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FLAT TIME-SPACE UNIVERSALITY

Emil Marinchev

Technical University of Sofia

Physics Department

8, Kliment Ohridski St.

Sofia-1000, BG

e-mail: [email protected]

Abstract. There is only one essential difference between newtonian mechanics and

Special Relativity. It is of kinematical nature, not of dynamical one. A very simple

new formulation of Special Relativity is presented.

Comments: 8 pages, 4 figuresSubj-class: General Physics

Key words: Universality, New Insight in Physics

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New Insight in Physics 2

1. Kinematics

1. A maximum limit speed exists and it is the same for all reference frames

(c = const = 3.108m/s). The speed of light in vacuum is equal to this speed.

This principle points out that the bodies cannot move with infinite speed for they

would be at an infinite distance from their initial place at the moment different from the

initial time. This points out that an upper limit speed exists (universal speed, perfect

measure, c = 1) and it is the same for all observers. This principle is valid in General

Relativity too. We consider two frames in relative motion along the x and x axes with

velocity V = (V, 0, 0) , fig.1. Let at the initial time t = 0 = t, O O.

Figure 1. Galilean transformations

Let us present the Galilean transformations r = r

+ ro in Cartesian coordinates:x = x + V.t

y = y

z = z

t = t

x = x V.ty = y

z = z

t = t

(1)

Let us emit pulse of light at t = t = 0, when the coordinate systems coincide, from a

point source at the common origin O O in the direction of the axes x and x, fig. 2.

Figure 2. Pulse of light

According to 1. the motion of light pulse in K and K

is described by x = c.t = tand x = c.t = t, (c = 1). Obviously from t = x = x = t t = t, i.e. the course of

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time is not absolute with velocity close to c. From t = t it follows that x = x + V.tand x = x V.t. Introducing a restoring factor we obtain the equations:

x = (x + V.t)

y = y

z = z

t = t

x = (x V.t)y

= yz = z

t = tLet us determine the factor from the transformations of the coordinates and the

motion of light pulse in the two frames.

x = (x + V.t)

x = (x V.t)We substitute x = t, x = t, which results in:

t = (t

+ V.t

) = (1 + V)t

t = (t V.t) = (1 V)tMultiplying these two equations and after dividing by tt, we determine , = V /c =

V, c = 1:

=1

1 2 (2)

Let us determine transformations of the time: we substitute x t, x t inx = (x t)x = (x + t),

which results in transformations of the time:

t = (t x)t = (t + x)

(3)

Combining we get:

1.1. Lorentz transformations

x = x+t

12y = y

z = z

t = t+x12

x = xt

12y = y

z = z

t = tx12

(4)

It is impossible for physical objects to move with speed exceeding c ! Lorentz

transformations turn into the Galilean ones, if V =

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1.2. Transformation of velocities

From Lorentz transformations, we have:

dx = dx+.dt

12dy = dy

dz = dz

dt = dt+.dx12

dx = dx.dt

12dy = dy

dz = dz

dt = dt.dx12

We divide the first three equations by the forth and we get transformation of velocities:

vx =vx+V1+.vx

vy =vy

12

1+.vx

vz =vz

12

1+.vx

vx =vxV1.vx

vy =vy

12

1.vx

vz =vz

12

1.vx

(5)

1.3. Corollaries

1.3.1. Length contraction: A rod travels with a velocity v = (v, 0, 0) relative to the

frame K, fig.3.

Figure 3. Length contraction

Let us link a frame K with the rod, V = v. The length of the rod in K is

l = l0 = x

2 x1. In K, the length of the rod is l = v.t, v and t are the velocity andthe time interval relative to an observer located at point P. The upper coordinates and

time are linked by Lorentz transformations:

x

2

= xpv.t2

12x

1

= xpv.t1

12l0 = x

2 x1 = v.(t1t2)12 =l12

l = l0

1 2

(6)

In the direction of the motion, length contracts and transverse dimensions

do not change!

1.3.2. Time dilation: Let l be the length of the trajectory that a moving body draws

relative to frame K,fig.4.

Let t be the time measured by a clock of the frame K and t be the time measured

by a clock stationary linked to the moving body. The length of the trajectory relative to

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Figure 4. Time dilation

the moving body will be; l

= l1 2 , as the vector of the velocity is always tangentto the trajectory and it is in relative motion with respect to the body with a velocity

v. It is easy to determine the relation between time intervals measured by the twoclocks:

t = l

v=

l

12

v

t < t(7)

Time dilation of the moving clocks!

t =t01

2

> t0, t = t0 = (8)

, is called the proper time!

The above results can easily be generalized for motion in an arbitrary manner on

the trajectory:

d = dl

v=

dl

12

v= dt

1 2

d = dt

1 2 =t0

dt

1 2(t)(9)

The time dilation has been directly confirmed experimentally. For example, the positive

charged pions have a mean lifetime 2.5.108s and they decay spontaneously into a muon

and a muon neutrino - +

+ + . Pions could travel a maximum distance for this

time interval limited by the maximum limit speed c = 1. But in the accelerators with

speed close to c they covered distances more than the maximum possible distance. The

explanation of this pseudo paradox is that 2.5.108s is the proper lifetime of pions. For

speed close to c, with respect to the accelerator:

smax = c.t0 < s = v.t =v.t01 2

, but s = v.t0 < smax with respect to the pion.

All subatomic particles of a given type are considered as identical (like

indistinguishable twins). But instead of the paradox of the twins we will retell oneold proverb of Shrimad Bhagavatam - one of the sacred writings of ancient India.

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Once a great raja took his daughter to the Creator Brama to ask whom to choose for

a good husband for his daughter. After arriving at the palace of Brama he waited some

time and made his asking. To his surprise Brama replied: Oh, king, when you return

to the Earth you will not find your people, neither your friends nor relatives, even yourcities and palaces. Although you came here for several seconds, these seconds are equal

to several thousands of years to the people on the Earth. When you return there will

be a new age and you will see the brother of god Krishna - Bala Rama who will be a

good husband to your daughter. When the king returned back to Earth after his several

minutes journey to Brama Loka, he saw a new world and a very different civilization,

people, culture and religion. On Earth several thousand years have passed although

they had traveled only several minutes. And so the daughter, born in the previous age,

married Bala Rama after thousands of years.

1.3.3. Simultaneity: Let us look upon two events: (t1, x1, y1, z1) and (t2, x2, y2, z2),

which are simultaneous in K (t1 = t2, t = 0). In K, t = x = 0. t = 0, ifand only if x = 0, the simultaneity of events is a relative term.

1.3.4. Causality: Let us look upon two events: (t1, x1, y1, z1) and (t2, x2, y2, z2), which

are related by cause in K, t2 > t1, t > 0. ds = d = dt

1 2 > 0 s = > 0 , events which have cause-and-effect relations are time-like.

1.4. Absolute quantities

Separated three dimensional space and time are not absolute but they can be joined

in a four dimensional space (time-space) which is absolute and pseudo-Euclidian. The

motion of a body can be looked upon from an arbitrary chosen reference frame. The

displacements dr, dt and the velocity v are different in different reference frames but

in each frame they are connected by their proper time which is unique and that is why

it is absolute. Let us introduce a new absolute quantity - four dimensional interval (in

short interval).

ds = 1.d = dt

1 2 =

dt2 dr2, dr = v.dt (10)The variable t has a behavior similar to that of the variables x, y and z. Let ussubstitute (t,x,y,z) with (x0, x1, x2, x3). We introduce a pseudo-Euclidian space (space

of Minkowski ) with a four dimensional radius-vector

r presented as (x0, x1, x2, x3) in a

four dimensional Cartesian coordinate system,

dr is the displacement in four dimensional

space. The magnitude of the displacement is independent of the choice of the reference

frame.

ds = | dr | =

dt2 dr2 = dt

1 v2 = d (11)The space of Minkowski is pseudo-Euclidian, i.e. instead of being added with positive

sign the space coordinates are added with negative sign.

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1.5. Four dimension velocity and acceleration

We determine the velocity four-vector as the derivative of the position four-vector

r

with respect to proper time .

u =

dr

d=

(dt,dr)

dt

1 2 = (1, v), |

u | =

2(1 v2) = 1 (12)

Every motion of mass objects in Minkowski space is with | u | = 1. That is why thefour-vector of acceleration will always be orthogonal to the four-vector of velocity.

1

2

d(

u )2

d= 0 =

ud

u

d=

u .

a ,

a =d

u

d a u (13)

Acceleration is not absolute (a = a)!

a = d

u

d= 4[a.b , a + b (ba)], a = dv/dt

(

a )2 = 6[a2 (b a)2] < 0, b =

= v(14)

a0 =dd

= ddt

= 4v.a = 4a.b, = 1/

1 v2a = d

dv = d

dtv = 2a + 4b(b.a)

a = 4[a + b(b.a) a2] = 4[a + b (b a)]

Acceleration is a space-like four-vector ((

a )2 > 0)), velocity is a time-like ((

u )2 > 0)).

Vector, whose magnitude is zero, is a light-like.

2. Dynamics

The first principle of Galilei can be generalized. Physical objects which do not

interact with other objects remain in their state of motion.This principle is

unversal and valid in curve time-spae too. Motion and rest are relative, depending

on the choice of reference frame. A reference frame in which the principle of

inertia is obeyed is called an universal one[1]. We call inertia the quality

of conservation of state of motion, and we characterize it by a set of conservative

quantities - momentum p, angular momentum L, energy Eand mass m. The four-vector

of the momentum is:

p = m

u = (m

1 2,

mv

1 2) (15)

Photons, although having no mass, have momentum and move by inertia if they

are not subject of influence.

The momentum is preserved if there is no interaction,

p =

const.

If there is interaction the state of motion is changed and the conservation quantities

are changed too. The rate of change of the conservation quantities defines quantitatively

the interaction with the surrounding objects, i.e. physical quantities: force, torque,

work and reactive force. The second law of Newton dp/dt = F remains valid but can

be generalized to four dimensional invariant.

d

pd

= F , m a = F , F = dpd

= dp1 2dt = F1 2 (16)

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The space part of 4D force F is expressed by Newtonian force F.

F . u = m a . u = 0 F0 = F.v = P = dEd

=dp0d

, (17)

P - power, E - total energy.E = p0 =

m12

p = p0.v = E.v

(

p )2 = E2 p2 = m2(18)

When p = 0, we get the rest energy E0 = m. This equation presents the

equivalence between rest energy and mass. Generally the equivalence is

between the mass and magnitude of the momentum four-vector, | p | = m ! In relativitythe laws of conservation of mass, energy and momentum are united in a general law of

conservation by the momentum four-vector. The total energy of a body can be presented

as a sum of the rest energy and kinetic energy - E = E0 + T.

T = E E0 = m1 2 m

m2

2=

mv2

2, if v

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