Transformaciones de Lorentz aplicadas a la función de onda electromagnética

[Richard R Richard]

Definiendo la función de onda electromagnética

Podemos escribir la función de onda electromagnética a partir del campo eléctrico o magnético

$$\displaystyle\frac{\partial^2E }{\partial x^2 }+\frac{\partial^2E }{\partial y^2}+\frac{\partial^2E}{\partial z^2 }=\frac{1}{c^2}\frac{\partial^2E }{\partial t^2 }$$

o

$$\displaystyle\frac{\partial^2B }{\partial x^2 }+\frac{\partial^2B }{\partial y^2}+\frac{\partial^2B}{\partial z^2 }=\frac{1}{c^2}\frac{\partial^2B }{\partial t^2 }$$

y sabemos que ambas soluciones se encuentran en diferencia de fase en plano perpendicular a la dirección de propagación.

Tambien que en terminos generales se puede escribir

$$\displaystyle\frac{\partial^2\phi }{\partial x^2 }+\frac{\partial^2\phi }{\partial y^2}+\frac{\partial^2\phi}{\partial z^2 }=\frac{1}{c^2}\frac{\partial^2\phi }{\partial t^2 }$$

Según la postura de Galileo dos observadores O y O' solidarios a dos sistemas de referencias inercial coincidentes en el origen de los tiempos y que se mueven con velocidad relativa $v$ debería observar la onda electromagnética y poder obtener una transformación de las lecturas de su sistema al otro.

El motivo del presente blog es analizar si las ecuaciones varían en función de la velocidad del observador.

Planteemos la relación entre los sistemas de referencia

La transformación de Lorentz es la relación entre las variables de un sistema de referencia que nos permite calcular las medidas tomadas por otro observador que se mueve a velocidad $v$ con respecto al primero

Llamemos $\gamma=\dfrac{1}{\sqrt{1-\frac {v^2}{c^2}}}$

la transformación se escribe

$$\boxed{\phi(x,y,z,t)=\left[\begin{aligned}x'&=\gamma(x-vt)\\y'&=y\\z'&=z\\t'&=\gamma\left ( t-\frac{xv}{c^2}\right )\end {aligned}\right.}$$

Ahora para tomemos derivadas parciales de las funciones que relacionan los sistemas de referencia en función de las variables del otro sistema de referencia.

$\dfrac{\partial x'}{\partial x}=\gamma$	$\dfrac{\partial x'}{\partial y}=0$	$\dfrac{\partial x'}{\partial z}=0$	$\dfrac{\partial x'}{\partial t}=-\gamma v$
$\dfrac{\partial y'}{\partial x}=0$	$\dfrac{\partial y'}{\partial y}=1$	$\dfrac{\partial y'}{\partial z}=0$	$\dfrac{\partial y'}{\partial t}=0$
$\dfrac{\partial z'}{\partial x}=0$	$\dfrac{\partial z'}{\partial y}=0$	$\dfrac{\partial z'}{\partial z}=1$	$\dfrac{\partial z'}{\partial t}=0$
$\dfrac{\partial t'}{\partial x}=-\dfrac{\gamma v}{c^2}$	$\dfrac{\partial t'}{\partial y}=0$	$\dfrac{\partial t'}{\partial z}=0$	$\dfrac{\partial t'}{\partial t}=\gamma$

Recordemos la regla de la cadena para derivadas ordinarias...

Sea $f=f(g(x))$

entonces

$$\dst \frac{\partial f(g(x))}{\partial x}=\frac{\partial f(g)}{\partial g}\frac{\partial g(x)}{\partial x}$$

o con otras notaciones

$$\dst \frac{\partial f}{\partial x}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}$$

Derivada total
recordemos el concepto de derivada total en función de las derivadas parciales

$$\dst \frac{\partial \phi}{\partial u}=\frac{\partial \phi}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial \phi}{\partial y}\frac{\partial y}{\partial u}+\frac{\partial \phi}{\partial z}\frac{\partial z}{\partial u}+\frac{\partial \phi}{\partial t}\frac{\partial t}{\partial u}<br />$$

Resolviendo la transformación de Lorentz

Aplicando los criterios anteriores a la transformación de Lorentz $\phi(x,y,z,t)$

Hacemos la primer derivada

$$\dst \frac{\partial \phi}{\partial x}=\frac{\partial \phi}{\partial x'}\frac{\partial x'}{\partial x}+\frac{\partial \phi}{\partial y'}\frac{\partial y'}{\partial x}+\frac{\partial \phi}{\partial z'}\frac{\partial z'}{\partial x}+\frac{\partial \phi}{\partial t'}\frac{\partial t'}{\partial x}<br />$$

remplazando por los valores de la tabla

$$\dst \frac{\partial \phi}{\partial x}=\frac{\partial \phi}{\partial x'}\gamma+\frac{\partial \phi}{\partial y'}0+\frac{\partial \phi}{\partial z'}0+\frac{\partial \phi}{\partial t'}\left (-\dfrac{\gamma v}{c^2}\right )$$

$$\dst \frac{\partial \phi}{\partial x}=\gamma\left[\frac{\partial \phi}{\partial x'}-\dfrac{ v}{c^2} \frac{\partial \phi}{\partial t'}\right ]$$

del mismo modo

$$\dst \frac{\partial \phi}{\partial y}=\frac{\partial \phi}{\partial x'}\frac{\partial x'}{\partial y}+\frac{\partial \phi}{\partial y'}\frac{\partial y'}{\partial y}+\frac{\partial \phi}{\partial z'}\frac{\partial z'}{\partial y}+\frac{\partial \phi}{\partial t'}\frac{\partial t'}{\partial y}<br />$$

$$\dst \frac{\partial \phi}{\partial y}=\frac{\partial \phi}{\partial x'}0+\frac{\partial \phi}{\partial y'}1+\frac{\partial \phi}{\partial z'}0+\frac{\partial \phi}{\partial t'}0$$

$$\dst \frac{\partial \phi}{\partial y}=\frac{\partial \phi}{\partial y'}$$

$$\dst \frac{\partial \phi}{\partial z}=\frac{\partial \phi}{\partial x'}\frac{\partial x'}{\partial z}+\frac{\partial \phi}{\partial y'}\frac{\partial y'}{\partial z}+\frac{\partial \phi}{\partial z'}\frac{\partial z'}{\partial z}+\frac{\partial \phi}{\partial t'}\frac{\partial t'}{\partial z}<br />$$

$$\dst \frac{\partial \phi}{\partial z}=\frac{\partial \phi}{\partial x'}0+\frac{\partial \phi}{\partial y'}0+\frac{\partial \phi}{\partial z'}1+\frac{\partial \phi}{\partial t'}0$$

$$\dst \frac{\partial \phi}{\partial z}=\frac{\partial \phi}{\partial z'}$$

pero con t

$$\dst \frac{\partial \phi}{\partial t}=\frac{\partial \phi}{\partial x'}\frac{\partial x'}{\partial t}+\frac{\partial \phi}{\partial y'}\frac{\partial y'}{\partial t}+\frac{\partial \phi}{\partial z'}\frac{\partial z'}{\partial t}+\frac{\partial \phi}{\partial t'}\frac{\partial t'}{\partial t}<br />$$

$$\dst \frac{\partial \phi}{\partial t}=\frac{\partial \phi}{\partial x'}(-\gamma v)+\frac{\partial \phi}{\partial y'}0+\frac{\partial \phi}{\partial z'}0+\frac{\partial \phi}{\partial t'}\gamma$$

$$\dst \frac{\partial \phi}{\partial t}=\gamma\left [-v\frac{\partial \phi}{\partial x'}+\frac{\partial \phi}{\partial t'}\right ]$$

Derivada segunda

Utilizando estos resultados para calcular la derivada total segunda...

$$\dfrac{\partial^2 \phi}{\partial x^2}= \dfrac{\partial\left ( \frac{\partial \phi}{\partial x} \right )}{\partial x'} \dfrac{\partial x'}{\partial x}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial x} \right )}{\partial y'} \dfrac{\partial y'}{\partial x}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial x} \right )}{\partial z'} \dfrac{\partial z'}{\partial x}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial x} \right )}{\partial t'} \dfrac{\partial t'}{\partial x}$$

recurriendo al resultado #10

$$\dfrac{\partial^2 \phi}{\partial x^2}= \gamma\left [\dfrac{\partial\left ( \frac{\partial \phi}{\partial x'} \right )}{\partial x}\right ]-\left (\dfrac{\gamma v}{c^2}\right )\left [\dfrac{\partial \left ( \frac{\partial \phi}{\partial t'} \right )}{\partial x}\right ]$$

$\dfrac{\partial^2 \phi}{\partial x^2}= \gamma\left [\dfrac{\partial^2\phi}{\partial x'^2} \dfrac{\partial x'} {\partial x}+\dfrac{\partial^2\phi}{\partial x'\partial y'} \dfrac{\partial y'} {\partial x}+\dfrac{\partial^2\phi}{\partial x'\partial z'} \dfrac{\partial z'} {\partial x}+\dfrac{\partial^2\phi}{\partial x'\partial t'} \dfrac{\partial t'} {\partial x}\right ]$$-\left (\dfrac{\gamma v}{c^2}\right )\left [\dfrac{\partial^2\phi}{\partial x'\partial t'} \dfrac{\partial x'} {\partial x}+\dfrac{\partial^2\phi}{\partial t'\partial y'} \dfrac{\partial y'} {\partial x}+\dfrac{\partial^2\phi}{\partial t'\partial z'} \dfrac{\partial z'} {\partial x}+\dfrac{\partial^2\phi}{\partial t'^2} \dfrac{\partial t'} {\partial x}\right ]$



$\dfrac{\partial^2 \phi}{\partial x^2}= \gamma\left [\dfrac{\partial^2\phi}{\partial x'^2} \gamma+\dfrac{\partial^2\phi}{\partial x'\partial y'} 0+\dfrac{\partial^2\phi}{\partial x'\partial z'} 0+\dfrac{\partial^2\phi}{\partial x'\partial t'} \left (-\dfrac{\gamma v}{c^2}\right )\right ]$$-\left (\dfrac{\gamma v}{c^2}\right )\left [\dfrac{\partial^2\phi}{\partial x'\partial t'} \gamma+\dfrac{\partial^2\phi}{\partial t'\partial y'} 0+\dfrac{\partial^2\phi}{\partial t'\partial z'} 0+\dfrac{\partial^2\phi}{\partial t'^2} \left (-\dfrac{\gamma v}{c^2}\right )\right ]$



$\dfrac{\partial^2 \phi}{\partial x^2}= \gamma^2\dfrac{\partial^2\phi}{\partial x'^2} -\dfrac{2v\gamma^2 }{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{\gamma^2 v^2}{c^4}\dfrac{\partial^2\phi}{\partial t'^2}$

$$\dfrac{\partial^2 \phi}{\partial x^2}= \gamma^2\left [\dfrac{\partial^2\phi}{\partial x'^2} -\dfrac{2v\ }{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{ v^2}{c^4}\dfrac{\partial^2\phi}{\partial t'^2} \right ]$$

luego

$$\dfrac{\partial^2 \phi}{\partial y^2}= \dfrac{\partial\left ( \frac{\partial \phi}{\partial y} \right )}{\partial x'} \dfrac{\partial x'}{\partial y}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial y} \right )}{\partial y'} \dfrac{\partial y'}{\partial y}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial y} \right )}{\partial z'} \dfrac{\partial z'}{\partial y}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial y} \right )}{\partial t'} \dfrac{\partial t'}{\partial y}$$

$$\dfrac{\partial^2 \phi}{\partial y^2}= \dfrac{\partial^2 \phi}{\partial x'\partial y'} \dfrac{\partial x'}{\partial y}+ \dfrac{\partial^2 \phi}{\partial y'^2} \dfrac{\partial y'}{\partial y}+ \dfrac{\partial^2 \phi}{\partial y'\partial z'} \dfrac{\partial z'}{\partial y}+ \dfrac{\partial^2 \phi}{\partial y'\partial t'} \dfrac{\partial t'}{\partial y}$$

$$\dfrac{\partial^2 \phi}{\partial y^2}= \dfrac{\partial^2 \phi}{\partial x'\partial y'} 0+ \dfrac{\partial^2 \phi}{\partial y'^2} 1+ \dfrac{\partial^2 \phi}{\partial x'\partial z'}0+ \dfrac{\partial^2 \phi}{\partial x'\partial t'} 0$$

$$\dfrac{\partial^2 \phi}{\partial y^2}= \dfrac{\partial^2 \phi}{\partial y'^2}$$

además

$$\dfrac{\partial^2 \phi}{\partial z^2}= \dfrac{\partial\left ( \frac{\partial \phi}{\partial z} \right )}{\partial x'} \dfrac{\partial x'}{\partial z}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial z} \right )}{\partial y'} \dfrac{\partial y'}{\partial z}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial z} \right )}{\partial z'} \dfrac{\partial z'}{\partial z}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial z} \right )}{\partial t'} \dfrac{\partial t'}{\partial z}$$

$$\dfrac{\partial^2 \phi}{\partial z^2}= \dfrac{\partial^2 \phi}{\partial x'\partial z'} \dfrac{\partial x'}{\partial z}+ \dfrac{\partial^2 \phi}{\partial y'\partial z'} \dfrac{\partial y'}{\partial z}+ \dfrac{\partial^2 \phi}{\partial z'^2} \dfrac{\partial z'}{\partial z}+ \dfrac{\partial^2 \phi}{\partial z'\partial t'} \dfrac{\partial t'}{\partial z}$$

$$\dfrac{\partial^2 \phi}{\partial z^2}= \dfrac{\partial^2 \phi}{\partial x'\partial z'} 0+ \dfrac{\partial^2 \phi}{\partial y'\partial z'} 0+ \dfrac{\partial^2 \phi}{\partial z'^2}1+ \dfrac{\partial^2 \phi}{\partial z'\partial t'} 0$$

$$\dfrac{\partial^2 \phi}{\partial z^2}= \dfrac{\partial^2 \phi}{\partial z'^2}$$

con respecto al tiempo tenemos

$$\dfrac{\partial^2 \phi}{\partial t^2}= \dfrac{\partial\left ( \frac{\partial \phi}{\partial t} \right )}{\partial x'} \dfrac{\partial x'}{\partial t}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial t} \right )}{\partial y'} \dfrac{\partial y'}{\partial t}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial t} \right )}{\partial z'} \dfrac{\partial z'}{\partial t}+\dfrac{\partial\left ( \frac{\partial \phi}{\partial t} \right )}{\partial t'} \dfrac{\partial t'}{\partial t}$$

o recurriendo al resultado #19

$$\dfrac{\partial^2 \phi}{\partial t^2}= -\gamma v\left [\dfrac{\partial\left ( \frac{\partial \phi}{\partial x'} \right )}{\partial t}\right ]+\gamma \dfrac{\partial \left ( \frac{\partial \phi}{\partial t'} \right )}{\partial t}$$

$\dfrac{\partial^2 \phi}{\partial t^2}= -\gamma v\left [\dfrac{\partial^2\phi}{\partial x'^2} \dfrac{\partial x'} {\partial t}+\dfrac{\partial^2\phi}{\partial x'\partial y'} \dfrac{\partial y'} {\partial t}+\dfrac{\partial^2\phi}{\partial x'\partial z'} \dfrac{\partial z'} {\partial t}+\dfrac{\partial^2\phi}{\partial x'\partial t'} \dfrac{\partial t'} {\partial t}\right ]+$$\gamma\left [\dfrac{\partial^2\phi}{\partial x'\partial t'} \dfrac{\partial x'} {\partial t}+\dfrac{\partial^2\phi}{\partial t'\partial y'} \dfrac{\partial y'} {\partial t}+\dfrac{\partial^2\phi}{\partial t'\partial z'} \dfrac{\partial z'} {\partial t}+\dfrac{\partial^2\phi}{\partial t'^2} \dfrac{\partial t'} {\partial t}\right ]$



$\dfrac{\partial^2 \phi}{\partial t^2}= -\gamma v\left [\dfrac{\partial^2\phi}{\partial x'^2} (-\gamma v)+\dfrac{\partial^2\phi}{\partial x'\partial y'} 0+\dfrac{\partial^2\phi}{\partial x'\partial z'} 0+\dfrac{\partial^2\phi}{\partial x'\partial t'} \gamma\right ]+$$\gamma\left [\dfrac{\partial^2\phi}{\partial x'\partial t'} (-\gamma v)+\dfrac{\partial^2\phi}{\partial t'\partial y'} 0+\dfrac{\partial^2\phi}{\partial t'\partial z'} 0+\dfrac{\partial^2\phi}{\partial t'^2} \gamma\right ]$

$$\dfrac{\partial^2 \phi}{\partial t^2}=\gamma^2\left [ v^2\dfrac{\partial^2\phi}{\partial x'^2} -2v\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{\partial^2\phi}{\partial t'^2}\right ]$$

dividiendo por $c^2$

$$\dfrac{\partial^2 \phi}{c^2\partial t^2}=\dfrac{\gamma^2}{c^2}\left [ v^2\dfrac{\partial^2\phi}{\partial x'^2} -2v\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{\partial^2\phi}{\partial t'^2}\right ]$$

resumiendo

(22)	$\dfrac{\partial^2 \phi}{\partial x^2}= \gamma^2\left [\dfrac{\partial^2\phi}{\partial x'^2} -\dfrac{2v\ }{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{ v^2}{c^4}\dfrac{\partial^2\phi}{\partial t'^2} \right ]$
(26)	$\dfrac{\partial^2 \phi}{\partial y^2}=\dfrac{\partial^2 \phi}{\partial y'^2}$
(30)	$\dfrac{\partial^2 \phi}{\partial z^2}=\dfrac{\partial^2 \phi}{\partial z'^2}$
(34)	$\dfrac{\partial^2 \phi}{c^2\partial t^2}=\dfrac{\gamma^2}{c^2}\left [ v^2\dfrac{\partial^2\phi}{\partial x'^2} -2v\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{\partial^2\phi}{\partial t'^2}\right ]$

Función de onda del observador O

si hacemos la suma de las ecuaciones 22,26,30 y restamos 34

por un lado debemos arribar al resultado nulo de acuerdo a la función de onda del observador O

$$\displaystyle\frac{\partial^2\phi }{\partial x^2 }+\frac{\partial^2\phi }{\partial y^2}+\frac{\partial^2\phi}{\partial z^2 }-\frac{1}{c^2}\frac{\partial^2\phi }{\partial t^2 }=0$$

reemplazando sus equivalentes

$$\gamma^2\left [\dfrac{\partial^2\phi}{\partial x'^2} -\dfrac{2v }{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{ v^2}{c^4}\dfrac{\partial^2\phi}{\partial t'^2} \right ]+\dfrac{\partial^2 \phi}{\partial y'^2}+\dfrac{\partial^2 \phi}{\partial z'^2}-\dfrac{\gamma^2}{c^2}\left [ v^2\dfrac{\partial^2\phi}{\partial x'^2} -2v\dfrac{\partial^2\phi}{\partial x'\partial t'} +\dfrac{\partial^2\phi}{\partial t'^2}\right ]=$$

distribuyendo

$$\gamma^2\dfrac{\partial^2\phi}{\partial x'^2} \cancel{-\dfrac{2v\gamma^2 }{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'}} +\dfrac{\gamma^2 v^2}{c^4}\dfrac{\partial^2\phi}{\partial t'^2} +\dfrac{\partial^2 \phi}{\partial y'^2}+\dfrac{\partial^2 \phi}{\partial z'^2}-\dfrac{\gamma^2 v^2}{c^2}\dfrac{\partial^2\phi}{\partial x'^2} \cancel{+\dfrac{2v\gamma^2}{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'}} -\dfrac{\gamma^2}{c^2}\dfrac{\partial^2\phi}{\partial t'^2}=$$

reagrupando

$$\dfrac{\partial^2\phi}{\partial x'^2} \left[\gamma^2-\dfrac{\gamma^2 v^2}{c^2} \right]+\dfrac{\partial^2 \phi}{\partial y'^2}+\dfrac{\partial^2 \phi}{\partial z'^2}+\frac{1}{c^2}\frac{\partial^2\phi }{\partial t'^2 }\left[\dfrac{\gamma^2 v^2}{c^2}-\gamma^2 \right]=$$

recordando que

$$\gamma^2=\dfrac{c^2}{c^2-v^2}$$

tenemos que

$$\gamma^2-\dfrac{\gamma^2 v^2}{c^2}=1$$

remplazando en la formula #38

$$\dfrac{\partial^2\phi}{\partial x'^2} 1+\dfrac{\partial^2 \phi}{\partial y'^2}+\dfrac{\partial^2 \phi}{\partial z'^2}+\frac{1}{c^2}\frac{\partial^2\phi }{\partial t'^2 }(-1)=$$

$$\boxed {\dfrac{\partial^2\phi}{\partial x'^2} +\dfrac{\partial^2 \phi}{\partial y'^2}+\dfrac{\partial^2 \phi}{\partial z'^2}-\frac{1}{c^2}\frac{\partial^2\phi }{\partial t'^2 }=0}$$

Conclusión

llegamos a la función de onda del observador O' y se observa que es independiente de su velocidad con respecto al observador O $\therefore$CQD

[pod]

dydy, abre un hilo en el foro.