Funiv

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 figures Subj-class: General Physics Key words: Universality, New Insight in Physics

2

New Insight in Physics 1. Kinematics

1. A maximum limit speed exists and it is the same for all reference frames (c = const = 3.108 m/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 x0 axes with velocity V = (±V, 0, 0) , fig.1. Let at the initial time t = 0 = t0 , O ≡ O 0 .

Figure 1. Galilean transformations

Let us present the Galilean transformations r = r0 + ro0 in Cartesian coordinates: x = x0 + V.t0 y = y0 z = z0 t = t0

x0 = x − V.t y0 = y z0 = z t0 = t

(1)

Let us emit pulse of light at t = t0 = 0, when the coordinate systems coincide, from a point source at the common origin O ≡ O 0 in the direction of the axes x and x0 , fig. 2.

Figure 2. Pulse of light

According to 1. the motion of light pulse in K and K 0 is described by x = c.t = t and x0 = c.t0 = t0 , (c = 1). Obviously from t = x 6= x0 = t0 ⇒ t 6= t0 , i.e. the course of

3

New Insight in Physics

time is not absolute with velocity close to c. From t 6= t0 it follows that x 6= x0 + V.t0 and x0 6= x − V.t. Introducing a restoring factor γ we obtain the equations: x = γ(x0 + V.t0 ) y = y0 z = z0 t 6= t0

x0 = γ(x − V.t) y0 = y z0 = z t0 6= t

Let us determine the factor γ from the transformations of the coordinates and the motion of light pulse in the two frames. x = γ(x0 + V.t0 ) x0 = γ(x − V.t) We substitute x = t, x0 = t0 , which results in: t = γ(t0 + V.t0 ) = γ(1 + V )t0 t0 = γ(t − V.t) = γ(1 − V )t Multiplying these two equations and after dividing by tt0 , we determine γ, β = V /c = V, c = 1: 1 γ=√ (2) 1 − β2 Let us determine transformations of the time: we substitute x ↔ t, x0 ↔ t0 in x0 = γ(x − βt) x = γ(x0 + βt0 ),

which results in transformations of the time: t0 = γ(t − βx) t = γ(t0 + βx0 )

(3)

Combining we get: 1.1. Lorentz transformations

0

0

x +βt x= √ 2

x0 = √x−βt 2

y = y0 z = z0 t0 +βx0 t= √

y0 = y z0 = z t0 = √t−βx 2

1−β

1−β 2

1−β

(4)

1−β

It is impossible for physical objects to move with speed exceeding c ! transformations turn into the Galilean ones, if V = β << c = 1.

Lorentz

4

New Insight in Physics 1.2. Transformation of velocities From Lorentz transformations, we have: 0

√+β.dt dx = dx 2

0

1−β 0

dy = dy dz = dz 0 0 0 dt = dt√+β.dx2 1−β

√ dx0 = dx−β.dt 2 1−β

dy = dy dz 0 = dz √ dt0 = dt−β.dx 2 0

1−β

We divide the first three equations by the forth and we get transformation of velocities: vx = vy = vz =

vx0 +V 1+β.v √ x0 vy 0 1−β 2 1+β.v √ x0 vz 0 1−β 2 1+β.vx0

v x0 = vy 0 = vz 0 =

vx −V 1−β.v √ x0 vy 1−β 2 1−β.v √ x0 vz 1−β 2 1−β.vx0

(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 0 with the rod, V = v. The length of the rod in K 0 is l0 = l0 = x02 − x01 . In K, the length of the rod is l = v.∆t, v and ∆t are the velocity and the time interval relative to an observer located at point P . The upper coordinates and time are linked by Lorentz transformations: xp −v.t2 xp −v.t1 x02 = √ x01 = √ 2 2 1−β

1−β

√ 1 −t2 ) = √ l l0 = x02 − x01 = v.(t 1−β 2 1−β 2 √ l = l0 1 − β 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, f ig.4. Let t be the time measured by a clock of the frame K and t0 be the time measured by a clock stationary linked to the moving body. The length of the trajectory relative to

5

New Insight in Physics

Figure 4. Time dilation

√ the moving body will be; l0 = l 1 − β 2 , as the vector of the velocity is always tangent to 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 two clocks: √ l 1−β 2 l0 0 ∆t = v = v (7) ∆t0 < ∆t Time dilation of the moving clocks! ∆t0 ∆t = √ > ∆t0 , ∆t0 = ∆t0 = ∆τ (8) 1 − β2 , ∆τ is called the proper time! The above results can easily be generalized for motion in an arbitrary manner on the trajectory: √ √ dl 1−β 2 dl0 dτ = v = = dt 1 − β2 v q (9) ∆t √ R dτ = dt 1 − β 2 ⇒ ∆τ = dt 1 − β 2 (t) 0

The time dilation has been directly confirmed experimentally. For example, the positive charged pions have a mean lifetime 2.5.108 s and they decay spontaneously into a muon and a muon neutrino - π + → µ+ + νµ . 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 2.5.108 s is the proper lifetime of pions. For speed close to c, with respect to the accelerator: v.∆t0 smax = c.∆t0 < s = v.∆t = √ 1 − β2

, but s0 = 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 one old proverb of Shrimad Bhagavatam - one of the sacred writings of ancient India.

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New Insight in Physics

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 your cities 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 0 , ∆t0 = −γβ∆x 6= 0. ∆t0 = 0, if and 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 = dτ = dt 1 − β 2 > 0 ⇒ ∆s = ∆τ > 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). q √ (10) ds = 1.dτ = dt 1 − β 2 = dt2 − dr2 , dr = v.dt The variable t has a behavior similar to that of the variables x, y and z. Let us substitute (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 = dτ (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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New Insight in Physics 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 τ . ◦→ ◦→

u =

dr (dt, dr) = γ(1, v), = √ dτ dt 1 − β 2

◦→

| u |=

q

γ 2 (1 − v 2 ) = 1

(12)

◦→

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

◦→

1 d( u )2 ◦→ d u ◦→ ◦→ =0= u = u . a, 2 dτ dτ Acceleration is not absolute (a 6= a0 )! ◦→

a =

(

◦→

d u dτ

◦→ 2 a

   

◦→

◦→

a =

= γ 4 [a.b , a + b × (b ×a)],

) = −γ 6 [a2 − (b × a)2 ] < 0,

du dτ

⇒

◦→

a ⊥u

(13)

a = dv/dt

(14)

→

b=β =v √ γ = 1/ 1 − v2

a0 = dτd γ = γ dtd γ = γ 4 v.a = γ 4 a.b, a = dτd γv = γ dtd γv = γ 2 a + γ 4 b(b.a)    a = γ 4 [a + b(b.a) − aβ 2 ] = γ 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 E and mass m. The four-vector of the momentum is: ◦→ m mv ◦→ p = m u = (√ √ , ) (15) 1 − β2 1 − β2 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. ◦→

◦→ dp = F, dτ

⇒

◦→

◦→

m a = F

, F=

dp dp F =√ =√ 2 dτ 1 − β dt 1 − β2

(16)

8

New Insight in Physics The space part of 4D force F is expressed by Newtonian force F . ⇒

◦→ ◦→

◦→ ◦→

F. u =m a . u =0

⇒

F0 = F.v = P =

dp0 dE = , dτ dτ

(17)

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

1−β 2

(18)

p = p0 .v = E.v ◦→ ( p )2 = E 2 − p 2 = m 2

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 relativity the 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 . mv2 m mβ 2 − m ≈ = , if v << c = 1 (19) 2 2 1 − β2 To consider a mechanical system with respect to a reference frame, in which p = 0: T = E − E0 = √

E0 = M =

X

E0i +

X

Ti + U =

X

mi +

X

Ti + U ,

U is the interaction energy. M=

X

mi +

X

Ti + U 6=

X

mi

(20)

Mass is absolute but not additive. Using the term ”relative mass” is a mistake and P P should be avoided! The binding energy is calculated by: Eb = Ti + U =M − mi The mass of stable nuclei is less than the mass of its constituent nucleons and this is only an experimental confirmation of the upper equations.

References [1]. Marinchev, E., Universality, LANL e-print: physics/0211106