Special Form Of Relativity – Definition And Explanations

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The formulas establish the transition between the coordinates (t, x) of an event in the fixed inertial system, for example that of the Earth (the Earth is the third planet in the solar system in order of distance…) and the coordinates (t ‘, x ‘) of the same event in the moving reference frame, say the rocket (Rocket can refer to:) moving along the x-axis with speed (We distinguish:) v.

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The origins of time are believed to coincide with

$t\,=\,t'\,=\,0$

We provide:

$\beta \,=\,v/c$

The invariant of special relativity

The following quantity is invariant when the coordinates change

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

and defines the proper time (In relativistic theory we call the proper time of a particle the time measured in…) $\,\dew\,.$

The angle parameter

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

$\beta \,=\,\tanh\theta$ either $\theta \,=\, \mathrm{atanh}\beta$

With 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

When the rocket clock measures the duration $\Delta\,t'$ between two events that occur in this rocket, i.e. are separated by a spatial distance $\Delta\,x'=0$ is the duration measured in the fixed laboratory on Earth

$\Delta t = \Delta t'\cosh\,\theta = \gamma \Delta t' = \frac{\Delta t'}{\sqrt{1 - (v^2/c^2)}}\,.$

The duration measured in an external benchmark is always greater than your 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 the hyperbolic functions of the angle θ, we obtain expressions analogous to the formulas for changing the coordinate axes by plane rotation:

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

In sense (SENS (Strategies for Engineered Negligible Senescence) is a scientific project with the aim of…) inverse (In mathematics, the inverse of an element x of a set, provided with a law of…)

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

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

Law of velocity composition

A grenade is fired at the rocket at a speed w ‘ relative to the reference of this rocket in the direction of movement. The velocity w of the envelope relative to the Earth is

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

Using angle parameters

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

we have the additive law

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

Length contraction

If the rocket has length (The length of an object is the distance between its two extreme ends…) in its own frame of reference L’, its length L is measured by the distance between the two points on Earth in accordance with the front – and back of the rocket at the same moment (on Earth), so accordingly $\Delta t\,=\,0$ is given by

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

The length measured on Earth is less than the rocket’s own length.

Kinetic energy

The kinetic energy (also called vis viva or living force in ancient writings) is…) of a particle is

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

For $v\ll c$

$K\,= \,(1/2) mv^2\,$

and for $v\simeq c$

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

The energy-momentum quadrivector

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

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

we have

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

At low speeds $v\,\ll\,c$

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

We still have the relationship

$p\,=\,\beta E/c\,.$

The following quantity is invariant when the reference changes

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

For a photon (in particle physics, the photon (often symbolized by the letter γ – gamma)…), m = 0 and

$E\,=\,pc$

Doppler-Fizeau effect

Doppler effect

$\naked\,$ is the frequency received on earth, $\naked'\,$ the frequency emitted by the source, $\theta'\,$ the angle that the photon makes with the Ox axis in the reference system of this source, $\theta\,$ the angle with the Ox axis in the terrestrial reference system, $v\,$ the speed of the source relative to the earth and $v_r\equiv v\cos\theta'$ we have the radial velocity

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

At low speeds $v\ll c$

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

As the star moves away, v is positive, cosθ’ is negative, $v_r\,=\,v\cos\theta'$ is negative, so the frequency decreases (The wavelength increases, this is the shift towards red).

Light aberration phenomenon:

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