfies the CC.
(3) Tab = 0 outside of O.
(4) Tab ≠ 0 at some point in O.

Then γ is timelike and can be reparametrized so as to be a geodesic. This might be paraphrased another way. Suppose that for some smooth curve γ , arbitrarily small bodies with energy-momentum satisfying conditions (1) and (2) can contain the image of γ in their worldtubes. Then γ must be a timelike geodesic (up to reparametrization). Bodies here are understood to be “free” if their internal energy-momentum is conserved (by itself). If a body is acted on by a field, it is only the composite energy-momentum of the body and field together that is conserved.

But, this formulation for granted that we can keep the background spacetime metric gab fixed while altering the fields Tab that live on M. This is justifiable only to the extent that we are dealing with test bodies whose effect on the background spacetime structure is negligible.

We have here a precise proposition in the language of matter fields that, at least to some degree, captures the interpretive principle. Similarly, it is possible to capture the behavior of light, wherein the behavior of solutions to Maxwell’s equations in a limiting regime (“the optical limit”) where wavelengths are small. It asserts, in effect, that when one passes to this limit, packets of electromagnetic waves are constrained to move along (images of ) null geodesics.

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# Dynamics of Point Particles: Orthogonality and Proportionality

Let γ be a smooth, future-directed, timelike curve with unit tangent field ξa in our background spacetime (M, gab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λa be a vector at p. Then there is a natural decomposition of λa into components proportional to, and orthogonal to, ξa:

λa = (λbξba + (λa −(λbξba) —– (1)

Here, the first part of the sum is proportional to ξa, whereas the second one is orthogonal to ξa.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λa relative to ξa (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξa is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

kab = ξa ξb —– (2)

hab = gab − ξa ξb —– (3)

then the decomposition can be expressed in the form

λa = kab λb + hab λb —– (4)

We can think of kab and hab as the relative temporal and spatial metrics determined by ξa. They are symmetric and satisfy

kabkbc = kac —– (5)

habhbc = hac —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′a. (Since ξa and ξ′a are both future-directed, they are co-oriented; i.e., ξa ξ′a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥hab ξ′b∥ / ∥kab ξ′b∥ —– (7)

For any vector μa, ∥μa∥ is (μaμa)1/2 if μ is causal, and it is (−μaμa)1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥kab ξ′b∥ = (kab ξ′b kac ξ′c)1/2 = (kbc ξ′bξ′c)1/2 = (ξ′bξb)

and

∥hab ξ′b∥ = (−hab ξ′b hac ξ′c)1/2 = (−hbc ξ′bξ′c)1/2 = ((ξ′bξb)2 − 1)1/2

so

v = ((ξ’bξb)2 − 1)1/2 / (ξ′bξb) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′bξb) = 1/√(1 – v2) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector Pa that is tangent to its worldline there. The length ∥Pa∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if Pa is timelike, we can write it in the form Pa =mξa, where m = ∥Pa∥ > 0 and ξa is the four-velocity of the particle. No such decomposition is possible when Pa is null and m = ∥Pa∥ = 0.

Suppose a particle O with positive mass has four-velocity ξa at a point, and another particle O′ has four-momentum Pa there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose Pa in terms of ξa. By equations (4) and (2), we have

Pa = (Pbξb) ξa + habPb —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = Pbξb. If O′ has positive mass and Pa = mξ′a, this yields, by equation (9),

E = m (ξ′bξb) = m/√(1 − v2) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc2 and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term habPb in the decomposition of Pa, i.e., the component of Pa orthogonal to ξa. It follows from equations (8) and (9) that it has magnitude

p = ∥hab mξ′b∥ = m((ξ′bξb)2 − 1)1/2 = mv/√(1 − v2) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξa. Just as we understand ξa to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξnnξa – the directional derivative of ξa in the direction ξa – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξa. (This is, since ξannξa) = 1/2 ξnnaξa) = 1/2 ξnn (1) = 0). The magnitude ∥ξnnξa∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field Fa on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξnnξa satisfies

Fa = mξnnξa —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields Fab. If a particle with mass m > 0, charge q, and four-velocity field ξa is present, the force exerted by the field on the particle at a point is given by qFabξb. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qFabξb = mξbbξa —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξa. But so is the force field on the left, since ξa(Fabξb) = ξaξbFab = ξaξbF(ab), and F(ab) = 0 by the anti-symmetry of Fab.

# Quantum Informational Biochemistry. Thought of the Day 71.0

A natural extension of the information-theoretic Darwinian approach for biological systems is obtained taking into account that biological systems are constituted in their fundamental level by physical systems. Therefore it is through the interaction among physical elementary systems that the biological level is reached after increasing several orders of magnitude the size of the system and only for certain associations of molecules – biochemistry.

In particular, this viewpoint lies in the foundation of the “quantum brain” project established by Hameroff and Penrose (Shadows of the Mind). They tried to lift quantum physical processes associated with microsystems composing the brain to the level of consciousness. Microtubulas were considered as the basic quantum information processors. This project as well the general project of reduction of biology to quantum physics has its strong and weak sides. One of the main problems is that decoherence should quickly wash out the quantum features such as superposition and entanglement. (Hameroff and Penrose would disagree with this statement. They try to develop models of hot and macroscopic brain preserving quantum features of its elementary micro-components.)

However, even if we assume that microscopic quantum physical behavior disappears with increasing size and number of atoms due to decoherence, it seems that the basic quantum features of information processing can survive in macroscopic biological systems (operating on temporal and spatial scales which are essentially different from the scales of the quantum micro-world). The associated information processor for the mesoscopic or macroscopic biological system would be a network of increasing complexity formed by the elementary probabilistic classical Turing machines of the constituents. Such composed network of processors can exhibit special behavioral signatures which are similar to quantum ones. We call such biological systems quantum-like. In the series of works Asano and others (Quantum Adaptivity in Biology From Genetics to Cognition), there was developed an advanced formalism for modeling of behavior of quantum-like systems based on theory of open quantum systems and more general theory of adaptive quantum systems. This formalism is known as quantum bioinformatics.

The present quantum-like model of biological behavior is of the operational type (as well as the standard quantum mechanical model endowed with the Copenhagen interpretation). It cannot explain physical and biological processes behind the quantum-like information processing. Clarification of the origin of quantum-like biological behavior is related, in particular, to understanding of the nature of entanglement and its role in the process of interaction and cooperation in physical and biological systems. Qualitatively the information-theoretic Darwinian approach supplies an interesting possibility of explaining the generation of quantum-like information processors in biological systems. Hence, it can serve as the bio-physical background for quantum bioinformatics. There is an intriguing point in the fact that if the information-theoretic Darwinian approach is right, then it would be possible to produce quantum information from optimal flows of past, present and anticipated classical information in any classical information processor endowed with a complex enough program. Thus the unified evolutionary theory would supply a physical basis to Quantum Information Biology.

# Gothic: Once Again Atheistic Materialism and Hedonistic Flirtations. Drunken Risibility.

The machinery of the Gothic, traditionally relegated to both a formulaic and a sensational aesthetic, gradually evolved into a recyclable set of images, motifs and narrative devices that surpass temporal, spatial and generic categories. From the moment of its appearance the Gothic has been obsessed with presenting itself as an imitation.

Recent literary theory has extensively probed into the power of the Gothic to evade temporal and generic limits and into the aesthetic, narratological and ideological implications this involves. Officially granting the Gothic the elasticity it has always entailed has resulted in a reconfiguration of its spectrum both synchronically – by acknowledging its influence on numerous postmodern fictions – and diachronically – by rescripting, in hindsight, the history of its canon so as to allow space for ambiguous presences.

Both transgressive and hybrid in form and content, the Gothic has been accepted as a malleable genre, flexible enough to create more freely, in Borgesian fashion, its own precursors. The genre flouted what are considered the basic principles of good prose writing: adherence to verisimilitude and avoidance of both narrative diversions and moralising – all of which are, of course, made to be deliberately upset. Many merely cite the epigrammatic power of the essay’s most renowned phrase, that the rise of the Gothic “was the inevitable result of the revolutionary shocks which all of Europe has suffered”.

The eighteenth-century French materialist philosophy purported the displacement of metaphysical investigations into the meaning of life by materialist explorations. Julien Offray de La Mettrie, a French physician and philosopher, the earliest of materialist writers of the Enlightenment, published the materialist manifesto L’ Homme machine (Man a Machine), that did away with the transcendentalism of the soul, banished all supernatural agencies by claiming that mind is as mechanical as matter and equated humans with machines. In his words: “The human body is a machine that winds up its own springs: it is a living image of the perpetual motion”. French materialist thought resulted in the publication of the great 28-volume Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des méttrie par une société de gens de lettres by Denis Diderot and Jean Le Rond d’ Alembert, and which was grounded on purely materialist principles, against all kinds of metaphysical thinking. Diderot’s atheist materialism set the tone of the Encyclopédie, which, for both editors, was the ideal vehicle […] for reshaping French high culture and attitudes, as well as the perfect instrument with which to insinuate their radical Weltanschauung surreptitiously, using devious procedures, into the main arteries of French Society, embedding their revolutionary philosophic manifesto in a vast compilation ostensibly designed to provide plain information and basic orientation but in fact subtly challenging and transforming attitudes in every respect. While materialist thinkers ultimately disowned La Mettrie because he ran counter to their systematic moral, political and social naturalism, someone like Sade remained deeply influenced and inspired for his indebtedness to La Mettrie’s atheism and hedonism, particularly to the perception of virtue and vice as relative notions − the result of socialisation and at odds with nature.

# Meillassoux’s Principle of Unreason Towards an Intuition of the Absolute In-itself. Note Quote.

The principle of reason such as it appears in philosophy is a principle of contingent reason: not only how philosophical reason concerns difference instead of identity, we but also why the Principle of Sufficient Reason can no longer be understood in terms of absolute necessity. In other words, Deleuze disconnects the Principle of Sufficient Reason from the ontotheological tradition no less than from its Heideggerian deconstruction. What remains then of Meillassoux’s criticism in After finitude: An Essay on the Necessity of Contigency that Deleuze no less than Hegel hypostatizes or absolutizes the correlation between thinking and being and thus brings back a vitalist version of speculative idealism through the back door?

At stake in Meillassoux’s criticism of the Principle of Sufficient Reason is a double problem: the conditions of possibility of thinking and knowing an absolute and subsequently the conditions of possibility of rational ideology critique. The first problem is primarily epistemological: how can philosophy justify scientific knowledge claims about a reality that is anterior to our relation to it and that is hence not given in the transcendental object of possible experience (the arche-fossil )? This is a problem for all post-Kantian epistemologies that hold that we can only ever know the correlate of being and thought. Instead of confronting this weak correlationist position head on, however, Meillassoux seeks a solution in the even stronger correlationist position that denies not only the knowability of the in itself, but also its very thinkability or imaginability. Simplified: if strong correlationists such as Heidegger or Wittgenstein insist on the historicity or facticity (non-necessity) of the correlation of reason and ground in order to demonstrate the impossibility of thought’s self-absolutization, then the very force of their argument, if it is not to contradict itself, implies more than they are willing to accept: the necessity of the contingency of the transcendental structure of the for itself. As a consequence, correlationism is incapable of demonstrating itself to be necessary. This is what Meillassoux calls the principle of factiality or the principle of unreason. It says that it is possible to think of two things that exist independently of thought’s relation to it: contingency as such and the principle of non-contradiction. The principle of unreason thus enables the intellectual intuition of something that is absolutely in itself, namely the absolute impossibility of a necessary being. And this in turn implies the real possibility of the completely random and unpredictable transformation of all things from one moment to the next. Logically speaking, the absolute is thus a hyperchaos or something akin to Time in which nothing is impossible, except it be necessary beings or necessary temporal experiences such as the laws of physics.

There is, moreover, nothing mysterious about this chaos. Contingency, and Meillassoux consistently refers to this as Hume’s discovery, is a purely logical and rational necessity, since without the principle of non-contradiction not even the principle of factiality would be absolute. It is thus a rational necessity that puts the Principle of Sufficient Reason out of action, since it would be irrational to claim that it is a real necessity as everything that is is devoid of any reason to be as it is. This leads Meillassoux to the surprising conclusion that [t]he Principle of Sufficient Reason is thus another name for the irrational… The refusal of the Principle of Sufficient Reason is not the refusal of reason, but the discovery of the power of chaos harboured by its fundamental principle (non-contradiction). (Meillassoux 2007: 61) The principle of factiality thus legitimates or founds the rationalist requirement that reality be perfectly amenable to conceptual comprehension at the same time that it opens up [a] world emancipated from the Principle of Sufficient Reason (Meillassoux) but founded only on that of non-contradiction.

This emancipation brings us to the practical problem Meillassoux tries to solve, namely the possibility of ideology critique. Correlationism is essentially a discourse on the limits of thought for which the deabsolutization of the Principle of Sufficient Reason marks reason’s discovery of its own essential inability to uncover an absolute. Thus if the Galilean-Copernican revolution of modern science meant the paradoxical unveiling of thought’s capacity to think what there is regardless of whether thought exists or not, then Kant’s correlationist version of the Copernican revolution was in fact a Ptolemaic counterrevolution. Since Kant and even more since Heidegger, philosophy has been adverse precisely to the speculative import of modern science as a formal, mathematical knowledge of nature. Its unintended consequence is therefore that questions of ultimate reasons have been dislocated from the domain of metaphysics into that of non-rational, fideist discourse. Philosophy has thus made the contemporary end of metaphysics complicit with the religious belief in the Principle of Sufficient Reason beyond its very thinkability. Whence Meillassoux’s counter-intuitive conclusion that the refusal of the Principle of Sufficient Reason furnishes the minimal condition for every critique of ideology, insofar as ideology cannot be identified with just any variety of deceptive representation, but is rather any form of pseudo-rationality whose aim is to establish that what exists as a matter of fact exists necessarily. In this way a speculative critique pushes skeptical rationalism’s relinquishment of the Principle of Sufficient Reason to the point where it affirms that there is nothing beneath or beyond the manifest gratuitousness of the given nothing, but the limitless and lawless power of its destruction, emergence, or persistence. Such an absolutizing even though no longer absolutist approach would be the minimal condition for every critique of ideology: to reject dogmatic metaphysics means to reject all real necessity, and a fortiori to reject the Principle of Sufficient Reason, as well as the ontological argument.

On the one hand, Deleuze’s criticism of Heidegger bears many similarities to that of Meillassoux when he redefines the Principle of Sufficient Reason in terms of contingent reason or with Nietzsche and Mallarmé: nothing rather than something such that whatever exists is a fiat in itself. His Principle of Sufficient Reason is the plastic, anarchic and nomadic principle of a superior or transcendental empiricism that teaches us a strange reason, that of the multiple, chaos and difference. On the other hand, however, the fact that Deleuze still speaks of reason should make us wary. For whereas Deleuze seeks to reunite chaotic being with systematic thought, Meillassoux revives the classical opposition between empiricism and rationalism precisely in order to attack the pre-Kantian, absolute validity of the Principle of Sufficient Reason. His argument implies a return to a non-correlationist version of Kantianism insofar as it relies on the gap between being and thought and thus upon a logic of representation that renders Deleuze’s Principle of Sufficient Reason unrecognizable, either through a concept of time, or through materialism.

# Solitude: Thought of the Day 18.0

A reason Nietzsche ponders solitude is that his is largely a philosophy of the future. There is heavy emphasis in Beyond Good and Evil on the temporal nature of the human condition. He posits that “the taste of the time and the virtue of the time weakens and thins down the will.” In order to surpass current modes and fashions in thinking, one must become removed from the present. The new philosopher is necessarily a man of tomorrow and the day after tomorrow and so he is solitary and in contradiction to the ideals of today. Fundamentally, Nietzsche sees current Europe (and especially Germany) as not yet prepared for an overturning of present morality. Although he does predict the time is approaching, there is the overarching sense throughout Beyond Good and Evil that Nietzsche expects (and even embraces) the fact that his philosophy needs a significant passage of time to be understood. His work is lonely. He labors to lay groundwork for the philosophers of the future who will continue on this path someday.

The life of the free spirit is solitary because it requires the recognition of the untruth of life in order to be beyond good and evil. Religion and democratic enlightenment in Europe have forged a herd mentality of mediocrity which has rejected such a possibility. In this society, everyone’s thoughts and morality are given equal merit. Nietzsche despises this because it forces us to reject our nature; both the ugliness and the beauty of it. He tells us that religion is able to teach even the lowliest of people how to place themselves in an illusory higher order of things so they may have the impression that they are content. This herd mentality protects the pack and also makes life palatable. It is also the first enemy of anyone looking to discover their own truths. Nietzsche concludes his book by reflecting on the wonders of solitude. For the free spirit, solitude is life-affirming because the absence of the stifling dogmas of the herd allows for the greatest expansion of one’s sense of self. To be truly beyond good and evil one must be removed from grappling with the order and morality imposed by democratic enlightenment and religion. Only when one stands alone vis-à-vis the herd is greatness and nobleness possible. Upon being removed from the seething torrent of austere and rigid thinking now strangling Europe, the free spirit foments his own morality and thrives.