We need a longitudinal magnetic field
On October 9, 2023 we derived the Lorentz transformation of the "electric field Eₓ" if we define the "electric field" through the acceleration of a test charge q. We noted that the ordinary electric field Eₓ does not contain enough information to derive the x' acceleration of q in another frame. This is because the acceleration of q depends on the velocity of the charge Q, and that velocity is not coded into the ordinary electric field Eₓ.
We suggested that one has to define a "longitudinal magnetic field" which would reveal the velocity v of Q, and allow the calculation of the effects of the inertia of q inside the electric field of Q.
The ordinary magnetic field B is
~ v × E,
but we need a new field C which is something like
~ v • E.
Turning mass flows in general relativity
1 3
\ / ^
\ / / v
\__________/ mass flow
2
^ V
|
• m test mass
On September 23, 2023 we showed that the metric of general relativity cannot handle a case where a mass flow turns between the parts 1 and 2, and 2 and 3.
We concluded that if we sum the metric perturbations for all the (very many) particles dm in the part 1, the summed metric is not aware of the way how the metric for each individual particle behaves dynamically with respect to time. The summed metric is time-independent, while the metric for each particle is very much time-dependent in every aspect.
Each particle dm in the part 1 moves toward the test mass m. There are dynamic effects with the increasing inertia as dm approaches m. But these effects are lost in the sum of metric perturbations.
This very much looks like the same problem as we had with Eₓ: the summed field has lost crucial information of the velocity of each charge element dq.
As each dm turns in the bends between 1, 2, and 3, there are further effects since the inertia of m held by dm accelerates in the turn. It is not clear if we can describe this effect with a field at all.
Birkhoff's theorem
Suppose that the gravity acceleration of a test mass m really does change when we manipulate the pressure inside the mass M. The field of M would then undergo a "longitudinal" change – something which is impossible in traditional electromagnetism.
This may be associated with the previous two sections. We have to analyze this in detail.
The Einstein field equations quite simply imply that the metric outside a spherically symmetric object cannot change. But it must change if the pressure trick affects the acceleration of m.
The Schwarzschild metric seems to include all the inertia effects of the field, in contrast to the Coulomb force. But somehow a summed metric then loses this ability.
Conclusions
This is a short note which explains how the Lorentz transformation problem of Eₓ, the gravitomagnetic problem of mass flows, and maybe even Birkhoff's theorem are related.
The traditional magnetic field B contains the required information to know a transverse "movement" of an electric field E. But it does not contain the crucial information which we need in a longitudinal movement of the electric field E.
The summed metric in general relativity, similarly, lacks the crucial information of a longitudinal movement of a gravity field. The information is present if we just have one mass element dm moving. But when we integrate over many such elements, the information may be lost.
Could this information loss explain why Birkhoff's theorem seems to fail?
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