s as the length of a xed segment relative to the length of the diameter of the unit ball

interseted by the half-line ontaining the segment

[O, P ]

^. ^This ^means ^that ^we ^an ^determine

the hange of the length of a diameter of a xed diretion if we hange the shape of the

body by the time. Consider a representation of the body by polaroordinates with respet to

its enter

O

^. ^Sine ^the ^boundary ^of ^the ^body ^is ^of ^lass

C ²

^, ^all ^of ^their ^oordinate ^funtions

have the analogous property. This funtion depends also on the time

τ

^, ^the ^hange ^of ^the

unit ball implies the hange of its oordinate funtions. We say that the trajetory

K(τ)

^is

a ontinuously dierentiable funtion if for a xed oordinate representation its oordinate

funtionsareontinuouslydierentiablefuntionsofthetime.Thisisequivalenttotheproperty

that the support funtion

h (K(τ))

^is ontinuously dierentiable as the funtion of the time

τ

The dierentiability property of the trajetory impliesthe analogous dierentiability property

of the hange of the norm of a x vetor sine the points of the boundary of the unit ballhas

anequation of the form

r ^τ = (r(ϕ 1 , · · · , ϕ n − 1 )) ^τ

^. ^W^e ^an ^onlude ^that ^if ^the ^trajetory

K(τ)

is a ontinuously dierentiable funtion, this holds also for the funtion

τ → p

[s, s] ^τ

^. ^In ^a

spae

S

^with ^an ^inner ^produt ^the ^polarity ^equation ^implies ^the ^required assumption. If

S

^is

(only)a smooth normedspae withasemi innerprodut, weneedfurther omments.Sinefor

adierentiable normfuntion MShane's equality holds, we have

[x, y] ^τ = k y k ^τ (( k · k ^τ ) ^′ _x (y)) = k y k ^τ ( k · k ^′ x (y)) ^τ .

On the other hand, the funtion

( k · k ^′ x (y)) ^τ

^is ^also ontinuously dierentiable funtion of

y

thusthe threadusingonthe normfuntionabove isappliableforit,too.Thismeansthat the

dierentiabilitypropertyofthetrajetoryimpliestheanalogousdierentiabilitypropertyofthe

funtion

τ → ( k · k ^′ x (y)) ^τ

^. ^Using ^the ^rule ^of ^the ^produt ^funtion ^we ^also^have ^that

τ → [x, y] ^τ

isontinuously dierentiable if the trajetory

τ → K (τ )

^holds ^this ^property.

We now dene a dierentiable trajetory through the points

(τ _i , K(τ _i ))

^. ^If

τ, τ _i ^′ ∈ [τ _i , τ _i+1 ]

denoteby

K Bezier (τ )

^the ^formal^Bezier^spline^of^seond^order^through^the ^points

(τ i , K(τ i ))

^and

(τ i+1 , K(τ i+1 ))

^with "tangents"through the point

(τ _i ^′ , L(τ _i ^′ ))

^. ^Thus^we ^have ^by ^denition

K Bezier (τ) :=

1 − τ − τ _i τ i+1 − τ i

2 K(τ i )+2

1 − τ − τ _i τ i+1 − τ i

τ − τ _i τ i+1 − τ i

L(τ i )+

τ − τ _i τ i+1 − τ i

2 K(τ i+1 ),

where the addition is the Minkowski addition and the produt is the respetive homotheti

mapping. If we assume that for all values of

i

⁽

1 < i < s

⁾ ^the ^body

K(τ i )

^is ^a ^Minkowski

onvex ombination of the bodies

L(τ _i ^′ )

^and

L(τ _i+1 ^′ )

^the ^funtion

K _Bezier (τ )

^is ^valid ^on ^the

wholeinterval

[τ ₁ , τ _s ]

^. ^Sine ^for ^positive ^onstants

α

β

^we ^have

h αK ^′ +βK ^′′ (x) = αh K ^′ (x) + βh K ^′′ (x),

we alsoget that

K Bezier (τ )

^is^a ontinuously dierentiable trajetory in its whole domain.We havetoproveyetthatforaxed

τ

^, ^the^set

K Bezier (τ )

^is^a^entrally^symmetri^onvex^ompat

body with

C ²

^-lass ^boundary ^but ^these ^statements ^follow immediately from the onept of Minkowski linear ombination.

Finally we normalize this trajetory under the volume funtion and extrat it to the whole

T

^. ^The ^funtion

K(τ )

^determines ^a ^required deterministi time-spae model if we dene it as

follows:

K(τ) =

 

 



 

 

n

q vol(B E )

vol(K Bezier (τ s )) K Bezier (τ s )

^if

τ s < τ

n

q vol(B E )

vol(K Bezier (τ)) K Bezier (τ )

^if

τ 1 ≤ τ ≤ τ s

n

q vol(B E )

vol(K Bezier (τ 1 )) K Bezier (τ 1 )

^if

τ < τ 1

An important onsequeneof this theorem that withoutlossof generalitywe an assume,that

the time-spae model isdeterministi.

APPENDIX A

Relativity theory in time-spae

Our model desribed in the previous setion an be onsidered also as a model of the universe 1

. The

deterministi variant obviously ontains as aspeial ase the model of Minkowski spae-time. On the other

hand it an be extended to ageneralization of the Robertson-Walkerspae-time, too. The advantage of our

modelthat

S

^an^be^onsidered^also^as^a^general^normed^spae^(without^inner^produt).

Thetime-spaean bedened in a moreonvenientway, using ashapefuntion. Itregulatesthemethods of

alulationsintime-spaeandgivesthepossibilitytorewritetheequalityofspeialandglobalrelativity.

A.1. Onthe formulasof speial relativitytheory

Considertheupperpartoftheimaginarysphereofparameter

c

ⁱⁿ^afour-dimensionaldeterministitime-spae model. Withouttheimaginaryunit sphereweonsider theimaginaryunit sphere

H c

^of^parameter

c

^with ^the

orrespondingprodut

[x ^′ , x ^′′ ] ^+,T := [s ^′ , s ^′′ ] ^τ ^′′ + c ² [τ ^′ , τ ^′′ ]

^. ^Pratially^the ^onstant

c

^an^be^onsidered ^as^the

speedofthelightinvauum.Assumethattheshape-funtionisatwo-timesontinuouslydierentiablefuntion.

Weneedtwoaxiomstointerpretintime-spaeoftheusualaxiomsofspeialrelativitytheory.Firstweassume

that:

Axiom A.1.1. The laws of physis are invariant under transformations between frames. The laws of physis

willbethe same whether you are testingthem in frame "at rest", ora frame moving witha onstant veloity

relative tothe "rest"frame.

AxiomA.1.2. The speedof lightinavauumismeasuredtobethe sameby allobserversin frames.

Thesetwoaxiomsanbetransformedintothelanguageofthetime-spaebythemethodofMinkowski[123℄.To

thisweuse

H c

^introdued^and^the^group

G c

^as^the^set^of^those^isometries^of^the^spae^whih^leave^invariant

H c

Suhanisometryanbeinterpretedasaoordinatetransformationofthetime-spaewhihsendstheaxisofthe

absolutetimeinto anothertime-axis

t ^′

^,^and^also^maps^theintersetionpointoftheabsolutetime-axiswiththe imaginarysphere

H c

^into^theintersetionpointofthenewtime-axisand

H c

^.^An^isometry ^of^the^time-spae^is

alsoahomeomorphismthusitmapsthesubspae

S

^into^a^topologial^hyperplane

S ^′

^of^the^embedding^normed

spae.

S ^′

^is^orthogonal^to^the^new^time-axisⁱⁿ^the^sense^that^its^tangent^hyperplane^at^the^origin^is^orthogonal

t ^′

^with^respet^to^the^produt^of^the^spae.^Of^ourse^the^new^spae-axes^areontinuouslydierentiableurves in

S ^′

^whih ^tangents^at ^the^origin^are^orthogonal ^to ^eah ^other. ^Sine^the ^absolute^time-axis^is ^orthogonal^to

the imaginary sphere

H c

^the ^new ^time-axis

t ^′

^must ^holds ^this ^property, ^too. ^Thus ^the investigations in the previoussetion are essentialfrom this point of view. Assumingthat the denition of the time-spae implies

thispropertyweangetsomeformulassimilartoofspeialrelativity.Wenotethat thefuntion

K(v, τ )

^holds

theorthogonalitypropertyof vetorsof

S

^and ^by^the^equality

[K(v, τ ), K(v, τ )] ^τ = k v k ² E

^we^an^see ^also^that

theformulason time-dilatation and length-ontrationare valid, too.This impliesthat usingthe well-known

notations

β = ^kvk _c ^E

γ = √ ¹

1−β ²

wegetthattheonnetionbetweenthetime

τ 0

^and

τ

^of^an^event^measuring^by

twoobserversoneofatrestandtheothermoveswithanonstantveloity

k v k ^E

^with^respet^to^the^time-spae

τ = γτ 0

^.^Consider^now^a^moving^rod^whih^points^move^onstant^veloity^with^respet^to^the^time-spae^suh

thatitisalwaysparalleltotheveloityvetor

K(v, τ)

^.^Then^we^have

k v k ^E = ^L _T ⁰

^where

T

^is^the^time^alulated

fromthelength

L 0

^and^the^veloity^vetor

v

^by^suh^an^observer^whih^moves^with^the^rod. ^Another^observer

analulatethelength

L

^from^the^measured^time

T 0

^and ^the^veloity

v

^by^the^formula

k v k ^E = _T ^L ₀

^.^Using^the

aboveformulaofdilatationwegettheknownFitzgeraldontrationof therod:

L = L 0

p 1 − β ² = ^L _γ ⁰

Lorentz transformationin timespae alsobasedontheusual experimentin whih wesend arayoflightto a

mirrorin diretionoftheunitvetor

e

^with^distane

d

^from^me.

Ifweatrestweandetermineintimespaethepoints

A

C

^and

B

^of^departure,^turn^and^arrival^of^the^ray^of

light, respetively.

A

^and

B

^are ^on^the^absolute ^time-axis^at^heights

τ A

^, ^and

τ B

^, respetively. Thepositionof

C

^is

(τ C − τ A )K(ce, τ C − τ A ) + τ C e 4 = τ B − τ A

2 K

ce, τ B − τ A

2 + τ B + τ A

2 e 4 ,

Inthisappendixwehektheusabilityouroneptinpratie.Despitetheontentofthisappendixbelongs

totheareaoftheoretialphysisitis stronglyonnetedtotheuselessofmymathematialinvestigations.

115

sineweknowthatthelighttaketheroadbakandforthoverthesametime.Weobservethatthenormofthe

spae-likeomponent

s C

^is

k s C k ^τ ^C = c ^τ ^B ^−τ ₂ ^A

^asⁱⁿ ^the^usual ^ase^of^spae-time.

The moving observer synhronized its lok with the observer at rest in the origin, and moves in the

dire-tion

v

^with ^veloity

k v k ^E

^. ^We ^assume ^that ^the ^moving ^observer^also ^sees ^the ^experiment ^thus ^its ^time-axis

orresponding to the vetor

v

^meats ^the ^world-line ^of ^the ^light ⁱⁿ ^two ^points

A ^′

^and

B ^′

positioning on the respetive urves

AC

^and

CB

^. ^This ^implies ^that ^the ^respetive ^spae-like ^omponents ^of ^the ^world-line ^of

the light and the world-line of the axis are parallels to eah other in every minutes. By formula we have:

k v k ^E K(e, τ ) = K(v, τ )

^.^F^rom^this^we^get^the^equality

τ A ^′ K(v, τ A ^′ ) + τ A ^′ e 4 = (τ A ^′ − τ A )K(ce, τ A ^′ − τ A ) + τ A ^′ e 4

1+β

^,^and^we^determine^the

newtimeoordinateofthepoint

C

^with^respet^to ^the^new^oordinate^system:

(τ C ) 0 = (τ A ^′ ) 0 + (τ B ^′ ) 0

On the other hand we also havethat the spae-likeomponent

((s C ) 0 ) S

^of ^the transformedspae-likevetor

(s C ) 0

^arise^also^from^a^vetor^parallel^to

e

^thus^it^is^of^the^form

K(((s C ) 0 ) S , τ ) = k ((s C ) 0 ) S k ^E K(e, τ )

^. ^F^or^the

world-lineofthelightinto

S

^with^the^denition

L b : K((s C ) 0 , (τ C ) 0 ) 7→ γ (K(s C , τ C ) − K(v, τ C )τ C ) .

Theonnetionbetweenthespae-likeoordinatesofthepointwithrespettothetwoframesnowhasamore

familiar form.Heneforth theLorentztransformationmeansforus theorrespondene:

s 7→ K(s \ ^′ , τ ^′ ) = γ (K(s, τ ) − K(v, τ )τ )

and theinverseLorentztransformationtheanotherone

K(s \ ^′ , τ ^′ ) 7→ K(s, τ) = γ (K(s ^′ , τ ^′ ) + K(v, τ ^′ )τ ^′ ) τ ^′ 7→ τ = γ τ ^′ + [K(s ^′ , τ ^′ ), K(v, τ ^′ )] ^τ ^′

c ²

! .

Firstnotethatweandeterminetheomponentsof

(s C ) 0

^with^respet^to^the^absolute^oordinate^system,^too.

Sine

(s C ) 0

^and

τK(v, τ ) + τ e 4

^are^orthogonal^to^eah ^other^we^get^that

In document Cvexiy ad E (Pldal 124-127)

s as the length of a xed segment relative to the length of the diameter of the unit ball

[O, P ]

O

C 2

τ

K(τ)

h (K(τ))

τ

r τ = (r(ϕ 1 , · · · , ϕ n − 1 )) τ

K(τ)

τ → p

[s, s] τ

S

S

[x, y] τ = k y k τ (( k · k τ ) ′ x (y)) = k y k τ ( k · k ′ x (y)) τ .

( k · k ′ x (y)) τ

y

τ → ( k · k ′ x (y)) τ

τ → [x, y] τ

τ → K (τ )

(τ i , K(τ i ))

τ, τ i ′ ∈ [τ i , τ i+1 ]

K Bezier (τ )

(τ i , K(τ i ))

(τ i+1 , K(τ i+1 ))

(τ i ′ , L(τ i ′ ))

K Bezier (τ) :=

1 − τ − τ i τ i+1 − τ i

2

K(τ i )+2

1 − τ − τ i τ i+1 − τ i

τ − τ i τ i+1 − τ i

L(τ i )+

τ − τ i τ i+1 − τ i

2

K(τ i+1 ),

i

1 < i < s

K(τ i )

L(τ i ′ )

L(τ i+1 ′ )

K Bezier (τ )

[τ 1 , τ s ]

α

β

h αK ′ +βK ′′ (x) = αh K ′ (x) + βh K ′′ (x),

K Bezier (τ )

τ

K Bezier (τ )

C 2

T

K(τ )

K(τ) =

 

 



 

 

n

q vol(B E )

vol(K Bezier (τ s )) K Bezier (τ s )

τ s < τ

n

q vol(B E )

vol(K Bezier (τ)) K Bezier (τ )

τ 1 ≤ τ ≤ τ s

n

q vol(B E )

vol(K Bezier (τ 1 )) K Bezier (τ 1 )

τ < τ 1

S

c

H c

c

[x ′ , x ′′ ] +,T := [s ′ , s ′′ ] τ ′′ + c 2 [τ ′ , τ ′′ ]

c

H c

G c

H c

t ′

C ²

r ^τ = (r(ϕ 1 , · · · , ϕ n − 1 )) ^τ

[s, s] ^τ

[x, y] ^τ = k y k ^τ (( k · k ^τ ) ^′ _x (y)) = k y k ^τ ( k · k ^′ x (y)) ^τ .

( k · k ^′ x (y)) ^τ

τ → ( k · k ^′ x (y)) ^τ

τ → [x, y] ^τ

(τ _i , K(τ _i ))

τ, τ _i ^′ ∈ [τ _i , τ _i+1 ]

(τ _i ^′ , L(τ _i ^′ ))

1 − τ − τ _i τ i+1 − τ i

1 − τ − τ _i τ i+1 − τ i

τ − τ _i τ i+1 − τ i

τ − τ _i τ i+1 − τ i

L(τ _i ^′ )

L(τ _i+1 ^′ )

K _Bezier (τ )

[τ ₁ , τ _s ]

h αK ^′ +βK ^′′ (x) = αh K ^′ (x) + βh K ^′′ (x),

C ²

[x ^′ , x ^′′ ] ^+,T := [s ^′ , s ^′′ ] ^τ ^′′ + c ² [τ ^′ , τ ^′′ ]

t ^′

S ^′

S ^′

t ^′

S ^′

t ^′

[K(v, τ ), K(v, τ )] ^τ = k v k ² E

β = ^kvk _c ^E

γ = √ ¹

1−β ²

k v k ^E

k v k ^E = ^L _T ⁰

k v k ^E = _T ^L ₀

p 1 − β ² = ^L _γ ⁰

k s C k ^τ ^C = c ^τ ^B ^−τ ₂ ^A

k v k ^E

A ^′

B ^′

k v k ^E K(e, τ ) = K(v, τ )

τ A ^′ K(v, τ A ^′ ) + τ A ^′ e 4 = (τ A ^′ − τ A )K(ce, τ A ^′ − τ A ) + τ A ^′ e 4

(τ C ) 0 = (τ A ^′ ) 0 + (τ B ^′ ) 0

K(((s C ) 0 ) S , τ ) = k ((s C ) 0 ) S k ^E K(e, τ )

s 7→ K(s \ ^′ , τ ^′ ) = γ (K(s, τ ) − K(v, τ )τ )