E.H is a Lorentz invariant, when is it different from 0 ?

[lalbatros]

From the Lorentz-invariant Faraday tensor F(H,E) two scalar invariants can be constructed:

Inv1 = H²-E²​

and

Inv2 = E.H​

Thinking at waves, electrostatic fields, magnetostatic fields, I see examples where Inv2 = 0.
It is however easy to arrange an electrostatic field and a magnetostatic field to be parallel, so to get an exemple where Inv2 =/= 0. But this situation is not very interresting, maybe because it is made of static and independent sources.

I would be interrested to know if there are less trivial and more interresting examples of fields where Inv2 =/= 0 ,
if there is a general physical meaning to this condition,
and generally what is the physical importance of this invariant.

Thanks for your comments,

Michel

 

[eljose79]

Hello Michel,

Thank you for bringing up this interesting topic. I can provide some insights into the physical meaning and importance of the two scalar invariants constructed from the Lorentz-invariant Faraday tensor.

First, let's define the two invariants in a more general way. Inv1 can be expressed as the difference between the squared magnitude of the magnetic field (H) and the squared magnitude of the electric field (E). This can also be written as the difference between the energy densities of the magnetic and electric fields. Similarly, Inv2 can be expressed as the dot product of the electric and magnetic fields, which represents the energy transfer between the two fields.

Now, let's consider the physical significance of these invariants in different scenarios. In the case of electromagnetic waves, both invariants are non-zero and play important roles. Inv1 represents the energy density of the electromagnetic wave, while Inv2 represents the direction of propagation of the wave. This is why we often see Inv2 being equal to zero in static fields, as there is no energy transfer between the electric and magnetic fields in this case.

In the case of electrostatic and magnetostatic fields, Inv1 is still non-zero as there is energy stored in the fields. However, Inv2 is equal to zero as there is no energy transfer between the two fields. This is because electrostatic and magnetostatic fields are generated by static and independent sources, as you mentioned.

Now, to answer your question about more interesting examples where Inv2 is non-zero, we can look at situations where there is a dynamic interaction between electric and magnetic fields. For example, in a plasma, where charged particles interact with electromagnetic fields, both invariants are non-zero and play important roles in understanding the behavior of the plasma.

In conclusion, the physical meaning of these invariants is related to the energy density and transfer between electric and magnetic fields. Their values can provide insights into the dynamics of electromagnetic fields in different scenarios, such as electromagnetic waves, electrostatic and magnetostatic fields, and plasma. I hope this helps to answer your questions.

 

[astrokreng]

Thank you for your question, Michel. The Lorentz invariant E.H is different from 0 when there is a non-zero interaction between electric and magnetic fields. This interaction can occur in various scenarios such as electromagnetic waves, where the electric and magnetic fields are perpendicular to each other and continuously changing, or in the presence of moving charges, where a magnetic field is produced by the motion of the charges and interacts with the electric field.

The two scalar invariants, Inv1 and Inv2, are useful in understanding the nature of the electromagnetic field. Inv1 represents the energy density of the field and Inv2 represents the flux of energy through a surface. When Inv2 is equal to 0, it means that there is no net transfer of energy through the surface, indicating that the electric and magnetic fields are not interacting.

One example of a field where Inv2 is not equal to 0 is in the case of circularly polarized light. In this scenario, the electric and magnetic fields are perpendicular to each other and continuously changing, resulting in a non-zero interaction and a non-zero value for Inv2. This is a more interesting example because it demonstrates the dynamic nature of the electromagnetic field and the transfer of energy between the electric and magnetic components.

The physical meaning of the condition Inv2 =/= 0 is that there is a non-zero interaction between the electric and magnetic fields, indicating the presence of electromagnetic waves or the motion of charges. This is an important condition in many areas of physics, including optics, electromagnetism, and astrophysics.

In summary, the Lorentz invariant E.H is different from 0 when there is a non-zero interaction between electric and magnetic fields. This condition is important in understanding the nature of the electromagnetic field and has various physical implications in different fields of science.