Continuous space-time transformations

Continuous space-time transformations

Clément de Seguins Pazzis, Peter Šemrl

KU Leuven 2016

The setting

V =R⁴equipped with the Lorentz quadratic form q : (x,y,z,t) 7→t²−x²−y²−z². Space-time events a and b are coherent iff

q(b−a) =0, they are adjacent iff coherent and a6=b.

Light cones

Basic light cone:

C(0) :={m∈V : q(m) =0}.

Light cone with vertex a∈V :

C(a) ={m∈V : q(m−a) =0}=a+C(0).

Critical note: the maximal coherent sets are lines in light cones.

Light cones

Basic light cone:

C(0) :={m∈V : q(m) =0}.

Light cone with vertex a∈V :

C(a) ={m∈V : q(m−a) =0}=a+C(0).

Critical note: the maximal coherent sets are lines in light cones.

Alexandrov’s problem

Q: What are the bijectionsφ:R⁴→R⁴that preserve coherency in both directions?

A: Only the standard maps (or Poincaré similarities):

m7→λu(m) +a

with a∈R⁴,λ∈R\ {0}, u :R⁴→R⁴a linear q-isometry.

Note: this generalizes to an n-dimensional Alexandrov space, i.e.

real quadratic space(V,q), with q regular (i.e. non-degenerate) of signature(1,n−1), with n≥4.

Alexandrov’s problem

Q: What are the bijectionsφ:R⁴→R⁴that preserve coherency in both directions?

A: Only the standard maps (or Poincaré similarities):

m7→λu(m) +a

with a∈R⁴,λ∈R\ {0}, u :R⁴→R⁴a linear q-isometry.

Note: this generalizes to an n-dimensional Alexandrov space, i.e.

real quadratic space(V,q), with q regular (i.e. non-degenerate) of signature(1,n−1), with n≥4.

Alexandrov’s problem

Q: What are the bijectionsφ:R⁴→R⁴that preserve coherency in both directions?

A: Only the standard maps (or Poincaré similarities):

m7→λu(m) +a

with a∈R⁴,λ∈R\ {0}, u :R⁴→R⁴a linear q-isometry.

Note: this generalizes to an n-dimensional Alexandrov space, i.e.

real quadratic space(V,q), with q regular (i.e. non-degenerate) of signature(1,n−1), with n≥4.

Matrix formulation

A model of 4-dimensional Alexandrov space:

H₂=

a c c b

|(a,b)∈R², c∈C

, with

q :M ∈ H₂7→det M.

A and B adjacent iff rk(A−B) =1.

Standard maps:

M7→ǫPMP^⋆+A, A∈ H₂, P∈GL₂(C), ǫ∈ {1,−1}

M7→ǫPM^TP^⋆+A, A∈ H2, P∈GL₂(C), ǫ∈ {1,−1}.

Matrix formulation

A model of 4-dimensional Alexandrov space:

H₂=

a c c b

|(a,b)∈R², c∈C

, with

q :M ∈ H₂7→det M.

A and B adjacent iff rk(A−B) =1.

Standard maps:

M7→ǫPMP^⋆+A, A∈ H₂, P∈GL₂(C), ǫ∈ {1,−1}

M7→ǫPM^TP^⋆+A, A∈ H2, P∈GL₂(C), ǫ∈ {1,−1}.

Problem

What if we relax the assumptions?

Drop bijectivity?

Assume the preservation of coherency in one direction only?

Problem

What if we relax the assumptions?

Drop bijectivity?

Assume the preservation of coherency in one direction only?

Problem

What if we relax the assumptions?

Drop bijectivity?

Assume the preservation of coherency in one direction only?

A theorem of Huang and Šemrl

Letφ:H2→ H2preserve adjacency in one direction only. Then:

eitherφis standard;

orφmapsH₂into a line (degenerate adjacency preserver).

Example of a degenerate preserver:

φ:A7→

tr(A) 0

0 0

.

The trace can be replaced with an injective map . . . Note: generalization to mapsHn→ Hn.

W.-l. Huang, P. Šemrl,

Adjacency preserving maps on hermitian matrices, Canad. J. Math. 60 (2008), 1050–1066.

A theorem of Huang and Šemrl

Letφ:H2→ H2preserve adjacency in one direction only. Then:

eitherφis standard;

orφmapsH₂into a line (degenerate adjacency preserver).

Example of a degenerate preserver:

φ:A7→

tr(A) 0

0 0

. The trace can be replaced with an injective map . . . Note: generalization to mapsHn→ Hn.

W.-l. Huang, P. Šemrl,

Adjacency preserving maps on hermitian matrices, Canad. J. Math. 60 (2008), 1050–1066.

A theorem of Huang and Šemrl

Letφ:H2→ H2preserve adjacency in one direction only. Then:

eitherφis standard;

orφmapsH₂into a line (degenerate adjacency preserver).

Example of a degenerate preserver:

φ:A7→

tr(A) 0

0 0

. The trace can be replaced with an injective map . . . Note: generalization to mapsHn→ Hn.

W.-l. Huang, P. Šemrl,

Adjacency preserving maps on hermitian matrices, Canad. J. Math. 60 (2008), 1050–1066.

Problem: what if we only assume that coherency is preserved (in one direction only)?

Very hard problem . . .

We add continuity.

Reformulation: describe the continuous mapsφ:V →V s.t.

∀(a,b)∈V², q(a−b) =0⇒q φ(a)−φ(b)

=0.

Problem: what if we only assume that coherency is preserved (in one direction only)?

Very hard problem . . . We add continuity.

Reformulation: describe the continuous mapsφ:V →V s.t.

∀(a,b)∈V², q(a−b) =0⇒q φ(a)−φ(b)

=0.

Problem: what if we only assume that coherency is preserved (in one direction only)?

Very hard problem . . . We add continuity.

Reformulation: describe the continuous mapsφ:V →V s.t.

∀(a,b)∈V², q(a−b) =0⇒q φ(a)−φ(b)

=0.

Problem: what if we only assume that coherency is preserved (in one direction only)?

Very hard problem . . . We add continuity.

Reformulation: describe the continuous mapsφ:V →V s.t.

∀(a,b)∈V², q(a−b) =0⇒q φ(a)−φ(b)

=0.

Main result

Theorem (dSP, Šemrl (2015))

Ifφis a continuous coherency preserver then:

eitherφis standard;

orφ(V)⊂ C(a)for some a∈V (degenerate preserver).

C. de Seguins Pazzis, P. Šemrl,

Continuous space-time transformations, arXiv: http://arxiv.org/abs/1502.01149

Surprise: Not all degenerate continuous coherency preservers map into a line!

Choose a sequence(Un)n∈Nof open subsets of V s.t.

∀(m,n)∈N², m6=n⇒ ∀(a,b)∈U_n×U_m, q(a−b)6=0.

Choose:

a∈V ;

For each n∈N, choose x_n ∈ C(0)withkxnk=1;

α:V →Rcontinuous with support in S

n∈N

U_n; Take

φ:m7→

(a+α(m)x_n if m ∈U_n

a otherwise

Surprise: Not all degenerate continuous coherency preservers map into a line!

Choose a sequence(Un)n∈Nof open subsets of V s.t.

∀(m,n)∈N², m6=n⇒ ∀(a,b)∈U_n×U_m, q(a−b)6=0.

Choose:

a∈V ;

For each n∈N, choose x_n ∈ C(0)withkxnk=1;

α:V →Rcontinuous with support in S

n∈N

U_n; Take

φ:m7→

(a+α(m)x_n if m ∈U_n

a otherwise

Example of a sequence ( U

)

With the standard Euclidian normk − konR⁴, Un:=Bo

(0R³,n),1 4

.

For(x,s)∈U_nand(y,t) ∈U_pwith n6=p (and(s,t) ∈R², (x,y)∈(R³)²),

(s−t)²> 1

4 >kx −yk² whence

q (x,s)−(y,t)

>0.

Theorem (dSP, Šemrl (2015))

Any degenerate continuous coherency preserver is of the previous type.

Letφ:V →V be a non-degenerate continuous coherency preserver.

Basic strategy: prove thatφpreserves adjacency;

Basic technique: analyze the action on light cones.

Letφ:V →V be a non-degenerate continuous coherency preserver.

Basic strategy: prove thatφpreserves adjacency;

Basic technique: analyze the action on light cones.

Letφ:V →V be a non-degenerate continuous coherency preserver.

Basic strategy: prove thatφpreserves adjacency;

Basic technique: analyze the action on light cones.

Step 1

φnever constant on a coherent line.

Consequence:

φmaps any coherent line into a unique coherent line.

Step 1

φnever constant on a coherent line.

Consequence:

φmaps any coherent line into a unique coherent line.

Step 2

An equivalence relation ofC(0)\ {0}:

a∼b ⇔

def ∃λ∈R^∗ : b=λa Set

Q:= (C(0)\ {0})/∼

Notes:

Qis a projective quadric, homeomorphic to the 2-sphere;

for each a∈V , we have a natural bijection fromQto the set of all lines inC(a).

For all a∈V ,

φ C(a)

⊂ C φ(a) yields

ϕa :Q → Q.

Then, one proves that

a∈V 7→ϕa ∈ C(Q,Q) is continuous.

Step 3

Ifϕanon constant and∃b∈ C(a)\ {a}s.t. φ(a) =φ(b), then φ⁻¹{φ(a)}has non-empty interior.

Same conclusion ifϕ_anonconstant and non-injective.

Definition

State a called generic ifφ⁻¹{φ(a)}has empty interior.

Note: there are generic points.

Step 3

Ifϕanon constant and∃b∈ C(a)\ {a}s.t. φ(a) =φ(b), then φ⁻¹{φ(a)}has non-empty interior.

Same conclusion ifϕ_anonconstant and non-injective.

Definition

State a called generic ifφ⁻¹{φ(a)}has empty interior.

Note: there are generic points.

Step 3

Ifϕanon constant and∃b∈ C(a)\ {a}s.t. φ(a) =φ(b), then φ⁻¹{φ(a)}has non-empty interior.

Same conclusion ifϕ_anonconstant and non-injective.

Definition

State a called generic ifφ⁻¹{φ(a)}has empty interior.

Note: there are generic points.

Reformulation of Step 3

If a generic andϕ_anonconstant then:

ϕainjective;

For each b adjacent to a,φ(b)adjacent toφ(a).

Step 4

There exists c ∈V generic s.t.ϕcnonconstant.

Consequences:

ϕ_c injective (Step 3);

ϕc a homeomorphism ofQ(invariance of domain theorem + compactness and connectedness ofQ);

ϕ_bnonconstant for all b∈ Q(homotopy theory of spheres).

Step 4

There exists c ∈V generic s.t.ϕcnonconstant.

Consequences:

ϕ_c injective (Step 3);

ϕc a homeomorphism ofQ(invariance of domain theorem + compactness and connectedness ofQ);

ϕ_bnonconstant for all b∈ Q(homotopy theory of spheres).

Step 4

There exists c ∈V generic s.t.ϕcnonconstant.

Consequences:

ϕ_c injective (Step 3);

ϕc a homeomorphism ofQ(invariance of domain theorem + compactness and connectedness ofQ);

ϕ_bnonconstant for all b∈ Q(homotopy theory of spheres).

Step 4

There exists c ∈V generic s.t.ϕcnonconstant.

Consequences:

ϕ_c injective (Step 3);

ϕc a homeomorphism ofQ(invariance of domain theorem + compactness and connectedness ofQ);

ϕ_bnonconstant for all b∈ Q(homotopy theory of spheres).

Conclusion of the proof

For all adjacent a,b with a generic,φ(a)andφ(b)adjacent (see Step 3).

Then, every point is generic and one concludes.

Conclusion of the proof

For all adjacent a,b with a generic,φ(a)andφ(b)adjacent (see Step 3).

Then, every point is generic and one concludes.

Possible further research

Extend the result to continuous coherency preserversHn→ Hn.

Consider a more general regular real quadratic space(V,q).

Possible further research

Extend the result to continuous coherency preserversHn→ Hn.

Consider a more general regular real quadratic space(V,q).