Black Hole Entropy in terms of Mass. Note Quote.

c839ecac963908c173c6b13acf3cd2a8--friedrich-nietzsche-the-portal

If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)

Thus

S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Gauge Fixity Towards Hyperbolicity: General Theory of Relativity and Superpotentials. Part 1.

Untitled

Gravitational field is described by a pseudo-Riemannian metric g (with Lorentzian signature (1, m-1)) over the spacetime M of dimension dim(M) = m; in standard General Relativity, m = 4. The configuration bundle is thence the bundle of Lorentzian metrics over M, denoted by Lor(M) . The Lagrangian is second order and it is usually chosen to be the so-called Hilbert Lagrangian:

LH: J2Lor(m) → ∧om(M)

LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds —– (1)

where

R = gαβ Rαβ denotes the scalar curvature, √g the square root of the absolute value of the metric determinant and ∧ is a real constant (called the cosmological constant). The coupling constant 1/2κ which is completely irrelevant until the gravitational field is not coupled to some other field, depends on conventions; in natural units, i.e. c = 1, h = 1, G = 1, dimension 4 and signature ( + , – , – , – ) one has κ = – 8π.

Field equations are the well known Einstein equations with cosmological constant

Rαβ – 1/2 Rgαβ = -∧gαβ —— (2)

Lagrangian momenta is defined by:

pαβ = ∂LH/∂gαβ = 1/2κ (Rαβ – 1/2(R – 2∧)gαβ)√g

Pαβ = ∂LH/∂Rαβ = 1/2κ gαβ√g —– (3)

Thus the covariance identity is the following:

dα(LHξα) = pαβ£ξgαβ + Pαβ£ξRαβ —– (4)

or equivalently,

α(LHξα) = pαβ£ξgαβ + PαβεξΓεαβ – δεβ£ξΓλαλ) —– (5)

where ∇ε denotes the covariant derivative with respect to the Levi-Civita connection of g. Thence we have a weak conservation law for the Hilbert Lagrangian

Div ε(LH, ξ) = W(LH, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(LH, ξ) = [Pαβ£ξΓεαβ – Pαε£ξΓλαλ – LHξε]dsε = √g/2κ(gαβgεσ – gσβgεα) ∇α£ξgβσdsε – √g/2κξεRdsε = √g/2κ[(3/2Rαλ – (R – 2∧)δαλλ + (gβγδαλ – gα(γδβ)λβγξλ]dsα —– (7)

W(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)gαβ)∇(αξβ)ds —– (8)

As any other natural theory, General Relativity allows superpotentials. In fact, the current can be recast into the form:

ε(LH, ξ) = ε'(LH, ξ) + Div U(LH, ξ) —– (9)

where we set

ε'(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)δαββ)dsα

U(LH, ξ) = 1/2κ ∇[βξα] √gdsαβ —– (10)

The superpotential (10) generalizes to an arbitrary vector field ξ, the well known Komar superpotential which is originally derived for timelike Killing vectors. Whenever spacetime is assumed to be asymptotically fiat, then the superpotential of Komar is known to produce upon integration at spatial infinity ∞ the correct value for angular momentum (e.g. for Kerr-Newman solutions) but just one half of the expected value of the mass. The classical prescriptions are in fact:

m = 2∫ U(LH, ∂t, g)

J = ∫ U(LH, ∂φ, g) —– (11)

For an asymptotically flat solution (e.g. the Kerr-Newman black hole solution) m coincides with the so-called ADM mass and J is the so-called (ADM) angular momentum. For the Kerr-Newman solution in polar coordinates (t, r, θ, φ) the vector fields ∂t and ∂φ are the Killing vectors which generate stationarity and axial symmetry, respectively. Thence, according to this prescription, U(LH, ∂φ) is the superpotential for J while 2U(LH, ∂t) is the superpotential for m. This is known as the anomalous factor problem for the Komar potential. To obtain the expected values for all conserved quantities from the same superpotential, one has to correct the superpotential (10) by some ad hoc additional boundary term. Equivalently and alternatively, one can deduce a corrected superpotential as the canonical superpotential for a corrected Lagrangian, which is in fact the first order Lagrangian for standard General Relativity. This can be done covariantly, provided that one introduces an extra connection Γ’αβμ. The need of a reference connection Γ’ should be also motivated by physical considerations, according to which the conserved quantities have no absolute meaning but they are intrinsically relative to an arbitrarily fixed vacuum level. The simplest choice consists, in fact, in fixing a background metric g (not necessarily of the correct Lorentzian signature) and assuming Γ’ to be the Levi-Civita connection of g. This is rather similar to the gauge fixing à la Hawking which allows to show that Einstein equations form in fact an essentially hyperbolic PDE system. Nothing prevents, however, from taking Γ’ to be any (in principle torsionless) connection on spacetime; also this corresponds to a gauge fixing towards hyperbolicity.

Now, using the term background for a field which enters a field theory in the same way as the metric enters Yang-Mills theory, we see that the background has to be fixed once for all and thence preserved, e.g. by symmetries and deformations. A background has no field equations since deformations fix it; it eventually destroys the naturality of a theory, since fixing the background results in allowing a smaller group of symmetries G ⊂ Diff(M). Accordingly, in truly natural field theories one should not consider background fields either if they are endowed with a physical meaning (as the metric in Yang-Mills theory does) or if they are not.

On the contrary we shall use the expression reference or reference background to denote an extra dynamical field which is not endowed with a direct physical meaning. As long as variational calculus is concerned, reference backgrounds behave in exactly the same way as other dynamical fields do. They obey field equations and they can be dragged along deformations and symmetries. It is important to stress that such a behavior has nothing to do with a direct physical meaning: even if a reference background obeys field equations this does not mean that it is observable, i.e. it can be measured in a laboratory. Of course, not any dynamical field can be treated as a reference background in the above sense. The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….

Object as Category-Theoretic or Object as Ontological: The Inadequacy of Implicitly Quantifying Over Elements. (2)

blue_morphism_by_eresaw-d3h1ef4

It will be convenient to use the term ‘object’ in two senses. First, as an object of a category, i.e. in a purely mathematical sense. We shall call this a C- object (‘C’ for category-theoretic). Second, in the sense commonly used in structural realist debates, and which was already introduced above, viz. an object is a physical entity which is a relatum in physical relations. We shall call this an O-object (‘O’ for ‘ontological’).

We will also need to clarify our use of the term ‘element’. We use ‘element’ to mean an element of a set, or as it is also often called, a ‘point’ of a set (indeed it will be natural for us to switch to the language of points when discussing manifolds, i.e. spacetimes.) This familiar use of element should be distinguished from the category-theoretic concepts of ‘global element’ and ‘generalized element’, which is introduced below.

Jonathan Bain’s first strategy for defending (Objectless) draws on the following idea: the usual set-theoretic representations of O-objects and relations can be translated into category-theoretic terms, whence these objects can be eliminated. In fact, the argument can be seen as consisting of two different parts.

In the first part, Bain attempts to give a highly general argument, in the sense that it turns only on the notion of universal properties and the translatability of statements about certain mathematical representations (i.e. elements of sets) of O-objects into statements about morphisms between C-objects. As Bain himself notes, the general argument fails, and he thus introduces a more specific argument, which is what he wishes to endorse. The specific argument turns on the idea of obtaining a translation scheme from a ‘categorical equivalence’ between a geometric category and an algebraic category, which in turn allows one to generalize the original C-objects. The argument is ‘specific’ because such equivalences only hold between rather special sorts of categories.

The details of Bain’s general argument can be reconstructed as follows:

G1: Physical objects and the structures they bear are typically identified with the elements of a set X and relations on X respectively.

G2: The set-theoretic entities of G1 are to be represented in category-theoretic language by considering the category whose objects are the relevant structured sets, and whose morphisms are functions that preserve ‘structure’.

G3: Set-theoretic statements about an object of a category (of the type in G2) can often be expressed without making reference to the elements of that object. For instance:

1. In any category with a terminal object any element of an object X can be expressed as a morphism from the terminal object to X. (So for instance, since the singleton {∗} is the terminal object in the category Set, an element of a set X can be described by a morphism {∗} → X.)

2. In a category with some universal property, this property can be described purely in terms of morphisms, i.e. without making any reference to elements of an object.

To sum up, G1 links O-objects with a standard mathematical representation, viz. elements of a set. And G2 and G3 are meant to establish the possibility that, in certain cases, category theory allows us to translate statements about elements of sets into statements about the structure of morphisms between C-objects.

Thus, Bain takes G1–G3 to suggest that: 

C: Category theory allows for the possibility of coherently describing physical structures without making any reference to physical objects.

Indeed, Bain thinks the argument suggests that the mathematical representatives of O- objects, i.e. the elements of sets, are surplus, and that category theory succeeds in removing this surplus structure. Note that even if there is surplus structure here, it is not of the same kind as, e.g. gauge-equivalent descriptions of fields in Yang-Mills theory. The latter has to do with various equivalent ways in which one can describe the dynamical objects of a theory, viz. field. By contrast, Bain’s strategy involves various equivalent descriptions of the entire theory.

Bain himself thinks that the inference from G1–G3 to C fails, but he does give it serious consideration, and it is easy to see why: its premises based on the most natural and general translation scheme in category theory, viz. redescribing the properties of C-objects in terms of morphisms, and indeed – if one is lucky – in terms of universal properties. 

First, the premise G1. Structural realist doctrines are typically formalized by modeling O-objects as elements of a set and structures as relations on that set. However, this is seldom the result of reasoned deliberation about whether standard set theory is the best expressive resource from some class of such resources, but rather the product of a deeply entrenched set-theoretic viewpoint within philosophy. Were philosophers familiar with an alternative to set theory that was at least as powerful, e.g. category theory, then O-objects and structures might well have been modeled directly in the alternative formalism. Of course, it is also a reasonable viewpoint to say that it is most ‘natural’ to do the philosophy/foundations of physics in terms of set theory – what is ‘natural’ depends on how one conceives of such foundational investigations.

So we maintain that there is no reason for the defender of O-objects to accept G1. For instance, he might try to construct a category such that O-objects are modeled by C-objects and structures are modeled by morphisms. For example, there are examples of categories whose C-objects might coincide with the mathematical representatives of O-objects. For instance, in a path homotopy category, the C-objects are just points of the relevant space, and one might in turn take the points of a space to be O-objects, as Bain does in his example of general relativity and Einstein algebras. Or he might take as his starting point a non-concrete category, whose objects have no underlying set and thus cannot be expressed in the terms of G1.

The premise G2, on the other hand, is ambiguous—it is unclear exactly how Bain wants us to understand ‘structure’ and thus ‘structure-preserving maps’. First, note that when mathematicians talk about ‘structure-preserving maps’ they usually have in mind morphisms that do not preserve all the features of a C-object, but rather the characteristic (albeit partial) features of that C-object. For instance, with respect to a group, a structure-preserving map is a homomorphism and not an isomorphism. Bain’s example of the category Set is of this type, because its morphisms are arbitrary functions (and not bijective functions).

However, Bain wants to introduce a different notion of ‘structure’ that contrasts with this standard usage, for he says:

(Structure) …the intuitions of the ontic structural realist may be preserved by defining “structure” in this context to be “object in a category”.

If we take this claim seriously, then a structure-preserving map will turn out to be an isomorphism in the relevant category – for only isomorphisms preserve the complete ‘structural essence’ of a structured set. For instance, Bain’s example of the category whose objects are smooth manifolds and whose morphisms are diffeomorphisms is of this type. If this is really what Bain has in mind, then one inevitably ends up with a very limited and dull class of categories. But even if one relaxes this notion of ‘structure’ to mean ‘the structure that is preserved by the morphisms of the category, whatever they happen to be’, one still runs into trouble with G3.

We now turn to the premise G3. First, note that G3 (i) is false, as we now explain. It will be convenient to introduce a piece of standard terminology: a morphism from a terminal object to some object X is called a global element of X. And the question of whether an element of X can be expressed as a global element in the relevant category turns on the structure of the category in question. For instance, in the category Man with smooth manifolds as objects and smooth maps as morphisms, this question receives a positive answer: global elements are indeed in bijective correspondence with elements of a manifold. This is because the terminal object is the 0-dimensional manifold {0}, and so an element of a manifold M is a morphism {0} → M. But in many other categories, e.g. the category Grp, the answer is negative. As an example, consider that Grp has the trivial group 1 as its terminal object and so a morphism from 1 to a group G only picks out its identity and not its other elements. In order to obtain the other elements, one has to introduce the notion of a generalized element of X, viz. a morphism from some ‘standard object’ U into X. For instance, in Grp, one takes Z as the standard object U, and the generalized elements Z → G allow us to recover the ordinary elements of a group G.

Second, while G3 (ii) is certainly true, i.e. universal properties can be expressed purely in terms of morphisms, it is a further – and significant – question for the scope and applicability of this premise whether all (or even most) physical properties can be articulated as universal properties.

Hence we have seen that the categorically-informed opponent of (Objectless) need not accept these premises – there is a lot of room for debate about how exactly one should use category theory to conceptualize the notion of physical structure. But supposing that one does: is there a valid inference from G1–G3 to C? Bain himself notes that the plausibility of this inference trades on an ambiguity in what one means by ‘reference’ in C. If one merely means that such constructions eliminate explicit but not implicit reference to objects, then the argument is indeed valid. On the other hand, a defense of OSR requires the elimination of implicit reference to objects, and this is what the general argument fails to offer – it merely provides a translation scheme from statements involving elements (of sets) to statements involving morphisms between C-objects. So, the defender of objects can maintain that one is still implicitly quantifying over elements. 

Relationist and Substantivalist meet by the Isometric Cut in the Hole Argument

General-Relativity-Eddington-eclipse

To begin, the models of relativity theory are relativistic spacetimes, which are pairs (M,gab) consisting of a 4-manifold M and a smooth, Lorentz-signature metric gab. The metric represents geometrical facts about spacetime, such as the spatiotemporal distance along a curve, the volume of regions of spacetime, and the angles between vectors at a point. It also characterizes the motion of matter: the metric gab determines a unique torsion-free derivative operator ∇, which provides the standard of constancy in the equations of motion for matter. Meanwhile, geodesics of this derivative operator whose tangent vectors ξa satisfy gabξaξb > 0 are the possible trajectories for free massive test particles, in the absence of external forces. The distribution of matter in space and time determines the geometry of spacetime via Einstein’s equation, Rab − 1/2Rgab = 8πTab, where Tab is the energy-momentum tensor associated with any matter present, Rab is the Ricci tensor, and R = Raa. Thus, as in Yang-Mills theory, matter propagates through a curved space, the curvature of which depends on the distribution of matter in spacetime.

The most widely discussed topic in the philosophy of general relativity over the last thirty years has been the hole argument, which goes as follows. Fix some spacetime (M,gab), and consider some open set O ⊆ M with compact closure. For convenience, assume Tab = 0 everywhere. Now pick some diffeomorphism ψ : M → M such that ψ|M−O acts as the identity, but ψ|O is not the identity. This is sufficient to guarantee that ψ is a non-trivial automorphism of M. In general, ψ will not be an isometry, but one can always define a new spacetime (M, ψ(gab)) that is guaranteed to be isometric to (M,gab), with the isometry realized by ψ. This yields two relativistic spacetimes, both representing possible physical configurations, that agree on the value of the metric at every point outside of O, but in general disagree at points within O. This means that the metric outside of O, including at all points in the past of O, cannot determine the metric at a point p ∈ O. General relativity, as standardly presented, faces a pernicious form of indeterminism. To avoid this indeterminism, one must become a relationist and accept that “Leibniz equivalent”, i.e., isometric, spacetimes represent the same physical situations. The person who denies this latter view – and thus faces the indeterminism – is dubbed a manifold substantivalist.

One way of understanding the dialectical context of the hole argument is as a dispute concerning the correct notion of equivalence between relativistic spacetimes. The manifold substantivalist claims that isometric spacetimes are not equivalent, whereas the relationist claims that they are. In the present context, these views correspond to different choices of arrows for the categories of models of general relativity. The relationist would say that general relativity should be associated with the category GR1, whose objects are relativistic spacetimes and whose arrows are isometries. The manifold substantivalist, meanwhile, would claim that the right category is GR2, whose objects are again relativistic spacetimes, but which has only identity arrows. Clearly there is a functor F : GR2 → GR1 that acts as the identity on both objects and arrows and forgets only structure. Thus the manifold substantivalist posits more structure than the relationist.

Manifold substantivalism might seem puzzling—after all, we have said that a relativistic spacetime is a Lorentzian manifold (M,gab), and the theory of pseudo-Riemannian manifolds provides a perfectly good standard of equivalence for Lorentzian manifolds qua mathematical objects: namely, isometry. Indeed, while one may stipulate that the objects of GR2 are relativistic spacetimes, the arrows of the category do not reflect that choice. One way of charitably interpreting the manifold substantivalist is to say that in order to provide an adequate representation of all the physical facts, one actually needs more than a Lorentzian manifold. This extra structure might be something like a fixed collection of labels for the points of the manifold, encoding which point in physical spacetime is represented by a given point in the manifold. Isomorphisms would then need to preserve these labels, so spacetimes would have no non-trivial automorphisms. On this view, one might use Lorentzian manifolds, without the extra labels, for various purposes, but when one does so, one does not represent all of the facts one might (sometimes) care about.

In the context of the hole argument, isometries are sometimes described as the “gauge transformations” of relativity theory; they are then taken as evidence that general relativity has excess structure. One can expect to have excess structure in a formalism only if there are models of the theory that have the same representational capacities, but which are not isomorphic as mathematical objects. If we take models of GR to be Lorentzian manifolds, then that criterion is not met: isometries are precisely the isomorphisms of these mathematical objects, and so general relativity does not have excess structure.

This point may be made in another way. Motivated in part by the idea that the standard formalism has excess structure, a proposal to move to the alternative formalism of so-called Einstein algebras for general relativity is sought, arguing that Einstein algebras have less structure than relativistic spacetimes. In what follows, a smooth n−algebra A is an algebra isomorphic (as algebras) to the algebra C(M) of smooth real-valued functions on some smooth n−manifold, M. A derivation on A is an R-linear map ξ : A → A satisfying the Leibniz rule, ξ(ab) = aξ(b) + bξ(a). The space of derivations on A forms an A-module, Γ(A), elements of which are analogous to smooth vector fields on M. Likewise, one may define a dual module, Γ(A), of linear functionals on Γ(A). A metric, then, is a module isomorphism g : Γ(A) → Γ(A) that is symmetric in the sense that for any ξ,η ∈ Γ(A), g(ξ)(η) = g(η)(ξ). With some further work, one can capture a notion of signature of such metrics, exactly analogously to metrics on a manifold. An Einstein algebra, then, is a pair (A, g), where A is a smooth 4−algebra and g is a Lorentz signature metric.

Einstein algebras arguably provide a “relationist” formalism for general relativity, since one specifies a model by characterizing (algebraic) relations between possible states of matter, represented by scalar fields. It turns out that one may then reconstruct a unique relativistic spacetime, up to isometry, from these relations by representing an Einstein algebra as the algebra of functions on a smooth manifold. The question, though, is whether this formalism really eliminates structure. Let GR1 be as above, and define EA to be the category whose objects are Einstein algebras and whose arrows are algebra homomorphisms that preserve the metric g (in a way made precise by Rosenstock). Define a contravariant functor F : GR1 → EA that takes relativistic spacetimes (M,gab) to Einstein algebras (C(M),g), where g is determined by the action of gab on smooth vector fields on M, and takes isometries ψ : (M, gab) → (M′, g′ab) to algebra isomorphisms ψˆ : C(M′) → C(M), defined by ψˆ(a) = a ◦ ψ. Rosenstock et al. (2015) prove the following.

Proposition: F : GR1 → EA forgets nothing.