Time is the Simplest (and Strongest) Thing

“Time, I’ll tell you about time. Time is the simplest thing.”

Clifford Simak, The Fisherman

Craig Callender

Philosophy, UCSD

ccallender@ucsd.edu

IntroductionWhat makes time different than space? Once a central question in metaphysics, this question has not been treated kindly by recent history. Why?

a. Methodological. One searched for statements S whose truth-values or meanings were not invariant under substitution of temporal and spatial terms. Finding {S}, one then consulted one’s intuitions to discern the difference between space and time. Compare: what makes protons different than neutrons? Science, not ordinary language and intuitions, gives the answer; and it’s not clear that there is the difference between the two.

b. Physical. Relativity seems to remove some of the motivation for the question and also obscure it. The difference between “spatialized” non-fundamental time and non-fundamental space doesn’t sound like it will yield unexpected riches. Furthermore, the question seems to presuppose that some features are intrinsic to time, yet relativity makes many features once considered intrinsic dependent on the contingent distribution of matter-energy--and so extrinsic.

Once the blows to the question are identified, they are easily parried.

Physical. There are still intrinsic metrical and topological features of relativistic spacetimes. Furthermore, relativity is not the only theory. To motivation, relativity still draws sharp and important distinctions between the timelike and spacelike directions on the manifold of events. And looking at physics as a whole, few eqs (e.g., uxx=utt) are invariant under a transformation of spatial and temporal variables--and no important ones. If modern physics is “spatializing” time, as Bergson charged, it’s going about the job awfully slowly.

Methodology. We need two changes. One, the data should be scientific, not linguistic. Second, we ought to admit that the goal of previous research may not exist: there may not be the difference between space and time. That doesn’t mean, however, that some differences aren’t more central and important to the concept of time than others.

Here I take up this rehabilitated project and propose a novel and I hope important distinction between space and time. Along the way, I hope we’ll learn something about the relationship among various temporal features too. At a time when researchers in QG propose speculative theories with no time in it at all, a better understanding of time is all the more important--if only to see what is lost by its potential absence.

Plan

1. Other Theories

2. Differences Between Time and Space in Contemporary Physics

3. General Proposal

4. Illustration

5. Argument for the Proposal

6. Conclusion: What is Time?

1. Other Theories • Metaphysics of time. Time is irreducibly tensed, flowing, becoming,

passing or branching; space is not. E.g., presentism, becoming, tenses, flow, etc. See, e.g., my “Shedding Light on Time” and “Time’s Ontic Voltage”

• Causal Theories of Time 1. Temporal relations are defined in terms of empirically accessible causal relations. E.g., Carnap, Reichenbach, Grünbaum, van Fraassen.– Founders on objections of detail and motivation

• Causal Theories of Time 2. Temporal relations are defined in terms of primitive causal relation. E.g., Mellor, Tooley.– Causation is primitive; obscure

• Laws of Nature. The laws of nature single out one dimension over the others in some way. E.g., Sider, Loewer, Skow.– If the laws primitive, then same objection as above.

• Skow 2005: “any direction …in which the laws govern the evolution of the world is a timelike direction.”

• He is then at pains to deny that imagined laws governing in a spacelike direction are really laws--to the extent that he denies theories of lawhood that might classify such generalizations as laws. But one doesn’t even need imaginary cases. E.g., Pauli exclusion principle. E.g., the 10 vacuum EFE separate into 6 “evolution” eqs Gij=0 and 4 “constraint” eqs, G00=0 and G0

i=0, with i=1,2,3. To decree that 4 of the 10 are not laws strikes me as unacceptable. E.g. two of Maxwell’s equations.

• Still, if sense can be made of the idea of the laws “preferring” time, I find the general idea attractive.

3. Time in Contemporary PhysicsPhysics is not invariant under a change of spatial and temporal directions. But writing a list of all such eqs would not be edifying. Instead, let’s think of those features commonly attributed to time but not space by our spacetime theories, and lump all the others that treat the two differently under the label “dynamics.” Lots of properties possibly “essential” to time, e.g., it being an ordering relation, will thus be ruled out b/c space also has these properties.

Inspection leads to at least four major differences:

Dynamics Temporal Direction Minus Sign Dimension

Orientable,HausdorfConnected, ordering…

Metrical Difference• The metrical structure of spacetime is arguably its most

fundamental and central feature. Both relativistic and classical metrics distinguish time from space. In Newton-Cartan spacetime there are two metrics, h, with signature (1,1,1,0) and t, with signature (0,0,0,1). Because it will play a role later, let’s focus on the relativistic metric.

“the world of Minkowski expresses the peculiarity of the time dimension mathematically by prefixing a minus sign to the time expression in the basic metrical formulae.” (Reichenbach, 112)

• In relativity, there is one Lorentzian metric. E.g., in Minkowski spacetime:

g = (dx1)2 + (dx2)2 + (dx3)2 - (dx4)2

• There are other coordinate systems that do not produce this asymmetry. Use “lightcone coordinates” as in string theory, or let =icx4. And of course, it’s conventional whether one uses (+++-) or (---+)--for this reason Skow 2005 discounts the importance of this feature.

Metrical Difference• But the signature of a metric is a geometrical invariant. Given g, we can find an orthonormal basis v1…vn of the tangent space at each point e of M. Let the number of basis vectors with g(v,v)=+1 be p and the number with g(v,v)=-1 be q. Then the metric has signature (p,q). In relativity, we assume the metric is nondegenerate, so p+q=n, where n=dim M. If M is connected, and g non-deg and continuous, the signature is an invariant.

•Positive Definite/Riemannian metric (manifold): the signature of g is (n,0) or (0,n)

•Lorentzian metric (manifold): the signature of g is (n-1,1) or (1, n-1)

•Note that for any semi-Riemannian metric g, a vector v is spacelike*, timelike* or null* depending on whether g(v,v) is positive, negative or zero.

Signature-Changing Spacetimes

• One can relax assumptions on g and have signature-changing spacetimes. These are generalizations of relativity).

• To get signature change, one

needs g to be either non-degenerate or discontinuous. (See, e.g, Dray, Ellis, Hellaby, Manague “Gravity and Signature Change”, 1996)

Simple example:

ds2=tdt2+a(t)2dx2

t

x

t=0=Σ

Lorentzian

Riemannian

Dimensionality

• In pre-classical, classical and relativistic physics, there is one time dimension. Classically, we can see this by simply grabbing the set of instants and showing that it forms a continuum under the earlier than or sim with relation, and that this relation determines the open sets that form a basis for the topological structure.

• What does this mean in relativity? Consider a point p on a timelike curve and a 4-velocity field va. Take a vector wa at p. Then wa can be composed into components parallel and orthogonal to va. The set of orthogonal (parallel) vectors forms a 3-dim (1-dim) subspace in the tangent space Mp at p.

Time

time

space

In speculative physics and speculative philosophy, there are models with more than one time dimension. In philosophy, Thompson 1965 and MacBeath 1993; in physics, (4,2) brane worlds, (11,2) F-theory, (3,3) Cole, and more.

But there are pressures against this possibility, and if my theory is right, it will explain these pressures. First, regarding the philosophical stories by MacBeath and Thompson, wherein it is allegedly plausible to posit two dimensions, notice that these very scenarios already happen in real life! MacBeath’s “thought experiment” is isomorphic to the situation with mesons entering the atmosphere. Yet science sticks with one time.

In physics all hell breaks loose with >1 timelike dimensions. ‘Hell’ means stability problems (Dorling 1967), causality and probability violations, and even observable causality and probability violations (Yndurain 1991: no compact timelike extra dim if their size is even 1/10th the Planck radius)

Direction

• Many equations in physics are ~TRI; few ~PRI. Equations not invariant under t -t but invariant under xi -xi single out time in some sense.

• But, arguably, directionality is a feature of processes in time, not time itself (e.g., Grünbaum, Horwich). Virtually all the fundamental equations of physics are TRI; directionality emerges at the macro-level from the behavior of ensembles with special initial conditions. Where this isn’t true, say, for neutral kaon decay, we can plausibly say that although it provides a difference between space and time, it isn’t a central and important over and above the general dynamical differences.

• That said, I want to acknowledge that temporal directionality is a central part of our concept of time, and that many have posited temporally asymmetric fields on space-time, or as part of the spacetime structure, in response to this centrality.

Matter fields

• All of the fundamental matter fields evolve differently in space than they do time.

• E.g.,

22

2

2

2

2

2

2

2

mzyxt

3. The TheoryMotivation. Recall from section 1 that I said it would be useful if the laws of nature approach could be given an empiricist slant. The best empiricist theory I know is that of MRL. The idea is:

Consider various deductive systems, each of which makes only true claims about what exists. The BEST SYSTEM is the deductive system that best balances simplicity and strength (and if probabilistic, also fit). Simplicity is measured with respect to a language that contains a primitive predicate for each natural property. Strength is informativeness about matters of particular fact.

And indeed, Loewer recently writes,

“In fact one might go so far as to say that what distinguishes the temporal dimension from the spatial ones is that it is the dimension picked out by the BEST THEORY for special treatment; in other words the distribution of fundamental properties is laid out in space-time in such a way that the theory that best combines simplicity and informativeness picks one of these dimensions for writing down equations that informatively describe that distribution. These remarks need more development and defense -especially in view of relativistic conceptions of space-time- then I will get into here but it is suggestive of the constructive potential of Lewis’ conception of laws.”

x0

x1

Fig1. Symmetric Two-dim World Fig2. Expanding FRW Model

The cosmological principle holds, but not the perfect cosmological principle

Isotropic

homogeneous

MRL Time?• The idea is vague as it stands and in need of

development. The fundamental concepts of MRL are murky, and it’s not clear how all of this can be translated into the setting of contemporary theories.

• But (a) arguably it answers our question without resort to a primitive and (b) the half about simplicity already fits with an attractive view of time, the idea that “time is the great simplifier” (MTW, Gravitation). Poincare, Reichenbach, Barbour all point out that duration is defined so as to make motion look simple.

• That said, nothing in what follows hangs on assuming MRL laws.

The general idea is that time is that direction on a spacetime manifold in which we can tell the strongest stories. Suppose we have an n-dimensional space-time <M,g>, where M is a diff manifold and g is the metric on M, then the general idea is:

Proposal. A temporal direction at point p on <M,g> is that direction in which the best theory tells the strongest, i.e., most informative, “story.”

1) The set of all such directions is the temporal direction. 2) In terms of MRL, time is the strongest direction of the on balance strongest and simplest theory. 3) To avoid terminological muddles, please remember that a priori, there is no reason to expect the timelike in the above sense to line up with the timelike* in the sense of g(v,v)<0 direction.

MOSAIC OF EVENTS

Another ConnectionIf an algorithm, given some input, could get back everything that happens, that would be best. A deterministic theory is maximally strong in this respect (it rules out all possibilities but one). Another way to put GC: time is that direction in which we can get as much determinism as possible.

Determinism. Call a history H a map from R to tuples of natural properties, where for any t in R, H(t) gives the state of the fundamental properties at t. Then a theory is deterministic iff for any pair of histories, H1 and H2, that satisfy the laws of physics, if H1(t) =H2(t) at one time t, then H1(t)=H2(t) for all t (Earman 2005)

This definition presupposes a time versus space split, and in fact it presupposes a global time function is definable (and also that time is orientable). A global time function is a smooth map t:M-->R such that for any events p,q in M, t(p) < t(q) iff there is a future directed temporal curve from p to q.

Idea: turn it around and define time as that mapping t such that for histories that satisfy the laws of physics, if any pair agree at one value of t then they agree for all values of t.

This idea works for some but not all spacetimes, e.g., there exist spacetimes with CTCs (and hence no global time fn) that are deterministic (Friedman 1994).

Further Comments

• Doesn’t commit me to determinism. Determinism is not a matter for armchair reflection, although it might be a regulative ideal.

• Distinguish “marks of strength” from “strength”. Being deterministic, being markovian, etc., are all marks of strength

•The vagueness of MRL might make one uneasy; and even if we scrap MRL, the vagueness of “information” makes the proposal lean on much that is murky. However, with suitable restrictions, and a precise sense of strength, we can prove that strength picks out something temporal. Moreover, we can display non-trivial connections between various features of time.

4. Illustration: The EquationsA very large class of important equations in mathematical physics are or can be approximated by linear PDE’s of second order. For an unknown function u(x1…xd), such an equation in Rd can be written generally in the form:

[ aij

x i

x j

bi

x i c]u0

i1

d

j1

d

i1

d

Scores of the most important equations of physics are of form (1): the wave equation, heat equation, Schrödinger equation, Klein-Gordon equation, Euler equation, Poisson equation, Dirac equation, linearized Einstein equation, Navier-Stokes equation, many equations in relativistic continuum mechanics including those describing elasticity, gas dynamics and magneto-fluid dynamics…

That’s an awful lot of physics…

(1)

StrengthThe non plus ultra in strength is having a well-posed Cauchy problem.

A PDE defined over a domain, supplemented by initial or boundary conditions, is well-posed if

1. There is a solution u for any choice of the data d, where d belongs to an admissible set X.

2. The solution u is uniquely determined within some set Y by the data d3. The solution u depends “continuously” on the data d, according to

some suitable topology

If the boundary conditions are a conjunction or linear combination of u and its normal derivative on the boundary, then it is a Cauchy or Mixed problem (as opposed to Dirichlet or Neumann)

Motivation•We should think of having a well-posed problem as a kind of methodological goal. There are plenty of questions mal posées used successfully in science every day.

•Existence, uniqueness, and continuity each make sense from the perspective of the Best System. They’re obviously also good things to have from the perspective of prediction, too.

•If u doesn’t depend continuously on d, then small errors in data can create large deviations in solution. Rounding off numbers, noise from perturbations, will dramatically affect the solution.

Illustration’s Specific Claim

• For systems governed by (1), a temporal direction at point p of n-dimensional <M,g> is that direction normal to the (n-1)-dimensional hypersurface intersecting p upon which Cauchy data can be prescribed to obtain a well-posed Cauchy problem.

• In other words, well-posed CP’s pick out time.

• To be plausible, the temporal directions picked out had better mesh well with the directions physics normally singles out as temporal, and they had better share many features normally attributed to time.

(n-1)-dim hypersurface;

u and udot prescribed

p

Time

Argument Sketch

• Equations of form (1) can be hyperbolic, elliptic, parabolic, and ultrahyperbolic, depending upon the number of positive and negative eigenvalues matrix aij has. Note that these classifications are coordinate-independent. With Z the number of zero eigenvalues of aij and P the number of positive eigenvalues…

Hyperbolic Z=0 & P=1 or Z=0 and P=d-1 E.g., Wave uyy=uxx

Parabolic Z>0 E.g., Heat uy=uxx

Elliptic Z=0 & P=d or Z=0 & P=0 E.g., LaPlace uyy=-uxx

Ultrahyperbolic Z=0 & 1<P<d-1

• (Courant & Hilbert 1962; Tegmark 1997) With Cauchy data on non-closed hypersurfaces, elliptic and parabolic eqs do not admit well-posed CP’s. Elliptic eqs suffer a variety of fates: non-unique solutions, lack of existence, lack of continuity. Parabolic eqs have too many solutions given Cauchy data.

[ aij

x i

x j

bi

x i c]u0

i1

d

j1

d

i1

d

Sketch, continued

• But ultrahyperbolic and hyperbolic (1) are finicky. Not as much in general is known about ultrahyperbolic eqs as hyperbolic eqs, but Asgeirsson’s theorem implies that these eqs do not possess the sort of hypersurface upon which data can be placed to get a WPCP.

• That leaves only hyperbolic versions of (1). For these eqs, the characteristic conoids consist of two sheets emanating from each point of the n-dim space. The sheets divide the space into three disjoint regions. Call surface elements at the vertex of these three regions spacelike if they lie in the region bounded by both sheets, and timelike if they point into one of the two regions bounded by a single sheet. AT implies only hyperbolic (1) with Cauchy data on spacelike surfaces give rise to WPCP.

• In fact, it’s a theorem that all linear hyperbolic second order systems have WPCP if data is so specified.

Intuitive Picture

characteristics

Γ

Cauchy surface

y

x

characteristics

Γ

Cauchy surface

y

• Q

• P

• Consider an arbitrary curve C and a point P not on C. Prescribe data on curve C. If we have a solution u(P), there will be characteristic curves intersecting P and C at points Q and R of C. u(P) will be determined by u and udot within “triangle” created by QP, PR, and RQ.

• One way to define “spacelike” here is that we mean as P tends to a point on C, the points Q and R also tend to that point on C.

• u(p) is consistent with the Cauchy data on C only if C is spacelike

• Define the timelike to be orthogonal to the spacelike, and the CP picks out time without putting time in.

P

Q RC

utt uxx f (x,t)

Fact 1

• The Cauchy surface Γ must be n-1 dimensional. Cauchy data specified on a n-2 dim or less submanifold Σ of M of dim n can never give a well-posed CP.

The timelike is one-dimensional!

Fact 2

• The signature of space-time is connected to the type of equation (if fundamental). For covariant field equations the matrix aij in (1) will have the same eigenvalues as the metric tensor.

• E.g. the Klein-Gordon equation looks the way it does (hyperbolic) b/c the signature of space-time is (+++-), whereas it would be elliptic if the signature were (++++) and ultrahyperbolic if it were (++--).

The Klein-Gordon equation possesses a well-posed CP. But if we changed the sign of the lhs, and it goes “LaPlacian”, it does not.

22

2

2

2

2

2

2

2

mzyxt

Fact 3

• The properties I’m identifying with the spacelike and timelike directions coincide, in relativity, with the relativistic sense of spacelike* (g(v,v)>0) and timelike* (g(v,v)<0).

• Hence the Cauchy data must be placed on the (+++) submanifold of a Lorentzian M, and “pushed” by the PDE orthogonally in the “(-) direction.”

Does This Time Correspond Well with the Temporal Directions in Physics?

• Fact 1 implies time is one-dimensional• Facts 2 and 3 imply that in Lorentzian manifolds time is given by

the “minus sign” direction

• Why does temporality supervene upon the “minus sign” direction? Why does physics tend to shun extra timelike dim? And what do the two have to do with one another?

• Answers: the timelike M has to be one-dim if we’re to get a well-posed CP; M should be Lorentzian if the PDE is hyperbolic, which it should be if we’re to get max strength; and the direction in which the 3-dim hypersurfaces march must be the one associated with one-dim.

• Is the Illustration merely a mathematical curiosity? After all, if we replaced our notion of strength with a well-posed Dirichlet problem, nothing temporal pops out.

– First, unlike for Tegmark, it just an illustration, not the general proposal. A world governed by a fundamental elliptic eq would have to find strength in some other mark of strength to get time; alternatively, such steady-state worlds might not be temporal.

– Second, the illustration does cover a lot of fundamental physics, and one can think of many other strong eqs as truncations of these, e.g., the elliptic Poisson eq is a truncation of the linear hyperbolic Maxwell eqs.

– Third, perturbing it slightly is okay. Higher-order eqs can be put in form (1) without loss with the help of auxiliary fields. Adding non-linear terms to (1) won’t make ill-posed problems well-posed.

– Fourth, there are results in the neighborhood of this. Geroch 1996 gives conditions for unique existence for first-order pde’s, which are capable of governing almost every system of physical interest, and one of these conditions is a “hyperbolization” -- a time/space split. So a time/space split is a piece of a sufficient condition for unique existence.

4. Back to the General Argument…

Is time the direction of strength in worlds like ours?

Against:

(a) Lots of strength in other directions, e.g., street signs, thermostatics, well-posed Dirichlet and Neuman problems, etc.

(b) Do I have access to some overall metric of strength covering everything?

(Heck, no.)

• Restrict matters to fundamental physics and worlds like ours.

Newtonian mechanics– Almost deterministic

Quantum mechanics– Deterministic if H is essentially

self-adjointGeneral relativity– Smattering of results (Wald,

Rendall, Anderson…); No “sideways” CP

• In all of these, strength is overwhelmingly dominant in one direction.

4th 5th

4th

6th

5th 6th

? ?

Virtues

• Fits beautifully with the MRL theory…

• Explains the difference between time and space in terms of distribution of fundamental properties and simplicity/strength, not a primitive

• Explains 1-dim of time, and to the extent that the Illustration is relevant, also (+++-)…

• Doesn’t prohibit laws holding across spacelike hypersurfaces

• Could get other aspects of time, too, e.g., past/future asymmetry, parameter v. coordinate time (Lautman 1936)

::

• The difference between space and time?

Putting it all together

Facts about the actual distribution of stuff

The PDE’s Strength

Implicit definition of time: a one-dim parameter in terms of which we can

tell maximally strong “stories”

Conclusion: What is Time?• Pointing out the difference between two

things is not the same as saying what each is. But it’s tempting to say…

• Call “space” <M,g>, the ultimate arena of all events. Then “time” is just the most informative direction of space. It is what space is, different only due to the fact that the distribution of matter, fields, etc., allow better prediction in that direction.

• If “strength” is defined Platonically, then the time/space split is perfectly objective; if not, if it’s strength-in-application for beings like us, then the difference between space and time, and hence time itself, partly depends on beings like us.

What is time?

St Augustine

Conclusion: What is Time?• Pointing out the difference between two things is not the same as saying what

each is. But…

• Call “space” <M,g>, the ultimate arena of all events. Then “time” is just the most informative direction of space. It is what space is, different only due to our interests.

• Aristotle– “Whether if soul did not exist time would exist or not, is a question that may

fairly be asked; for if there cannot be someone to count there cannot be anything counted…”

• Kant– Time is a form of intuition, a condition of sensible perception

• One shouldn’t exaggerate the anthropic basis of the division on my theory. As Aquinas remarked about Aristotle, still a world with motion might be countable; so too a world devoid of systematizers might have a BT because it is systematizable. Even bigger difference: <M,g> is objective; that is, the topological and metrical features are objective.

22

2

2

2

2

2

2

2

mzyxt

The Klein-Gordon equation possesses a well-posed IVF. But if we changed the sign of the lhs, and it goes “LaPlacian”, it does not.

More general than you might think…

Theorem. Consider a linear, diagonal second order hyperbolic system. Let (M, g) be a globally hyperbolic region of an arbitrary spacetime. Let Σ be a smooth spacelike Cauchy surface. Then one has a well-posed IVF.

(Hawking and Ellis, 1973; Wald 1984, 250-1 for details)

Furthermore, this theorem holds locally for quasi-linear diagonal second order systems, where quasi-linear means linear in the highest derivative terms.

Probably more general than anyone could have guessed…

First order (only first derivatives of the fields) quasi-linear systems of PDE’s are “sufficiently broad to include virtually all equations of physical interest” (Geroch “Partial Differential Equations of Physics”, 2002). Geroch shows that if 3 conditions are met, then the system enjoys an IVF. On of these conditions necessitates a timelike versus spacelike split.

Then we have the well-known classification via the discriminants D=B2-AC. Equation 2 is elliptic, parabolic or hyperbolic if D<0, D=0, or D>0, respectively. D is evaluated at a point, so eq may change type if A, B, C aren’t constants.

Elliptic Parabolic Hyperbolic

D<0 D=0 D>0

LaPlace eq Heat eq Wave eq

Klein-Gordon eq

Equilibriumsteady state

diffusion evolution

The type of PDE is coordinate-independent.

As an example, consider the most studied PDE or them all, the one-dim wave equation

a2uxx – utt = 0

The characteristics are defined by

a2dt2 – dx2 = 0

So the characteristics are the lines x+at=const=ξ and x-at=const=η.

ξ η

x+at=const x-at=const

Domain of influence Domain of influence

All of this generalizes to arbitrary finite dimension: characteristic surfaces, hypersurfaces, etc., and these surfaces describe a domain of dependency for the PDE.

IV. The Theory• Mathematicians have implicitly defined the “timelike” and “spacelike”

directions for decades via the nature of PDE’s and Cauchy problems (see, e.g. John, 1982, 27-28). I want to take this seriously:

The timelike direction just is that direction on M in which one has a (well-posed) IVF (in our fundamental theories).

• Note: the concepts I’ve reviewed do not depend on time being one of the variables. Whether a PDE is hyperbolic, elliptic, etc., is a truth in “Plato’s heaven”, whether there is time or not; the family of characteristics is what it is, come what may; and PDE’s either have unique solutions continuously depending on surface data or not. Time does not enter into any of this. Yet time is implicitly defined by these notions. Let’s see how this works.

III. The Three C’s of PDE’sMost important equations in mathematical physics are or can be approximated by linear PDE’s of second order. For an unknown function u(x1…xd), such an equation in Rd can be written generally in the form:

[

d

i

d

j

d

i ii

jiij uc

xb

xA

1 1 1

0][

where A is a matrix, b a vector, c a scalar. We can classify (1) as elliptic, parabolic, hyperbolic or ultrahyperbolic depending upon how many positive negative and zero eigenvalues the matrix A has. But for introducing these ideas, it won’t hurt to simplify (1) and consider an unknown function of two variables, x and y. Then (1) can be written as

022

22

2

2

y

uC

yx

uB

x

uA

(1)

CharacteristicsWe seek the “characteristic directions” along which the PDE involves only total differentials. Given a PDE, we get a family of characteristic lines (the number of real characteristics is connected to the type of PDE).

Consider the equation

3.1 a(x,y)ux + b(x,y)uy = f(x,y,u).

Now grab an arbitrary curve C on the plane and parameterize is via parameter s. Then x=x(s) and y=y(s) and u(x,y) goes to u(x(s),y(s)). We can now define the directional derivative of u on C as

s

y

y

u

s

x

x

u

ds

du

),();,( yxbds

dyyxa

ds

dx

Comparing the lhs of 3.1 with the rhs of 3.2, we see that along a special family of curves, C*--found by integrating the ODE’s

--3.1 can be replaced by the ODE du/ds = f(x,y,u). This family of curves C* are the characteristic curves of 3.1.

Fact 1

• Cauchy surfaces, the surfaces upon which we place “initial” data, cannot be placed willy-nilly on a manifold. E.g. typically, to get a well-posed IVF, the “initial” surface Γ must be nowhere parallel to the characteristic surfaces.

characteristics

ΓCauchy surface

y

x

characteristics

ΓCauchy surface

y

• Q

• P

Here a characteristic emanating from Γ at point P intersects point Q. This characteristic becomes tangent to Γ at Q. Conflict may arise. The solution u at Q is at once determined by u at P and by the Cauchy data assigned at Q on Γ. In such a case, we won’t be able to specify arbitrary data and get solution; and if we get solution, we tend to get many (not unique). Hence, the family of characteristic surfaces constrains where the “initial” values can be put if one is to have a well-posed IVF.

Objection 1• There are well-posed boundary value problems. Doesn’t focusing

on initial value formulations presuppose a time/space split?

• Reply

• First, a BV problem is not simply a spatial version of an IV problem. If it were, this objection would be correct. In a BV problem, one typically gets data on different d-1 hypersurfaces, rather than on one d-1 hypersurface, as in IV problem.

• Second, that said, BV problems can be strong and simple. However, (i) data on two slices is harder to come by than data on one, and (ii) if time is the dimension of change, then well-posed elliptic eqs—the ones for which there tend to be well-posed BV problems—describe situations in equilibrium or steady state.

Objection 2There are IVF’s besides Cauchy’s.

(One is the “characteristic IVF”, where one gives data on one or more null hypersurfaces, e.g., on the light cone.)

Reply.

• Yes, these are interesting cases to examine. I could enlarge the spirit of my project to include these, and then it helps undermine the tensed theory in some respects, i.e, seeing the universe “unfold” along null as opposed to spacelike hypersurfaces doesn’t sit well with tenses.

• As I understand matters, at least for GR, one has local existence theorems, not well-posed-ness (in most cases). And in some of these cases, one only gets existence by reducing the problem to a Cauchy problem. (Dossa, M., Ann. Inst. Henri Poincare, A, 66, 37-107, (1997); Chrusciel, P.T., Commun. Math. Phys., 137, 289-313, (1991); Rendall, A.D., Proc. R. Soc. London, Ser. A, 427, 221-239, (1990). But there are IVF’s for initial surfaces formed by two intersecting null surfaces (Sachs 1962; Muller zum Hagen and Seifert 1977).

Objection 3 We had no problem distinguishing time in Newtonian mechanics,

and it doesn’t have a well-posed IVF.

True: N>3: Mather and McGee, Gerver, Xia, (see Earman’s Primer); N=4, Saari, global existence for almost all initial conditions; for all we know, for N≥5, the set of initial conditions leading to catastrophe may be full measure!

But look at all this work; clearly, well-posed IVF is a goal here. Clearly, most experts suspect that what Saari proved will be true of N>4. And we do have lots and lots of PDE’s with specified forces, etc., where we have existence and uniqueness. That is, we have an IVF, but not a well-posed IVF, for classical particle mechanics –see Coddington and Levinson 1955.

Natural Kinds

y

x

y0

dxyx 20 |),(|

dyyx 20 |),(|

x0

1

2

If 1 is constant for all values of y0, and 2 behaves badly for x0, then y is the time coordinate. But this presupposes a lot already. Belot and Earman 2001, Unruh 1988.