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🔎 The model of special relativity

🔎 The model of special relativity

introduction

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Electromagnetic
Quantum physics
Thermodynamics
Fluid mechanics (Fluid mechanics is the branch of physics that studies fluid flows…)
mechanical (In common parlance, mechanics is the field of machines, engines, vehicles, organs, etc.)
Relativity (Special relativity is the formal theory developed by Albert Einstein…)
Black hole (In astrophysics, a black hole is a massive object whose gravitational field is extremely strong…)
Vector analysis (Vector analysis is a branch of mathematics that studies areas…)

Ratings

The formulas define the passage between coordinates (R, s ) for an event in the inertial frame PinnedLet’s say it Land (Earth is the third planet in the solar system in terms of distance…)and coordinates (R’, s ) for the same event in the tag MovingLet’s say rocket (Rocket may refer to:)which moves along an axis s with Speed (We distinguish:) Fifth.

It is assumed that the origins time (Time is a concept that humans developed to understand…) Coinciding with the

is put :

\beta\,=\,v/c

Constants of special relativity

there amount (Quantity is a general metrological term (account, amount); numerical,…) Next is fixed in changing coordinates

c^2\tau^2\,=\,c^2t^2 - (x^2 + y^2 + z^2) = \,c^2t'^2 - (x'^2 + y'^ 2 + d'^2)

And determines Clean time (In the theory of relativity, we call the time of a particle the time measured by…) \,\Tao\,.

angle parameter

To simplify the formulas, it is useful to introduce the angular parameter defined by the following formulas:

\beta\,=\,\tanh\theta also \theta \,=\, \mathrm{atanh}\beta

Using this parameter we can write:

    \gamma = (1 - \beta^2)^{-1/2} = (1 - \tanh^2\, \theta)^{-1/2} = \cosh \,\theta
\beta\gamma \,=\, \beta (1 - \beta^2)^{-1/2} = \tanh\theta \,\cosh \,\theta = \sinh\,\theta

Time dilation

If a rocket clock measures duration \delta\,t' Between two events that occur in this rocket, they are therefore separated by a spatial distance \Delta\,x'=0the duration measured in the fixed Earth laboratory is

\Delta t = \Delta t'\cosh\,\theta = \gamma \Delta t' = \frac{\Delta t'}{\sqrt{1 - (v^2/c^2)}}\,.
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The duration measured by an external standard is always greater than the own duration.

Lorentz transformations

    \begin{cases}ct = \gamma (ct'+ \beta x')\\ x = \gamma (x' + \beta ct')\\ y = y'\\ z = z' \end{cases}

Which gives in matrix form (easier to visualize):

    \begin{pmatrix} ct\\x\\y\\z \end{pmatrix} = \begin{pmatrix} \gamma & \beta\gamma & 0 & 0\\ \beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix} \begin{pmatrix} ct'\\x'\\y'\\z' \end{pmatrix}

Using hyperbolic functions forcorner (In geometry, the general concept of angle is divided into several concepts…) θwe obtain expressions similar to the formulas for changing the coordinate axes by rotating the plane:

    \begin{cases} ct= ct'\cosh\,\theta + x'\sinh\,\theta \\ x = ct' \sinh\,\theta + x'\cosh\,\theta \end{cases}

in sense (SENS (Engineered Neglected Aging Strategies) is a scientific project aiming to…) reflects (In mathematics, the inverse of the x element of the set is given by a law…)

    \begin{cases}ct' = \gamma (ct - \beta x)\\ x' = \gamma (x - \beta ct )\\ y' = y\\ z' = z \end{cases}

or

    \begin{cases} ct'= \ct\cosh\,\theta - x\sinh\,\theta \\ x' = -ct \sinh\,\theta + x\cosh\,\theta \end{cases}

Law of formation of velocities

A projectile is fired at the missile quickly Th Relative to the reference of this missile, in the direction of movement. Speed Th of shell relative to Earth

w \,=\, \frac{w'+v}{1 + (w' v/c^2)}\,.

Using angle parameters

\alpha'\,=\,\mathrm{atanh}(w'/c)
\alpha\,=\,\mathrm{atanh}(w/c)
\theta\,=\,\mathrm{atanh}(v/c)

We have the additional law

\alpha\,=\,\alpha'+\theta

Length contraction

If the missile Length (The length of a body is the distance between its two extremes…) to’ In its own standard, its length to It is measured by the distance between two points on the ground that correspond to the front and rear of the missile at the same time Moment (An instant defines the smallest component of time. A moment is not…) (in the ground), so it corresponds to \Delta T\,=\,0given before

L = L'/\gamma = L' \sqrt{1 - (v^2/c^2)}\,.

The length measured on the ground is smaller than the length of the rocket.

Kinetic energy

to’Kinetic energy (Kinetic energy (also called in ancient writings vis viva, or living force) is…) Of particles it is

    K\, = E - mc^2 \,=\,mc^2\left( \frac{1}{\sqrt{1 - (v^2/c^2)}} - 1\right)\,.

to v\ll c

K\,= \,(1/2) mt^2\,

And for v\simeq c

K \simeq E \simeq pc = \frac{mc^2}{\sqrt{2(1-\beta)}}\equiv \frac{mc^2}{\sqrt{2[1-(v/c)]}}\,.

Quadruple energy driving force

    \begin{cases} p_t=E/c = mc\dt/d\tau\\ p_x = m\ dx/d\tau\\ p_y =m\ dy/d\tau\\ p_z=m\ dz/d\ Tao\,.  \end{cases}

like

dt/d\tau\,=\,\gamma \equiv (1-\beta^2)^{-1/2}\,,

we’ve got

E\,=\,\gamma mc^2
p \equiv (p_x^2 + p_y^2 + p_z^2)^{1/2} = m\gamma\beta c= mv /\sqrt{1 - (v^2/c^2)}

At low speeds ice

E\,=\, mc^2 + (1/2) mv^2\,.

We still have the relationship

P\,=\,\beta E/c\,.
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The following quantity is constant in changing the reference

E^2 - p^2c^2\,=\,m^2c^4

to Photon (In particle physics, a photon (often symbolized by the letter γ — gamma)…), M = 0 f

e\,=\,pc

Doppler-Fizeau effect

Doppler effect

\naked\, It is the frequency that is received on the ground, \naked\, The frequency emitted by the source, \theta'\, The angle that the photon makes with the axis bull In reference to this source, \theta\, Angle with axis bull In terrestrial landmarks, Fifth\, The speed of the source relative to the ground and v_r\equiv v\cos\theta' Radial velocity, we have

\nu\,=\,\gamma\,(1 + \beta\cos\theta') \nu'\equiv \frac{1 + (v_r/c)}{\sqrt{1 - (v^2/c) ) ^2)}}\,\nu'

At low speeds v\ll c

\frac{\Delta\nu}{\nu} \equiv \frac{\nu - \nu'}{\nu'} \simeq \frac{\nu - \nu'}{\nu} = \frac{v_r }{against}\,.

ifa star (A star is a celestial body that emits light independently, such as…) walks away, Fifth is positive, cosθ’ is negative, v_r\,=\,v\cos\theta' is negative, so that frequency Decreases (LengthWave (A wave is the propagation of a disturbance that results in a difference in its passage…) increases, this is a red shift).

Light diffraction phenomenon:

\begin {states}\cos\theta = (\beta + \cos\theta')/(1 + \beta\cos\theta')\\ \sin\theta = \gamma^{-1}\sin\theta '/(1+\beta\cos\theta') \end{cases}