Applicability of E and B field Transformations

[tade]

I have a question regarding these transformation formulas:

$\begin{align} & E'_x = E_x & \qquad & B'_x = B_x \\ & E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\ & E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\ \end{align}$

Suppose we have E and B vector fields in one frame, and we wish to transform them in another frame.

To what extent are the above formulae applicable? Can they be applied to E and B fields of any shape or form, or are they applicable only under certain circumstances?

 

[dextercioby]

They are applicable in the same manner that Special Relativity is.

 

[tade]

They are applicable in the same manner that Special Relativity is.

Nooo, you have to stop talking like that.

 

[dextercioby]

OK, let me elaborate, then :) Those formulas are particular cases of the very general one:

$$F'^{\mu\nu}(x) = \Lambda^{\mu}_{~\alpha}\Lambda^{\nu}_{~\beta} F^{\alpha\beta}(x)$$

if you plug in a particular form of the Lorentz transformation. You can also consider the GR transformation formulas for the double contravariant components of the em tensor if you want. Just replace $\Lambda$ by the partial derivatives.

 

[tade]

OK, let me elaborate, then :) Those formulas are particular cases of the very general one:

$$F'^{\mu\nu}(x) = \Lambda^{\mu}_{~\alpha}\Lambda^{\nu}_{~\beta} F^{\alpha\beta}(x)$$

if you plug in a particular form of the Lorentz transformation. You can also consider the GR transformation formulas for the double contravariant components of the em tensor if you want. Just replace $\Lambda$ by the partial derivatives.

So you were just trolling? Damn.

What if we consider flat spacetime only?

 

[robphy]

I have a question regarding these transformation formulas:

$\begin{align} & E'_x = E_x & \qquad & B'_x = B_x \\ & E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\ & E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\ \end{align}$

This assumes a boost along the x-direction.

 

[tade]

This assumes a boost along the x-direction.

Yeah. But can they be applied to E and B fields of any shape or form, say spherical wavefronts? Or only to plane waves?

 

[dextercioby]

So you were just trolling? Damn.

What if we consider flat spacetime only?

No trolling, just wanting you to do some thinking/textbook research by yourself. The Lorentz transformations are applicable to flat space-time. There's a theorem in differential geometry (proved in an archaic version by Riemann himself) which states that if a space-time manifold is globally flat, then at each point of it you can set the components of the metric tensor to +1 or -1. With this choice of metric, you necessarily have SR with its Lorentz transformations between inertial frames.

The shape of the wavefronts is not important. What is important is how the system of coordinates is chosen.

 

[robphy]

Yeah. But can they be applied to E and B fields of any shape or form, say spherical wavefronts? Or only to plane waves?

Yes... apply it at each point in space [in an inertial frame] to an arbitrary E and B field.
For a plane wave, it is simple because of the symmetry of that configuration.
In other configurations, it is more complicated.

If one is trying to highlight a general feature, it might be good to choose a highly symmetrical situation first... or else your message might be clouded by the complications of a less symmetric choice. Certainly, you can [with more work] show that the feature persists in a less symmetrical situation.

 

[PeterDonis]

To what extent are the above formulae applicable?

To the extent that you are applying a Lorentz boost in the $x$ direction. More general Lorentz transformations required more complicated formulas.

 

[dextercioby]

https://en.wikipedia.org/wiki/Lorentz_transformation This page, even though it badly mixes +--- with -+++ convention for the metric (different writers, different books as sources) pretty much exhausts the subject.

 

[tade]

To the extent that you are applying a Lorentz boost in the $x$ direction. More general Lorentz transformations required more complicated formulas.

In that case might I direct you to an earlier thread of mine: (post #31 onwards, full convo also posted below)
https://www.physicsforums.com/threa...vs-momentum-models.890533/page-2#post-5607457

Note that the image posted in that thread has the fields as a function of (x) and (x'). That wasn't my intention, but was in the image I plucked from the net. I'm still thinking about the question I posed in #31.

In that same paper, Einstein derived an expression for relativistic KE, which has been applied to classical massless particles.
Einstein did this by using the E and B field transformations:

He plugged these transformations into the Lorentz force law in order to determine how a force transforms between frames.

From that he was able to derive an expression for KE.How might this derivation be modified in order to accurately account for the Doppler shifts of spherical wavefronts instead of just plane waves?

Probably start with the E and B field for your spherical wavefront and transform those instead. Maybe the standard dipole field wold be best. The math would get annoying pretty fast.

Do these apply to spherical wavefronts?

Not as written since the fields are written only as functions of x. I would construct the EM tensor and use that instead.

 

[PeterDonis]

In that case might I direct you to an earlier thread of mine: (post #31 onwards, full convo also posted below)

That conversation is about whether the fields, in a given frame, are a function of $x$ only or of all the spatial coordinates.

What I posted earlier in this thread is about what kind of Lorentz transformation you have to make to transform from one frame to another. That's a separate question.

 

[tade]

That conversation is about whether the fields, in a given frame, are a function of $x$ only or of all the spatial coordinates.

What I posted earlier in this thread is about what kind of Lorentz transformation you have to make to transform from one frame to another. That's a separate question.

Sorry, I guess my question wasn't stated specifically enough.

So are the fields a function of $x$ only or of all the spatial coordinates?

 

[PeterDonis]

So are the fields a function of $x$ only or of all the spatial coordinates?

It depends on the situation.

 

[robphy]

So are the fields a function of $x$ only or of all the spatial coordinates?

In physics, a vector field is an assignment of a vector to every point in space.
It may turn out that because of certain symmetries, there is no variation in the vectors when one moves along (say) the y-axis.
So, we can say, in that case, that "it doesn't depend on [the specific value of] the y-coordinate"... but it's still technically a function of all of the spatial coordinates.

 

[tade]

It depends on the situation.

do you want to move back to the older thread?

 

[PeterDonis]

do you want to move back to the older thread?

Regarding the spherical wavefront situation you were asking about in the older thread, your question was already answered by Dale.

 

[tade]

Regarding the spherical wavefront situation you were asking about in the older thread, your question was already answered by Dale.

Are these two sets of transformations considered identical, or is there a slight difference?

$\begin{align} & E'_x = E_x & \qquad & B'_x = B_x \\ & E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\ & E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\ \end{align}$

 

[tade]

Regarding the spherical wavefront situation you were asking about in the older thread, your question was already answered by Dale.

I do apologize for being vexing, I am easily confused, so I spend a lot of effort trying to get to the bottom of things.

 

[PeterDonis]

Are these two sets of transformations considered identical

They don't look identical, do they? The second one specifies that the fields being transformed are functions of $x$ (or $x'$); the first doesn't say anything about what they are functions of, but it is implicit that we are transforming the fields at a particular event (point in spacetime). So without knowing what $x$ (and $x'$) represent, how can we know whether the two are identical?

 

[Dale]

So are the fields a function of x only or of all the spatial coordinates?

Sorry it seems like I may have caused you some confusion in the other thread.

The fields for a dipole are functions of t, x, y, and z. All I meant was that they didn't match the form as written. I didn't mean to imply that they were otherwise wrong.

However, as I mentioned before, I would not use this approach anyway, I would use the EM tensor.

 

[tade]

They don't look identical, do they? The second one specifies that the fields being transformed are functions of $x$ (or $x'$); the first doesn't say anything about what they are functions of, but it is implicit that we are transforming the fields at a particular event (point in spacetime). So without knowing what $x$ (and $x'$) represent, how can we know whether the two are identical?

I think our confusion stems from the fact that we were on different wavelengths regarding which set of transformations we were supposed to discuss. For this I am partly responsible, and I apologize once again.

As I mentioned earlier:

Note that the image posted in that thread has the fields as a function of (x) and (x'). That wasn't my intention, but was in the image I plucked from the net. I'm still thinking about the question I posed in #31.

I meant to use the top set instead of the bottom set. Dale has answered my question for the bottom set, which I did not want; what's the answer for the top set? My objective is to answer the question I posed in #31.

Are these two sets of transformations considered identical, or is there a slight difference?

$\begin{align} & E'_x = E_x & \qquad & B'_x = B_x \\ & E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\ & E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\ \end{align}$

 

[tade]

Sorry it seems like I may have caused you some confusion in the other thread.

The fields for a dipole are functions of t, x, y, and z. All I meant was that they didn't match the form as written. I didn't mean to imply that they were otherwise wrong.

However, as I mentioned before, I would not use this approach anyway, I would use the EM tensor.

That's fine, I made a mistake too.

If we had considered the top set (#23, the post above), without the (x) and (x'), instead, what would have been your answer to #31?
https://www.physicsforums.com/threa...vs-momentum-models.890533/page-2#post-5607457

 

[Dale]

what would have been your answer to #31?

Then I would just have answered that I wouldn't do it this way. I would use tensors.

I prefer to use tensors specifically so that I don't have to go through and check complicated expressions on a component by component basis.

 

[tade]

Then I would just have answered that I wouldn't do it this way. I would use tensors.

I prefer to use tensors specifically so that I don't have to go through and check complicated expressions on a component by component basis.

Can I say that using a tensor will lead to a slightly different transformation of E and B, (as compared to the component by component basis) which leads to different results for the Doppler shift?

 

[PeterDonis]

what's the answer for the top set?

It depends on the situation. As I said before:

the first doesn't say anything about what they are functions of

That means the first set, as given, does not contain enough information to answer your question. The only thing we can know from the first set is that it applies to a Lorentz boost in the $x$ direction. It tells us nothing about how the fields vary with the coordinates.

 

[PeterDonis]

Dale has answered my question for the bottom set, which I did not want

My objective is to answer the question I posed in #31.

In your post #31 in the other thread, you referenced the bottom set, not the top set.

You might want to take a step back at this point. If you are trying to figure out how Lorentz transformations act on E and B fields that describe spherical wavefronts, your first step should be to find expressions for the E and B fields, as a function of the coordinates in a given inertial frame, that describe spherical wavefronts. (Which actually don't exist, as someone pointed out in the other thread: that is, there is no spherically symmetric solution of the source-free Maxwell Equations. But you might merely be interested in, say, a solution describing omnidirectional radiation from an antenna, which will not be exactly spherically symmetric.)

Also, you might observe that, in both of the sets of transformation equations you gave, there are no derivatives of the fields; only values of the fields at the chosen point. So to apply the transformation at a single point of spacetime, you actually don't even need to know how the fields vary with the coordinates. But you probably are interested in how the transformation acts at multiple points of spacetime.

 

[Dale]

Can I say that using a tensor will lead to a slightly different transformation of E and B, (as compared to the component by component basis) which leads to different results for the Doppler shift?

If you do the component by component approach correctly then you will get the same answer as with the tensors.

It is just easier to remember the equation and get to the answer with tensors. In fact, with tensors you can use arbitrary coordinates, which is often convenient for taking advantage of symmetries. Also, tensors don't make the artificial distinction between the E and B field that components do.