# AN ENTERTAINING SIMULATION OF THE SPECIAL THEORY OF RELATIVITY USING METHODS OF CLASSICAL PHYSICS

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# In lieu of a preface. The essence of the simulation

The behavior of objects that, being slow-moving, nonetheless act in accordance with laws similar to those of the special theory of relativity are examined in the booklet.

Individual barges and groups of barges located on the surface of a flat-bottomed water body with a depth of h, filled with slack water, will serve as the objects of our conceptual observation. The barges are equipped with hardware components that perform metrological operations. The hardware components have at their disposal speedboats that whisk over the water surface between the barges and high-speed underwater shuttles that travel between the barges and the bottom. The velocity of the speedboats and the shuttles equals V and is unapproachable for the other watercraft, i.e., the velocity of the barges, v, which do not fall into the category of high-speed watercraft, correspond to the inequality v < V. Each barge is equipped with a clock, during which a high-speed shuttle that continuously moves along a plumb line (relative to this barge) between the barge and the bottom performs the function of the pendulum. Each shuttle trip to the bottom and back requires a time of Δt(1) = 2h/V_{Z} , where V_{Z} – the speed of submersion and surfacing of the underwater shuttle, and is accompanied by the replacement of the clock readings with a standard unit quantity that is uniform for the barge. This standard quantity for the barges both at rest and in motion equals 2h/V. The shuttle clock “mechanism” controls not only the clock’s hands, but also all the barge hardware components, thereby ensuring the proportionality of their working speed to the clock’s working speed. We assumed that the t time scale for barges at rest relative to the water equaled the time scale of our conventional “terrestrial” clock, as well as that the reading replacement rate on the barges at rest and our clock was identical.

During phase one, we examined a group of barges at rest. Here, we assumed that the clock readings on the this group’s various barges were not synchronized, i.e., when the working speeds of the clocks on each barge in the group are identical, their readings at a given moment in time might be different.

Assuming that the barges might change their position due to certain external causes (for example, because of the wind), we entrusted the hardware components with the function of maintaining the distance between this group’s barges by means of interaction between the barges using the speedboats.

The procedure for maintaining distance consists of the following.

A speedboat is dispatched from each barge to the neighboring barge, and upon reaching it, the boat goes back again. The barge’s hardware components, using their clocks, measure the boat’s time of movement to the neighboring barge and back, then approach or move away from the neighboring barge as necessary in order to maintain this time and the immutability of the “radar” distance. This method for maintaining the “radar” distance between barges does not require the synchronization of the readings on the different barges and makes it possible to track the distance of the neighboring barges from each of the barges independently, without resorting to the measurement of the boat movement time from one barge to the another using synchronously running clocks on these barges.

We then examined a group of barges located at the points of intersection of an imaginary coordinate grid of the K' coordinate system. The group is initially at rest on the water body surface, after which it is transformed, together with the K' coordinate system that belongs to it, from a state of rest to a state of motion at a velocity of v in the direction of the X' axis (this axis lies on the water surface). As the group of barges accelerates to a velocity of v, the speed at which the clocks tick and the response time of the hardware components on the barges decreases. This occurs due to the fact that as the barges move at a velocity of v, the VZ submersion and surfacing speed of a shuttle moving through the water between a barge and the bottom along the hypotenuses of right-angled triangles is equal to . Time on the moving barges, which we named simulated time, t', also flows more slowly than our time, t, by times.

The lateral dimensions of the group are retained in this instance.

Indeed, let us assume that a boat is sent out from and returns to a barge, r'o', that is moving within the R' group and that is located at the origin of coordinates of the K' system along the Y' axis at a point with a coordinate of y' relative to the neighboring barge, r'y', in this same group. If the Y' axis is located on the water surface perpendicular to the X' axis, the boat then moves over the water surface along the hypotenuses of right-angled triangles at a velocity of V. This corresponds to the movement of a boat along the Y' axis at a velocity of V_{Y} in our time scales and with a length equaling . Since the t' time equals , the simulated time of movement of the boat from barge r'_{o'} to barge r'_{y'} and back, Δt', is independent of the speed of movement of the R' group, as well as the distance between barges r'_{o'} and r'_{y'}, and the hardware components perceive the group as unchanged when the velocity changes.

However, the longitudinal dimensions (in the direction of the X' axis) of the barge group in motion are contracted for the following reason.

In negotiating the distance, l_{o′x′}, between barge r'_{o'}, which is located at the origin of coordinates, O', and barge r'_{x'}, which is located on the X' axis at a point with a coordinate of x', the boat needs a Δt_{1} time that equals l_{o′x′}/(V – v) in order to move from barge r'_{o'} to barge r'_{x'}, and a Δt2 time that equals l_{o′x′}/(V + v) for the trip back. The total time of movement, Δt_{1} + Δt_{2}, from barge r'_{o'} to barge r'_{x'} and back comes to 2l_{o′x′}V/(V2 – v2). According to the barge’s slow clocks, the Δt'_{1} + Δt'_{2} time is times shorter and comes to .

If the hardware components do not maintain the distance between barges, the instruments on barge r'_{o'} would then perceive this as an increase in the distance between the barges in the direction of the X' axis by times. But the instruments, in tracking the distance between the barges using the radar technique, keep the radar distance unchanged, which we perceive as the contraction of l_{o′x′} by times. We called the physical quantities expressed through simulated distances and times the simulated quantities.

We then made the transition to an examination of the synchronization of the clocks of two groups of barges – group R and group R' – and the related coordinate systems, K and K'. The R group and the K system are at rest on the water, while the R' group and the K' system are in motion on the water and relative to the R group at a velocity of v.

Imagine that at a certain moment in time when the origin of coordinates and the axes of the K and K' coordinate systems coincide, the readings on all the barges of the barge groups at rest and in motion were reset to zero. From that moment in time forward, the synchronous change in the readings on all the barges of the barge group in motion occurs more slowly than the synchronous change in the readings on the barges of the group at rest.

If the hardware components on the barges of the group at rest, R, track the clock of barge r' of the group in motion, R', which is moving past them, they then record the slowness of the clock rate on moving barge r'. If the hardware components on the barges of the group in motion, R', track the clock on barge r of the R group, which moving past them, but is at rest relative to water, they then record the fastness of the clock rate on barge r. There is no symmetry of any kind. What we have here is the asymmetry of the clock movement speeds on the barges at rest and in motion. The simulated time readings of the group in motion are linked to the time readings of the group at rest by the transformations and . The coordinate transformations take the form of , and y' = y, where the primed quantities are expressed in the distance and time scales of the barge group in motion.

It is clear that if the hardware components of the group in motion, R', now measure the speed of movement of a boat from one of the barges in their group to another barge in this same group using the synchronously running clocks on these barges, they will then find that the boat’s speed of movement in the barge group’s direction of movement, which we see from the outside, and opposite its direction of movement, are different.

We then assumed that the hardware components on the R and R' group barges are not in contact with the water and have no information concerning their motion or rest relative to the water. Not finding a basis for synchronization during which the velocity of a boat back and forth is assumed to be different, the hardware components resynchronize the clocks within the group of barges in motion, R', so that the boat’s speed of movement there becomes identical to the boat’s speed of movement back. In this instance, the time after resynchronization, t'' – we named this t'' time the double simulated time – is linked to the simulated time, t', by the correlation , while the coordinates and the clock readings are linked by the transformations

,

and , as well as

.

Here, the quantities with double primes are expressed through the double simulated time. The transformations obtained are consistent with the direct and inverse Lorentz transformations to notational accuracy. In particular, this results in the fact that by tracking the clock rate of barge r at rest in the water, which is motionless relative to the water, but moves past a barge of the group in motion, the hardware components on the group of barges in motion, R', detect the slowness of the clock rate on barge r. The results of the measurements made by the hardware components on the barges of the groups in motion and at rest become symmetrical. The same thing is true of the distances.# How to buy the book

From the authors In lieu of a preface. The essence of the simulation |