Link:     Back     Contents     Next

7. PHYSICAL MEANING OF NEW TRANSFORMATIONS OF COORDINATES AND TIME

For determination of the length of a moving body let us consider the same two inertial reference frames A and B, which we have considered in chapter 3. Let us suppose, that in the inertial reference frame B, with respect to which some body is at rest, the coordinates of the front end and of the rear end of the body are equal to  x₁' = 0 and x₂' = L_o, where L_o is a proper length of the body (the length of a body in a reference frame, relatively which the body is at rest).

For determination of a length of this moving body using the first method we shall use the second equation from system (6.9) (because the body is at rest in the inertial reference frame B). During the motion of this body together with reference frame B relatively reference frame A the ccordinates of the body ends in the reference frame  A are determined according to formulas

x₁ = x₁'/Г + u t_1, (7.1)

x₂ = x₂'/Г + u t_2,   (7.2)

where  t₁, t₂ are time moments (according to clocks of the reference frame A), at which positions of the   front and  the rear ends of the  moving body are marked. Substracting termwise the expression (7.1) from expression (7.2) we have

x₂ - x₁ = (x₂' - x₁')/Г + u (t₂ - t₁). (7.3)

The value (x₂ - x₁) in the left-hand part of the expression (7.3) will be the length L of a moving body in the inertial reference frame A only under condition, if in the right-hand part of the expression (7.3)

(t₂ - t₁) = 0, (7.4)

i. e. if positions of the both ends of the body in the inertial reference frame A are marked at the same time moment according to clocks of the reference frame A. As a result we have

L = L_o/Г = L_o(1 + u²/c_o²) ^-1/2. (7.5)

Expression (7.5) is the formula for calculation of a length of a moving body in a  reference frame, relatively which this body is moving at a speed u if its proper length is L_o.

For determination of a moving body length using the second method, let us find, at what time moments of the reference frame A the points x₁' = 0 and x₂' = L_o of the reference frame B coincide with the origin of the reference frame A.

The origins of the inertial reference frames A and B (points x₁' = 0 and x₁ = 0) coincide with each other at the time moment t₁' = t₁ = 0. So, point x₁' = 0 coincides with the origin of the reference frame A at time moment t₁ = 0.

Substituting into the expression (7.2) (since the body is at rest in the reference frame B, we must use transformations (6.9), from which the expression (7.2) was derived) the values x₂ = 0 and x₂' = L_o, we shall have that point x₂' = L_o coincides with the origin of the reference frame A at time moment (according to clocks, which are at rest in the reference frame A)

t₂ = - L_o/(u Г)_. (7.6)

So, a time interval between the moments of coincidence of points x₁' = 0 and  x₂' = L_o of the reference frame B with the origin of the reference frame A is equal to

t₁ - t₂ = L_o/(u Г)_. (7.7)

Having multipied the time interval (7.7) by the speed u of the body motion relatively the reference frame A we have the length of a moving body in the reference frame A meassured using  the second method

L = u (t₁ - t₂) = L_o/ Г= L_o(1 + u²/c_o²) ^-1/2. (7.8)

Comparing the expression (7.5) with the expression (7.8) we can see, that the both measuring methods result in the same formula for the length of a moving body, in accordance with which the length of a moving body is less than the proper length of the body.

According to the lorentz transformations from the special relativity theory (SRT) the both methods result in the formula ([53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977, p. 70).

L = L_o(1 - V²/c_o²)^1/2. (7.9)

Comparing the expression (7.9) with the expression (7.8) we can note, that both in the SRT and in the new space-time theory the length of a moving body decreases with increase of the moving body speed. It means that qualitatively the dependence of the length of a moving body is the same in the both theories. The difference is only quantitative.

It is quite in place here to explane, why in the lorentz transformations (6.13) and (6.14) from the special theory of relativity we have designated the speed of motion of one inertial reference frame relatively other by  letter   "V", and in the new transformations of coordinates and time for designation of the speed of motion of one inertial reference frame relatively other we have used letter "u".

The truth is that both  the parameter

b = V/c_o   (7.10)

from the lorentz transformations, and the parameter

b = u/c_u  (7.11)

from the new transformations change within a range from zero to one. This allows us to make an assumption that this is the same parameter. For designation of this parameter we used therefore the same letter b in the both  space-time theories. Then, having equated the right-hand parts of expressions (7.10) and (7.11), we have

V = u (1 + u²/c_o²) ^-1/2. (7.12)

Having solved the expression (7.12) with respect to variable quantity u, we have

u = V (1 - V²/c_o²) ^-1/2. (7.13)

In expressions (7.12) and (7.13): u is the speed of motion of one inertial reference frame relatively other one in the new space-time theory, V is the speed of motion of one inertial reference frame relatively other one in  the special relativity theory.

From expressions (7.12) and (7.13) it follows that the area of admissible values of "speed V"  lies within limits from zero to the speed of light in vacuum c_o, and the area of admissible values of "speed u"  lies within limits from zero to infinity. From this we can conclude, that either the "speed u" from the new space-time theory, or the "speed V" from the special theory of relativity is not a physically measured speed of motion. For various reasons we can suppose that the "speed u" from the new space-time theory is a physically measured speed of motion of one inertial reference frame relatively another one. It is not difficult to note that expressions (7.12) and (7.13) can be obtained equating the right-hand parts of formulas (7.8) and    (7.9).

7.2. Disappearance of time paradoxes

According to the special theory of relativity a moving chronometer loses with respect to an immovable chronometer because of time dilation effect.  The time dilation effect resulting in the well-known time paradox of the special relativity theory (clock paradox) can be explained as follows.

Let us consider a chronometer of the following design. Between two parallel mirrors immovable relatively each other and separated by a distance of D_o   a light pulse is moving alternately reflecting from each of mirrors. On one of mirrors a photocell is mounted and on the other mirror a laser amplifier is mounted so, that such clock could operate infinitely. A pulse counter is connected to the photocell. At reflection of the light pulse from the mirror with the photocell some part of the incident light pulse comes to the photocell and an electric pulse appears at its output.   A counter counting  the quantity of pulses from the photocell output counts thereby a quantity of time intervals, during each of which the light flies twice the distance between the mirrors (from the mirror with the photocell to the mirror with laser amplifier aand back).  Chronometers of such design, which are at rest in the reference frame B, we shall call B-chronometers, and the chronometers of the same design, which are at rest in the reference frame A, we shall call A-chronometers. In the inertial reference frame B a time interval between two neighbouring pulses at the output of the B-chronometer photocell is equal to

D_BB = 2 D_o/c_o . _. (7.14)

In the inertial reference frame A a time interval between two neighbouring pulses at the output of the A-chronometer photocell is equal to

D_AA = 2 D_o/c_o . _.  (7.15)

Fig. 7.1. Mutual arrangement of B-chronometer mirrors and reference frame A coordinate axes.

If the planes of B-chronometer mirrors are perpendicular to X axis of the reference frame A (see Fig. 7.1), then according to the special relativity theory in the inertial reference frame A (with respect to which the B-chronometer moves at a speed of u along the X axis) the distance between two mirrors of the B-chronometer is equal to

D₁^SRT = D_o(1 - u²/c_o²)^1/2_.  (7.16)

As a result a time interval between two neighbouring pulses at the output of the B-chronometer photocell according to A-chronometers will be equal to (according to thethe special relativity theory)

D_BA^SRT = D₁^SRT/(c_o - u) + D₁^SRT / (c_o + u) = 2 D_o/[ c_o (1 - u²/c_o²)^1/2]_. (7.17)

Let us assume, that such B-chronometer in the inertial reference frame A moves from point M to point N during a time interval

Dt_A= L_A/u , (7.18)

where L_A is the distance between points М and N measured using a scale, which is at rest in the reference frame A; Dt_A is a time interval, during which the B-chronometer moves from point M to point N according to A-chronometers, which are at rest in points M and N and which are synchronised with each other using Eistein's  method and light signals from a source being at rest in the reference frame A.

Then an observer, who is at rest in the reference frame A and who have measured the distance L_A and the speed u of B-chronometer motion, will calculate, that during the B-chronometer motion from point M to point N the A-chronometers of the reference frame A will count a number of pulses equal to

n₁ = Dt_A / D_AA= L_Ac_o/(2 u D_o). (7.19).

The same observer (for whom the time interval between two neighbouring pulses at the output of the B-chronometer photocell is determined by the expression (7.17)), who is at rest in the reference frame A, can calculate, that during the time of B-chronometer motion from point M to point N this B-chronometer will count the quantity of pulses equal to

n₂ = Dt_A / D_BA^SRT = n₁(1 - u²/c_o²)^1/2 < n₁. (7.20)

And namely on the basis of calculations according to the formula (7.20) this observer, who is at rest in the reference frame A, will conclude, that according to the special relativity theory the moving B-chronometer loses against immovable (with respect to the observer resting in the reference frame A) A-chronometers.

In the formulas (7.16), (7.17) and (7.20) under the square root we have written the real speed u of B-chronometer motion, because namely at the speed u  the B-chronometer moves relatively the reference frame A.

So, from the expression (7.20) it follows, that according to the special relativity theory if in points M and N of the reference frame A there are A-chronometers synchronised with each other and if the B-chronometer moving from point M to point N at a constant speed u in point M had a reading, which coincided with a reading of the A-chronometer from point M, then during the B-chronometer   motion from point M to point N the readings of A-chronometers will increase by a number n₁, and the reading of B-chronometer will increase by a number n₂ < n₁. This means that according to the special relativity theory, even if the B-chronometer reading was in point M equal to the reading of A-chronometer, which is at rest in point M, then at the moment of this B-chronometer arrival to point N the reading of this B-chronometer will not be equal to the reading of the A-chronometer, which is at rest in point N, but will be less than the reading of A-chronometer of point N. Namely in this sence  they say that according to the special relativity theory a moving chronometer loses relatively immovable chronometers.

Now let us consider the same situation in that case, when  the square-law dependence c_u = c_o(1 + u²/c_o²)^1/2 exists in the nature. In this case a distance between the B-chronometer mirrors (equal to D_o in the inertial  reference frame B) in the inertial reference frame A will be equal to (see formulas (7.5) and (7.8))

D₂ = D_o(1 + u²/c_o²)^-1/2. (7.21)

And according to the new space-time theory a time interval between two neighbouring pulses at the output of the B-chronometer photocell according to A-chronometers of the reference frame A will be equal to

D_BA = D₂/(c_u - u) + D₂/(c_u + u) = 2 D_o/c_o. (7.22)

Then the observer, who is at rest in the reference frame A and who have measured the distance L_A and the speed u of the B-chronometer motion, will calculate, that during the motion of the B-chronometer from point M to point N this B-chronometer will count a number of pulses equal to

n₃ = Dt_A / D_BA = n₁.  (7.23)

This means, that according to the new space-time theory the moving B-chronometer will count during its motion from point M to point N exactly the same number of pulses, which will be counted by  A-chronometers, which are at rest in points M and N.

Thus, according to the space-time theory, which follows from the square-law dependence c_u = c_o(1 + u²/c_o²)^1/2, a moving chronometer neither loses nor gains relatively to the immovable chronometers, the rate of the moving chronometer reading change is exactly equal to the rate of   immovable chronometers readings change.  That is why in the new space-time theory any time paradoxes are absent. By the way, the absence of time dilation in moving reference frames follows directly from the relativity principle.  Indeed, the very dependence c_u = c_o(1 + u²/c_o²)^1/2 was obtained in section 2 only thanks to the fact, that from the relativity principle we have managed to obtain the assertion

"The laws, according to which the indication of a clock changes, do not depend upon which of two coordinate systems moving uniformly and rectilinearly each relatively other  these changes of indication are referred to."

Link:     Back     Contents     Next