# The Affinity of Mirror Symmetry to Algebraic Geometry: Going Beyond Formalism  Even though formalism of homological mirror symmetry is an established case, what of other explanations of mirror symmetry which lie closer to classical differential and algebraic geometry? One way to tackle this is the so-called Strominger, Yau and Zaslow mirror symmetry or SYZ in short.

The central physical ingredient in this proposal is T-duality. To explain this, let us consider a superconformal sigma model with target space (M, g), and denote it (defined as a geometric functor, or as a set of correlation functions), as

CFT(M, g)

In physics, a duality is an equivalence

CFT(M, g) ≅ CFT(M′, g′)

which holds despite the fact that the underlying geometries (M,g) and (M′, g′) are not classically diffeomorphic.

T-duality is a duality which relates two CFT’s with toroidal target space, M ≅ M′ ≅ Td, but different metrics. In rough terms, the duality relates a “small” target space, with noncontractible cycles of length L < ls, with a “large” target space in which all such cycles have length L > ls.

This sort of relation is generic to dualities and follows from the following logic. If all length scales (lengths of cycles, curvature lengths, etc.) are greater than ls, string theory reduces to conventional geometry. Now, in conventional geometry, we know what it means for (M, g) and (M′, g′) to be non-isomorphic. Any modification to this notion must be associated with a breakdown of conventional geometry, which requires some length scale to be “sub-stringy,” with L < ls. To state T-duality precisely, let us first consider M = M′ = S1. We parameterise this with a coordinate X ∈ R making the identification X ∼ X + 2π. Consider a Euclidean metric gR given by ds2 = R2dX2. The real parameter R is usually called the “radius” from the obvious embedding in R2. This manifold is Ricci-flat and thus the sigma model with this target space is a conformal field theory, the “c = 1 boson.” Let us furthermore set the string scale ls = 1. With this, we attain a complete physical equivalence.

CFT(S1, gR) ≅ CFT(S1, g1/R)

Thus these two target spaces are indistinguishable from the point of view of string theory.

Just to give a physical picture for what this means, suppose for sake of discussion that superstring theory describes our universe, and thus that in some sense there must be six extra spatial dimensions. Suppose further that we had evidence that the extra dimensions factorized topologically and metrically as K5 × S1; then it would make sense to ask: What is the radius R of this S1 in our universe? In principle this could be measured by producing sufficiently energetic particles (so-called “Kaluza-Klein modes”), or perhaps measuring deviations from Newton’s inverse square law of gravity at distances L ∼ R. In string theory, T-duality implies that R ≥ ls, because any theory with R < ls is equivalent to another theory with R > ls. Thus we have a nontrivial relation between two (in principle) observable quantities, R and ls, which one might imagine testing experimentally. Let us now consider the theory CFT(Td, g), where Td is the d-dimensional torus, with coordinates Xi parameterising Rd/2πZd, and a constant metric tensor gij. Then there is a complete physical equivalence

CFT(Td, g) ≅ CFT(Td, g−1)

In fact this is just one element of a discrete group of T-duality symmetries, generated by T-dualities along one-cycles, and large diffeomorphisms (those not continuously connected to the identity). The complete group is isomorphic to SO(d, d; Z).

While very different from conventional geometry, T-duality has a simple intuitive explanation. This starts with the observation that the possible embeddings of a string into X can be classified by the fundamental group π1(X). Strings representing non-trivial homotopy classes are usually referred to as “winding states.” Furthermore, since strings interact by interconnecting at points, the group structure on π1 provided by concatenation of based loops is meaningful and is respected by interactions in the string theory. Now π1(Td) ≅ Zd, as an abelian group, referred to as the group of “winding numbers”.

Of course, there is another Zd we could bring into the discussion, the Pontryagin dual of the U(1)d of which Td is an affinization. An element of this group is referred to physically as a “momentum,” as it is the eigenvalue of a translation operator on Td. Again, this group structure is respected by the interactions. These two group structures, momentum and winding, can be summarized in the statement that the full closed string algebra contains the group algebra C[Zd] ⊕ C[Zd].

In essence, the point of T-duality is that if we quantize the string on a sufficiently small target space, the roles of momentum and winding will be interchanged. But the main point can be seen by bringing in some elementary spectral geometry. Besides the algebra structure, another invariant of a conformal field theory is the spectrum of its Hamiltonian H (technically, the Virasoro operator L0 + L ̄0). This Hamiltonian can be thought of as an analog of the standard Laplacian ∆g on functions on X, and its spectrum on Td with metric g is

Spec ∆= {∑i,j=1d gijpipj; pi ∈ Zd}

On the other hand, the energy of a winding string is (intuitively) a function of its length. On our torus, a geodesic with winding number w ∈ Zd has length squared

L2 = ∑i,j=1d gijwiwj

Now, the only string theory input we need to bring in is that the total Hamiltonian contains both terms,

H = ∆g + L2 + · · ·

where the extra terms … express the energy of excited (or “oscillator”) modes of the string. Then, the inversion g → g−1, combined with the interchange p ↔ w, leaves the spectrum of H invariant. This is T-duality.

There is a simple generalization of the above to the case with a non-zero B-field on the torus satisfying dB = 0. In this case, since B is a constant antisymmetric tensor, we can label CFT’s by the matrix g + B. Now, the basic T-duality relation becomes

CFT(Td, g + B) ≅ CFT(Td, (g + B)−1)

Another generalization, which is considerably more subtle, is to do T-duality in families, or fiberwise T-duality. The same arguments can be made, and would become precise in the limit that the metric on the fibers varies on length scales far greater than ls, and has curvature lengths far greater than ls. This is sometimes called the “adiabatic limit” in physics. While this is a very restrictive assumption, there are more heuristic physical arguments that T-duality should hold more generally, with corrections to the relations proportional to curvatures ls2R and derivatives ls∂ of the fiber metric, both in perturbation theory and from world-sheet instantons.

# Killing Fields

Let κa be a smooth field on our background spacetime (M, gab). κa is said to be a Killing field if its associated local flow maps Γs are all isometries or, equivalently, if £κ gab = 0. The latter condition can also be expressed as ∇(aκb) = 0.

Any number of standard symmetry conditions—local versions of them, at least can be cast as claims about the existence of Killing fields. Local, because killing fields need not be complete, and their associated flow maps need not be defined globally.

(M, gab) is stationary if it has a Killing field that is everywhere timelike.

(M, gab) is static if it has a Killing field that is everywhere timelike and locally hypersurface orthogonal.

(M, gab) is homogeneous if its Killing fields, at every point of M, span the tangent space.

In a stationary spacetime there is, at least locally, a “timelike flow” that preserves all spacetime distances. But the flow can exhibit rotation. Think of a whirlpool. It is the latter possibility that is ruled out when one passes to a static spacetime. For example, Gödel spacetime, is stationary but not static.

Let κa be a Killing field in an arbitrary spacetime (M, gab) (not necessarily Minkowski spacetime), and let γ : I → M be a smooth, future-directed, timelike curve, with unit tangent field ξa. We take its image to represent the worldline of a point particle with mass m > 0. Consider the quantity J = (Paκa), where Pa = mξa is the four-momentum of the particle. It certainly need not be constant on γ[I]. But it will be if γ is a geodesic. For in that case, ξnnξa = 0 and hence

ξnnJ = m(κa ξnnξa + ξnξanκa) = mξnξa ∇(nκa) = 0

Thus, J is constant along the worldlines of free particles of positive mass. We refer to J as the conserved quantity associated with κa. If κa is timelike, we call J the energy of the particle (associated with κa). If it is spacelike, and if its associated flow maps resemble translations, we call J the linear momentum of the particle (associated with κa). Finally, if κa is spacelike, and if its associated flow maps resemble rotations, then we call J the angular momentum of the particle (associated with κa).

It is useful to keep in mind a certain picture that helps one “see” why the angular momentum of free particles (to take that example) is conserved. It involves an analogue of angular momentum in Euclidean plane geometry. Figure below shows a rotational Killing field κa in the Euclidean plane, the image of a geodesic (i.e., a line) L, and the tangent field ξa to the geodesic. Consider the quantity J = ξaκa, i.e., the inner product of ξa with κa – along L, and we can better visualize the assertion. Figure: κa is a rotational Killing field. (It is everywhere orthogonal to a circle radius, and is proportional to it in length.) ξa is a tangent vector field of constant length on the line L. The inner product between them is constant. (Equivalently, the length of the projection of κa onto the line is constant.)

Let us temporarily drop indices and write κ·ξ as one would in ordinary Euclidean vector calculus (rather than ξaκa). Let p be the point on L that is closest to the center point where κ vanishes. At that point, κ is parallel to ξ. As one moves away from p along L, in either direction, the length ∥κ∥ of κ grows, but the angle ∠(κ,ξ) between the vectors increases as well. It should seem at least plausible from the picture that the length of the projection of κ onto the line is constant and, hence, that the inner product κ·ξ = cos(∠(κ , ξ )) ∥κ ∥ ∥ξ ∥ is constant.

That is how to think about the conservation of angular momentum for free particles in relativity theory. It does not matter that in the latter context we are dealing with a Lorentzian metric and allowing for curvature. The claim is still that a certain inner product of vector fields remains constant along a geodesic, and we can still think of that constancy as arising from a compensatory balance of two factors.

Let us now turn to the second type of conserved quantity, the one that is an attribute of extended bodies. Let κa be an arbitrary Killing field, and let Tab be the energy-momentum field associated with some matter field. Assume it satisfies the conservation condition (∇aTab = 0). Then (Tabκb) is divergence free:

a(Tabκb) = κbaTab + Tabaκb = Tab∇(aκb) = 0

(The second equality follows from the conservation condition and the symmetry of Tab; the third follows from the fact that κa is a Killing field.) It is natural, then, to apply Stokes’s theorem to the vector field (Tabκb). Consider a bounded system with aggregate energy-momentum field Tab in an otherwise empty universe. Then there exists a (possibly huge) timelike world tube such that Tab vanishes outside the tube (and vanishes on its boundary).

Let S1 and S2 be (non-intersecting) spacelike hypersurfaces that cut the tube as in the figure below, and let N be the segment of the tube falling between them (with boundaries included). Figure: The integrated energy (relative to a background timelike Killing field) over the intersection of the world tube with a spacelike hypersurface is independent of the choice of hypersurface.

By Stokes’s theorem,

S2(Tabκb)dSa – ∫S1(Tabκb)dSa = ∫S2∩∂N(Tabκb)dSa – ∫S1∩∂N(Tabκb)dSa

= ∫∂N(Tabκb)dSa = ∫Na(Tabκb)dV = 0

Thus, the integral ∫S(Tabκb)dSa is independent of the choice of spacelike hypersurface S intersecting the world tube, and is, in this sense, a conserved quantity (construed as an attribute of the system confined to the tube). An “early” intersection yields the same value as a “late” one. Again, the character of the background Killing field κa determines our description of the conserved quantity in question. If κa is timelike, we take ∫S(Tabκb)dSa to be the aggregate energy of the system (associated with κa). And so forth.

# 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.

# Ricci-flow as an “intrinsic-Ricci-flat” Space-time.

A Ricci flow solution {(Mm, g(t)), t ∈ I ⊂ R} is a smooth family of metrics satisfying the evolution equation

∂/∂t g = −2Rc —– (1)

where Mm is a complete manifold of dimension m. We assume that supM |Rm|g(t) < ∞ for each time t ∈ I. This condition holds automatically if M is a closed manifold. It is very often to put an extra term on the right hand side of (1) to obtain the following rescaled Ricci flow

∂/∂t g = −2 {Rc + λ(t)g} —– (2)

where λ(t) is a function depending only on time. Typically, λ(t) is chosen as the average of the scalar curvature, i.e. , 1/m ∱Rdv or some fixed constant independent of time. In the case that M is closed and λ(t) = 1/m ∱Rdv, the flow is called the normalized Ricci flow. Starting from a positive Ricci curvature metric on a 3-manifold, Richard Hamilton showed that the normalized Ricci flow exists forever and converges to a space form metric. Hamilton developed the maximum principle for tensors to study the Ricci flow initiated from some metric with positive curvature conditions. For metrics without positive curvature condition, the study of Ricci flow was profoundly affected by the celebrated work of Grisha Perelman. He introduced new tools, i.e., the entropy functionals μ, ν, the reduced distance and the reduced volume, to investigate the behavior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricci flow with surgery, leading to solutions of the Poincaré conjecture and Thurston’s geometrization conjecture.

In the general theory of the Ricci flow developed by Perelman in, the entropy functionals μ and ν are of essential importance. Perelman discovered the monotonicity of such functionals and applied them to prove the no-local-collapsing theorem, which removes the stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using such monotonicity, he further proved the pseudo-locality theorem, which claims that the Ricci flow can not quickly turn an almost Euclidean region into a very curved one, no matter what happens far away. Besides the functionals, Perelman also introduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-time admits a remarkable comparison geometry picture, which is the foundation of his “local”-version of the no-local-collapsing theorem. Each of the tools has its own advantages and shortcomings. The functionals μ and ν have the advantage that their definitions only require the information for each time slice (M, g(t)) of the flow. However, they are global invariants of the underlying manifold (M, g(t)). It is not convenient to apply them to study the local behavior around a given point x. Correspondingly, the reduced volume and the reduced distance reflect the natural comparison geometry picture of the space-time. Around a base point (x, t), the reduced volume and the reduced distance are closely related to the “local” geometry of (x, t). Unfortunately, it is the space-time “local”, rather than the Riemannian geometry “local” that is concerned by the reduced volume and reduced geodesic. In order to apply them, some extra conditions of the space-time neighborhood of (x, t) are usually required. However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to have some new tools to balance the advantages of the reduced volume, the reduced distance and the entropy functionals.

Let (Mm, g) be a complete Ricci-flat manifold, x0 is a point on M such that d(x0, x) < A. Suppose the ball B(x0, r0) is A−1−non-collapsed, i.e., r−m0|B(x0, r0)| ≥ A−1, can we obtain uniform non-collapsing for the ball B(x, r), whenever 0 < r < r0 and d(x, x0) < Ar0? This question can be answered easily by applying triangle inequalities and Bishop-Gromov volume comparison theorems. In particular, there exists a κ = κ(m, A) ≥ 3−mA−m−1 such that B(x, r) is κ-non-collapsed, i.e., r−m|B(x, r)| ≥ κ. Consequently, there is an estimate of propagation speed of non-collapsing constant on the manifold M. This is illustrated by Figure We now regard (M, g) as a trivial space-time {(M, g(t)), −∞ < t < ∞} such that g(t) ≡ g. Clearly, g(t) is a static Ricci flow solution by the Ricci-flatness of g. Then the above estimate can be explained as the propagation of volume non-collapsing constant on the space-time. However, in a more intrinsic way, it can also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced volume. On the Ricci flat space-time, Perelman’s reduced volume has a special formula

V((x, t)r2) = (4π)-m/2 r-m ∫M e-d2(y, x)/4r2 dvy —– (3)

which is almost the volume ratio of Bg(t)(x, r). On a general Ricci flow solution, the reduced volume is also well-defined and has monotonicity with respect to the parameter r2, if one replace d2(y, x)/4r2 in the above formula by the reduced distance l((x, t), (y, t − r2)). Therefore, via the comparison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat” space-time. However, the disadvantage of the reduced volume explanation is also clear: it requires the curvature estimate in a whole space-time neighborhood around the point (x, t), rather than the scalar curvature estimate of a single time slice t.

# Geometric Structure, Causation, and Instrumental Rip-Offs, or, How Does a Physicist Read Off the Physical Ontology From the Mathematical Apparatus? The benefits of the various structuralist approaches in the philosophy of mathematics is that it allows both the mathematical realist and anti-realist to use mathematical structures without obligating a Platonism about mathematical objects, such as numbers – one can simply accept that, say, numbers exist as places in a larger structure, like the natural number system, rather than as some sort of independently existing, transcendent entities. Accordingly, a variation on a well-known mathematical structure, such as exchanging the natural numbers “3” and “7”, does not create a new structure, but merely gives the same structure “relabeled” (with “7” now playing the role of “3”, and visa-verse). This structuralist tactic is familiar to spacetime theorists, for not only has it been adopted by substantivalists to undermine an ontological commitment to the independent existence of the manifold points of M, but it is tacitly contained in all relational theories, since they would count the initial embeddings of all material objects and their relations in a spacetime as isomorphic.

A critical question remains, however: Since spacetime structure is geometric structure, how does the Structural Realism (SR) approach to spacetime differ in general from mathematical structuralism? Is the theory just mathematical structuralism as it pertains to geometry (or, more accurately, differential geometry), rather than arithmetic or the natural number series? While it may sound counter-intuitive, the SR theorist should answer this question in the affirmative – the reason being, quite simply, that the puzzle of how mathematical spacetime structures apply to reality, or are exemplified in the real world, is identical to the problem of how all mathematical structures are exemplified in the real world. Philosophical theories of mathematics, especially nominalist theories, commonly take as their starting point the fact that certain mathematical structures are exemplified in our common experience, while other are excluded. To take a simple example, a large collection of coins can exemplify the standard algebraic structure that includes commutative multiplication (e.g., 2 x 3 = 3 x 2), but not the more limited structure associated with, say, Hamilton’s quaternion algebra (where multiplication is non-commutative; 2 x 3 ≠ 3 x 2). In short, not all mathematical structures find real-world exemplars (although, for the minimal nominalists, these structures can be given a modal construction). The same holds for spacetime theories: empirical evidence currently favors the mathematical structures utilized in General Theory of Relativity, such that the physical world exemplifies, say, g, but a host of other geometric structures, such as the flat Newtonian metric, h, are not exemplified.

The critic will likely respond that there is substantial difference between the mathematical structures that appear in physical theories and the mathematics relevant to everyday experience. For the former, and not the latter, the mathematical structures will vary along with the postulated physical forces and laws; and this explains why there are a number of competing spacetime theories, and thus different mathematical structures, compatible with the same evidence: in Poincaré fashion, Newtonian rivals to GTR can still employ h as long as special distorting forces are introduced. Yet, underdetermination can plague even simple arithmetical experience, a fact well known in the philosophy of mathematics and in measurement theory. For example, in Charles Chihara, an assessment of the empiricist interpretation of mathematics prompts the following conclusion: “the fact that adding 5 gallons of alcohol to 2 gallons of water does not yield 7 gallons of liquid does not refute any law of logic or arithmetic [“5+2=7”] but only a mistaken physical assumption about the conservation of liquids when mixed”. While obviously true, Chihara could have also mentioned that, in order to capture our common-sense intuitions about mathematics, the application of the mathematical structure in such cases requires coordination with a physical measuring convention that preserves the identity of each individual entity, or unit, both before and after the mixing. In the mixing experiment, perhaps atoms should serve as the objects coordinated to the natural number series, since the stability of individual atoms would prevent the sort of blurring together of the individuals (“gallon of liquid”) that led to the arithmetically deviant results. By choosing a different coordination, the mixing experiment can thus be judged to uphold, or exemplify, the statement “5+2=7”. What all of this helps to show is that mathematics, for both complex geometrical spacetime structures and simple non-geometrical structures, cannot be empirically applied without stipulating physical hypotheses and/or conventions about the objects that model the mathematics. Consequently, as regards real world applications, there is no difference in kind between the mathematical structures that are exemplified in spacetime physics and in everyday observation; rather, they only differ in their degree of abstractness and the sophistication of the physical hypotheses or conventions required for their application. Both in the simple mathematical case and in the spacetime case, moreover, the decision to adopt a particular convention or hypothesis is normally based on a judgment of its overall viability and consistency with our total scientific view (a.k.a., the scientific method): we do not countenance a world where macroscopic objects can, against the known laws of physics, lose their identity by blending into one another (as in the addition example), nor do we sanction otherwise undetectable universal forces simply for the sake of saving a cherished metric.

Another significant shared feature of spacetime and mathematical structure is the apparent absence of causal powers or effects, even though the relevant structures seem to play some sort of “explanatory role” in the physical phenomena. To be more precise, consider the example of an “arithmetically-challenged” consumer who lacks an adequate grasp of addition: if he were to ask for an explanation of the event of adding five coins to another seven, and why it resulted in twelve, one could simply respond by stating, “5+7=12”, which is an “explanation” of sorts, although not in the scientific sense. On the whole, philosophers since Plato have found it difficult to offer a satisfactory account of the relationship between general mathematical structures (arithmetic/”5+7=12”) and the physical manifestations of those structures (the outcome of the coin adding). As succinctly put by Michael Liston:

Why should appeals to mathematical objects [numbers, etc.] whose very nature is non-physical make any contribution to sound inferences whose conclusions apply to physical objects?

One response to the question can be comfortably dismissed, nevertheless: mathematical structures did not cause the outcome of the coin adding, for this would seem to imply that numbers (or “5+7=12”) somehow had a mysterious, platonic influence over the course of material affairs.

In the context of the spacetime ontology debate, there has been a corresponding reluctance on the part of both sophisticated substantivalists and (R2, the rejection of substantivalist) relationists to explain how space and time differentiate the inertial and non-inertial motions of bodies; and, in particular, what role spacetime plays in the origins of non-inertial force effects. Returning once more to our universe with a single rotating body, and assuming that no other forces or causes, it would be somewhat peculiar to claim that the causal agent responsible for the observed force effects of the motion is either substantival spacetime or the relative motions of bodies (or, more accurately, the motion of bodies relative to a privileged reference frame, or possible trajectories, etc.). Yet, since it is the motion of the body relative to either substantival space, other bodies/fields, privileged frames, possible trajectories, etc., that explains (or identifies, defines) the presence of the non-inertial force effects of the acceleration of the lone rotating body, both theories are therefore in serious need of an explanation of the relationship between space and these force effects. The strict (R1) relationists face a different, if not less daunting, task; for they must reinterpret the standard formulations of, say, Newtonian theory in such a way that the rotation of our lone body in empty space, or the rotation of the entire universe, is not possible. To accomplish this goal, the (R1) relationist must draw upon different mathematical resources and adopt various physical assumptions that may, or may not, ultimately conflict with empirical evidence: for example, they must stipulate that the angular momentum of the entire universe is 0.

All participants in the spacetime ontology debate are confronted with the nagging puzzle of understanding the relationship between, on the one hand, the empirical behavior of bodies, especially the non-inertial forces, and, on the other hand, the apparently non-empirical, mathematical properties of the spacetime structure that are somehow inextricably involved in any adequate explanation of those non-inertial forces – namely, for the substantivalists and (R2) relationists, the affine structure,  that lays down the geodesic paths of inertially moving bodies. The task of explaining this connection between the empirical and abstract mathematical or quantitative aspects of spacetime theories is thus identical to elucidating the mathematical problem of how numbers relate to experience (e.g., how “5+7=12” figures in our experience of adding coins). Likewise, there exists a parallel in the fact that most substantivalists and (R2) relationists seem to shy away from positing a direct causal connection between material bodies and space (or privileged frames, possible trajectories, etc.) in order to account for non-inertial force effects, just as a mathematical realist would recoil from ascribing causal powers to numbers so as to explain our common experience of adding and subtracting.

An insight into the non-causal, mathematical role of spacetime structures can also be of use to the (R2) relationist in defending against the charge of instrumentalism, as, for instance, in deflecting Earman’s criticisms of Sklar’s “absolute acceleration” concept. Conceived as a monadic property of bodies, Sklar’s absolute acceleration does not accept the common understanding of acceleration as a species of relative motion, whether that motion is relative to substantival space, other bodies, or privileged reference frames. Earman’s objection to this strategy centers upon the utilization of spacetime structures in describing the primitive acceleration property: “it remains magic that the representative [of Sklar’s absolute acceleration] is neo-Newtonian acceleration

d2xi/dt2 + Γijk (dxj/dt)(dxk/dt) —– (1)

[i.e., the covariant derivative, or ∇ in coordinate form]”. Ultimately, Earman’s critique of Sklar’s (R2) relationism would seem to cut against all sophisticated (R2) hypotheses, for he seems to regard the exercise of these richer spacetime structures, like ∇, as tacitly endorsing the absolute/substantivalist side of the dispute:

..the Newtonian apparatus can be used to make the predictions and afterwards discarded as a convenient fiction, but this ploy is hardly distinguishable from instrumentalism, which, taken to its logical conclusion, trivializes the absolute-relationist debate.

The weakness of Earman’s argument should be readily apparent—since, to put it bluntly, does the equivalent use of mathematical statements, such as “5+7=12”, likewise obligate the mathematician to accept a realist conception of numbers (such that they exist independently of all exemplifying systems)? Yet, if the straightforward employment of mathematics does not entail either a realist or nominalist theory of mathematics (as most mathematicians would likely agree), then why must the equivalent use of the geometric structures of spacetime physics, e.g., ∇ require a substantivalist conception of ∇ as opposed to an (R2) relationist conception of ∇? Put differently, does a substantivalist commitment to whose overall function is to determine the straight-line trajectories of Neo-Newtonian spacetime, also necessitate a substantivalist commitment to its components, such as the vector d/dt along with its limiting process and mapping into ℜ? In short, how does a physicist read off the physical ontology from the mathematical apparatus? A non-instrumental interpretation of some component of the theory’s quantitative structure is often justified if that component can be given a plausible causal role (as in subatomic physics)—but, as noted above, ∇ does not appear to cause anything in spacetime theories. All told, Earman’s argument may prove too much, for if we accept his reasoning at face value, then the introduction of any mathematical or quantitative device that is useful in describing or measuring physical events would saddle the ontology with a bizarre type of entity (e.g., gross national product, average household family, etc.). A nice example of a geometric structure that provides a similarly useful explanatory function, but whose substantive existence we would be inclined to reject as well, is provided by Dieks’ example of a three-dimensional colour solid:

Different colours and their shades can be represented in various ways; one way is as points on a 3-dimensional colour solid. But the proposal to regard this ‘colour space’ as something substantive, needed to ground the concept of colour, would be absurd.

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

# Speculative String-Cosmologies as Geodesically Incomplete. Penrose diagrams of de Sitter space in the flat (left) and static (right) slicings that each cover only part of the whole de Sitter space, and that are both geodesically incomplete.

Alan Guth, Alvin Borde and Alexander Vilenkin have argued that within the framework of a future-eternal inflationary multiverse, as well as some more speculative string-cosmologies, all worldlines are geodesically incomplete and, thus, the multiverse has to have a beginning. Unfortunately, if future-eternal inflation is true, all “hypotheses about the ultimate beginning of the universe would become totally divorced from any observable consequences. Since our own pocket universe would be equally likely to lie anywhere on the infinite tree of universes produced by eternal inflation, we would expect to find ourselves arbitrarily far from the beginning. The infinite inflating network would presumably approach some kind of steady state, losing all memory of how it started […] Thus, there would be no way of relating the properties of the ultimate origin to anything that we might observe in today’s universe.” (Guth).

On the other hand, Andrei Linde has argued that the multiverse could be past-eternal, because either all single world lines might have to start somewhere, but not the whole bundle of them (Linde), or there could even exist some (albeit strange) space-times with single past-eternal world lines.

This issue is not settled, and even in those scenarios a global arrow of time may not necessarily exist. However, there are other frameworks possible – and they have even already been developed to some extent, where a future-eternal inflationary multiverse is both not past-eternal and beginningless but arise from some primordial vacuum which is macroscopically time-less. Thus, again, the beginning of some classical space-times is not equivalent with the beginning of everything.

We can even imagine that there is no multiverse, but that the whole (perhaps finite) universe – our universe – once was in a steady state without any macroscopic arrows of time but, due to a statistical fluctuation above a certain threshold value, started to expand  – or to contract, bounce and expand – as a whole and acquired an arrow of time. In such a case the above-mentioned reply, which was based on the spatial distinction of a beginning of some parts of the world and the eternity of the world as a whole, would collapse.

Nevertheless it is necessary to distinguish between the different notions and extensions of the term “universe”. In the simplest case, Kant’s antinomy might be based on an ambiguity of the term “world” (i.e. the difference between “universe” and “multiverse”), but it does not need to; and it was not assumed here that it necessarily does. The temporal part of Kant’s first antinomy was purely about the question whether the macroscopic arrow of time is past-eternal or not. And if it is not past-eternal this does not mean that time and hence the world has an absolute beginning in every respect – it is still possible that there was or is a world with some underlying microscopic time. (By the way, one can also imagine that, even if our arrow of time is past- and/or future-eternal, there might exist “timeless islands” someday: for instance isolated black holes if they would not ultimately radiate away due to quantum effects, or empty static universes if they could split off of our space-time.)

Of course it is possible that firstly a natural principle of plentitude is realized and different multiverses (sets of universes) exist totally independent from each other, and secondly that some of them are truly past-eternal while others have an absolute beginning and others have only local starting points of local arrows of time as it was suggested here. If so, we might not be able to tell in what kind we live in. And this would be irrelevant in the end, because then every possible world is actual and probably exists infinitely often. But we do not know whether such an extreme principle of plentitude does apply or if cosmology is ultimately just and only a matter of pure logical consistency, allowing us finally to calculate the complete architecture of the world by armchair-reasoning.