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Crash » srb
Question: Does special relativity have anything to do with extra dimensions of time (Dimensions)?

Answer: Yes, let's consider it on the example of a passenger on a train (A) moving at speed v and an observer standing still on the platform (B), on the upper image, or its link with Einstein's interpretation (1905). The speed of light does not depend on the speed of the source, and motion is relative.
Exactly in the middle of the carriage, a passenger (A) struck a match, and its light (lower, blue arrows) simultaneously reached the right and left on the front and rear walls of the carriage. However, concerning the observer from the platform (B), the wagon crossed the path vt during the time t and moved the back wall of the wagon by that amount where the light of the match arrived before. Therefore, the simultaneous arrival of light on the walls of the car for observer A is not simultaneous for observer B.
If A measured the vertical displacement of light by length ct0, it will be a longer path for B for the length ct. From the right triangle in the picture, we calculate:
(ct0)² = (ct)² - (vt)²,
\[ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}. \]It is the dilation (stretching) of the unit of train time relative to the observer from the platform, which means that time in the train flows that much more slowly. Let's look at those known phenomena now with an unknown consequence.
Let's say that a light trigger is placed in the middle of the wagon that can be triggered by a photon pulse from the rear of the wagon, but such a trigger would remain upright, balanced, and unactivated when the light from the front reached it at the same time. When, by activating the trigger, some significantly different events occur (bomb explosion, release of poisonous gas, etc.), then the experiences of passengers on the train and observers from the platform will become significantly different. Let's then imagine the simultaneous movement of light from two walls of the car, say, in relation to the observer from the platform. The question arises of how the passengers from the train and the observer from the platform will be able to have the same present, the same experience of the event that followed.
After an unpleasant event occurs for a passenger on the train (A), which is not noticed by the observer from the platform (B), their two realities (A and B) flow along different time paths. However, after the train stops and the passengers disembark on the floor of the platform, those passengers are not from present A, but from B. To them and to those who watched the events from the moving train while standing still on the ground — the present are the same.
It is a situation similar to that known with two twin brothers, one of whom remains on Earth while the other travels through space. When the traveling brother appears again on earth, it will be noticed that he has aged less than the brother who did not travel. That twin paradox (1911) has been known for a long time, but I emphasize that it is not well understood. He is actually a proof of pseudo-reality, the reality of the traveler's brother, who would consider the one from the earth to be younger, i.e., additional dimensions of time. The "bonus" below is the similar example (Fizeau).
Therefore, special relativity has to do with additional dimensions of time (Dimensions), it proves them, but we have not been ready to notice it since its discovery.
Defect » srb
Question: Are those unusual formulas (3.33) also valid outside of gravity, which you stated in the book Sprega (3.2.5 Defect) and, second question, what kind of "defect" is it?

Answer: Of course, they were done for the centrally symmetric form of Einstein's general equations, i.e., Schwarzschild's solution, where I put:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{2GM}{rc^2}}}, \]with G the gravitational constant and M the mass of the gravitating body, r the distance from its center, and c the speed of light in vacuum. The second parameter χ is just so "arbitrary" that we can also define the possible speed of movement of a material point in such a gravitational field, which otherwise must be added separately to the Schwarzschild metric. Those transformations:
\[ dr = \chi \ dr' - \gamma^{-1}\sqrt{1 - \gamma^2\chi^2}\ d\tau' \] \[ d\tau = \gamma\sqrt{1 - \gamma^2\chi^2}\ dr' + \gamma^2\chi \ d\tau' \]are with τ = ict, imaginary unit i² = -1, at time t. Without gravity, it will be γ = 1, and for radial movement with speed v, it will be:
\[ \chi = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}. \]That's it, so we calculate:
\[ \sqrt{1 - \chi^2} = \frac{iv/c}{\sqrt{1 - v^2/c^2}}, \] \[ dr = \frac{dr' + vdt'}{\sqrt{1 - \frac{v^2}{c^2}}}, \quad dt = \frac{dt' + \frac{v}{c^2}dr'}{\sqrt{1 - \frac{v^2}{c^2}}}, \]and these are the Lorentz transformations (Einstein's theory of relativity) for uniform inertial motion with speed -v along the r-axis.
Another is the issue of "defect." This refers to the change in the direction of the vector \(\vec{v}\), in the upper picture on the right, written in green color in seven places, which we move (translate) in parallel along any closed curve by the gravity-curved space. The starting and ending positions are not the same vectors \(\vec{v}_1 \ne \vec{v}_7\). Applied to momentum or to vector quantities in physics in general, that change in vector direction means a defect in energy, or whatever, that does not occur in the absence of gravity.
It is paradoxical, and I state one of the solutions using "dark matter." It would be the release of "trace masses" from the body's present to form their past, which in return and partially from the past, could act on their present.
Reality III » srb
Question: What is "reality" and what is "fiction" that you write about in the book Sprega Informacija (Coupling of Information)?

Answer: The word "real" is, first of all, what someone somewhere can perceive and that we can perceive directly or indirectly. This is because the structure of space, time, and matter, their body, tissue, and ingredients are information, and their essence is uncertainty. This means that the power to communicate with everyone is not mandatory either, but it is about a chain of possible intermediaries A1 ↔ A2 ↔ ... ↔ An, when we say that all participants are mutually real. The sustainability of such chains is sought.
The second condition of "reality" is permanence. Now we are talking about the law of conservation, say, of energy that can change forms but not the total amount. Information is also indestructible, the amount of uncertainty though not the forms it will take, such as momentum, spin, and similar physical quantities. When we talk about the reality of physics, let's note that it cannot lie and does not look back at lies; it does not see them. Namely, if we prove that something is really not true, it will not even happen in the experiment.
The third condition of "reality" is truthfulness. By this, we mean that we will also treat mathematical truths and exact abstractions similar to them as types of reality. Moreover, we will try to classify them under the types of information transmitted by, say, action quanta (energy changes over time). From this it follows, for example, that the eternal duration of the law is extremely attractive (to those who could communicate with such). In other words, the constant quantum of physical action "hoovers" some around it when it has little or no energy.
These are the three conditions of "reality" around which the entire book "The Connection of Information" revolves. In addition, we note the enormous importance of multiplicity for uncertainty, then the diversity and layering of reality. Call the image link, then read the "2.4.2 Uniqueness" section. Subjects who perceive the environment are unrepeatable, and deductive theories and their forms and parts, which perceive cognitively, — they repeat themselves endlessly. I'm paraphrasing, "The truth is everyone's, and a lie is always someone else's."
This brings us to the other side of the coin: that what exists and is not "reality" must be "fiction." Therefore, fictitious is that whose eventual chains of perception mentioned above are not sustainable, it just does not last, that is, it is not true. That's why living beings know how to lie, because they are a combination of reality and fiction.
Snell » srb
Question: Space, time, and matter are woven from information whose essence is uncertainty, and "real" is what is 1. perceptible, 2. sustainable, and 3. true. Did I understand correctly?
Answer: Yes, that's right. Therefore, the spontaneous drive to do less makes more sense. Nature is all woven from information, from its constant amounts and indestructible uncertainties, which it pours "from hollow to empty," doing everything as if it would get rid of them first. That is why there is a "minimalism of space," demonstrated by the refraction of light in the first following example, and a "minimalism of time" in the example of refraction.

In the picture on the right, light travels from point A to point B, reflecting off a mirror, a line p at point P, so A-P-B is the shortest path because B' over axis p is the symmetric image of point B while point A-P-B' lies on the same line.
Indeed, if the point Q ∈ p was the reflecting point of the ray, from the triangle AQB' we see:
AQ + QB = AQ + QB' ≥ ≥ AB' = AP + PB' = AP + PB,
AQ + QB ≥ AP + PB,
which means that the deflection at point P ∈ p really the shortest path. The inequality becomes an equality if and only if Q = P. From the picture, we can also see that the angles of incidence and reflection are equal because the angles of the θ ray and the normal at p are also equal.

The picture on the left is the Cartesian (Oxy) system through which the particle wave passes from the point P(0, b) to the point Q(a, -c) in the shortest time. The first quadrant is the medium where it has a constant speed v1 and the fourth where it has a constant speed v2 with the point x abscissa breaks. There are the incident φ1 and the angle of refraction φ2 of the ray path normal to the boundary surface of velocities.
The paths by which the particle wave passed through the I and IV quadrants are, respectively:
\[ \ell_1 = \sqrt{b^2 + x^2}, \quad \ell_2 = \sqrt{c^2 + (a-x)^2}. \]Road quotients ℓk/vk = tk with corresponding speeds, for both k = 1, 2, give the elapsed times, and the sum of these times is the function f(x) = t1 + t2 whose minimum, derivative at the stationary point, gives us the trajectory of the shortest duration:
\[ f(x) = \frac{\sqrt{b^2 + x^2}}{v_1} + \frac{\sqrt{c^2 + (a-x)^2}}{v_2}, \] \[ f'(x) = \frac{1}{v_1}\frac{x}{\sqrt{b^2 + x^2}} - \frac{1}{v_2}\frac{a-x}{\sqrt{c^2 + (a - x)^2}} = 0, \] \[ \frac{\sin\varphi_1}{v_1} - \frac{\sin\varphi_2}{v_2} = 0. \]This is Snell's law of refraction. It demonstrates the "shortest time" in the example of refraction, as opposed to the previous example, the reflection of light, which represents "shortest paths."
Information is a weaving of space, time, and matter, and its essence is uncertainty. However, there is nature's aversion to uncertainty, which can be read in the more frequent realization of more probable events and in the tendency of the system towards less informative states, or to less physical actions, inertness. This is how nature should disappear, but the laws of conservation, i.e. reality, do not allow it to do so.
Brachistochrone » srb
Question: Explain to me the brachistochronic curve problem?

Answer: Brahistochrones (ancient Greek βράχιστος χρόνος meaning "shortest time") are curves that represent the movement of a point from rest to the desired point in the shortest time if we ignore friction and air resistance. Figure right, middle, red curve. It is the problem of the fastest gravitational descent between two given points.
A classic example of the calculus of variations is finding the brachistochrone, defined as a smooth curve joining two points A and B (not one below the other) along which a particle will slide pulled by gravity, and for the shortest possible time. Otherwise, in the calculus of variations, the following so-called fundamental lemma.
Lemma. Let y(x) be continuous on the interval [a, b] and suppose that for each function η(x) ∈ C2[a, b], such that η(a) = η(b) = 0, we have:
\[ \int_a^b y(x)\eta(x) \ dx = 0. \]Then y(x) = 0 for all x ∈ [a, b].
Proof: Suppose the opposite, that for some ξ ∈ [a, b] we have y(ξ) > 0. For continuity, it will be y(ξ) > 0 and in some interval, the neighborhood of the ξ. Product y(x)η(x) > 0 is then in that interval, and everywhere outside it is zero, so the integral of the product is greater than zero — which contradicts the assumption. ∎
The search for the path of minimum time, in addition to this, requires the law of conservation of energy to limit the speed v of a particle of mass m dragged along x by gravitational acceleration g. Thus, from mv²/2 = mgx follows \(v = \sqrt{2gx}\), so we find the elapsed time with the given conditions y(0) = 0 and y(h) = a:
\[ T(y) = \int_A^B dt = \int_A^B \frac{ds}{ds/dt} = \int_A^B \frac{ds}{v} = \int_0^h \frac{\sqrt{1 + (y')^2}}{\sqrt{2gx}}\ dx, \]where ds² = dx² + dy² defines the infinitesimal length of the interval. The one we are looking for is the extremum of this functional and therefore satisfies the Euler-Lagrange equation, and hence:
\[ \frac{d}{dx}\left(\frac{\partial T}{\partial y'}\right) - \frac{\partial T}{\partial y} = 0, \quad y(0) = 0, \quad y(h) = a, \] \[ \frac{d}{dx}\left(\frac{y'}{\sqrt{2gx(1 + (y')^2)}}\right) = 0, \] \[ \frac{y'}{\sqrt{2gx(1 + (y')^2)}} = c. \]This c is the integration constant, and let's rearrange that expression:
\[ y' = \frac{dy}{dx} = \frac{\sqrt{x}}{\sqrt{\alpha - x}}, \quad \alpha = \frac{1}{2gc^2}, \]solving by y'. Substituting x = α sin²θ and integrating we find:
\[ y = \int\frac{\sqrt{x}}{\sqrt{\alpha - x}}\ dx = \int\frac{\sin\theta}{\cos\theta} 2\alpha \sin\theta \cos\theta\ d\theta = \int \alpha(1 - \cos 2\theta)\ d\theta, \] \[ y = \frac{\alpha}{2}(2\theta - \sin 2\theta) + c_1, \]where c1 is the new integration constant. Returning x and using the condition y(0) = 0 to find c1 = 0, we get:
\[ y(x) = \alpha \arcsin\left(\sqrt{\frac{x}{\alpha}}\right) - \sqrt{x}\sqrt{\alpha - x}. \]This curve is called a cycloid. The cycloid, with the peaks upwards, is the curve of fastest descent under uniform gravity and is the sought-after brachistochrone.
The constant α is determined implicitly by the remaining boundary condition y(x) = a. The equation of a cycloid is often given in the following parametric form (which can be obtained by substitution in the integral):
\[ x(\theta) = \frac{\alpha}{2}(1 - \cos 2\theta), \quad y(\theta) = \frac{\alpha}{2}(2\theta - \sin 2\theta). \]It is constructed by the trace, the initial contact point, when a circle of radius α/2 rolls along a straight line (turned by an angle of 2θ). A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.
A cycloid is a circle whose center moves and, in that sense, is similar to a hammer that rotates around the putter in the stride for better swing and higher weight speed for greater range. It is the gain of the maximum through the minimum (the competitor's strength), as opposed to the decline where the minimum (time) arises from the maximum. Brahistochrona is a beautiful, though more complex, example of the principles of minimalism.
Fizeau » srb
Question: What about Snell's Law in a moving medium?

Answer: In 1851, Hippolyte Fizeau performed experiments measuring the relative speed of light in moving water, like the diagram on the left. According to the knowledge of the time, light traveling through a moving medium would be dragged through the medium, so the measured speed of light would be the sum of the speed of the medium and the speed of light through the medium. Fizeau detected a drag effect, but it was far less than expected. Later (1905), this deficiency was explained with the theory of relativity and Lorenz transformations.
According to the Lorentz transformations (Defect):
\[ dx = \gamma(dx' + vdt'), \quad dz = dz', \quad dt = \gamma(dt' + \frac{v}{c^2}dx'), \]where now the Lorentz coefficient γ = 1/(1 - v²/c²)-1/2. Let's say that the light from the origin along the x-axis encounters water (or glass, and transparent medium), through which it passes with a speed c' = c/n.
Then the components of the speed of light in a moving medium are:
\[ u_x = \frac{dx}{dt} = \frac{\gamma(dx' + vdt')}{\gamma(dt' + \frac{v}{c^2}dx')} = \frac{\frac{dx'}{dt'} + v}{1 + \frac{v}{c^2}\frac{dx'}{dt'}} = \frac{c'_x + v}{1 + \frac{c'_xv}{c^2}}, \] \[ u_z = \frac{dz}{dt} = \frac{dz'}{\gamma(dt' + \frac{v}{c^2}dx')} = \frac{c'_z\sqrt{1 - \frac{v^2}{c^2}}}{1 + \frac{c'_xv}{c^2}} = 0. \]When the light hits the surface of the medium at an angle of incidence α = 0, the output is then β = 0, there is no deflection towards the normal, and c'z = 0. Therefore, with a smaller media velocity v or a higher refractive index n, it will pull higher and closer to the sum of the speeds of the medium and the speed of light through the medium.
Example 1. In the following picture, let's show that for the path of light through the medium (Δℓ), the continuation of the path along the abscissa (Δx0) and the descent by height (Δ z0) applies:
Δℓ0 = Δs0/cos β0, Δx0 = Δs0/cos α0, Δz0 = Δs0 (tg α0 - tg β0) cos α0,
where Δs0 is the thickness of the optical media layer. The media is at rest .

Solution: The layer thickness Δs0 = NQ is taken along the normal to the surface of the medium, the dashed line NQ in the first point. The continuation of the abscissa path Δx0 = NM is the hypotenuse of the right triangle MNQ of the angles ∠Q = 90° and ∠N = α0. Hence Δx0 = Δs0 / cos α0 and finally, the length MQ = Δs0⋅tg α0.
Triangle PNQ is also right-angled, with angle ∠N = β0, so the path of light through the medium is NP = Δℓ0 = Δs0 / cos β0 and PQ = Δs0 ⋅ tg β0. With the height Δz0 = CA drawn from the point M we get a right angled triangle with angle ∠ M = α0, and hence:
Δz0 = MP ⋅ cos α0 = (MQ - PQ) ⋅ cos α0 = Δs0 ⋅ (tg α0 - tg β0) ⋅ cos α0,
and that is the last of the required equalities. ∎
As a bonus to this one, here's another example. Wave refraction is the change in direction that occurs when waves travel from one medium to another. Refraction always goes with a change in wavelength and speed. Diffraction is the bending of waves around obstacles and apertures. The amount of diffraction increases with increasing wavelength.

In the picture on the right, a ray of light comes from the left with speed c and refracts through the middle (brown strip), which it passes for the observer at rest with speed c' = c/n < c, represented by the vector \(\vec{u}_0\), to continue at the speed c after leaving the point P0 until the point A0. It is an image of the static refraction medium.
However, if the medium moves with speed v, parallel to the abscissa, it drags the light with it at the speed of intensity:
\[ u = \frac{c' + v}{1 + \frac{v}{c'}\frac{1}{n^2}},\quad n > 1, \]which at the exit continues again at speed c. This is the velocity after refraction, whose direction at rest is at an angle θ0 = α0 - β0 according to the x-axis (abscissa), and θ for the observer who sees the movement. The projections of the speed of the first on the x-axis, y-axis, and z-axis, respectively, are:
ux = u⋅cos(α0 - β0), uy = 0, uz = u⋅sin(α0 - β0).
Contraction of the length of the medium by the direction of movement gives Δx = Δx0/γ, the length it sees movement observer, while there is none along the vertical axes. That's why θ > θ0, so the exit point from the media is no longer P0, but P, and the final one becomes A. With the thickness of the medium, the gap AA0 increases.
As in the above (Crash), the question is what will happen to an observer who moves along with the medium, versus one who sees them moving at v, if in position A there is a light switch that could activate a very different environment inside the room. The answer is the same as the previous one, that this moving belongs to a different reality (Dimensions) than the one in which one descends from the moving space to the space of the relative observer of their movement at speed v . See also "Angles" further down in the blog.
Lorentz » srb
Question: Do you have a simpler derivation of Lorentz transformations than the one above (Defect)?

Answer: Yes, these are the well-known tasks of Einstein's special theory. It is based on two principles, the first of which concerns the relativity of motion: "All the laws of physics are invariant, i.e., equal, relative to observers from inertial reference frames.” The second principle concerns the universal speed of light. The speed of light (c = 299 792 458 m/s, in a vacuum) is constant for any inertial observer. The continuation is from one of my recent (2017) books (Space-Time, Example 1.4.1.).
1. Example. Let's derive the Lorentz transformations:
x' = γ(x - βct), y' = y, z' = z, ct' = γ(ct - βx),
where are the Lorentz coefficient gamma and parameter beta, respectively:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}, \quad \beta = \frac{v}{c}. \]We use two principles of Einstein's special relativity.
Derivation: In the general case, for a constant velocity v along the abscissa, the x-axis, we have linearity of transformations:
x' = γ(x - βct), y' = y, z' = z, ct' = act - bx,
where γ, β, a, b are unknown constants that have yet to be determined. They are reduced to classical, Galilean ones, with γ = 1, β = v/c, a = 1 and b = 0.
The origin O', position x' = 0, moves with a constant speed v along the abscissa of the system O, so that x = vt, hence β = v/c. The inverse transformations have the same form, with the velocity of the opposite sign, so x = γ(x' + βct'), where x = ct whenever x' = ct', and we have:
x' = γ(x - βct) = γ(x - βx) = γ(1 - β)x, x = γ(x' - βct') = γ(x' - βx') = γ(1 - β)x',
xx' = γ²(1 - β²)xx',
\[ \gamma^2 = \frac{1}{1 - \beta^2} = \frac{1}{1 - \frac{v^2}{c^2}}. \]Time transformation also follows from the condition x' = ct' ⇔ x = ct, whence ct' = γ(ct - βx), which is what was requested. ∎
The Interval is given by the square, by a pseudo-Euclidean expression:
(ds)² = (dx)² + (dy)² + (dz)² - (dct)²,
which is also written without these brackets. This form is an invariant of Lorentz transformations. Namely,
(ds)² = [dγ(x' + βct')]² + (dy')² + (dz')² - [dγ(ct' + βx')]² =
= γ²(1 - β²)(dx')² + (dy')² + (dz')² - γ²(1 - β²)(dct')² = (ds')²,
considering the values of the gamma and beta parameters. The same, assumed, is used for another way of performing Lorentz transformations, further from the assumption of their linearity and symmetry of space and time coordinates:
x' = γ(x - βct), y' = y, z' = z, ct' = γ(ct - βx).
Time dilation follows this invariance. The distance dℓ that is covered at the speed v in the time dt is the length dℓ² = dx² + dy² + dz². By equalizing the intervals ds² = dℓ² - c²dt² = dℓ'² - c²dt'² = ds'², and in one of the two systems the speed of the observed point is zero, so we find:
\[ dt = \frac{dt_0}{\sqrt{1 - \frac{v^2}{c^2}}}. \]Here t0 is the elapsed time in which the given point is at rest, and t is the time as measured by the observer of the motion.
Contraction length. We observe the length x'b - x'a = Δℓ0 which is at rest in a moving system and Lorentz transformations we calculate for Δℓ as it is seen by a relative observer from another system:
Δℓ0 = x'b - x'a = γ(xb - βctb) - γ(xa - βcta) =
= γ(xb - xa) - γβc(tb - ta) = γ(xb - xa) = Δℓ, tb - ta = 0.
We consider the ends of the length to be simultaneous (tb = ta), so it is:
\[ \Delta \ell = \Delta \ell_0 \sqrt{1 - \frac{v^2}{c^2}}, \]the length Δℓ0 placed along the direction of motion at speed v as seen by the observer of the motion. The length Δℓ0 is proper, while Δℓ is its relative.
Note. A slower flow of time, seen as a slow frequency of events and then a smaller amount of uncertainty in the sense of greater directionality and therefore increased inertia concerning the time of a relative observer, would give her a greater relative energy, E = E 0⋅γ, proportional to time. According to a global, statistical observation, as opposed to the local ΔE⋅Δt = h, for a photon, for example.
This can also be understood as an increase in energy, i.e., a greater mass of a body that falls freely from a position outside the gravitational field of a body of mass M at a distance r from its center:
\[ m = \frac{m_0}{\sqrt{1 - \frac{2GM}{rc^2}}} \]in the book Sprega Informacije, 114th example. Another question is whether the stationary mass, hypothetically introduced there (3.22), can be understood similarly; it is less than this but larger for a parked body at a smaller distance from the center.
Angles » srb
Question: How to perform the contraction of the length and the change of the angle of inclination of a rod inclined to the axis of movement by means of Lorentz transformations?

Answer: In the picture on the right, there is such a rod AB in the Oxy plane of the rectangular Cartesian coordinate system (Oxyz). The ends of the stick are points A(xA, yA) and B(xB, yB) with a slope to abscissa θ. The rod's proper (relative to a calm observer) length is Δℓ0, and its projections are on the x and y-axes, respectively: Δx0 = Δℓ0⋅cos θ and Δy0 = Δℓ0⋅sin θ.
Parallel to the abscissa (x-axis), the rod moves with speed v, and (only) in that direction the lengths shorten. The projections on the axes to a relative observer, who sees the movement, amount to:
\[ \Delta x = \Delta x_0\sqrt{1 - \frac{v^2}{c^2}}, \quad \Delta y = \Delta y_0 \]and there are none along the z-axis. The relative length of the rod is:
\[ (\Delta \ell)^2 = (\Delta x)^2 + (\Delta y)^2 = (\Delta x_0)^2 + (\Delta y_0)^2 - \frac{v^2}{c^2}(\Delta x_0)^2 = \] \[ = (\Delta \ell_0)^2 - \frac{v^2}{c^2}(\Delta \ell_0 \cos \theta)^2 = (\Delta \ell_0)^2 \left(1 - \frac{v^2}{c^2}\cos^2\theta\right), \] \[ \Delta \ell = \Delta \ell_0 \sqrt{1 - \frac{(v\cos \theta)^2}{c^2}}. \]Here vℓ = v cos θ is the projection of the velocity onto the rod direction. The proper length and angle are Δℓ0 and θ0, and the relative Δℓ and θ, where the tangent of this angle is:
\[ \text{tg }\theta = \frac{\Delta y}{\Delta x} = \frac{\Delta y_0}{\Delta x_0\sqrt{1 - \frac{v^2}{c^2}}} = \frac{\text{tg }\theta_0}{\sqrt{1 - \frac{v^2}{c^2}}}. \]The tangent of the relative angle θ is the magnified tangent of the proper θ0 in proportion to time dilation. To a relative observer, the stick is shortened but also straightened.
Note. Light that is refracted through a moving optical medium (Fizeau) is refracted even more to a relative observer. Therefore, a vertical gap appears between the own (A0) and the relative (A) blind point at the exit, in places where light switches can be placed, which would lead the participants to different futures.
Proper Velocity » srb
Question: What is "proper velocity"?

Answer: That is the intrinsic velocity, seen from the system in rest, which the object sees from its frame. It is Einstein's (picture left) velocity that was measured by its proper time τ, which is slowed down from relative:
\[ \Delta t = \frac{\Delta\tau}{\sqrt{1 - \frac{v^2}{c^2}}}. \]The relativistic dilation of the time interval is the inherent interval Δτ, the time passed by the body itself (or by the observer standing next to the body), increased to the relative interval Δt, observed from motion. The own observer moves away from the relative one at a speed v, while the relative one moves away from the resident one at a speed -v; these are exactly the movements at "proper speeds" in the opposite direction.
1. Expressed in terms of its proper velocity, the object's momentum:
\[ p = \frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}}, \]with its proper mass m0, the one it has at rest, p becomes a physical quantity for which the first postulate of relativity applies (laws of physics are invariant in relation to observers from inertial reference systems, i.e. frames). Such "momentum" grows as time slows down.
Let's note how well the annotations in the answer "Lorentz" align with this. A slower flow of time, if it is a slower frequency of events with a reduced bulk of uncertainty, most directionality, therefore larger inertia, represents increased relative energy, E = E0⋅γ, proportional to time.
2. Accordingly, we define the relativistic force:
\[ F = \frac{d}{dt}(\gamma m_0 v) = \frac{d}{dt}\frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}}. \]We use force to define relativistic kinetic energy (EK). It is the difference between the final and initial energy of a body that, under the influence of a given force (F) from a state of rest, reached a certain speed (v). Let's integrate that sum:
\[ E_K = \int_0^v F\ dx = \int_0^v \frac{dp}{dt}\ dx = \int_0^v \frac{d(\gamma m_0 v)}{dt}\ dx = \] \[ = m_0 \int_0^v \frac{d}{dt}\left(\frac{v}{\sqrt{1 - \frac{v^2}{c^2}}}\right)\ \frac{dx}{dt}dt \] \[ = m_0\left(v\cdot \frac{v}{\sqrt{1 - \frac{v^2}{c^2}}} - \int \frac{dv}{dt}\cdot \frac{v}{\sqrt{1 - \frac{v^2}{c^2}}}\ dt \right)_0^v \]It is a partial integration, with dx/dt = v. Continuing it, we find:
\[ E_K = m_0\left( \frac{v^2}{\sqrt{1 - \frac{v^2}{c^2}}} - \int \frac{v\ dv}{\sqrt{1-\frac{v^2}{c^2}}}\right)_0^v = \] \[ = m_0\left( \frac{v^2}{\sqrt{1 - \frac{v^2}{c^2}}} + c^2\sqrt{1 - \frac{v^2}{c^2}}\right)_0^v \] \[ = m_0\left(\frac{v^2}{\sqrt{1 - \frac{v^2}{c^2}}} + c^2\sqrt{1 - \frac{v^2}{c^2}} - c^2\right) \] \[ = m_0\left[\frac{v^2 + c^2\left(1 - \frac{v^2}{c^2}\right)}{\sqrt{1 - \frac{v^2}{c^2}}} - c^2\right], \] \[ = m_0\left[\frac{v^2 + c^2\left(1 - \frac{v^2}{c^2}\right)}{\sqrt{1 - \frac{v^2}{c^2}}} - c^2\right], \]EK = m0c²(γ - 1).
3. We have obtained a general expression for the relativistic kinetic energy:
\[ E_K = \frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}} - m_0c^2. \]It is consistent with the "proper velocity" and that EK is the exact value of the kinetic energy for all speeds v. In the case of low speeds (v/c → 0), this EK becomes classical kinetic energy. Namely, by developing γ = (1 - v²/c²)-1/2 in Taylor's Power Series:
\[ (1 - x)^{-1/2} = 1 + \frac12x + \frac38x^2 + \frac{5}{16}x^3 + ..., \]putting x = (v/c)², when v << c, ignoring x of higher powers, will be:
\[ E_K \approx m_0c^2\left(1 + \frac12\frac{v^2}{c^2}\right) - m_0c^2 = \frac12m_0v^2. \]As a special case, we recognize the kinetic energy of classical physics.
Massless » srb
Question: Are there massless particles that have momentum and energy?

Answer: Yes, there are physical particles that are massless. These are: gluon, photon, and graviton. First, gluons are massless particles that hold together atomic nuclei. I wrote about photons in a recent script (Sprega Informacije, 3.2.1 Light) in one way, but here I will do the same a little differently, to avoid repetition. In the image link, read: "How Do Massless Particles Experience Gravity?".
In the explanation of massless particles that have momentum and energy, let's start now from the expression for kinetic energy of the previous answer (Proper Velocity, 3):
EK = γm0c² - m0c²,
E = γm0c² = EK + m0c².
1. That E = γm0c² is the total relativistic energy of the system. If the velocity is v = 0, we get the energy of the system at rest E0 = m0c², where m0 is its rest mass, and c ≈ 300,000 km/s is the speed of light in a vacuum. Considering the Lorentz coefficient γ, we further find:
\[ E^2 = \left(\frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}}\right)^2, \] \[ E^2\left(1 - \frac{v^2}{c^2}\right) = m^2_0c^4, \] \[ E^2 - \frac{E^2v^2}{c^2} = m_0^2c^4, \]and again because of Ev = γm0vc² = pc², we further find:
E² - p²c² = m0²c4,
E² = m0²c4 + p²c².
2. We get for the total relativistic energy:
\[ E = \sqrt{m_0^2c^4 + p^2c^2}. \]It is a very important term for modern physics. Here are some simpler uses. When the particle is at rest, v = 0, momentum is p = 0, and the total energy is E = m0c². Another example: if the speed is close to light, v → c, but the mass of the particle increases indefinitely, it will be pc >> m0c² and E ≈ pc.
3. The third example, if m0 → 0, we again have:
E = pc,
(γm0c²) = (γm0v)c,
v = c.
This is the answer to the question: that a massless particle can have momentum and energy, but then it must move at the speed of light.
From this also follows, for example, p = h/λ, the familiar Louis de Broglie relation with h Planck's constant and λ wavelength. A photon of frequency ν has energy E = hν = pc, and λν = c.
4. Note. With the theory of information, I go a step further than this. I also consider states analogous to particles, but without mass, momentum, and energy. I've written some of these before, either for private use (like these blogs) or publicly (with reviews), citing them as abstract statements of mathematics or timeless truths.
Maupertuis » srb
Question: What is Maupertuis's principle?

Answer: Maupertuis (1747) is one of the discoverers of the principle of the smallest effect, completely misunderstood at the time. He called:
S0 = mvs
effect, action. Simply considering the greater action (S0) such that happens to a greater mass m, when it has a greater speed v and it's on the long way s. Nature tends to keep this action constant, he found.
1. Let's note that by maintaining such a size, the body, reflecting off the surface, will achieve the same effect as light reflecting off a mirror. This is the case with the body in the above image on the left that passes through point A, bounces off hitting an obstacle (abscissa) at point C to end up at point B, arriving as short as possible via A-C-B, with the requested condition. At the same time, it is the shortest travel time, but also with the smallest change of "action" (S0).
Knowing that the law of momentum conservation applies, \(\vec{p} =\) const, we easily find that the projections of the traveled paths of the body on the x and y-axis in equal times also be conserved, Δx, Δy = const, and therefore the angle of incidence must be equal to reflective, α = β. It is the result of the reflection of light (Snell), where we have constant momentus to and from the abscissa to and after point C.
Namely, for the vector \(\vec{p} = m\vec{v}\), as well as for each of its components \(\vec{p} = \vec{p}_x + \vec{p}_y\) , conservation laws apply. Thus, this first (\(\vec{p}_x\)) before and after the collision remains the same, and the second (\(\vec{p}_y\)) changes direction, in the exchange of action and reaction with a zero-impulse background, after elastic rejection (the more you give, the more you get).

2. In the picture on the right, we see the trajectory of a body descending the y-axis at a constant speed vy, at the same time moving from the point A to C horizontally with speed vx = v sin α, and from points C to B continues with a lower speed v'x = v' sin β, so overall it has the smallest change in Maupertuis action (S0). This is equivalent to the refraction of light (Snell), but here we do not observe Fermat's principle of shortest time because the law of conservation of energy takes precedence.
3. Namely, from the parts of the actions of the form:
\[ dS_0 = mv\ dx = mv\frac{dx}{dt}\ dt = 2\frac12mv^2\ dt = 2E_k \ dt, \]the least total action is the sum of the least changes in it along the path:
\[ \delta\int dS_0 = 0, \] \[ 0 =\delta\int (2E_k)\ dt = \delta\int (E_k + E - E_p)\ dt = \delta \int (E_k - E_p)\ dt, \]because the total energy E is a constant sum of kinetic (Ek) and potential (Ep). However, the last one is the integral of the Lagrangian (L = Ek - Ep), for which the principle of least action of physics applies:
\[ \delta \int L(x, \dot{x}, t)\ dt = 0. \]Therefore, the condition of conservation of energy (E = const.), the Hamiltonian principle (Ek + Ep = const .), leads directly to Maupertuis's principle of the Lagrangian (Ek - Ep = const.), i.e. the principle of least action which is more common today we use
For example, see problem 19 in Sprega Informacija, that the central permanent forces (gravity, Coulomb) decrease with the square of the distance, that they drive charges along the trajectories of ellipses, parabolas, or hyperbolas and reach them in equal times sweeps equal surfaces. This last of the series of results, by Lagrange's principle of least action, connects seemingly unrelated physical phenomena.
4. Note. The Maupertuis's action, say along the abscissa, can be written using the momentum along that axis: Sx = mvx = px. Then it retains the same meaning in the case of massless particles—waves, for example, in the case of light. Up to at least their wavelengths λ, and when x = λ, then p = h/λ, where h is Planck's constant. At that extreme, Maupertuis's action becomes a quantum of action. Conservation laws apply to quanta of actions, and hence, collectively in macrobodies, nature tends to keep Mauperty's actions constant.
However, quanta of action are equivalent to physical information, so it is more practical to write action as the product of the change in energy (ΔE) over time (Δt), because Δ E⋅Δt = Δp⋅Δx = h, where Δp is also the change, but the momentum during path Δx. Particle waves are always oscillations in which the mentioned quantities alternate. In the macro world, in which the time intervals Δt can be considered equal, the law of conservation of action h becomes the law of conservation of energy.
Projectile » srb
Question: Explain Maupertuis's principle to me on the example of an oblique shot?

Answer: In the continuation of the previous one, I explained that Maupertuis's "action" is not so much practical as it is principled, but here is also that "school example" of application in the verification of Newton's law, how force gives acceleration to mass. The task is an oblique shot, and the principle of least action will confirm something about force and acceleration (\(F = m\ddot{y}\)).
In the picture on the left and the attachment, the orthos (unit vectors) x and y-axes are \(\hat{i}\) and \(\hat{j}\). The constant acceleration, \(\ddot{y} = -g\), has the opposite direction to the ordinate. Still, it is not along the abscissa \(\ddot{x} = 0\), and we are left to deal only with vertical movement and the relationship between force and acceleration.
A body of mass m had an initial vertical velocity v0. It reached its maximum height H after time T and then fell symmetrically. We ignore both mass change with velocity and air resistance. The height and velocity of a thrown body are functions of time, respectively:
\[ y = v_0t - \frac12gt^2, \quad v = v_0 - gt. \]By eliminating time, we find the velocities projected on the ordinate:
\[ v = \pm \sqrt{v_0^2 - 2gy}, \]when climbing (+) and falling (-). Maupertuis's action is total:
\[ S = 2\int_0^H mv\ dy = 2m\int_0^H \sqrt{v_0^2 - 2gy}\ dy = \] \[ = -\frac{m}{g}\left[\frac{(v^2_0 - 2gy)^{3/2}}{3/2}\right]_{y=0}^H = \frac{m}{g}\frac{2v_0^3}{3}, \]because at the highest point the speed is zero, v0² - 2gH = 0. Note that mv⋅dy has the physical value of the product of momentum and time; however, the value of E⋅dt is the product of energy and time.
The total energy of the body is E = Ek + Ep = mv²/2 + Fy. It is the sum of kinetic and potential, in other words, the half product of the mass and the square of the velocity of the body and the work of the force on the road. These two energies are variable in height y or time t, so we have to break them apart and integrate them in order to get their total Maupertuis actions. These are now the products of energy and time (Et = py), so we integrate over time and take the mean value of the result during the period 2T, how long the body spent in flight.
The climbing and descending energies are symmetrical, and we calculate the averages for each of them:
\[ \mu(E_k) = \frac{m}{2T}\int_0^T (v_0 - gt)^2\ dt = \frac{m}{2T}\left[-\frac{1}{g}\frac{(v_0 - gt)^3}{3}\right]_{t=0}^T = \frac{mv_0^2}{6}, \] \[ \mu(y) = \frac{1}{T}\int_0^T y\ dt = \frac{1}{T}\int_0^T(v_0t - \frac{a}{2}t^2)\ dt = \frac{v_0^2}{3g}, \] \[ \mu(E_p) = F\frac{v_0^2}{3g}, \]because v0 - gT = 0, and the force is constant; otherwise, it is negative, so the average potential energy is equal to the product of such a force and the average height, μ(Ep) = -F⋅μ(y). We further use the already calculated action:
\[ S = \frac{2v_0^3}{3g} \]in the average total energy, and we find its optimum:
\[ \mu(E) = \mu(E_k) + \mu(E_p) = \frac{mv_0^2}{6} + F\frac{v_0^2}{3g} = \frac{mv_0^2}{6} + F\frac{S}{2v_0}, \] \[ \frac{\partial \mu(E)}{\partial v_0} = \frac{m}{3}v_0 - \frac12 F S v_0^{-2} = 0, \] \[ F = \frac{2mv_0^3}{3S} = \frac{2mv_0^3}{3}\frac{3g}{2v_0^3} = mg. \]So, Newton would be right when he said that "a force gives acceleration to a mass," more precisely F = mg.
Nothing new, we will say, because we have known this for a long time. In the discussion about the need for all this, we also referred to the statement of the Hungarian mathematician George Pólya that "Mathematics consists of proving the most obvious things in the least obvious ways." However, checking and doubting is a researcher's normal job; as it sometimes happens, it pays off. For example, we now know that force, thus the product of mass and acceleration, is the result of the principle of least action — without mentioning the law of conservation of energy.
Note. At the end of the previous answer, I stated that the law of conservation of energy follows from the law of conservation of (quantum) action — in conditions where equal time intervals can be counted on. Now we have that priority should be given to conservation of action. I am considering the possibility that time is only regulated in such a way, i.e. calculated, that a closed system, from the smallest to the largest, has its units of duration for which both conservations, energy, and action would be equally valid. But about that later.
Doppler » srb
Question: How is it possible that both action and energy are conserved and the latter is a factor of the former? Can you explain it to me with the example of light?

Answer: You asked a tough question. It is a raw topic and all the more challenging. Especially because light is very important to us and is so special. First of all, it has no time of its own; its time is so slowed down (stands still) that its continuity comes from ours, and it doesn't really have a "velocity of its own" (Proper Velocity).
What we see as the speed of light in a vacuum c ≈ 300,000 km/s should be understood as the "speed" of the progression of our present into our future. I emphasize "our," referring to each subject's own. But, on the other hand, it makes just as much sense to talk about the energy average (E) or the time average (Δt) of the photon E = hf, with h ≈ 6.626×10-34 J⋅Hz-1 Planck constant and f = 1/Δt frequency light. I proposed it (the book "Space-Time", 1.3.3 Red shift, 2017) as a tentative assumption.
The idea is that we do not define light by what it "could be," but by how it is seen. The beginning is the relativistic Doppler Shift with two continuations. Let's treat the interpretation of frequencies separately from wavelengths. We calculate the average of the frequencies when we use the Maupertuis action, and we see the wavelength simply as the uncertainty of the position in the direction of movement.

The light source moves along the lower line at speed v, while on the line at an angle θ to it, we record two signals, hence the moments t1 and t2 in the period Δt = t2 - t1, during which the source crosses the path Δx = v⋅Δt . By the time the second signal arrives, the first one has already traveled the path Δx' = Δx⋅cos θ, relative to us, the external observer.
Because of the size of the distance ℓ from the second signal to the observer versus the distance Δx, we consider the angles θ to be equal, as well as the places of reception of the signal at moments t'1 and t'2. Time and duration Δt' = t'2 - t'1 those receptions are:
t'2 = t2 + ℓ/c, t'1 = t1 + (ℓ + v⋅Δt⋅cos θ)/c,
Δt' = Δt⋅(1 - β⋅cos θ), β = v/c.
However, the source's time and ours do not flow at the same speed because:
\[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}, \]where Δt0 is the light source's own (proper) time, so the previous is:
\[ \Delta t' = \frac{\Delta t_0}{\sqrt{1 - \beta^2}}\cdot(1 - \beta \cdot \cos\theta), \quad \beta = \frac{v}{c}. \]Therefore, the general frequency formula for the observer is reciprocal:
\[ f' = f_0 \cdot \frac{\sqrt{1 - \beta^2}}{1 - \beta \cos\theta}, \]when f0 is the proper frequency of the light source (as seen by an observer at rest relative to the source). I repeat, frequency is the number of blinks or periods in a unit of time.
1. Example. Let's prove the relativistic formula of the Doppler effect for the "redshift" of the observer on the line of motion from the light source:
\[ f'_+ = f_0 \sqrt{\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}. \]Proof: Then θ = 180°, so for the general formula we find cos θ = -1 and:
\[ f'_+ = f_0\cdot \frac{\sqrt{1 - \frac{v^2}{c^2}}}{1 + \frac{v}{c}} = f_0 \sqrt{\frac{\left(1 - \frac{v}{c}\right)\left(1 + \frac{v}{c}\right)}{\left(1 + \frac{v}{c}\right)\left(1 + \frac{v}{c}\right)}} = f_0 \sqrt{\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}, \]and hence follows the given formula. ∎
2. Example. Let's prove the relativistic formula of the Doppler effect for the "blue shift" of the observer on the line of motion towards the light source:
\[ f'_- = f_0 \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}. \]Proof: Then θ = 0°, so for the general formula we find cos θ = +1 and:
\[ f'_- = f_0\cdot \frac{\sqrt{1 - \frac{v^2}{c^2}}}{1 - \frac{v}{c}} = f_0 \sqrt{\frac{\left(1 - \frac{v}{c}\right)\left(1 + \frac{v}{c}\right)}{\left(1 - \frac{v}{c}\right)\left(1 - \frac{v}{c}\right)}} = f_0 \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}, \]and hence the given formula. ∎
Because the speed of light does not depend on the speed of the source and λf = c, the general formula for the wavelength of light will be:
\[ \lambda' = \lambda_0 \cdot \frac{1 - \beta\cos\theta}{\sqrt{1 - \beta^2}}, \quad \beta = \frac{v}{c}, \]where again θ is the angle of the direction of movement of the light source of wavelength λ0 towards the observer, who sees it as λ'. The speed of light is c ≈ 300,000 km/s, and the speed of its source is v. We have long known this, as well as the transferal relativistic Doppler effect (θ = 90°):
\[ f'_{\perp} = f_0 \sqrt{1 - \frac{v^2}{c^2}}, \quad \lambda'_{\perp} = \frac{\lambda_0}{\sqrt{1 - \frac{v^2}{c^2}}},\]which is otherwise absent in the non-relativistic, classic Doppler effect of sound, nor do we take it today for other, slow waves.
For the mean values of the outgoing and incoming light sources, we find:
\[ f'_{\pm} = \frac12(f'_+ + f'_-), \quad \lambda_\pm = \frac12(\lambda'_+ + \lambda'_-), \] \[ f'_{\pm} = \frac{f_0}{\sqrt{1 - \frac{v^2}{c^2}}}, \quad \lambda_{\pm} = \frac{\lambda_0}{\sqrt{1 - \frac{v^2}{c^2}}}, \]which is easy to check by adding the roots and editing the expression. So much for familiar things; now let's get back to the question I was asked.
The average state of frequencies taken over the whole space would again be f± and therefore could be taken to be the "real" frequency of light, which photons don't actually have. What moves at the speed of light is not a subject but a series of objects perceived by participants in the course of events. Consistency in the movement of such perceived entities is a consequence of the consistency of the world of those subjects.
In accordance with this, the energy calculation is based on the smaller acton, so for the photon energy we find E = hf±. It is an unusual idea, but it is consistent with the understanding of objects as mean values of the experiences of the subjects around them. Deep in the foundations of information theory is that to be means to communicate, last, and establish. Until further notice, therefore, I would not lightly dismiss that unusual thought about the mean value of action.
Note. The light source that comes to us, like all the objects that are coming to us, comes from the future into the ever-closer present (because time runs slowly for them), so that at the moment of passing we would have the same present. The opposite is the case with a source that is receding and moving into our increasingly distant past (our time is running faster). Shortening the wavelengths of the light of the incoming source thus becomes, in this interpretation, a reduction of uncertainty in the future.
Least Action » srb
Question: That oblique shot (Projectile), you say "school example" of physics, is part of the general principle of information theory. How?

Answer: A stone thrown as an oblique shot is a textbook example of a physics lesson. Here, from that problem, we only extract heights (y-axis) along the ordinate, not distances (x-axis), but over time (t-axis) from left to right along the abscissa. See the markings of that vertical shot in the graphic to the right.
1. The red path is a function of height y(t) over time t, and it is, say, the line of least action. At an arbitrary moment of the stone's flight, t ∈ [t1, t2], we also note the heights q (t) = y(t) + η(t), which deviate by some arbitrarily chosen η(t) from the optimal height y(t). All that is needed is included in the account of "action," for which we find the minimum. The task is to vary it in very general ways to reach the principle minimum action. Among other things, that is why neither the starting y(t1) nor the final y(t2) height are now not the same.
2. Every physical action conveys some information, but also every physical information makes some action. In short, it is the meaning of the equivalence of physical action and information. That's where we start, for now only in my information theory, not in physics. Additionally, we will notice that more likely events are realized more often, and that such events are less informative. When we know something is going to happen and it happens, then it's not news. The two:
((physical action) ⇆ (information)) ∧ (minimalism of information),
are the basis of the further story.
3. As we have seen, physical action is the product of mass, velocity, and path (S = mvs), and hence of momentum and path (S = px, projected onto the same axis), because momentum is the product of mass and velocity. If the path action can be changed, then we must observe the bits (ΔS = Δp⋅Δx) to be added, but definitely notice (when Δx → 0):
\[ dp\cdot dx = \frac{dp}{dt}\ dt\ dx = F\ dx \ dt = dE\ dt. \]This at the infinitesimal level of magnitude means that the change in momentum over time is a force (Δp : Δt = F), that the action of a force on a path does work, a change in energy (F Δx = ΔE). Finally, the impact we are talking about, that is, physical action, is the change in energy over time (S = Et). A change in momentum along a path or a change in energy over time are therefore equal equivalents to physical information.
4. Our perceptions are always in finite packages, because infinity (then in divisibility) defines the property that it can be a proper subset of itself, and then there is nothing from the law of conservation. I derive that from Neter's theorem (e.g., Information Stories, 1.14 Emmy Neter, 2021). Therefore, the world of perceptions is finite, wherever the law of conservation applies. That's why theorems, or proofs, are our understanding of reality and truths, like the quantum of physical action — always in steps, that is, in the perception of final contents. I am not establishing that our perceptions are all reality.
That's why we have Heisenberg's uncertainty relations according to which the product of uncertainty of position and momentum, or product of uncertainty of time and energy, is at least of the order of Planck's constant. They require a minimum amount of physical action, and such is limited on the lower side. I say consistently that information is the tissue of space, time, and matter, and uncertainty is its essence, without which nature cannot do and which it would like to get rid of.
5. Let us now consider variations around stationary points, when δ∫S = 0.

In the picture above, we can see that these are extremes, or inflexions. On the left is the local maximum when the function rises and falls, that its change is (δf = f2 - f1) positive and then negative, between which is the "stationary point" when that change is zero (δf = 0). Around the local minimum it falls and then rises, and again in between is a stationary point where the change in the value of the function is zero. However, there are also points of "bending," where the function falls, stops, then continues to fall, so that at the point of stagnation its change is zero. Such is also the reverse inflection, when the function increases, stops, and increases again. We search for stationary sites of action, δ∫S = 0, targeting the minimum.
6. In the variation δ∫S = 0, we include the action values and continue with it of the form δ∫mvds = 0. As the speed is the distance traveled in the elapsed time, it is ds = vdt, so we have δ∫mv²dt = 0. The product of the mass and the square of the velocity is then twice the kinetic energy and is δ∫2Ek dt = 0. However, the sum of the kinetic and potential is the total energy, E = Ek + Ep. Hence the kinetic Ek = E - Ep, so we have δ∫(Ek + E - Ep) dt = 0. We further divide our variation into two additions:
δ∫(Ek - Ep) dt + δ∫E dt = 0,
δ∫(Ek - Ep) dt + δ(Et) = 0,
δ∫(Ek - Ep) dt + Eδt + tδE = 0.
In larger bodies it makes sense to vary the time-energy product, δ(Et), which at the smallest value levels would be quanta, but the energy variations are zero, tδE = 0, because we assume that the law of conservation of energy applies. Next is:
δ∫(Ek - Ep) dt = -Eδt,
δ∫(Ek - Ep) dt = 0,
because we observe macro bodies in equal time intervals when time variations are zero. The difference between kinetic and potential energy, L = Ek - Ep, is the Lagrangian with which the previous variation becomes:
δ∫L dt = 0.
That expression dS = Ldt and the way of writing the principle of least action was discovered by Hamilton (1834) and worked out Lagrange, after which this energy difference got its name. About the application of the Lagrangian in mechanics, it is good to first try some school examples (Space-Time, 1.2.6 Lagrangian) for easier understanding of the continuation.
In the note (at the end of Projectile's), it is once again noted that conservation of energy could be reduced to conservation of action, so that from ΔE = const, followed by Δt = const. Here it says the same: tδE = 0 ⇒ Eδt = 0. To take equal time intervals, because then we can always do it.
7. Now let's go back to the picture of throwing a rock from the beginning of this answer to find the optimal path y(t) using this old-new Hamiltonian principle. With various variations of η, this path is not optimal but becomes q(t) = y(t) + η(t). These variations are small, and so are the deviations:
δS = S[q(t)] - S[y(t)] = 0.
Then, there are some examples and applications.
1. Example. Let us show that for the kinetic energy Ek = mv²/2 of the function q the approximate equality applies
\[ E_k \approx \frac12m\left(\frac{dy}{dt}\right)^2 + m\frac{dy}{dt}\frac{d\eta}{dt}. \]Proof: We calculate Ek = mv²/2, respectively:
\[ E_k = \frac12m\left(\frac{dq}{dt}\right)^2 = \frac12m\left[\frac{d}{dt}(y + \eta)\right]^2 = \] \[ = \frac12m\left(\frac{dy}{dt}\right)^2 + m\frac{dy}{dt}\frac{d\eta}{dt} + \frac12m\left(\frac{d\eta}{dt}\right)^2, \]and due to small increments η we ignore the last square. ∎
Developing the potential in series, we find:
Ep(q) = Ep(y + η) = Ep(y) + ηE'p(y) + η²E''p(y) + ...
Ep(q) ≈ Ep(y) + ηE'p(y).
From the examples, the following are summaries of actions:
\[ S[q(t)] = \int_{t_1}^{t_2}[E_k(q) - E_p(q)]\ dt = \] \[ = \int_{t_1}^{t_2} \left[\frac12m\left(\frac{dy}{dt}\right)^2 + m\frac{dy}{dt}\frac{d\eta}{dt} - E_p(y) - \eta E'_p(y)\right]\ dt \] \[ = \int_{t_1}^{t_2} \left[\frac12m\left(\frac{dy}{dt}\right)^2 - E_p(y)\right]\ dt + \int_{t_1}^{t_2}\left[m \frac{dy}{dt}\frac{d\eta}{dt} - \eta E'_p(y)\right]\ dt. \] \[ S[y(t)] = \int_{t_1}^{t_2} [E_k(y) - E_p(y)]\ dt = \int_{t_1}^{t_2}\left[\frac12\left(\frac{dy}{dt}\right)^2 - E_p(y)\right]\ dt. \]Therefore, the action variation, δ S = S[q(t)] - S[y(t)] = 0, becomes:
\[ \delta S = \int_{t_1}^{t_2} \left(m\frac{dy}{dt}\frac{d\eta}{dt} - \eta E'_p(y)\right)\ dt = 0, \]and this, by partial integration, gives:
\[ \delta S = \int_{t_1}^{t_2} \left[-m \frac{d^2y}{dt^2} - E'_p(y)\right]\eta\ dt = 0, \quad \forall \eta. \]Using the fundamental lemma of the calculus of variations, the arbitrary factor η makes the expression in square brackets of this integral zero:
\[ -m \frac{d^2y}{dt^2} - E'_p(y) = 0, \]-ma + F(y) = 0,
F = ma.
So, we got as in the previous solution (Projectile) that "the force gives acceleration to the mass," more precisely F = mg. The previous solution used the "Maupertuis's action" (product of mass, speed, and path), and here used the "Lagrangian" and Hamilton's principle of least action.
8. The optimal action is stationary and minimal, and that which leads nature to a smaller product of momentum and distance traveled, i.e., energy multiplied by the elapsed time, within the framework of classical mechanics will be classical mechanical, i.e., Newtonian force (product of mass and acceleration). It is the approximation to which we regularly reduce the results of both quantum and relativistic physics. However, the principles of least action also apply to those two new areas of physics. I'll show that another time.
It is more important for me to point out here the greater frequency of more probable outcomes, and therefore the tendency of the state towards less informative, the principle of minimalism, which I also consider as the force of probability. In such an approximation, the reluctance of the state of nature of uncertainty is a "force." It is the same primordial, Newtonian force, which through evolution has developed into "fear of uncertainty," or the need for submission (oneself to others or others to oneself), hence the tendency to exchange excess freedom for security and efficiency.
Virtual » srb
Question: How do you explain virtual photons?

Answer: To me, virtual particles are similar to those introduced into physics by Feynman (1948) with his diagrams. Minor changes are due to some of its inconsistencies, and the rest because information theory is not just physics.
In the picture on the left, we see that one classical quantum-mechanical hopping virtual photon (γ) from one electron (e- ) to the second, when the photon stops being virtual and becomes real in order to transfer momentum and spin from the first electron to the second. For example, when the spin of the first electron is +1/2 and the second is -1/2, a photon of spin +1 will leave the first with spin -1/2 and the second with +1/2. The sum of the momentum vectors before and after the transmission also remains the same.
Let's say, due to some inconsistencies, the first questions that bothered me at the time (Space-Time, 2017), in short, the success of that random shooting. Where does the burst energy of a photon that does not hit anything go, why is there no loss of the electrons left behind, or how come the photon becomes real only after the interaction? Attached is a script that explains parts of these questions using "virtual spheres" instead of linear photons.
Virtual photons are electron-derived concentric spheres from which they spread out and away at the speed of light. They are waves of constant wavelength λ that carry equal uncertainties (energy E, momentum p, spin s) but of decreasing amplitudes a whose modulus squares a*a = |a| ² are the chances of interaction with another possible electron. In that part, I (almost) agree with official quantum physics.

In the image on the right, the upper electron emitted a virtual sphere moving at the speed of light c, both with an arrow to the right, so that the lower one would reach the second electron at a distance ℓ. With the eventual interaction of two electrons, that photon γ becomes real; it is included in "the perceptible world of conservation laws and truths" and leaves its uncertainty to others. The story continues physically.
Photon energy E = hf. There h is Planck's constant, frequency f = 1/τ, and τ is the period of oscillation. Action Eτ = h is again the product of the photon's momentum and its wavelength, pλ = h. Hence E = pc, the photon energy is equal to the product of the momentum and the speed of light c = λf. The conservation of momentum p of photons and electrons can be seen on the right of that image, for example, where M is the mass of the electron and v is the speed of its movement due to the interaction of photons. So:
Mv = p, ℓ = c⋅Δt,
\[ M\frac{\Delta x}{\Delta t} = \frac{E}{c}, \quad \Delta t = \frac{\ell}{c}, \] \[ M\Delta x = \frac{E}{c^2}\ell. \]Let's continue with this interesting lecture Famous Equation, whose proof may look like "Circulus vicious" (due to the momentum of the photon p = E/c), but it is still correct.
We put ourselves in the place of an observer who is at rest in relation to the center of mass, M of electrons and fictitious mass, m of photons, events before and after the interaction, where the second electron can be much further away of this length ℓ:
\[ \frac{M x_1 + m x_2}{M + m} = \frac{M(x_1 - \Delta x) + m(x_2 + \ell)}{M + m}, \]M x1 + m x2 = M(x1 - Δx) + m(x2 + ℓ),
M Δx = m ℓ.
We got an equality like the balance of Archimedes' lever. However, from the above, M Δx = E ℓ/c² so incorporating that we find:
\[ \frac{E}{c^2}\ell = m\ell, \]E = mc².
This fictitious photon mass suddenly becomes real m = E/c². Real energy E, i.e., mass m, is transferred by this interaction. The center of mass is also respected; the relations between and properties of elementary wavelengths, frequencies, and momentums are valid.
The last part of this story, the answer to the question, is non-physical and concerns my theory of information. More precisely, it builds on the idea that there is a pair of reality and fiction where the latter are inconsistent both in perceptibility, persistence, and accuracy.
There, the equivalence of the worlds of truth and falsehood is one of the obligations. Then, there is also the possibility of connecting those others with the first when they become a coupling between other things and vitality. The power of such associations is to choose beyond the principle of the least action of inanimate physical matter, to defy or lie and ride on physical reality, as mortal and miserable as they are, yet manipulate the otherwise indestructible (inanimate) reality — demonstrating the power and powerlessness of uncertainty. Thus, the further story becomes a topic for itself.
Opposites » srb
Question: Is it and how true is it that opposites attract?

Answer: In principle, according to the theory of information that I am developing, opposites attract. As the field is crowded, among the invited there are also those who are pushed out and appear as if they are rejecting each other.
1. Since it is the most informative uniform distribution (Extremes), among the distributions of limited intervals, and nature spontaneously tends toward reduced information, the more it goes towards diversity. But there is evidence (Least Action, 1st example) that walking towards less force (F = ma). It is a spontaneous force of nature with which she would avoid uncertainties, but in vain, as everything is woven from uncertainty. What makes her job even more tough is the absence of losses in total quantities due to the law of conservation.
2. The force changes the momentum with time (Δp = F⋅Δt) and does work on the path by consuming energy (ΔE = F ⋅Δx), from which follows:
\[ \frac{\Delta p}{\Delta t} = \frac{\Delta E}{\Delta x}, \]Δp⋅Δx = ΔE⋅Δt,
that "spontaneous force" tends to be a smaller action, the product of the change in momentum along the way, that is, the change in energy over time. Due to the equivalence of action and information, this means that information emissions change both the momentums and the energies of communicating subjects.
This is not something we do not know, saying "all physical actions convey some information, but also all physical information makes some actions" (Least Action, 2). We know that the computer works with energy; our brain also; however, we now emphasize the "world of fiction," which exists so that it is equivalent to the "world of reality," but it is imperceptible, baseless, and false. That "pseudo-reality" combined with "real reality" gives the latter vitality, the possibility of freer action (outside the principle of least action), and at least some kind of longevity, duration.
3. Fleeing from more information, nature retreats before a greater force and opposes a lesser one. The news is that we can also see it from the top image on the right. Namely, a quantum state is an interpretation of a vector. The sum of many vectors is thus a macro-state of objects with independent observables as a linearly independent basis of the vector space. These two are indicative:
\[ \vec{a} = a_x\vec{i} + a_y\vec{j}, \quad \vec{b} = b_x\vec{i} + b_y\vec{j}. \]They are divided into sums along the abscissa and ordinate lines. The unit vector of the first vector and the vector of the projection of the second onto the first are shown:
\[ \vec{n}_a = \frac{\vec{a}}{|\vec{a}|}, \quad \overrightarrow{OB'} = \vec{n}_a\cdot |\vec{b}|\cos\theta. \]The Oxy system is rectangular, so for scalar products:
\[ \vec{a}\cdot\vec{b} = \langle a, b\rangle = a_xb_x + a_yb_y = |\vec{a}||\vec{b}|\cos\theta, \]because such a product of orthogonal of the same name is one, of different, is zero. When the vectors are individual states, the scalar product will be able to interpret the range of their mutual interaction.
4. That's pretty much all we need to know for the following remarks, except that the number of macro-state dimensions (observables) is huge. However, when the base is quite small, then these coefficients could also be complex numbers. The squares of the modulus of the vector coefficients are the probabilities of occurrence of those observables (base vectors) in the interactions. However, the scalar product is the sum of multiplied pairs of corresponding components (which can be mutually observed).
In the script Sprega Informacija (1.1.5 Boundary Information), you will find a useful observation. Summaries of Shannon information, -pk⋅log pk, with k = 1, 2,..., n, in the case of a very large number of additions n → ∞, can roughly be considered probabilities. This is an extension of the scalar product to interpretations of sum of products information up to probabilities themselves. Of course, these are not the only representations of vector spaces within this information theory.
5. After the above brief introduction, here is how we find that the lesser action goes to the greater, and the greater to the lesser, as regards the spontaneity of (still) nature and the described interpretation by vectors. For example, we multiply the vectors \(\vec{a} = (1, 3, 4)\) and \(\vec{b} = (2, 3, 5)\) both with increasing coefficients, and then the first same a others with decreasing coefficients:
1⋅2 + 3⋅3 + 4⋅5 = 2 + 9 + 20 = 31,
1⋅5 + 3⋅3 + 4⋅2 = 5 + 9 + 8 = 22.
We see how the conjunction "smaller with smaller and greater with greater" (31) gives a greater sum than the conjunction "smaller with greater and greater with smaller" (22). In short, nature retreats before the stronger and advances against the weaker — to reduce this coupling, when the sum of the products becomes "information of perception". The link is to the book where you can find more detailed explanations.
However, from the top image on the right, it is also visible why the perception information would be reduced when the smaller force (smaller components, influences) goes to the larger one and the larger one to the smaller one. Namely, then the angle θ between the states is larger, so the cosine of the angle is smaller and the scalar product is smaller. Note that "pseudo reality" added to "real reality" makes the latter vital making a coupling that can defy the principle of least effect, with greater powers of choice than dead matter, expanding its freedom of movement in defiance of the upper one. Then the ability to oppose "bigger with bigger and smaller with smaller" is precisely the measure of that unnaturalness, that is, vitality.
6. In the aforementioned vector interpretation, the above "spontaneous force" opens the angle θ between the vectors by rearranging the components of the vectors. Smaller ones gravitate to larger ones and vice versa, like opposites that attract. Fleeing from excess information, nature evades a greater effect, that is, a force, to go against a lesser one. What else can I say except that opposites attract? But, in addition to pushing out the weaker from the stronger in that crowd and diversity (which is essentially uncertainty), there is also vitality with exactly the opposite aspirations.
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