# Electromagnetic waves relativistic

#### Petar Mali

If we have case $$\rho=0$$, $$\vec{j}=\vec{0}$$

then we have this equations

$$\Delta\vec{E}-\frac{1}{c^2}\frac{\partial^2 \vec{E}}{\partial t^2}$$

$$\Delta\vec{B}-\frac{1}{c^2}\frac{\partial^2 \vec{B}}{\partial t^2}$$

Particular solutions of this equation

$$\vec{E}=\vec{E}_0 e^{i(wt-\vec{k}\cdot\vec{r})}$$

$$\vec{B}=\vec{B}_0 e^{i(wt-\vec{k}\cdot\vec{r})}$$

If we have coordinate frames $$S,S'$$

System $$S'$$ has relative velocity $$\vec{u}$$ in $$x$$ direction compared with $$S$$

System $$S'$$

$$\vec{E}'=\vec{E}_0' e^{i(w't'-\vec{k}'\cdot\vec{r}')}$$

$$\vec{B}'=\vec{B}'_0 e^{i(w't'-\vec{k}'\cdot\vec{r}')}$$

System $$S$$

$$\vec{E}=\vec{E}_0 e^{i(wt-\vec{k}\cdot\vec{r})}$$

$$\vec{B}=\vec{B}_0 e^{i(wt-\vec{k}\cdot\vec{r})}$$

Why phase

$$wt-\vec{k}\cdot\vec{r}$$

must be invariant?

#### phyzguy

Think of it this way. The phase determines where in the cycle you are. If I choose a point in space where the electric field is a maximum, all observers should agree that this is the maximum point. They may disagree on the the value of the spacetime coordinates, the value of the E-field, the frequency of the wave, the wavelength of the wave, etc, but they should all agree that this point in space is the peak. In order for this to happen, the phase needs to be a scalar invariant.

#### Petar Mali

Why they must agree that in this point is peak?

If I choose a point in space where the electric field is a maximum, all observers should agree that this is the maximum point.

Why? That is my question!

#### Phrak

If you want a solid derivation, the electric and magnetic fields transform under Lorentz as elements of the Faraday Tensor. Otherwise, you could do it an easier way and look-up the equations for the Lorentz transformation of the electric and magnetic fields.

But
$$wt-\vec{k}\cdot\vec{r}$$
appears to be Lorentz invariant, in general, for any wave in Special Relativity, which may help.

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#### Phrak

In fact,

$D^\mu = (ct, \vec{r})$
$K^\nu = (c\omega, \vec{k})$

$D_\mu = (-ct, \vec{r})$
$K_\nu = (-c\omega, \vec{k})$

$\phi = *(D_\mu \wedge (* K_\nu)) = wt-\vec{k}\cdot\vec{r}= scalar$

Scalars are constant under Lorentz Transformations. So the invariance of
$wt-\vec{k}\cdot\vec{r}$
is true for a wave at any velocity.

Group velocity could be more interesting.

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#### Petar Mali

Ok I can tell

$$x^{\mu}=(\vec{r},ct)$$

$$k^{\mu}=(\vec{k},c\omega)$$

$$x_{\mu}=(-\vec{r},ct)$$

$$k_{\mu}=(-\vec{k},c\omega)$$

$\phi = *(D_\mu \wedge (* K_\nu)) = wt-\vec{k}\cdot\vec{r}= scalar$

What is $$*$$ and what is $$\wedge$$?

#### Phrak

hmm... this are something of an advanced topic called variously the exterior calculus or differential form (or just forms, or p-forms). But the mathematicians here, are comfortable with it. It is really the only way to go with electromagnetism as a four dimensional theory, or anytime one impresses a vector field on a Riemann manifold.

* = Hodge duality operator
/\ = wedge product

add the exterior derivative "d" and Stokes Theorem, and the set is, for the most part, complete.

The general form of your equation
$\phi = wt-\vec{k}\cdot\vec{r}$
keeps coming up--twice this week, even. With other variables/operators, it's the charge continuity equation, or the redefinition of the metric.

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#### Petar Mali

hmm... this are something of an advanced topic called variously the exterior calculus or differential form (or just forms, or p-forms). But the mathematicians here, are comfortable with it. It is really the only way to go with electromagnetism as a four dimensional theory, or anytime one impresses a vector field on a Riemann manifold.

* = Hodge duality operator
/\ = wedge product

add the exterior derivative "d" and Stokes Theorem, and the set is, for the most part, complete.

Is it hard? What do this operators?

#### Mentz114

Gold Member
Petar Mali said:
Why? That is my question!
phase must be invariant under LT otherwise a paradox with interference patterns can happen.

If an event is triggered by a dark line falling on a detector, then in some frames the event is triggered, in others not. This is obviously because a phase change displaces interference lines.

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#### Phrak

phase must be invariant under LT otherwise a paradox with interference patterns can happen.

If an event is triggered by a dark line falling on a detector, then in some frames the event is triggered, in others not. This is obviously because a phase change displaces interference lines.
Nicely thought and put.

It begs the question," In the case of wave phenomena, why should the relative covariance between the propagation vector and displacement have a correspondence to what we would call objective observations?"

#### Mentz114

Gold Member
Nicely thought and put.

It begs the question," In the case of wave phenomena, why should the relative covariance between the propagation vector and displacement have a correspondence to what we would call objective observations?"
Thanks.

I'm not sure what your question means, but this correspondence is only true for a certain choice of gauge constraint, which is usually a wave equation.

#### Phrak

Thanks.

I'm not sure what your question means, but this correspondence is only true for a certain choice of gauge constraint, which is usually a wave equation.
I'm not sure what you mean by gauge constraint. But, often in the course of studying relativity you see objects like

$\partial_t \phi - \nabla \cdot \vec{J}=constant$

This happens to be the charge continuity equation. The constant in this case is zero, which can be traced back to the fact that the dual 4-vector of current density is an exact 1-form, expressed as a function of the Faraday tensor, J = -d*F, and therefore J=-dd*F=0. (All exact forms are closed.) If neither is an exact form, the constant will, in general, be nonzero.

It should be a little more interesting if the angular frequency and propagation vector given by the OP are converted to energy and momentum through the Einstein and deBroglie equations respectively. Then we get

[itex]tE - \vec{x}\cdot \vec{p} = \hbar \phi [/tex]

This is also equal to the proper time times intrinsic mass.

But the more interesting part I referred to is the grouping of conjugate variables.

By gauge constraint, were you referring to U(1) the phase invariance of the photon?

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#### Mentz114

Gold Member
Phrak,

I have to admit that in this sentence

"why should the relative covariance between the propagation vector and displacement have a correspondence to what we would call objective observations?"

the bolded terms mean nothing to me. I should have kept quiet instead of shooting in the dark.

#### Phrak

No, no. Thanks for your indulgence, Mentz. I appreciate any input into this corner of relativity physics involving differential p-forms that are not too popular and even less appreciated. I see these fascinating patterns emerge and wonder why. You see, through my series of mathematical gyrations, I obtain something that makes sense, while you, through admirable insight present a solid thought experiment, and obtain the same result. Why? Why are these conclusions the same?? I've been trying--and failing--all day to see the connection.

Two different observers in relative motion will disagree on such things as the magnitude of an electric field. They will not disagree on the phase of a wave, or the interference pattern developed on a piece of film, or a physical event. So I ask, what is the fundamental distinction in these two classes.

#### Mentz114

Gold Member
Phrak said:
Two different observers in relative motion will disagree on such things as the magnitude of an electric field. They will not disagree on the phase of a wave, or the interference pattern developed on a piece of film, or a physical event. So I ask, what is the fundamental distinction in these two classes.
I always had a quick answer to this question - counting does not depend on relative simultaneity and so is invariant, but measuring always involves simultaneity, so measurements are always frame dependent.

To labour the point, when a bell rings once, it is a coming together of the worldlines of the clapper and the bell ( an event x,t ) and everyone agrees (a) that the bell rang, and (2) on the number of times the bell chimes along a segment of its worldline. But the time between events depends on two chimes, not just one, and so involves at least 2 events.

The transformation between frames of the delta x and delta t between the points will mix space and time, electricity and magnetism, but none of this happens if there's only one event involved.

#### TrickyDicky

No, no. Thanks for your indulgence, Mentz. I appreciate any input into this corner of relativity physics involving differential p-forms that are not too popular and even less appreciated. I see these fascinating patterns emerge and wonder why. You see, through my series of mathematical gyrations, I obtain something that makes sense, while you, through admirable insight present a solid thought experiment, and obtain the same result. Why? Why are these conclusions the same?? I've been trying--and failing--all day to see the connection.

Two different observers in relative motion will disagree on such things as the magnitude of an electric field. They will not disagree on the phase of a wave, or the interference pattern developed on a piece of film, or a physical event. So I ask, what is the fundamental distinction in these two classes.
I think this observation, the agreement on the phase of a wave for different observers, is key to understand relativity and its connection with the electromagnetic interaction, and it's telling us something deep about the true nature of the spacetime structure in our universe.
Thinking out loud, this points to a determinate topological property of the space where EM wavefronts have to be agreed by observers, well this seems to arise from the Lorentz invariance of Maxwell equations. The fact that other components of the Faraday tensor like the magnitude of the electric field are coordinate-dependent is compatible with the observation of the observers disagreement on this.
So perhaps the distinction you seek is just that the constant phase or wavefront of EM radiation is defined by the trace of the Faraday tensor and therefore is Lorentz invariant, but I believe this is what has been said better than me in the previous posts by you and phizguy.
Ultimately why is this so can only be traced to the question why does empty space follows minkowski metric. And as I said that is a topological property of the vacuum.
I don't know if this helps or further confuses.

#### TrickyDicky

I always had a quick answer to this question - counting does not depend on relative simultaneity and so is invariant, but measuring always involves simultaneity, so measurements are always frame dependent.

To labour the point, when a bell rings once, it is a coming together of the worldlines of the clapper and the bell ( an event x,t ) and everyone agrees (a) that the bell rang, and (2) on the number of times the bell chimes along a segment of its worldline. But the time between events depends on two chimes, not just one, and so involves at least 2 events.

The transformation between frames of the delta x and delta t between the points will mix space and time, electricity and magnetism, but none of this happens if there's only one event involved.
This is a clever way of explaining it.

#### Mentz114

Gold Member
Thanks, Tricky. I didn't mean to sound so certain.

TrickyDicky said:
Ultimately why is this so can only be traced to the question why does empty space follows minkowski metric. And as I said that is a topological property of the vacuum.
That's interesting. I presume topology comes into it because it determines things like

1. How many times inertial observers can get close to each other
2. How many geodesics can go through two points ( hmm, not sure about that).
3. any offers ?

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#### TrickyDicky

I was thinking about a space with certain properties invariant under specific homeomorphisms.
For instance EM wave phase invariance could be achieved in a spacetime that allowed foliations by constant phase plane waves in every direction,in every point in space and for any Lorentz boost, this would make every point in the space a confluence of the plane waves and of the wave vectors K of every wave (that would make it isotropic). Theoretically such a space would be filled with a pervading radiation that could be detected in the apropriate wavelength with an antenna.

#### TrickyDicky

Why phase

$$wt-\vec{k}\cdot\vec{r}$$

must be invariant?
Certainly in science questions that start with a why are interesting but hard to answer, generally they are given "how" answers, like mathematical derivations of the result.
In this case I think it would be important to try and give a reason why, I hinted at a solution based on constructing a spacetime manifold with certain topological features that ensure the congruence of electromagnetic waves phase for all observers, but I lack the ability to develop it, maybe someone has some other idea to construct such spacetime, or any other way to give a causation to the OP question.

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