space-time diagram(final)

Space-time diagram in the unit disc From non-Euclidean models to space-time diagram in the unit disk

3/25/2015

cross ratio unit diskPoincare disk model

hyperbolicgeometry

specialrelativity

Minkowskidiagram

conformal coordinates system psi-model

space - timediagram

conformaloptics

in the unit disk

1829

1905

1908

Varichak 1921

Fig.1- Historical background

Abstract

A space-time diagram in the unit disk is presented by use of developed non-Euclidean models.

Keywords

Space-time diagram, unit disk, cross ratio, non-Euclidean models

Introduction

1- Poincare presented his model in the unit disk and proved that the hyperbolic geometry is

equivalent with his model.(Ref.1)

2- Varichak proved (1921) that the hyperbolic geometry (1829) is equivalent with the theory of

special relativity (1905).(Ref.2)

3- Minkowski presented his diagram (1908) and proved that the theory of special relativity is

equivalent with his diagram.(Ref.3)

From 1, 2 and 3 we conclude that it should be a representation of space-time in the unit disk

that is equivalent with the Minkowski diagram at least locally.

In this paper such a diagram will be presented.

In section-1 all initial definitions are presented. In section-2 the 2-d unit disk equipped with four degrees of

freedom is presented. In section-3 cross ratio matrix of space-time interval is defined. In sections-4 and 5

Minkowski metric, addition law of velocities and time dilation are derived for antimatter and matter respectively.

Psi-model (mentioned in fig.1) a novel non-Euclidean model in the unit disk is presented in appendix-1. Conformal

optics (mentioned in fig.1) a new approach to study classical optics extracted from present paper naturally and is

presented in Appendix-2.

Section-1 (Initial definitions)

Definition one: An event (world-point) in 4-dimensional space-time is shown by a point inside

the unit disk. (See fig.2)

Definition two: A world-line in 4-dimensional space-time is represented with a continuous curve

inside the unit disk. (See fig.3)

Definition three: An inertia world-line in 4-dimensional space-time is represented with part of a

circle inside the unit disk. (See fig.3)

Let define velocity of moving particle as follows:

v = −i tanψ 1

Where v represents velocity of moving particle and ψ represents angle between an inertia

world-line (of moving particle) and the unit disk at their intersection points.

Definition four: A ψ-model is defined in appendix-1.

w

Fig.2- An event is shown by a point “W” in the unit disk Fig.3- A world-line in the unit disk

W2

W1

Fig.4- Three types of inertia world-lines Fig.5- Inertia world-lines between two points

In fig.4 all types of inertia world-lines are shown. Note that circle inside the unit disk represents

particle with pure imaginary velocity. In fig.5 a circle is shown so that W1W2 is its diameter and

all inertia world-lines between W1 and W2 are inside it. So the unit disk may be separated into

three different regions as figure-6.

Imaginary

V>1

V>1

V>1

45°

45°

V

0<V<1

0<V<1

Fig.6- Three regions in the unit disk: 0<V<1(matters), pure imaginary V (anti-matters) and V>1 (tachyons)

Section-2 (Degrees of freedom)

w1

w2

w3

w6

w5

w4

d

w8

w7Ø

psi

tetha

Fig.7: Degrees of freedom are ψ, φ, ϴ and unit vector W1W6 (direction of P-line from W1 toward W2)

In fig.7 4-dimensional unit disk is presented. Note that two variables ψ and φ represent the ψ-

lines and φ-lines that pass through points (events) W1 and W2. In fact you need to know at

least four variables to move from given event W1 and pass through given event W2. It is clear

that dimension of such space-time is equal to four. One may increase dimensions of space-time

by adding needed variables. In present paper we just study 4-dimensional space-time. Anyway

we define conformal coordinates system as set of angular variables {ψ, φ, α, iϴ}. Angles ψ, φ

and α represent spatial coordinates and iϴ represents time. (Angle ψ is shown by “psi” in fig.7)

Section-3 (cross ratio matrix of space-time interval)

There are exactly 360 cross ratios for four arbitrary points among six points of W1, W2, W3,

W4, W7 and W8. But they are not independent. In fact distinct members reduce up to 15

(360/4! =15). One may arrange them in symmetric matrix 4*4 as follows:

(W1W2, W3W4), (W2W3, W4W7), (W2W4, W7W8), (W1W2, W3W8)



(W1W2, W4W7), (W1W4, W7W8), (W2W4, W3W8), exp(𝑖𝛼)

Where “α” represents direction of Poincare-line from W1 toward W2

Cross ratio matrix Ϯ may be divided in two symmetric matrices as follows:

Amplitude matrix Ϯ𝐴 = 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑊𝑖𝑊𝑗 ,𝑊𝑝𝑊𝑞 = 𝐴𝑏𝑠[ WiWp

W𝑗W𝑝

WjWq

W𝑖W𝑞 ] 2

Phase matrix Ϯ𝑃 =(𝑊𝑖𝑊𝑗 ,𝑊𝑝𝑊𝑞)

𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒(𝑊𝑖𝑊𝑗 ,𝑊𝑝𝑊𝑞) 3

Simple calculations yield:

Ϯ𝑃 1,1 = 1 Ϯ𝑃 1,1 = 𝑒𝑖𝜓 Ϯ𝑃 1,1 = 𝑒𝑖𝜑 Ϯ𝑃 1,1 = 𝑒𝑖𝛳

Ϯ𝑃 1,1 = 𝑒𝑖𝜓 Ϯ𝑃 1,1 = 1 Ϯ𝑃 1,1 = 𝑒𝑖𝛳 Ϯ𝑃 1,1 = 𝑒𝑖𝜑

Ϯ𝑃 1,1 = 𝑒𝑖𝜑 Ϯ𝑃 1,1 = 𝑒𝑖𝛳 Ϯ𝑃 1,1 = 1 Ϯ𝑃 1,1 = 𝑒𝑖𝜓

Ϯ𝑃 1,1 = 𝑒𝑖𝛳 Ϯ𝑃 1,1 = 𝑒𝑖𝜑 Ϯ𝑃 1,1 = 𝑒𝑖𝜓 Ϯ𝑃 1,1 = 𝑒𝑖𝛼

Fig.8: Symmetric 4*4 phase matrix of cross ratio

Calculations of amplitude matrix are more complicated. Referring to fig.7 results are as follows:

Ϯ𝐴 1,1 =1−𝑡𝑎𝑛 𝜓 𝑡𝑎𝑛 (𝑑 𝑐𝑜𝑠 𝜓)

1+𝑡𝑎𝑛 𝜓 𝑡𝑎𝑛 (𝑑 𝑐𝑜𝑠 𝜓) 4

Ϯ𝐴 3,3 =1−𝑡𝑎𝑛 𝜑 𝑡𝑎𝑛 (𝑑 𝑐𝑜𝑠 𝜑)

1+𝑡𝑎𝑛 𝜑 𝑡𝑎𝑛 (𝑑 𝑐𝑜𝑠 𝜑) 5

ϮA 4,4 = 1 6

Ϯ𝐴 2,2 = W3W4, W7W8 = (W3W7

W3W8)(

W4W8

W4W7) 7

W3W7 =

2 1 − cos𝜓 cos𝜑{ 1 − 𝑑2 (cos𝜓)2 tan𝜓 + 𝑑 cos𝜓 1 − 𝑑2 (cos𝜑)2 tan𝜑 + 𝑑 cos𝜑 + 1 − 𝑑2 (cos𝜓)2 − 𝑑 sin𝜓 1 − 𝑑2 (cos𝜑)2 − 𝑑 sin𝜑 }

W3W8 =

2 1 − cos𝜓 cos𝜑{ 1 − 𝑑2 (cos𝜓)2 tan𝜓 + 𝑑 cos𝜓 1 − 𝑑2 (cos𝜑)2 tan𝜑 + 𝑑 cos𝜑 − 1 − 𝑑2 (cos𝜓)2 − 𝑑 sin𝜓 1 − 𝑑2 (cos𝜑)2 − 𝑑 sin𝜑 }

lim𝑑→0 Ϯ𝐴 2,2 = [cos (

𝜓−𝜑

2)

cos (𝜓+𝜑

2)]2 8

limd→0 ϮA 1,1 = 1 − 2d sinψ tan β

β , β = d cosψ 9

W2W3 =cos (ψ+d cos ψ)

cos ψ 10

W2W4 =cos (ψ−d cos ψ)

cos ψ 11

W2W7 =cos (φ−d cos φ)

cos φ 12

W2W8 =cos (φ+d cos φ)

cos φ 13

ϴ = ϴ1 + ϴ 2 , (sinϴ1/2 = d cosψ), (sinϴ 2/2 = d cosφ) 14

Section-4 (space interval and time interval)

The most important part of this paper is defining time and spatial intervals correctly. In addition

any model of 4-dimensional space-time in the unit disk must have Minkowski metric locally.

4-1) time interval Δt

Let define time interval Δt between events W1 and W2 as follows:

Δt = iϴ 15

Where ϴ represents intersection angle of ψ-line with φ-line pass through events 𝑊1 and 𝑊2

i = −1

Let assume that variables are time “t” and angle “ψ”. So for small enough “d” we have:

Δt = 2id cos𝜓 16

Where “d” represents Euclidean distance between W1 and W2 (without any restriction W1 is

located at the center of the unit disk)

And “ψ” represents cut angle between the unit disk and ψ-lines path through W1 and W2.

Note that exactly two ψ-lines path through W1 and W2.

4-2) spatial interval Δσ

Let define spatial interval Δσ between events W1 and W2 as follows:

Δσ = − log[Ϯ𝐴 1,1 ] = −log(1−𝑡𝑎𝑛 𝜓 𝑡𝑎𝑛 (d 𝑐𝑜𝑠 𝜓)

1+𝑡𝑎𝑛 𝜓 𝑡𝑎𝑛 (d 𝑐𝑜𝑠 𝜓)) 17

Note that Poincare metric is derived from equation 17 when ψ=𝛑/2. Anyway for small

enough “d” we have:

Δσ = 2d sin𝜓 18

4-3) Minkowski metric Δs:

From equations (16) and (18) we have locally:

Δσ2 − Δt2 = 4d2 = (Δ𝑠)2 19

Equation 19 states that Minkowski metric has hold in the unit disk locally.

4-3-1) Velocities adding law

Velocity is defined as follows:

v = Δ𝜎/𝛥𝑡 20

From (16) and (18) we have:

v = −i tanψ 21

We had defined such velocity before in (1). Now let calculate adding of velocities. Equation (21)

means that

v 𝜓1 + 𝜓2 = v 𝜓1 +v 𝜓2

1+v 𝜓1 v 𝜓2 22

4-3-2) time dilation


Δt =Δ𝜏

1−v2 23

Where Δτ represents the proper time

Section-5 (Pure imaginary angle - matters)

Equation (21) states that for real angle of ψ velocity is pure imaginary. Let assume that mass of

“m” has real value. Then kinetic energy of 1

2mv2 is negative. Negative energy refers to

antimatter. Now let assume that angle of ψ is pure imaginary. Hence velocity of (21) and kinetic

energy of 1

2mv2 have real values. From (21), (16) and (18) we have:

v = tanhψ 24

Δt = 2id cosh𝜓 25

Δσ = 2id sinh𝜓 26


Δσ2 − Δt2 = 4d2 = (Δ𝑠)2

Also from (24) we have:

v 𝜓1 + 𝜓2 = v 𝜓1 +v 𝜓2

1+v 𝜓1 v 𝜓2

And finally from (24) and (25) we have:

Δt =Δ𝜏

1−v2

Section-6 (Conclusion)

It is proved that a representation of space-time in the unit disk exists. Also the space-time

diagram in the unit disk has Minkowski metric locally. The unit disk separated the three regions,

matter region, antimatter region and tachyon region. Since the Minkowski diagram does not

contain the antimatter region, the diagram presented here has more advantages than the

Minkowski one. Space-time interval, addition law of velocities and time dilation equations has

hold for matter and antimatter symmetrically. All results obtained free from customary

coordinates systems. Novel coordinates system applied here called conformal coordinates

system is just in the basis of angles. So all obtained results are invariant under conformal

mappings from physical scene (Appendix-2) to the unit disk.

References

Ref.1 – Poincare disk model, Wikipedia

Ref.2 - Varichak, Wikipedia

Ref.3 – Minkowski diagram, Wikipedia

Appendices

Appendix-1

A non-Euclidean model in the unit disk

Appendix-2

Theory of conformal optics

space-time diagram(final)

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