Wave Equation & Wave Displacement Invariance: Modern Physics

In summary: So we can mix primes and un-primes within the same d'Alembertian?...It is acceptable to use any of the six E and B components in the d'Alembertian.
  • #1
greswd
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This question concerns a section from the book Modern Physics by James Rohlf.

http://srv3.imgonline.com.ua/result_img/imgonline-com-ua-twotoone-Bs4zgy7pruqG.png

He shows that the form of the Wave equation for light remains invariant under a Lorentz boost (4.42):

##\frac{∂^2F}{∂x'^2}+\frac{∂^2F}{∂y'^2}+\frac{∂^2F}{∂z'^2}=\frac{1}{c^2}\frac{∂^2F}{∂t'^2}##

What I am confused about is the lack of F' in these derivations. There is no F', only F.

I always thought that if one wants to show that the form of the Wave equation remains invariant under a Lorentz boost, shouldn't the final equation be:

##\frac{∂^2F'}{∂x'^2}+\frac{∂^2F'}{∂y'^2}+\frac{∂^2F'}{∂z'^2}=\frac{1}{c^2}\frac{∂^2F'}{∂t'^2}## ?


Why is it F instead of F'?


Also, I have to apologize for all the poor concepts and erroneous physics that I post on these forums.
 
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  • #2
F is a scalar field.
 
  • #3
robphy said:
F is a scalar field.
But can't a scalar field also transform between frames?
 
  • #4
Like this:
##\begin{align}
& E'_x = E_x & \qquad & B'_x = B_x \\
& E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\
& E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\
\end{align}##
 
  • #5
Here, ##F## is any function that obeys the wave equation. In one coordinate system it is a function of ##(x,y,z,t)##, while in the other it is a function of ##(x',y',z',t')##, but in both systems it is the same function.
 
  • #6
James R said:
Here, ##F## is any function that obeys the wave equation. In one coordinate system it is a function of ##(x,y,z,t)##, while in the other it is a function of ##(x',y',z',t')##, but in both systems it is the same function.
That appears to be what James Rohlf (James R? Same as you! o0)) is saying.

I've always thought that all the quantities in the primed frame should be primed as well. Especially when we're talking about an EM wave.

In his seminal paper on relativity, On the Electrodynamics of Moving Bodies, Einstein makes everything in the primed frame prime when utilizing Maxwell's equations:

img78.gif

img86.gif


img79.gif
img93.gif
The E and B components (or F in Rohlf's example) are transformed in the manner I brought up in #4:
greswd said:
Like this:
##\begin{align}
& E'_x = E_x & \qquad & B'_x = B_x \\
& E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\
& E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\
\end{align}##
 
  • #7
greswd said:
I've always thought that all the quantities in the primed frame should be primed as well. Especially when we're talking about an EM wave.
When you talk about an EM field you are essentially talking about a rank two tensor field. Generally a tensor's components depends on the basis you use. When you talk about ##F## above, it is a scalar field. By definition, scalars (rank zero tensors) are invariant under coordinate transformations.
 
  • #8
Orodruin said:
When you talk about an EM field you are essentially talking about a rank two tensor field. Generally a tensor's components depends on the basis you use. When you talk about ##F## above, it is a scalar field. By definition, scalars (rank zero tensors) are invariant under coordinate transformations.
Is it acceptable to plug any of the 6 E and B components into the d'Alembertian? Wouldn't that make F just a stand in for any component of the E or B field?
 
  • #9
greswd said:
Is it acceptable to plug any of the 6 E and B components into the d'Alembertian? Wouldn't that make F just a stand in for any component of the E or B field?

This adds further complication. You would also have to take into account that the components transform between frames. You can say that the E' components following the wave equation in S' will imply that the E' components will follow the wave equation in S (keeping talking about components of the E-field relative to the primed basis). It is then a matter of collecting the E' and B' components into the E and B components in order to conclude that the E and B components (relative to the unprimed system) also satisfy the wave equation.
 
  • #10
Orodruin said:
This adds further complication. You would also have to take into account that the components transform between frames. You can say that the E' components following the wave equation in S' will imply that the E' components will follow the wave equation in S (keeping talking about components of the E-field relative to the primed basis). It is then a matter of collecting the E' and B' components into the E and B components in order to conclude that the E and B components (relative to the unprimed system) also satisfy the wave equation.
So we can mix primes and un-primes within the same d'Alembertian? Cool..
 
  • #11
greswd said:
So we can mix primes and un-primes within the same d'Alembertian? Cool..
The d'Alembertian is a linear operator. This should not come as a surprise.
 
  • #12
Orodruin said:
When you talk about an EM field you are essentially talking about a rank two tensor field. Generally a tensor's components depends on the basis you use. When you talk about ##F## above, it is a scalar field. By definition, scalars (rank zero tensors) are invariant under coordinate transformations.
Rohlf has shown that it ##F## can satisfy the d'Alembertian in both primed and unprimed frames.
We can replace ##F## with ##E_x##, since ##E_x'=E_x##, and get a similar result.

Am I right to say that we need the transformations in #4 to satisfy Maxwell's equations, but we don't need them to satisfy the d'Alembertian?
 
  • #13
Orodruin said:
The d'Alembertian is a linear operator. This should not come as a surprise.
Not really a surprise, but an interesting result. Especially to an ignoramus like myself, who finds it difficult to obtain answers.
 
  • #14
The d'Alembertian is a differential operator. It is not something you can "satisfy".

It is involved in the wave equation, which is something a quantity can satisfy.

In general, you know that E and B are linear combinations of E' and B'. Thus, if E' and B' satisfy the wave equations, then so do E and B by the linear property of the d'Alembertian.
 
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Likes greswd
  • #15
Orodruin said:
The d'Alembertian is a differential operator. It is not something you can "satisfy".

It is involved in the wave equation, which is something a quantity can satisfy.

In general, you know that E and B are linear combinations of E' and B'. Thus, if E' and B' satisfy the wave equations, then so do E and B by the linear property of the d'Alembertian.
Thanks for the clear answer.

To reiterate my question in #12, am I right to say that we need the transformations in #4 to satisfy Maxwell's equations, but we don't need them to satisfy the wave equation (for E and B fields)?
 
  • #16
greswd said:
Thanks for the clear answer.

To reiterate my question in #12, am I right to say that we need the transformations in #4 to satisfy Maxwell's equations, but we don't need them to satisfy the wave equation (for E and B fields)?
They will automatically satisfy the wave equation of the original fields do.
 
  • #17
Orodruin said:
They will automatically satisfy the wave equation of the original fields do.
?
Which is 'they'?
 

1. What is the wave equation?

The wave equation is a mathematical expression that describes the propagation of a wave through a medium. It is typically written as d2y/dt2 = c2(d2y/dx2), where y represents the wave displacement, t represents time, x represents position, and c is the wave speed. This equation can be used to model a variety of waves, including sound waves, light waves, and water waves.

2. What is wave displacement invariance?

Wave displacement invariance is a principle in modern physics that states that the shape of a wave is unchanged when it is observed from different reference frames. This means that the wave equation and other wave properties, such as frequency and wavelength, will remain the same regardless of the observer's position or velocity. This principle is important in understanding the behavior of waves in the universe, such as the Doppler effect.

3. How is the wave equation derived?

The wave equation is derived from the fundamental laws of physics, such as Newton's second law and the conservation of energy. By considering the forces acting on small elements of a wave, such as tension or pressure, and applying these laws, the wave equation can be derived. It is also derived using calculus techniques, such as the wave function and Taylor series expansions.

4. What are some real-world applications of the wave equation?

The wave equation has numerous real-world applications in fields such as acoustics, optics, and electromagnetism. It is used to model the behavior of sound waves in musical instruments, the propagation of light in optical fibers, and the behavior of electromagnetic waves in communication systems. It is also used in seismology to study earthquake waves and in fluid dynamics to model water waves.

5. Are there any limitations to the wave equation?

The wave equation is a simplified model that does not take into account all factors that may affect the behavior of a wave, such as dissipation, dispersion, and non-linearity. In some situations, these factors may significantly impact the wave's behavior and cause the wave equation to be inaccurate. Additionally, the wave equation only applies to waves in a homogenous and isotropic medium, meaning that the properties of the medium do not change with position or direction. In reality, most mediums are not completely homogenous and isotropic, so the wave equation may not accurately describe the behavior of waves in these situations.

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