Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

FREDERIC P. SCHULLER

Max Planck Institute for Gravitational Physics

Spacetimes beyond Einstein

Kavli Institute for the Physics and Mathematics of the Universe Tokyo, Japan

SPACETIME

教科書

TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

教科書

SET

TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

教科書

SET

TOPOLOGY

TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

LORENTZIAN METRIC

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

LORENTZIAN METRIC

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

LORENTZIAN METRIC

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915 教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915 教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

MAXWELL ELECTRO-

MAGNETISM

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880

(CO-) VECTOR FIELD

Ein

stei

n 1

90

5

Ku

char+Teitelb

oim

19

72

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

QM

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880

(CO-) VECTOR FIELD

Ein

stei

n 1

90

5

Ku

char+Teitelb

oim

19

72

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

Wigner 1930

QM

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

Wigner 1930

QM

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

QM

TIME ORIENTATION

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

QM

TIME ORIENTATION

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

TENSORS | SPINORS

QM

TIME ORIENTATION

Wign

er

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

TENSORS | SPINORS

STANDARD MODEL

DYNAMICS Q

M

TIME ORIENTATION

Wign

er

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

1880 Ei

nst

ein

19

05

K

uch

ar+Teitelbo

im 1

97

2 LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Einstein 1915

TENSORS | SPINORS

STANDARD MODEL

DYNAMICS Q

M

TIME ORIENTATION

Wign

er

1979

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

2010

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Wigner 1930

TIME ORIENTATION

TENSORS | SPINORS

STANDARD MODEL

DYNAMICS 1880

教科書

SET

TOPOLOGY

SMOOTH ATLAS TIME ORIENTATION

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

2010

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

Wigner 1930

TIME ORIENTATION

TENSORS | SPINORS

STANDARD MODEL

DYNAMICS 1880

教科書

TIME ORIENTATION

SET

TOPOLOGY

SMOOTH ATLAS

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

STANDARD

MATTER

DARK MATTER

DARK

ENERGY

教科書

TIME ORIENTATION

SET

TOPOLOGY

SMOOTH ATLAS

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

STANDARD

MATTER

DARK MATTER

DARK

ENERGY

教科書

TIME ORIENTATION

SET

TOPOLOGY

SMOOTH ATLAS

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

STANDARD

MATTER

DARK MATTER

DARK

ENERGY

教科書

TIME ORIENTATION

SET

TOPOLOGY

SMOOTH ATLAS

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

STANDARD

MATTER

DARK MATTER

DARK

ENERGY

教科書

TIME ORIENTATION

SET

TOPOLOGY

SMOOTH ATLAS

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS 教科書

SPACETIME KINEMATICS

修正

SET

TOPOLOGY

SMOOTH ATLAS

TIME ORIENTATION

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

TIME ORIENTATION

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

TIME ORIENTATION

STANDARD MODEL

DYNAMICS

LORENTZIAN METRIC

EINSTEIN GRAVITY

DYNAMICS

TENSOR AND SPINOR FIELDS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

TENSOR FIELDS

PREDICTIVE FIELD

EQUATIONS

SET

TOPOLOGY

SMOOTH ATLAS

TENSOR FIELDS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

TENSOR FIELDS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELDS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELDS

OBSERVATIONS

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

Algebraic Geometry

Modern PDE Theory

Convex Analysis

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

Rät

zel,

Riv

era,

FP

S 2

01

0

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

Algebraic Geometry

Modern PDE Theory

Convex Analysis

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

predictive

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

predictive

hyperbolic polynomial

TENSOR FIELD

[Hörmander]

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

predictive

hyperbolic polynomial

TENSOR FIELD HYPERBOLIC POLYNOMIAL

[Hörmander]

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

predictive

[Hörmander]

hyperbolic polynomial

Example: Klein-Gordon

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

P-null surface in -space

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

P-null surface in -space

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

P-null surface in -space

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

P-null surface in -space

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

E

piercing direction

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold

P-null surface in -space

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

E

piercing direction

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS hyperbolic polynomial

(deg P)–fold piercing directions

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

ALL

hyperbolicity cones

„co-normals to initial data surfaces“

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Corollary: massless particle action

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

eliminate [real algebraic geometry]

Corollary: massless particle action

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

eliminate [real algebraic geometry]

Corollary: massless particle action

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

eliminate [real algebraic geometry]

Corollary: massless particle action

Theorem: hyperbolic

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

eliminate [real algebraic geometry]

Corollary: massless particle action

Theorem: hyperbolic

dual polynomial exists / unique*

Proof: Hyperbolicity saves the day over

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS Theorem: massless dispersion

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

eliminate [real algebraic geometry]

Corollary: massless particle action

Theorem: hyperbolic

dual polynomial exists / unique*

Proof: Hyperbolicity saves the day over

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem: massless dispersion

eliminate [real algebraic geometry]

Corollary: massless particle action

Theorem: hyperbolic

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem: hyperbolic

exist unique* dual polynomial :

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem: hyperbolic

exist unique* dual polynomial :

cotangent space

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem: hyperbolic

exist unique* dual polynomial :

cotangent space tangent space

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem: hyperbolic

exist unique* dual polynomial :

cotangent space tangent space

massless momenta velocities

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem: hyperbolic

exist unique* dual polynomial :

cotangent space tangent space

Bi-hyperbolic

Algebraic geometry

massless momenta velocities

dual polynomial exists / unique*

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Postulate: massive dispersion

hyperbolic hyperbolic AND

only for

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Postulate: massive dispersion

hyperbolic

Theorem: massive particle action

hyperbolic AND

only for

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Postulate: massive dispersion

hyperbolic

eliminate [convex analysis]

Theorem: massive particle action

hyperbolic AND

only for

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Postulate: massive dispersion

hyperbolic

eliminate [convex analysis]

Theorem: massive particle action

massive momenta velocities

hyperbolic AND

only for

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Postulate: massive dispersion

hyperbolic

eliminate [convex analysis]

Theorem: massive particle action

Bi-hyperbolic

Convex analysis

massive momenta velocities

hyperbolic AND

only for

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Postulate: massive dispersion

hyperbolic

eliminate [convex analysis]

Theorem: massive particle action

Bi-hyperbolic

Convex analysis

massive momenta velocities

hyperbolic AND

only for

HYPERBOLIC POLYNOMIAL

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

3+1 DISPERSION

3-space

time Application: 3+1 decomposition

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

3+1 DISPERSION

Corollary: modified dispersion E(p)

3-space

time Application: 3+1 decomposition

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

3+1 DISPERSION

solve for E [Galois]

Corollary: modified dispersion E(p)

3-space

time Application: 3+1 decomposition

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

3+1 DISPERSION

solve for E [Galois]

Corollary: modified dispersion E(p)

3-space

time Application: 3+1 decomposition

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

3+1 DISPERSION

hard! solve for E

[Galois]

Corollary: modified dispersion E(p)

3-space

time Application: 3+1 decomposition

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

cotangent space tangent space

HYPERBOLIC POLYNOMIAL

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

cotangent space tangent space

HYPERBOLIC POLYNOMIAL

future-directed observers

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

cotangent space tangent space

pos/neg energy cones

HYPERBOLIC POLYNOMIAL

future-directed observers

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

cotangent space tangent space

pos/neg energy cones

HYPERBOLIC POLYNOMIAL

future-directed observers

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

cotangent space tangent space

pos/neg energy cones

HYPERBOLIC POLYNOMIAL

future-directed observers

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL

Theorem:

cotangent space tangent space

pos/neg energy cones

disjointly cover massless momenta

HYPERBOLIC POLYNOMIAL

hyperbolicity cone of

future-directed observers

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Theorem: pos/neg energy cones

disjointly cover massless momenta

hyperbolicity cone of

cotangent space tangent space

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Theorem: pos/neg energy cones

disjointly cover massless momenta

hyperbolicity cone of

(Einstein)

cotangent space tangent space

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Theorem: pos/neg energy cones

disjointly cover massless momenta

hyperbolicity cone of

(Einstein)

cotangent space tangent space

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Theorem: pos/neg energy cones

disjointly cover massless momenta

hyperbolicity cone of

(Einstein)

cotangent space tangent space

Application: timeenergy orientation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Theorem: pos/neg energy cones

disjointly cover massless momenta

hyperbolicity cone of

no other geometries predictive

and time-orientable

(Einstein)

cotangent space tangent space Theorem:

Application: timeenergy orientation

SPACETIME DYNAMICS

修正

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

Rät

zel,

Riv

era,

FP

S 2

01

0

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Representation Theory

Functional Analysis

Refined Kinematics

Giesel, FP

S, Wo

hlfarth

PREDICTIVE GRAVITY

DYNAMICS

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

initial data

surface

vital injection from physics:

Legendre map tells geometrodynamics

about the matter it must carry

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

deformation

initial data

surface

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL co-normal

normal

normal = Legendre(co-normal)

deformation

initial data

surface

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

deformation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

DIVINE VIEW

deformation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

DIVINE VIEW

pull-back

deformation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

DIVINE VIEW

pull-back

deformation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

DIVINE VIEW

pull-back

deformation

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

DIVINE VIEW

deformation operators

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

DIVINE VIEW

deformation operators

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

DIVINE VIEW

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

DIVINE VIEW

hard (NON-LIE ALGEBRA) REPRESENTATION THEORY PROBLEM

[Rätzel, Rivera, FPS]

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

DIVINE VIEW

hard (NON-LIE ALGEBRA) REPRESENTATION THEORY PROBLEM

only a HOMOGENEOUS

LINEAR PDE PROBLEM

[Rätzel, Rivera, FPS] [Giesel, FPS, Witte, Wohlfarth]

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

HUMAN VIEW

DIVINE VIEW

hard (NON-LIE ALGEBRA) REPRESENTATION THEORY PROBLEM

only a HOMOGENEOUS

LINEAR PDE PROBLEM

[Rätzel, Rivera, FPS] [Giesel, FPS, Witte, Wohlfarth]

Search for modified geometrodynamics

existence/uniqueness/solution of homogeneous linear PDE

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

Rät

zel,

Riv

era,

FP

S 2

01

0

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Spacetime had been tailored to

to apples for two centuries,

to light for one century

One can tailor it to any matter.

Rät

zel,

Riv

era,

FP

S 2

01

0

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

OBSERVATIONS

TENSOR FIELD

SET

TOPOLOGY

SMOOTH ATLAS

PREDICTIVE FIELD

EQUATIONS

TENSOR FIELD

OBSERVATIONS

HYPERBOLIC POLYNOMIAL HYPERBOLIC POLYNOMIAL

Giesel, FP

S, Wo

hlfarth

PREDICTIVE

GRAVITY DYNAMICS

TAKE-HOME MESSAGE

結果

• Lorentzian spacetimes are tailored to Maxwell theory.

• Lorentzian spacetimes are tailored to Maxwell theory.

Other geometries tailored to other matter severely restricted

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• central insight, impact on several fields

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• central insight, impact on several fields

@ classification of physical dispersion relations

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• central insight, impact on several fields

@ classification of physical dispersion relations

@ identification of all classical spacetime geometries

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• central insight, impact on several fields

@ classification of physical dispersion relations

@ identification of all classical spacetime geometries

@ construction of all classical gravity dynamics

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• central insight, impact on several fields

@ classification of physical dispersion relations

@ identification of all classical spacetime geometries

@ construction of all classical gravity dynamics

@ constraints for classical limits of quantum gravity

• Lorentzian spacetimes are tailored to Maxwell theory.

bi-hyperbolic

Other geometries tailored to other matter severely restricted

• good mathematical control over all bi-hyperbolic spacetimes.

• central insight, impact on several fields

@ classification of physical dispersion relations

@ identification of all classical spacetime geometries

@ construction of all classical gravity dynamics

• It is all very simple. Mathematical certainty.

@ constraints for classical limits of quantum gravity

Example 1: Metric geometry carrying Maxwell theory

Example 1: Metric geometry carrying Maxwell theory

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Hypersurface deformation algebra

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Hypersurface deformation algebra

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Hypersurface deformation algebra

Representation on 3-geometry phase space

Example 1: Metric geometry carrying Maxwell theory

Maxwell

Hypersurface deformation algebra

Representation on 3-geometry phase space

Example 2: Metric geometry carrying quartically interacting Proca

Proca

Hypersurface deformation algebra

Representation on 3-geometry phase space

solve

good cosmology!

Example 3: Area-metric geometry

Example 3: Area-metric geometry

Example 3: Area-metric geometry

Hehl-Obukhov

Example 3: Area-metric geometry

Hehl-Obukhov

Example 3: Area-metric geometry

Hehl-Obukhov

Example 3: Area-metric geometry

Hehl-Obukhov

Example 3: Area-metric geometry

Hehl-Obukhov

Hypersurface deformation algebra

Example 3: Area-metric geometry

Hehl-Obukhov

Hypersurface deformation algebra

Example 3: Area-metric geometry

Hehl-Obukhov

Hypersurface deformation algebra

Representation on 3-geometry phase space

Example 3: Area-metric geometry

Hehl-Obukhov

Hypersurface deformation algebra

Representation on 3-geometry phase space

Example 3: Area-metric geometry

Hehl-Obukhov

Hypersurface deformation algebra

Representation on 3-geometry phase space

solve good cosmology!

General case: your favourite candidate geometry

General case: your favourite candidate geometry

General case: your favourite candidate geometry

your favourite matter

General case: your favourite candidate geometry

your favourite matter

General case: your favourite candidate geometry

your favourite matter

General case: your favourite candidate geometry

your favourite matter

Hypersurface deformation algebra

General case: your favourite candidate geometry

your favourite matter

Hypersurface deformation algebra

Representation on 3-geometry phase space

General case: your favourite candidate geometry

your favourite matter

Hypersurface deformation algebra

Representation on 3-geometry phase space

solve