Time is the Simplest (and Strongest) Thing

1

• Time is the Simplest (and Strongest) Thing

Time, Ill tell you about time. Time is the
simplest thing. Clifford Simak, The
Fisherman Craig Callender Philosophy,
UCSD ccallender_at_ucsd.edu
2
Introduction

• What makes time different than space? Once a
  central question in metaphysics, this question
  has not been treated kindly by recent history.
  Why?
• 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 ones 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 its not clear
  that there is the difference between the two.
• 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
  doesnt 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.

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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.,
uxxutt) are invariant under a transformation of
spatial and temporal variables--and no important
ones. If modern physics is spatializing time,
as Bergson charged, its 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 doesnt mean, however, that some
differences arent 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 well 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.
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Plan

• Other Theories
• Differences Between Time and Space in
  Contemporary Physics
• General Proposal
• Illustration
• Argument for the Proposal
• Conclusion What is Time?

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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
  Times 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.

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• 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 doesnt even need imaginary
  cases. E.g., Pauli exclusion principle. E.g.,
  the 10 vacuum EFE separate into 6 evolution eqs
  Gij0 and 4 constraint eqs, G000 and G0i0,
  with i1,2,3. To decree that 4 of the 10 are not
  laws strikes me as unacceptable. E.g. two of
  Maxwells equations.
• Still, if sense can be made of the idea of the
  laws preferring time, I find the general idea
  attractive.

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3. Time in Contemporary Physics
Physics is not invariant under a change of
spatial and temporal directions. But writing a
list of all such eqs would not be edifying.
Instead, lets 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.
Orientable,Hausdorf Connected, ordering
Inspection leads to at least four major
differences Dynamics Temporal Direction Minus
Sign Dimension
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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, lets 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, its conventional whether one uses
  (-) or (---)--for this reason Skow 2005
  discounts the importance of this feature.

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Metrical Difference

• But the signature of a metric is a geometrical
  invariant. Given g, we can find an orthonormal
  basis v1vn 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 pqn, where ndim 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.

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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)

t
Riemannian
t0S
Lorentzian
x
Simple example ds2tdt2a(t)2dx2
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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.

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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! MacBeaths 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 gt1
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)
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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 isnt true, say, for
  neutral kaon decay, we can plausibly say that
  although it provides a difference between space
  and time, it isnt 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.

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Matter fields

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

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3. The Theory

• Motivation. 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.

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x0
Isotropic homogeneous
x1
Fig2. Expanding FRW Model The cosmological
principle holds, but not the perfect cosmological
principle
Fig1. Symmetric Two-dim World
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MRL Time?

• The idea is vague as it stands and in need of
  development. The fundamental concepts of MRL are
  murky, and its 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.

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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 ltM,ggt, 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 ltM,ggt 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)lt0 direction.
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MOSAIC OF EVENTS
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Another Connection
If 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 tM--gtR such that
for any events p,q in M, t(p) lt 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).
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Further Comments

• Doesnt 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.

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4. Illustration The Equations
A very large class of important equations in
mathematical physics are or can be approximated
by linear PDEs of second order. For an unknown
function u(x1xd), such an equation in Rd can be
written generally in the form
(1)
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 Thats an awful lot of
physics
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Strength

• The 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
• There is a solution u for any choice of the data
  d, where d belongs to an admissible set X.
• The solution u is uniquely determined within some
  set Y by the data d
• 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)

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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.
  Theyre obviously also good things to have from
  the perspective of prediction, too.
• If u doesnt 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.

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Illustrations Specific Claim

• For systems governed by (1), a temporal direction
  at point p of n-dimensional ltM,ggt 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 CPs 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.

Time
p
(n-1)-dim hypersurface u and udot prescribed
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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 Z0 P1 or Z0 and Pd-1 E.g., Wave
  uyyuxx
• Parabolic Zgt0 E.g., Heat uyuxx
• Elliptic Z0 Pd or Z0 P0 E.g., LaPlace
  uyy-uxx
• Ultrahyperbolic Z0 1ltPltd-1
• (Courant Hilbert 1962 Tegmark 1997) With
  Cauchy data on non-closed hypersurfaces, elliptic
  and parabolic eqs do not admit well-posed CPs.
  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.

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Sketch, continued

• But ultrahyperbolic and hyperbolic (1) are
  finicky. Not as much in general is known about
  ultrahyperbolic eqs as hyperbolic eqs, but
  Asgeirssons 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, its a theorem that all linear
  hyperbolic second order systems have WPCP if data
  is so specified.

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Intuitive Picture
characteristics
y
Q
characteristics
y
G
P
G
Cauchy surface
x
Cauchy surface
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• 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.

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Fact 1

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

The timelike is one-dimensional!
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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.
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Fact 3

• The properties Im identifying with the spacelike
  and timelike directions coincide, in relativity,
  with the relativistic sense of spacelike
  (g(v,v)gt0) and timelike (g(v,v)lt0).
• Hence the Cauchy data must be placed on the ()
  submanifold of a Lorentzian M, and pushed by
  the PDE orthogonally in the (-) direction.

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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
  were to get a well-posed CP M should be
  Lorentzian if the PDE is hyperbolic, which it
  should be if were to get max strength and the
  direction in which the 3-dim hypersurfaces march
  must be the one associated with one-dim.

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• 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) wont 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 pdes, 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.

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4. Back to the General Argument

• Restrict matters to fundamental physics and
  worlds like ours.
• Newtonian mechanics
• Almost deterministic
• Quantum mechanics
• Deterministic if H is essentially self-adjoint
• General relativity
• Smattering of results (Wald, Rendall, Anderson)
  No sideways CP
• In all of these, strength is overwhelmingly
  dominant in one direction.

• 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.)

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?
?
4th
5th
6th
4th
5th
6th
37
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 (-)

• Doesnt 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?

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Putting it all together
Facts about the actual distribution of stuff
The PDEs
Strength
Implicit definition of time a one-dim parameter
in terms of which we can tell maximally strong
stories
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Conclusion What is Time?

• Pointing out the difference between two things is
  not the same as saying what each is. But its
  tempting to say
• Call space ltM,ggt, 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 its 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
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Conclusion What is Time?

• Pointing out the difference between two things is
  not the same as saying what each is. But
• Call space ltM,ggt, 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 shouldnt 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 ltM,ggt is
  objective that is, the topological and metrical
  features are objective.

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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
S 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 PDEs 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.
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Then we have the well-known classification via
the discriminants DB2-AC. Equation 2 is
elliptic, parabolic or hyperbolic if Dlt0, D0, or
Dgt0, respectively. D is evaluated at a point, so
eq may change type if A, B, C arent constants.
The type of PDE is coordinate-independent.
43
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 xatconst? and x-atconst?.
xatconst
x-atconst
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.
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IV. The Theory

• Mathematicians have implicitly defined the
  timelike and spacelike directions for decades
  via the nature of PDEs 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 Ive reviewed do not depend on
  time being one of the variables. Whether a PDE
  is hyperbolic, elliptic, etc., is a truth in
  Platos heaven, whether there is time or not
  the family of characteristics is what it is, come
  what may and PDEs 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. Lets
  see how this works.

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III. The Three Cs of PDEs
Most important equations in mathematical physics
are or can be approximated by linear PDEs of
second order. For an unknown function u(x1xd),
such an equation in Rd can be written generally
in the form
(1)
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 wont hurt to simplify (1) and consider an
unknown function of two variables, x and y. Then
(1) can be written as
46
Characteristics

• We 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 xx(s) and
  yy(s) and u(x,y) goes to u(x(s),y(s)). We can
  now define the directional derivative of u on C
  as

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 ODEs
--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.
47
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 G must be nowhere
  parallel to the characteristic surfaces.

characteristics
y
G
Cauchy surface
x
48
Here a characteristic emanating from G at point P
intersects point Q. This characteristic becomes
tangent to G 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 G. In
such a case, we wont 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.
characteristics
y
Q
G
Cauchy surface
P
49
Objection 1

• There are well-posed boundary value problems.
  Doesnt 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 eqsthe ones for which there tend to be
  well-posed BV problemsdescribe situations in
  equilibrium or steady state.

50
Objection 2

• There are IVFs besides Cauchys.
• (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 doesnt 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 IVFs for initial
  surfaces formed by two intersecting null surfaces
  (Sachs 1962 Muller zum Hagen and Seifert 1977).

51
Objection 3

• We had no problem distinguishing time in
  Newtonian mechanics, and it doesnt have a
  well-posed IVF.
• True Ngt3 Mather and McGee, Gerver, Xia, (see
  Earmans Primer) N4, Saari, global existence
  for almost all initial conditions for all we
  know, for N5, 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
  Ngt4. And we do have lots and lots of PDEs 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.

52
Natural Kinds
y
1
y0
2
x
x0
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.