RELATIVITY THEORIES ON x-t-DIAGRAMS
Uzghorod National University, 46, Pidhirna Str., 88 000, Uzhgorod, Ukraine
The way of representation of events on x-t-diagrams for Galileo’s relativity theory, and also for the theories, in which the division value of the axes of space and time coordinates depends on the motion state, is stated. The comparison with Minkowski’s diagrams is given.
In the literature, Minkowski’s diagrams are considered as quite a vivid way of the demonstration of the essence of special relativity theory (SRT) and as a persuasive proof of its trueness. The detailed analysis shows, however, that not everything is so evident and simple.
2. Essence of x-t-Diagrams
If in the real physical world something took place somewhere (for example, a light flash appeared), this fact is called an event. The event is characterized by the set of four numbers – by the values of space coordinates (x, y, z) and time moment (t). The primary idea consists in the following fact: to represent the physical world of events on the diagram, in which one of the axes denotes space, and the second – time. For the simplicity, space is represented by one axis, e.g., x axis. In such a simplified case, the set of two numbers, which are some coordinates on space and time axes, is understood as an event.
Since these coordinates do not depend on each other, they can be used for representing any event, i.e. an event that occurred at any point of space and at any moment of time. The fact that the set of numbers (x, t) can characterize though an arbitrary, but a concrete event, is often perceived intuitively as the interdependence of space and time coordinates. Mathematical relations, that allow calculating the space and time coordinates of an arbitrary event from one frame into another, are called space-time transformations.
In the literature, the space axis is usually represented by the abscissa, and the time – by the ordinate. Due to the independence of these quantities, the measuring units of space and time can be selected in such a way, that light velocity would be equal to a unit. In this case, the graph of the photon’s motion in the coordinates x-t will be a straight line located at the angle of 45 degrees to the space axis – world lines 1 and 2, Fig. 2.1. The primary diagram is rational to make up in such a way that the zero marks on the space and time axes would coincide, Fig. 2.1, a. Obviously, that the graph of the object resting at point x1 will have the form of a vertical line 5 with the origin at point x1 – the time is passing, but the position of the material object is not changing. The graphs of the objects moving uniformly will be represented by straight lines 3, in this case, the higher the object’s velocity is, the greater the graph’s slant is, but for a material object, the slant cannot exceed the angle of 45 degrees. The graphs of bodies moving non-uniformly will be curves whose slant on any of their part cannot exceed 45 degrees either to one, or to the other side.
All events occurred at the time moment t=0, are depicted by the points located on x axis. All points situated above x axis are future events. If point A in Fig. 2.1, a has coordinates (x, t), then this event has not occurred yet, it will occur at point x after time interval t. If point B has coordinates (x, -t), then this event had occurred t time before at the point with coordinate x.
By means of this graph, one can also demonstrate, what the situation for a resting observer will be at the time moment t. For this purpose, the origin of the space axis should be shifted at t value up along the time axis, Fig. 2.1, b, since the ordinate itself, i.e. the time axis (graph 4) is the graph of the motion of the space axis’ origin (the world line). Evidently, that at an arbitrary moment t the origins of the space and time axes will not coincide already. In contrast to the space axis’ origin, the time axis’ origin does not move together with the observer – it “stays in the past”. As seen from Fig. 2.1, b, now event A for the immovable observer is present already, and event B – at the distance 2t in the past. Space and time in a definite sense have opposite properties – on the space axis, there cannot be two identical coordinates, and the time cannot be different. In this sense they supplement each other.
Probably this is the whole useful information that can be taken from Fig. 2.1. Everything stated above had no relation to the relativity theory yet, since we did not yet collated the viewpoints of observers from different coordinate frames at one or another event because of that simple reason that we analyzed the situation in one coordinate frame.
Fig. 2.1. Depicting of the same events on x-t-diagrams at different time moments:
a) – at the moment t=0. In the given case, the world lines are the prognosis for the future, since the laws of the objects’ motion are known. 1,2 – the world lines of the photons, emitted from the coordinate origin in opposite directions, 3 – the world line of the body moving uniformly from the coordinate origin, 4 – the world line of the coordinate origin (this line coincides with the time axis), 5 – the world line of the body resting at the point with coordinate x1 . Event A is in the future, it will occur after t time. All events on x axis (for example, C and D) are in the present, event B – in the past. The origins of the space 0s and time 0t coordinate axes coincide – point 0;
b) – at the moment t . The origin of the resting space coordinate moves along the time axis (along the origin’s world line), therefore, positions 0s and 0t do not coincide. Event A now appears in the present time, events C, D and B – in the past. The world lines 1-4 take their origin from point 0t .
3. Light Cones
Now let us analyze the question of the so-called light cone located between the photons’ world lines, Fig. 3.1. It is considered [3, 4], that the region of events of the upper cone is the region of the “absolute future”, and the region of the lower cone is the region of the “absolute past”, and the rest part is the absolutely indifferent region. Such a classification is connected with the supposition that in the physical world, it is impossible to send a signal with the velocity higher than light velocity.
Fig. 3.1. The present, past and future events on the x-t-diagrams. Light cones.
All points on the space axis x are present events; all points at t > 0 are future events; all points at t < 0 are past events.
As seen from the figure, event A with space coordinate x1 is in the future, but from the coordinate origin, it is impossible to send an electromagnetic signal to point x1 in such a way that it would reach this point at the moment t1, – the electromagnetic signal will reach point x1 at the moment t2 , but it will be some event B already. A rocket starting in the direction to x1 will reach this point still later (the dashed line), at the moment t3 , and the reaching moment will be event C. If from point x1 at the moment t1 an electromagnetic signal is sent in the direction to the coordinate origin, it will reach this point at the moment t4 . If tachyons are found out in the future, or torsion fields appear to be real, then it will be also possible to send the signal from the coordinate origin to point A.
Event ^ is in the past. Its electromagnetic signal comes to the origin of the space coordinate in the past, too, at the moment –t1. Moreover, from all events located inside the lower cone, the electromagnetic signals have already come in the past. However, if one uses slower communication aids, for example, a rocket, then the information about these events can be transmitted to the coordinate origin at the moment t=0 (dot-and-dashed world line), or at the moment t >0 , if one uses a still slower rocket.
In the literature, this question, probably, was not considered – maybe due to the absence of necessity, maybe simply nobody was interested in this question. However, if one has some way of representing the relativity theory, this way must be suitable and useful for any variant of the theory, including Galileo’s theory. Assume that the coordinate frame K’ moves with the velocity V with respect to the immovable frame K. There appears the question: what will be the situation (what will be the events’ coordinates) from the viewpoint of the moving observer at different moments of time? For the present we have found out, that for this purpose, on the figure, the axis x should be transmitted at the distance t along the time axis. Now the motion along the space axis takes place in addition to it. These motions taken together represent the world line.
In all theories, for convenience and simplification, it is assumed that at the moment t=0 the origins of the space 0s and time 0t axes of both frames coincided, Fig. 4.1, a. In Galileo’s theory, the division values on the axes of the moving frame K’ (thin lines) are the same as in the immovable K. Therefore, we have a conclusion that for K’ the situation (position of point А) at the moment t=0 is exactly the same as for K.
Fig.4.1. Galileo’s relativity theory on x-t-diagrams. The quantities belonging to Galileo’s transformations are denoted by braces:
a) – time moment t=t’=0. Event A for both observers will occur after t time. Events C and D for both observers are in the time present for them, and they are simultaneous. Event B is in the past at the distance t. Four coordinate axes origins (the origins of space ones 0s and 0s’, and also time ones 0t and 0t’) are situated at the common point 0;
b) – time moment t=t’ 0. The origin of the space coordinate of the moving frame is shifted at both coordinates to point 0s’. The time coordinate of the moving frame is shifted along the space axis at the value Vt (the chronometer located at the space axis origin of the moving frame is moving together with this origin).
At the arbitrary moment of time t, the origin of the immovable frame will be transmitted along its own world line at the distance t, and the origin of the moving frame will be also transmitted along the space coordinate in addition. Finally, the transmission of the origin of the moving frame will take place along the oblique line, in accordance with the equation of the coordinate origin motion x=Vt , Fig.4.1, b. As seen from the figure, event A (with time coordinate t), which was future before, now became present for both observers. The fact, that the event became present for both observers, may lead to deception, as if t=0 at this moment for both observers. However, one should remember that the moment t=0 (moment of the chronometer synchronization) stayed in the past. So long as we cannot stop the time, we cannot be located with the origin of our space coordinate all the time opposite the mark t=0 on the time scale, i.e. opposite 0t. Thus, the chronometers of both observers will show the same value of time, i.e. t’=t. This simple relation is just the mathematical writing of Galileo’s time transformation.
In contrast to the time coordinate, both the immovable observer and the moving one can be located near zero mark of his space coordinate all the time. Therefore, event ^ for the moving observer at the moment t will appear to be nearer in distance at the value Vt . Since the division values on the space axes (the moving and the immovable) are identical in Galileo’s theory, then, from Fig. 4.1, b, we have: x’=x-Vt , i.e. we have obtained Galileo’s space transformation. Thus, we have an equation system, in which x has the meaning of the coordinate of an arbitrarily selected point (x=const), and x’ has the meaning of the coordinate of the same point in the moving frame (x'const) at the time moment t, moreover, the time moment is also selected arbitrarily, out of any dependence on the quantity x.
Evidently, that so long as the graphs are the same mathematics, but represented by other means – not symbolic, but vivid ones, on the x-t-diagrams, the mathematical connections describing the analyzed process must be clearly seen. The quantities depicted in the figure by braces illustrate Galileo’s space transformation.
In the works [1, 2] the relativity theory is proposed in the assumption that all elementary particles are soliton formations of the light-carrying ether (“soliton relativity theory”, SolRT). In this case, the division value both on the space axis and on the time one of the moving frame depends on the frame velocity:
The first equation of the system (5.1) coincides with Lorentz’s space transformation both in form and in the meaning of the quantities belonging to the formulae. Coefficient G, which can be called the “non-Galileoity coefficient”, belongs to the equations of (5.1) in a principally different way. The mentioned asymmetry is connected with the fact, that as the velocity V increases, the division value of the space axis x’ decreases (Fitzgerald–Lorentz shortening), and the division value on the axis t’ increases, which corresponds to the decrease of the rate of physical processes in the moving frame – the so-called “time dilation”.
According to the correspondence principle, any new theory must mathematically turn into old one in those conditions, when the old one is agreed with the experiment. In the mathematical language it means that in case of low velocities, the transformations of any new relativity theory must turn into Galileo’s transformations, if one neglects the separately situated terms of higher orders, in the given concrete case, if the quantity V2/C2 is neglected. As seen, the system (5.1) turns into the system (4.1) in these conditions, i.e. SolRT turns into Galileo’s theory, and the correspondence principle is not violated. Since the graphics is the same mathematics, the correspondence principle should also be manifested in the graphical representation of the theory. In this case, the x-t-diagram of the new theory should not strongly differ from the old one, and the way, how the diagram of the new theory turns into the diagram of the old one, should be manifested.
In Fig. 5.1, a, the moment, when the origins of the immovable frame and the moving one coincide, is shown. The moving frame is depicted with thin lines. The division value on the moving space coordinate is lower than that on the immovable one (moving bodies are shortening, coefficient G is in the denominator of the space transformation). In accordance with the time transformation, the division value on the moving time coordinate is higher than that on the immovable one (moving chronometers tick slower). At the moment t=0, all events on the axis х are simultaneous in both frames, but the distance to them in the space coordinates is different in different frames. For event D, in accordance with Fig. 5.1, а, we have 2.5 spatial marks in frame K and 5 marks in frame K’.
Fig. 5.1. Soliton relativity theory on the x-t-diagrams:
a) – moment t = 0. Event A is in the future time, but the distance to it, according to the measurements of different observers, is different, both at the time coordinate and at the space one. Events C and D are simultaneous for both frames, but the spatial distance to them is different, due to the decrease of the division value of the moving space coordinate axis – due to the shortening the solitons’ dimensions (and physical bodies in the whole) in the direction of motion;
b) – time moment t. The situation is highly analogous to Galileo’s theory in Fig. 4.1. The basic difference consists in the fact, that the marks on the space axis of the moving frame are located denser (moving bodies are shortening), and on the time axis, on the contrary, marks are located sparser (the chronometers’ ticking rate decreases). As the difference between the marks decreases, Fig. 5.1 turns into Fig. 4.1 step by step, and SolRT – into Galileo’s relativity theory. 1 – the world line of the coordinate origin of the moving frame.
Event A for both observers is in the future, but this event, in the moving observer’s opinion, will occur sooner (in accordance with the figure – after two units of time), and in the moving observer’s opinion – after four units of time. It is obvious, however, that the data about event A will be taken in both frames simultaneously. The fact, that the results of measurements will be different, is quite another thing, and it simply cannot be called the relativity of simultaneity.
In Fig. 5.1, b, it is shown, what the situation for both observers at the time moment t will be (this moment is simultaneously one of the characteristics of event ^ ). As seen, now event A is also located in the present time, though the characteristic of this present is different in different frames – in the moving frame, it is lower, only 2 marks, in comparison with 4 marks in the immovable frame. The chronometers in both frames were started (their indications were zeroed) simultaneously, at the moment t = t’ = 0. The chronometers’ indications were also taken simultaneously, at the moment of the occurrence of event A (at the moment t), but the presence of difference in the chronometers’ indications, t’ t, does not yet mean that the indications are taken not simultaneously, that the events were not simultaneous.
The spatial distance from the coordinate origin to point ^ for the immovable observer remained the same, but for the moving one, it decreased at value Vt. From the comparison of Fig. 4.1 and 5.1 it is seen that, as the difference in the division values on the coordinate axes decreases, the SolRT diagram differs from the diagram of Galileo’s RT less and less. It serves as the vivid (graphical) demonstration of carrying out the correspondence principle.
The fact, that the coordinate frames are not of equal rights, is well illustrated on x-t-diagrams. In accordance with Fig. 5.1, the velocity of frame K’, according to the measurements of K, is equal to
However, according to the measurements of K’, it is equal to
If G=0.5, these results correspond to formula (5.3) from SolRT, [1,2].
For convenience, let us represent Lorentz’s space transformation as follows:
This expression represents more vividly that fact, that the division value on x’ axis is lower than on x axis, and also that the quantity Vt/G corresponds to a simple shift along x’ axis, Fig. 5.1, b.
Let us also represent Lorentz’s time transformation in the same way:
Since х has the meaning of the coordinate of an arbitrarily selected point, the low value of the quantity V/C2 can be always compensated by the quantity х. It means that in the procedure of simplifying the equation (6.2), one must not neglect the quantity Vx/C2 (for transition to low velocities) already, and it means in turn that in case of low velocities, Lorentz’s time transformation does not turn into Galileo’s transformation. Thus, we have an obvious violation of Bohr’s principle.
In this case, one usually denies that in space and time transformations, the quantity t denotes not a time interval, but a time moment. It is easy to see, however, that the notion of a time moment cannot be mathematically represented in another way than through the notion of a time interval, and the notion of a point’s space coordinate – in another way than through the notion of spatial distance. In all equations, which contain the quantities x and t as space and time coordinates, x always denotes the number of marks on the space axis, and t always denotes the number of marks on the time axis, i.e. the time interval.
The second usually advanced denial consists in the fact, that for the transition from Lorentz’s time transformation to Galileo’s transformation, one proposes to direct light velocity to infinity, as they say, in Newtonian mechanics or in Galileo’s relativity theory, light velocity is equal to infinity. In reality, neither Newtonian mechanics, nor Galileo’s relativity theory append any limits to light velocity – light velocity simply does not belong to Galileo’s transformations. The fact, that the chronometer’s synchronization in Galileo’s theory could be done by means of an indefinitely quick transmission of information, does not yet mean that light velocity must be equal to infinity, even if we have no other way out. In the theory, the synchronicity of the chronometers’ ticking is postulated, and in reality, the making of concrete synchronization is the problem for experimenters.
From (6.2), one can see one discouraging thing consisting in the fact, that in reality, Lorentz’s time transformation contains the information not about “time dilation”, but about its “acceleration”! In accordance with (6.2), marks on axis t’ must be located denser than on the axis t, but not sparser, as it is confirmed in SRT. Evidently, that the quantity Vx/C2G denotes only the shift along the time axis. What we have said can be illustrated by means of conventional calculations done in accordance with (6.2). In Table 1, the calculations for the time t’ in a moving frame, if V=2108 , x=108 at different moments of time t, and also the time t* in the moving frame without subtracting the correction for the local time t*=t/G.
t 0.0000 10.0000 20.0000 30.0000
t’ –0.2981 13.1186 26.5347 39.9511
t* 0.0000 13.4164 26.8328 40.2492
As seen from the table, in case of velocity 2108, the chronometer in the moving frame, according to Lorentz’s time transformation, is ticking in the rate accelerated approximately at one third (opposite 10th mark of the absolute time, 13th mark of the “moving time” is located, etc.), and it has a sharp contradiction with the declared “time dilation”. In case of the given velocity, at the distance 108, the correction for the local time is only one third of a second.
One more unpleasant specificity of the SRT time transformation (in contrast to Galileo’s or SolRT transformation) consists in the fact, that the quantity t’ depends on the coordinate x of an arbitrarily selected point. One can make a few suppositions about the cause of the similar dependence.
1. In the time transformation, the creators of SRT understood not the coordinate of an arbitrary point as the quantity x (like in the space transformation), but the coordinate of a point, to which the origin of the moving frame is transmitted during the time t, i.e. x=Vt. In this case, we deal with an elementary mathematical error, when different quantities are denoted by the same symbol. There is no physical error in this case, since it is easy to show that in this case, t’ has the meaning of the time in the whole moving frame.
2. It is understood that the quantity x has the meaning of the coordinate of an arbitrarily selected point indeed. Therefore, in Lorentz’s transformation, the quantity t’ has another physical sense than in Galileo’s transformation – it determines the value of the time not in the whole moving frame, but only at that point on the space axis, which is located opposite point x at the observed moment. In (6.1), it is enough to assume that t=0, and we will see that at the moment when all the chronometers in frame K show t=0, in frame K’ the moving chronometers show different time – that one located on the right from point x=0 shows the “negative time”, and that one located on the left from x=0 goes ahead, (6.3).
The principal difference of the space and time coordinates consists in the fact, that, when introducing these notions as the basic ones, we affirm that on the space axis, two or more identical coordinates do not exist, but the time at the given moment must be one and the same on the whole infinite space axis. These are the basic positions, basic suppositions of any theory, in which the notions of a space and time coordinate are used. If the time is allowed to be not one and the same on the whole space axis, it is equivalent, as a matter of fact, to the situation, when different (or even all) points are allowed to have the same coordinate on the space axis. Therefore, now we deal with a gross physical error. Lorentz, certainly, saw this disagreement, therefore, he called t’ “local time”, “mathematical trick”, which he hoped to liquidate later.
We saw that in case of low velocities, SRT does not turn into Galileo’s theory, however, the numerical value of the disagreement is also interesting to some extent. Let us estimate this value. For simplicity, let us take V=900 (the velocity of a modern middle fighter) and determine the chronometers’ indications of the moving frame at different points of space at the moment, when all the chronometers in the immovable frame show zeros, t=0. In these conditions, in accordance with Lorentz’s time transformation (6.3), we have, Table 2.
t' 100 10 1 0 -1 -10 -100
x -1016 -1015 -1014 0 1014 1015 1016
As seen from the table, at the moment, when all the chronometers in K showed zeros, all the chronometers in the moving frame showed different time, and only the chronometer at x=0 showed zero. In order that the theory would give the result agreeing with the idea of light velocity invariance, the incorrectly adjusted devices should be used. Thus, in order that light velocity, measured according to the moments of passing the coordinate x’=0 and x’=1016 by the photon, will appear to be the same as in the immovable frame, the chronometer opposite x=1016 should be corrected at 100 seconds (of course, for the mentioned velocity of the frame). In the process of measuring the light velocity in the moving frame, the photon will catch up the necessary mark on the axis x’, the marks are located on the axis denser than on the axis x, the moving chronometer ticks faster than in frame K, but finally, taking also the correction for the local time into account, the light velocity appears to be numerically equal to C again.
Let us represent this process on the x-t-diagram. Note that if we put the points t’=0 on the diagram, we will obtain the line, whose slant to axis x will turn to be the same as the slant of the world line of the moving frame’s origin to the axis t . In this case, this line should pass through the origin of frame K – we have some analogy with Minkowski’s diagram, but now not the axis x’ is slanted, but the line of zeros on the axes of local time, which, in contrast to the zone local time, changes continuously with the change of the space coordinate. With respect to the other coordinate, not the axis t’ is slanted, but the world line of the moving frame’s origin. The axes of the moving frame remain parallel to the corresponding axes of the immovable frame, Fig. 6.1.
For proving this statement, one should compare the corrections Vt and Vx/C 2 for identical segments on the axes x and t, Fig. 6.1. In accordance with the figure, x=Ct, therefore:
After substituting x=Ct , we have С=1, and it corresponds to the condition of making up Fig. 6.1 – to the equality of the space and time units.
Fig. 6.1. SRT on x-t-diagrams (scale is not carried out).
1 – the line of zeros of the time axes 3 of the local time. 2 – the world line of the coordinate origin of frame K’. The quantities belonging to Lorentz’s transformations are denoted by braces. t* is the time in the moving frame without correction for the local time. On this figure, the situation x/t=c=1, x’/t’=c’=1 is depicted.
As seen from the figure, if we subtract Vt from x and divide the obtained quantity by the coefficient G, we will obtain the quantity x’ – as before, we have the vivid representation of Lorentz’s space transformation, according to which, the division value of the space axis of the moving frame decreases.
If we subtract the correction for the local time Vx/C 2 from t and divide the obtained quantity by G, we will obtain t’ in Lorentz’s time transformation. Therefore, three factors taken together – the decrease of the length, the increase of the ticking rate of the moving chronometer, and introduction of the correction for the local time – lead to the preservation of the same numerical value of light velocity. If the photon moves to the opposite side, the appropriate corrections are not subtracted, but added, and finally, C is obtained again.
Certainly, first appears the following question: how in SRT the “time dilation” (which, by the way, is “experimentally confirmed” in investigations of muons and pi-mesons) was obtained till nowadays, if the transformation itself contains the information about “time acceleration”? The answer to this question is not simple. Probably it is impossible to answer it without x-t-diagrams. First one should notice, how in SRT the formula t=tG is obtained, and illustrate this process on the x-t-diagram.
In the literature, for determining the duration of some process according to the data of the measurements in different coordinate frames, one calculates the difference of the chronometers’ indications at the end and the beginning of the process, . It is supposed that the chronometers are located at the origin of the moving frame. For frame K, the process begins at point x1 and ends at point x2, moreover, x2–x1=V t. The connection between t2 and t’2 , and also t1 and t’1 is calculated by means of Lorentz’s transformations:
Fig. 6.2. On the derivation of the relation of time intervals in SRT (scale is not carried out).
1 – world line of the beginning of the moving frame. 2 – the line of zeros of the time axis of the local time. 0s – the beginning of the space axis, 0t – the beginning of the time axis of local time.
The process of the derivation of (6.6) is graphically represented in Fig. 6.2. As seen from the figure, the difference between t’2 and t’1 is not equal to the depicted meaning of t’, which corresponds to the interval t in the immovable frame. The calculation quantity is less than t’, since from t’2 one subtracts a larger value than needed for obtaining t’ (for convenience, the division value of the moving frame’s axes is shown halved on the figure). This is just the cause of that fact, that the calculation quantity t’ appeared to be numerically (in the quantity of marks on its axis) less than t, though the division value on the axis t’ is less than on the axis t. Strictly speaking, this is just the whole “secret” of deriving the relation of time intervals in SRT. Without x-t-diagrams, it would be very difficult to discover this secret, if possible at all. The abstractness of the symbolic mathematics often does not allow seeing the essence of mathematical operations. Thus, we have the vivid demonstration of the usefulness of x-t-diagrams.
In reality, the relation between time intervals should be obtained from Lorentz’s time transformation in the differential form, by collating the change dt of the chronometer resting at an arbitrary point x with the change of the moving chronometer indications. From (6.2) we have:
Taking into account, that x=const, we have dx=0, therefore
After integration, we have:
As we see, in reality, Lorentz’s time transformation contains the information about the acceleration of chronometers’ ticking rate, as the velocity increases, but not about dilation, as it is affirmed in SRT. Certainly, this result is agreed with (6.2). In Fig. 6.1, the quantity t’ is denoted as t*, and the quantity t –as t .
Fig. 6.3. On the derivation of the relation of time intervals, case V=0.866 C, scale is carried out.
1 – world line of the moving frame’s origin. 2 – line of zeros of the time axis of the local time.. W1, W2 – corrections for the local time. t* – time interval in the moving frame without taking the corrections for the local time into account.
In Fig. 6.3, the same situation is represented as in Fig. 6.2, but scale is carried out. The terms of Lorentz’s time transformation are represented in the following form:
If the velocity is V=0.866 C , the division value of the space axis on the moving frame is halved, i.e. G=0.5. As seen from the figure, for this velocity, the corrections for the local time W1=Vx1/C2G and W2=Vx2/C2G significantly exceed the meanings of t’1 and t’2. In spite of the halving of the division value, taking of the local time into account leads to the fact, that t’ in the given example appeared to be equal to only one decreased mark, and it is agreed with formula t’=Gt, but contradicts to the situation depicted in Fig. 6.2. In reality, due to the increase of the moving chronometer’s ticking rate, at the moving frame’s axis, the time interval must be numerically doubled, t*=4. In the figure, the quantity t’1 is depicted twice: for the first time – at its own place, for the second time – for comparison with the quantity t’2 and demonstration of the way of obtaining the quantity t’ mathematically. As seen from Fig. 6.3, in order that t’ would be equal to a unit, in the moving frame, the measurements must be started not at the moment t1, as it must be in accordance with the measurement conditions, but at some moment t3 . The incorrectness of similar “measurements” is evident.
Since Fig. 6.3 is made up in accordance with Lorentz’s transformations, it is seen that in reality, these transformations also contain the information that the coordinate frames are of equal rights. Approximately one may consider that x1=2 and x2=4 (in reality, these points are located somewhat to the right, opposite the intersection of the axes x’ with the straight line x=Ct). It is seen from the figure that x2 , t2 , x’–4 , t*4, therefore, V’=–V1. However, if one uses the calculation meaning t’=1, which takes the correction for the local time into account, we will obtain V’= –4.
Mathematically, it is not difficult to obtain V’=–V. For this purpose, one should take Lorentz’s transformations in the differential form, and take into account, that the arbitrarily selected point is at rest, i.e. x=const, or dx=0.
It is easy to see that in the procedure of deriving (6.13), the information about the local time was lost, and (6.12) contains the information not about dilation, but about “acceleration” of time. The condensation of the marks on the space axis is accompanied by the condensation of marks on the time axis, therefore, V’=–V is obtained. This conclusion is correct within the limits of SRT, though it contradicts to the experiment. However, if one tries to obtain (6.13) through “deltas”, then, taking into account the generally accepted in SRT (6.6), and also the well-known x=x/G , we have:
In reality, the result (6.15) indicates the fact, that the expression (6.6) contradicts to Lorentz’s transformations, i.e. within the limits of SRT, (6.6) is incorrect.
On the diagrams suggested by Minkowski, in a moving frame, the space and time axes are simply slanted at the equal angle, which depends on the moving frame’s velocity, [3-5]. In this case, the division values on the space and time axes change in the equal proportion, moreover, to the increasing side, Fig. 7.1. The necessity of motion along the world line is attributed to all material objects, but it does not concern the moving frame for some reason – it is enough to turn the coordinate axes.
Fig. 7.1. SRT on Minkowski’s diagram, .
As the frame’s motion velocity increases, its axes are slanted at a greater angle, and finally they begin to coincide with the world line of the photon 1. The division values of the axes x’ and t’ are the same. For determining the division value on the scale t’, the hyperbola t2-x2 =1 should be used. In this case, the division value of the moving axis t’ appears to be increased in comparison with the axis t . On the space axis, the division value is also increased, and it contradicts to Lorentz’s space transformation and Fitzgerald-Lorentz’s shortening).
In SRT, as if nobody denied that moving bodies decrease their dimensions (otherwise Michelson’s experiments cannot be explained), and chronometers decrease their ticking rate (otherwise the experiments with muons), but on Minkowski’s diagrams, a significantly different picture is shown – the decrease of the chronometers’ ticking rate is accompanied not by the decrease, but by the increase of the moving bodies’ dimensions. The division value on the time axis is determined by means of the hyperbola t2 –x2 =1, which in reality also corresponds to the increase of the division value on axis t’, i.e. the “time dilation”, but in this case, the division value of the space axis is considered as equal to the division value of the time axis (otherwise one cannot obtain the same numerical meaning for the photons’ velocity, as in the immovable frame, C=x’/t’), but it will already correspond not to the bodies’ shortening, but to their lengthening.
1. On x-t-diagrams, one can represent any relativity theory; in this case, one succeeds in representing the essence of the theory vividly.
2. On x-t-diagrams, one can succeed in vivid representing Bohr’s principle concerning relativity theories.
3. Soliton relativity theory is agreed with Bohr’s principle. In this case, all bodies are really shortened, and all processes, including the moving chronometer, decrease their going rate.
4. The representation of SRT on the x-t-diagram vividly demonstrates the physical meaning of Lorentz’s time transformation and the contribution for the local time. In reality, the moving chronometer, in accordance with Lorentz’s transformations, must increase its ticking rate, as the frame’s velocity increases, and it contradicts to the experiments with muons and pi-mesons.
5. Special relativity theory contradicts to Bohr’s principle – at large distances, the correction for the local time turns to be quite significant even in case of low velocities.
6. On the x-t-diagrams, SRT is represented by Minkowski incorrectly, at least, because of the fact, that the division value on the space axis is depicted not decreased, but increased, and it contradicts both to Lorentz’s transformations and to Michelson’s experiments.
1. Chavarga N. Relative Motion of Solitons in the Light-Carrying Ether. // Uzhgorod University Scientific Herald, series “Physics”, Issue 7, 2000. – P. 174–194.
3. Taylor E.F., Wheeler J.A. Spacetime Physics. – San Francisco and London: W.H. Freeman and Company, 1966. – 320 p.
4. Born M. Einstein’s Theory of Relativity. – New York: Dover Publications, Inc. 1962. – 368 p.
5. Õiglane H.H. In the World of High Velocities. – Moscow: Nauka, 1967. – 364 p.
6. Landsberg G.S. Optics. – Moscow: Nauka, 1967. – 928 p.
Uzhgorod University Scientific Herald, series “Physics”, Issue 15, 2004 p. 187-193.
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