Electromagnetic Interference in Einestien Field Equation

[Mueiz]

According to EM theory,in the absence of any interaction with matter, destructive interference in one location will always be balanced by constructive interference in another location keeping the law of conservation which means that there is no local conservation law for energy flux regarding this phenomena (energy is created in some locations and destroyed in another location).
But the local conservation of energy is one of the basis of EFE (vanishing of the divergence of both sides of the equation).
Is there any paradox to resolve?

 

[Bill_K]

As you point out, energy is locally conserved. Interference does not suddenly take place at the screen, it happens way back at the slit(s), where the energy emerging from the slit is diverted in certain directions.

 

[jtbell]

According to EM theory,in the absence of any interaction with matter, destructive interference in one location will always be balanced by constructive interference in another location keeping the law of conservation which means that there is no local conservation law for energy flux regarding this phenomena (energy is created in some locations and destroyed in another location).

(I added the boldface to indicate the point that I am specifically addressing below.)

No. In classical EM theory, we not only have the concept of an energy density associated with the magnitudes of the E and B fields, we also have the concept of an energy flux density (Poynting vector) which represents the "flow" of energy through a point and which is associated with the vector product \vec E \times \vec B. Any change of the total energy inside some volume is accompanied by a net inward or outward flux of the Poynting vector through the surface of that volume.

 

[Dale]

According to EM theory,in the absence of any interaction with matter, destructive interference in one location will always be balanced by constructive interference in another location keeping the law of conservation which means that there is no local conservation law for energy flux regarding this phenomena (energy is created in some locations and destroyed in another location).

This is not correct at all. See http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

You are making the very mistaken assumption that the mere fact that something has a nonlocal explanation necessarily implies that it does not have a local explanation.

 

[Mueiz]

OK..I am not familiar with this " Poynting vector" but I think (but not sure) I can take it away from this discussion If we apply these rules in a small region of interference of two similar beams of light but traveling in an opposite direction so there will not be any Poynting vector.
If so can you answer the question without Poynting vector

 

[Dale]

The Poynting vector is generally non-zero in the case of two beams of light interfering.

 

[Mueiz]

The Poynting vector is generally non-zero in the case of two beams of light interfering.

Ok generally ..but what about the special case of beams traveling in opposite direction.

 

[Dale]

In that special case it is also generally non-zero. By "generally" I mean that there may be some specific events where the Poynting vector is zero, but in the rest of the spacetime it is non-zero.

 

[Mueiz]

In that special case it is also generally non-zero. By "generally" I mean that there may be some specific events where the Poynting vector is zero, but in the rest of the spacetime it is non-zero.

but according to the definition of Poynting vector in the paper you posted to me it (as a vector) must equal zero in a location of meeting of two similar beams of light traveling in opposite direction .
I do not know why you want to speak about the rest of spacetime while we are discussing the local application of conservation of energy.

 

[Dale]

but according to the definition of Poynting vector in the paper you posted to me it (as a vector) must equal zero in a location of meeting of two similar beams of light traveling in opposite direction .
I do not know why you want to speak about the rest of spacetime while we are discussing the local application of conservation of energy.

The beams of light exist at more than one event, at most of those events the Poynting vector is non-zero.

You are making some unfounded assumptions here. I would strongly recommend that you actually work through an example. The simplest would be for a pair of lineraly polarized monochromatic plane waves. Working through that should clear up a lot of misconceptions you are having. If you get stuck just post what you have and I would be glad to help.

 

[Mueiz]

The beams of light exist at more than one event, at most of those events the Poynting vector is non-zero.

If we take for example a beam of light coming from a distant star then all the rays of the beam have the same phase and direction in a small region ..so when this beam interfere with another one coming from the opposite direction with the same density and frequency from another distant star the Poyting vector is equal zero in all the events of this region of spacetime.

 

[Dale]

You should work this out for yourself in detail. What you are saying is simply incorrect.

The Poynting vector is non-zero at most events in that region of spacetime (for linearly polarized waves).

 

[Mueiz]

You should work this out for yourself in detail. What you are saying is simply incorrect.

The Poynting vector is non-zero at most events in that region of spacetime.

Does it need working out to know that the sum of two vectors that are equal in magnitude and opposite in direction is zero? Is it me who make unfounded assumptions and need to clear up a lot of misconceptions?

 

[Dale]

Remember that the Poynting vector is a vector field and that it is not constant even for a single beam. It varies over time and space. Your assumption that it simply cancels out is wrong (for linearly polarized waves).

Why are you so reluctant to work it out? It is not that difficult and you will learn a lot about interference.

 

[Mueiz]

Remember that the Poynting vector is a vector field and that it is not constant even for a single beam. It varies over time and space. Your assumption that it simply cancels out is wrong.

According to the conditions I mentioned in my post #11 the vector field is constant.

Why are you so reluctant to work it out? It is not that difficult and you will learn a lot about interference.

Ok I will do it now:
A + ( - A ) = O

 

[Dale]

According to the conditions I mentioned in my post #11 the vector field is constant.

If it is constant in space and time then it is not a wave and is not a solution to Maxwell's equations in vacuum.

Ok I will do it now:
A + ( - A ) = O

No wonder you are confused, that is not the equation of a wave.

 

[chrisbaird]

I think you are confusing the time-averaged Poynting vector with the instantaneous Poynting vector. The time-averaged Poynting vector of a monochromatic plane wave is indeed constant over all space and time, and thus can be represented by a single, constant, vector. But that's just because it is an average. The actual Poynting vector depends on space and time.

Two monochromatic, in-phase, plane waves traveling in opposite directions creates a standing wave. A standing wave has a time-averaged Poynting vector of zero everywhere. This means there is no net energy flow in a standing wave as we expect. But the actual Poynting vector of a standing wave is non-zero. This is demonstrated in detail http://faculty.uml.edu/cbaird/all_homework_solutions/7standing_wave.pdf" .

 

[Matterwave]

The electro-magnetic energy-stress-momentum tensor is just like any other stress-energy tensor in its conservation properties, so I don't think there's anything funny happening.

 

[Mueiz]

If it is constant in space and time then it is not a wave and is not a solution to Maxwell's equations in vacuum.

It is constant as a wave because of the conditions I mentioned in #11

No wonder you are confused, that is not the equation of a wave.

A here is the Poynting vector of the first beam in any event of the spacetime and -A is Poynting vector of the other beam at the same event.
You do all this exposed camouflage to get away from the clear result of vanishing of Poynting vector in the case which I described clearly in post #11.

 

[Dale]

It is constant as a wave because of the conditions I mentioned in #11

The conditions you mentioned in #11 can't overrule Maxwell's equations.

By the way, your conditions in post #11 are wrong even without reference to Maxwell's equations. When you want to look at a function in a small region you make the assumption that that the function is linear over that small region, not that it is constant.

A here is the Poynting vector of the first beam in any event of the spacetime and -A is Poynting vector of the other beam at the same event.

The red highlighted text is wrong. They only cancel out at a few selected events in the spacetime for linearly polarized waves, as I have mentioned several times. In general they do not cancel out.

Please stop being lazy and just work the problem I suggested earlier in post #10. You will find that it is as I have been describing.

 

[Mueiz]

The electro-magnetic energy-stress-momentum tensor is just like any other stress-energy tensor in its conservation properties, so I don't think there's anything funny happening.

Is there interference in any other stress-energy tensor ?

 

[Matterwave]

All stress energy tensors obey:

\nabla_\mu T^{\mu\nu}=0

The Electro-magnetic tensor is not special in that respect. That's all I meant.