Transformaciones de Galileo 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.

También que en términos 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

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

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

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

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

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

Reolviendo la transformación de Galileo
Aplicando este criterio a la transformación de Galileo $\phi(x,y,z,t)$

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

remplazando por los valores de la tabla

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

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

del mismo modo

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

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

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

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

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

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

$$\dst \frac{\partial \phi}{\partial t}=-v\frac{\partial \phi}{\partial x'}+\frac{\partial \phi}{\partial t'}$$

Derivada Segunda

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

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

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

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

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

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 #18

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

$\dfrac{\partial^2 \phi}{\partial t^2}= -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 ]+$$\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}= -v\left [\dfrac{\partial^2\phi}{\partial x'^2} (-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'} 1\right ]+$$\left [\dfrac{\partial^2\phi}{\partial x'\partial t'} (-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} 1\right ]$

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

Análisis

si suponemos que la ecuación #3 se cumple para el observador O'

$$\dst\frac{\partial^2\phi }{\partial x'^2 }+\frac{\partial^2\phi }{\partial y'^2}+\frac{\partial^2\phi}{\partial z'^2 }=\frac{1}{c^2}\frac{\partial^2\phi }{\partial t'^2 }$$

reemplazando en la anterior

$$\dst\frac{\partial^2\phi }{\partial t^2 }=v^2\dfrac{\partial^2\phi}{\partial x'^2} -2v\dfrac{\partial^2\phi}{\partial x'\partial t'} +c^2\left[\dst\frac{\partial^2\phi }{\partial x'^2 }+\frac{\partial^2\phi }{\partial y'^2}+\frac{\partial^2\phi}{\partial z'^2 }\right]<br />$$

usando las relaciones #22,#26 y #30 y diviviendo por $c^2$

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

si la transformación es independiente de la velocidad del sistema de referencia de#3 se cumpliría la igualdad del primer miembro de la izquierda

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

pero pasando de lado los miembros de#36 claramente se observa que no hay independencia del valor de la velocidad del sistema de referencia.

$$\dst\frac{1}{c^2}\frac{\partial^2\phi }{\partial t^2 }-\left[\dst\frac{\partial^2\phi }{\partial x^2 }+\frac{\partial^2\phi }{\partial y^2}+\frac{\partial^2\phi}{\partial z^2 }\right]=\frac{v^2}{c^2}\dfrac{\partial^2\phi}{\partial x'^2} -\frac{2v}{c^2}\dfrac{\partial^2\phi}{\partial x'\partial t'} \neq 0$$