UPDATE November 25, 2023: We have to assume that the repulsion between two negative charges Q and q increases the inertia of q, that is, we flip the sign in the Hypothesis below. We also have to assume that the momentum exchange is "paradoxical", that is, has an unintuitive sign.
----
Our explanation is a "unified field theory" in the sense that it reveals the analogy between the Coulomb force and the Newton/Einstein gravity force.
We already showed on August 4, 2023 that if q comoves with a moving charge Q, then the Lorentz transformation of the electric field, which includes a "magnetic field", explains why the acceleration is by a factor 1 - v² / c² slower in the laboratory frame. The concept of a "magnetic field" in this case is devised to explain the effects of a Lorentz transformation. We will check below that this explains all movements of q where q moves tangentially relative to Q, that is, the initial velocity of q is tangential to Q.
If there is a radial movement, then there is an acceleration which is due to q having a different inertia relative to Q close to Q. The concept of a "magnetic field" is devised to explain this acceleration.
Note that in the Schwarzschild orbit, there are two things which increase the inertia of a test mass m relative to M in the field of a large mass M:
1. a movement of m ships energy around in the common field of m and M (for a negative mass M, the movement would ship negative energy);
2. m gains kinetic energy when it approaches M.
The same effects exist for electric charges, but in practical applications, the electric field usually is negligible. When electrons are flowing in a wire, the positive charges of nuclei cancel the electric field. This is the reason why we do not encounter item 2 when working with magnetism. This explains why we on August 5, 2023 concluded that the gravitomagnetic effect of a moving mass M is double of its electric analogue. The effect actually is the same, but a half of the electromagnetic effect is canceled because there is no gain of kinetic energy.
Hypothesis. All magnetic effects come from the Lorentz transformation plus the fact that a test charge q has different inertia when it is close to another charge Q.
We conjecture that the hypothesis also holds for the gravity analogue of m and M. The Coulomb force and the newtonian gravity force are completely analogous in this respect.
For gravity, there are no negative masses which would cancel the newtonian attraction and make gravitomagnetic effects clearly visible.
A test charge q close to another charge Q
We assume here that the fields are weak.
Hypothesis about the the movement of a test charge q close to another charge Q. The movement can be calculated with the Schwarzschild solution constants of motion, but one has to remember the rules:
1. Q and q attract each other if they have opposite charges, and repulse if they have charges of the same sign, and
2. the inertia of q relative to Q decreases in the field of Q if q and Q have the same sign, and the inertia is increased if they have the opposite sign.
3. Kinetic energy has a gravity charge but not an electric charge. The electric potential stays as the Coulomb potential 1 / r, and does not grow steeper like the Schwarzschild gravity potential ~ sqrt(1 - r_s / r) does, relative to the newtonian 1 / r potential.
The conversion factor between a mass m and a charge q is obtained from
G m M ~ 1 / (4 π ε₀) q Q
=>
m ~ sqrt(1 / (G * 4 π ε₀)) q.
Hypothesis about the movement of a test charge q close to many other charges Q₀, Q₁, Q₂, ... (fields are "private"). The accelerations of q have to be calculated individually for each pair q, Q_i, and summed. However, the privacy does not hold for the calculation of the kinetic energy of the test charge. The kinetic energy changes caused by the charges Q₀, Q₁, Q₂, ... collectively are visible to all. Also, the collective inertia that Q₀, Q₁, Q₂, ... give on q is visible to all. Privacy is needed to calculate magnetic effects.
Lorentz covariance. All movements of the test charge q are Lorentz covariant.
Note that the electric charge of a test particle does not increase when it gains kinetic energy, but the gravity charge increases.
If a test mass m approaches another mass M, m gains inertia relative to M equally from two sources: it ships field energy around and gains more kinetic energy. When we study electromagnetic fields, there usually is no gain of kinetic energy, and the magnetic effect appears to be 1/2 of that in gravity.
Above we have two equations which we used on August 10, 2023 to calculate Schwarzschild orbits. In both equations, the key is that the proper time τ of the test mass slows down as it descends. This causes a correction to a newtonian orbit. The rightmost term in the second equation becomes relevant if the tangential velocity of the test mass m is significant relative to radial velocity. However, we will below show that it must be ignored in electromagnetism.
What does more or less inertia "relative to" M or Q mean?
It means the inertia which an observer measures if he stands on M or Q and tries to move a test mass m or a test charge q.
For an outside observer, the inertia of the whole combined system M & m or Q & q cannot change. If it did, that would break conservation of the center of mass.
In this blog we have repeatedly asked the question: what is the inertia of an electron under the electric field of another charge Q If the hypothesis above is true, then the inertia changes by the value -W / c², where W is the potential energy of the electron in the field of Q. Previously, we thought that the inertia always increases by |W| / c².
It is strange if an electron could have negative inertia when it is close to negative charges. In this blog post we study weak fields, and this does not happen. We have to investigate this further in another blog post.
It might be that the true carriers of charge in metal are positively charged pseudoparticles. Then we do not need to assume anything about the inertia of an electron relative to a negative charge.
In the Klein paradox, an electron appears to have negative inertia if it meets a potential wall which requires more than 511 keV of kinetic energy to climb.
What does it mean that a field is "private"?
Suppose that we have particles 0, 1, 2, ... which have velocities v₀, v₁, v₂, ... and generate fields f₀ v₀, f₁ v₁, f₂ v₂, at the test particle P.
If the acceleration of P depends on f₀ v₀², etc., then the sum field f₀ v₀ + f₁ v₁ + f₂ v₂ + ... may miscalculate the acceleration. The accelerations must be calculated "privately" for each particle pair, and summed.
In the case of the rotating disk, the parts dm in it are under an acceleration, and it might happen that the linearly summed metric for all dm loses the information which was required to calculate the acceleration.
General relativity may be missing the concept of an "accelerating" metric. The first derivatives in the geodesic equation maybe cannot capture what happens when the source of gravity is accelerated.
An electric current in a straight metal wire
When an electric current, denoted by I, flows in a metal wire, there are actually two charged cylinders: the cylinder of static, positively charged nuclei, and the moving cylinder of electrons, which moves to the opposite direction of the current.
ρ = electron charge / length
<-- v <-- v electrons
dq' I --> dq
====•===============•==== wire
/|
/ | β = angle
y axis vs. (q, dq)
^ V = (V_x, V_y)
/
•
q negative test charge
m mass of test charge
R = distance (q, wire)
^ y
|
-------> x
The parts dq' and dq are electrons flowing at a velocity v to the left. The positive nuclei are static in the laboratory frame. The angle, relative to the y axis from q to dq' is -β and to dq β.
We assume that the positive and negative charge densities in the wire exactly cancel each other in the laboratory frame. In a frame which moves in the x direction, charge densities do not cancel.
The collective inertia given to q by all the charges in the wire is zero in the laboratory frame.
We determine the acceleration caused by each dq in a manner similar to our August 10, 2023 calculation. We work in the comoving frame of the electrons in the wire, and then transform the acceleration to the laboratory frame.
The velocity of q relative to dq is
V - v = (V_x + v, V_y).
The tangential velocity of q relative to dq is
V_t = (V_x + v) cos(β) - V_y sin(β).
The radial velocity is
V_r = (V_x + v) sin(β) + V_y cos(β),
where a decreasing radial distance is counted as positive.
We will integrate using the angle β and the charge density measured in the laboratory frame, and check that using the corresponding angle β' of the moving frame would change the result negligibly.
Calculating the x acceleration of the test charge
Tangential acceleration
Let the test charge q approach dq a distance Δr in a time Δt. The inertia of the test charge q is reduced by W / c², where
W = q Δr * 1 / (4 π ε₀) * dq / r²
is the work we do when we push q closer to dq. Consequently, the tangential speed of q increases by
ΔV_t = V_t W / (m c²).
The tangential acceleration is
a_t = V_t q Δr * 1 / (4 π ε₀)
* dq / r² * 1 / (m c²) * 1 / Δt
= q / m * V_t V_r * μ₀ / (4 π) * dq / r².
The symmetry with dq' cancels many of the terms in V_t V_r, and leaves:
(V_x + v) V_y * (2 cos²(β) - 1).
We have
r = R / cos(β)
and
dq = ρ R / cos²(β) * dβ.
Then
dq / r² = ρ / R * dβ.
The x component of a_t is
q / m * V_y (V_x + v) * (2 cos²(β) - 1)
* μ₀ / (4 π) * ρ / R * cos(β) dβ.
The y component of a_t contributes nothing because sin(β) flips sign for dq' and dq.
The integral
π / 2
2 * ∫ 2 cos³(β) - cos(β) dβ
0
π / 2
= 2 * / 2 sin(β) - 2/3 sin³(β) - sin(β)
0
= 2/3.
We obtain a contribution to the x acceleration:
a_x = q / m * V_y
* (V_x + v) * 2/3 μ₀ / (4 π) * ρ / R.
Radial acceleration
The rightmost term probably does not appear in the electromagnetic analogy because there is no steepening of the Coulomb potential. Let us assume that the test mass is on a circular orbit.
Then dt / dτ is zero and in a newtonian field, the centrifugal pseudoforce cancels the force of gravity:
G M m / r² = m v² / r
= L² / (m r³)
<=>
G M m / r² = L² / (m r²).
The derivative of the gravity potential with respect to r is
d ( m c² sqrt(1 - r_s / r) ) / dr
= m c² * -1/2 * 1 / sqrt(...)
* -2 G M / c² * 1 / r²
= G M m / r² * 1 / sqrt(1 - r_s / r).
Gravity relative to the newtonian one is stronger by a factor
1 / sqrt(1 - r_s / r) = 1 + 1/2 r_s / r,
for small r_s. This implies a correction
L² / (m r²) * 1/2 * 2 G M / c² * 1 / r
= G M L² / (c² m r³),
which is the rightmost term.
We do not think that there is a steepening of the Coulomb potential, and drop the rightmost term.
Let the test charge q approach dq a distance Δr in a time Δt. The inertia of the test charge q is reduced by W / c².
The radial acceleration from this effect is
a_r = q / m * V_r² * μ₀ / (4 π) * dq / r².
We have
V_r² = (V_x + v)² sin²(β) + V_y² cos²(β)
+ 2 (V_x + v) sin(β) V_y cos(β).
To get the x component of a_r, we have to multiply a_r by sin(β) and integrate. The first two terms contribute nothing because sin(β) flips sign at 0. We have:
dq / r² = ρ / R * dβ.
The integral is:
π / 2
a_x' = ∫ q / m * μ₀ / (4 π) * 2 (V_x + v) V_y
-π / 2
* sin²(β) cos(β) ρ / R * dβ
= q / m * μ₀ / (4 π) * 2 (V_x + v) V_y
* 2/3 ρ / R.
The contribution to the x acceleration is
a_x' = q / m * V_y
* (V_x + v) * 4/3 μ₀ / (4 π) * ρ / R.
The sum of x accelerations:
a_x + a_x' = q / m * V_y
* (V_x + v) μ₀ / (2 π) * ρ / R.
It corresponds to a magnetic field
B = (V_x + v) μ₀ / (2 π) * ρ / R,
which is the field of an electric current
I = (V_x + v) ρ.
The part V_x ρ is canceled by the effect of the positively charged nuclei.
The y acceleration is:
π / 2
a_y = ∫ q / m * μ₀ / (4 π) * 2 (V_x + v) V_y
-π / 2
* sin(β) cos²(β) ρ / R * dβ
= 0.
Removing the terms which come from V_x: only the current I matters in x acceleration
Above we calculated the x acceleration which comes from the relative x velocity V_x + v of the electrons in the wire and the test charge q.
Next we should calculate the x acceleration caused by the positive nuclei in the wire. The calculation is essentially the same, but the relative x velocity is only V_x, and the effect is opposite.
Thus, we can remove the V_x terms from the x acceleration, and conclude that
B = v μ₀ / (2 π) * ρ / R,
or
B = I μ₀ / (2 π) * 1 / R,
corresponds to the x acceleration of the test charge q. We still have to check that the y acceleration matches the same magnetic field B.
Calculating the y acceleration of the test charge
In section 13-6 of his magnum opus, Richard Feynman (1964) derived one aspect of the magnetic force from the different length contraction of cylinders containing negative electrons versus positive nuclei. The test charge q is moving at a velocity v₀ relative to the wire, parallel to the wire. The electrons in the wire move at a velocity v.
In our own example, the test charge q is moving with a velocity V_x relative to the wire. Let us calculate if the different charge densities in the comoving frame of q explain the magnetic field
B = v μ₀ / (2 π) * ρ / R.
The density of the negative electron charges in the comoving frame of q is
ρ * 1 / sqrt(1 - (v + V_x)² / c²)
/
1 / sqrt(1 - v² / c²)
= ρ (1 + 1/2 (v + V_x)² / c² - 1/2 v² / c²)
= ρ (1 + 1/2 V_x² / c² + v V_x / c²).
The density of positive charges of nuclei, as seen in the comoving frame of q, is
-ρ (1 + 1/2 V_x² / c²).
The test charge q sees an excess negative charge density of
ρ v V_x / c²,
whose electric field has the field strength
E = 2 / (4 π ε₀) * ρ v V_x / c² * 1 / R
at a distance R. The repulsive force on the negative test charge q is
F = q V_x * v μ₀ / (2 π) * ρ / R
= q V_x B.
The repulsive force matches what the magnetic field B would produce.
The x and y accelerations of the test charge q match the ones that a magnetic field of the electric current I is expected to produce. We derived the effect of the magnetic field from Lorentz transformations, and from the change of the inertia of q inside an electric field. The change in the inertia matches the one in the Schwarzschild solution of gravity.
We were able to explain the magnetic field from "first principles". This is a new result in physics.
What about Lorentz transformations of the accelerations?
We still have to check that Lorentz transformations of various accelerations, and the distortion of the angle β in a moving frame do not affect our results. We did not yet (September 6, 2023) do a thorough analysis and checks of the proofs.
Is magnetism simply a Lorentz transformation of the electric field?
The answer is no. Above, the x acceleration of the test charge q comes from changes in the inertia of q in an electric field. We do not know any way to derive the inertia using special relativity. Inertia is like "syrup". You cannot derive a syrup from special relativity. It is a property of the interaction.
The y acceleration comes from a Lorentz transformation. We could say that one half of magnetism is from a Lorentz transformation – the other half is from inertia.
To our knowledge, no one has realized this before us. Richard Feynman in his Lectures (1964) vaguely states that magnetism is just Lorentz transformed Coulomb forces, but does not offer any proof. On the Internet, one finds conflicting views. Some people agree with Feynman's 1964 opinion, others oppose it.
We settled the question.
Empirical tests of our model
It probably is possible to measure the inertia of an electron under various electric fields. We did write about one such practical experiment a couple of years ago. The researchers got a result that the inertia of an electron does not change in an electric potential. But we analyzed their test configuration and noticed that conducting materials close to the electrons hide any change in the inertia of electrons.
The experiment should be repeated very carefully.
Basically, it would be very surprising if there is no change in inertia. If we believe the Poynting vector, field energy does move around if q moves relative to Q.
Is the gravity potential really steeper than the newtonian 1 / r potential?
General relativity implies the Schwarzschild solution, and the gravity potential there is steeper than in newtonian gravity. If we lower a test mass m hanging from a rope to a black hole of a mass M, we have recovered all m c² of the energy of the test mass when it is at the Schwarzschild radial coordinate
r_s = 2 G M / c².
In newtonian gravity, the coordinate would be
r = G M / c².
We are not sure if general relativity calculates the coordinate right. The "privacy" principle of the interaction between particles suggests that there should not be any steepening of the potential. Particles are so lightweight that their Schwarzschild radius is negligibly short.
Is there any empirical evidence that gravity steepens? Data from black holes has very wide margins of uncertainty. If the horizon would be at half a distance, would that show up in any empirical data?
Certain equivalence principles imply that gravity has to steepen, but since general relativity does not obey even the weak equivalence principle (see our August 23, 2023 post), it is not clear why we should believe those equivalence principles.
Why does special relativity match with the change of inertia so well that they form a common magnetic field B?
Our blog post on August 25, 2023 offers a model which explains this. A "particle" with a rest mass really is a box where massless subparticles are bouncing around.
When a rocket flies at a great speed relative to us, clocks slow down inside the rocket relative to us. This is because when we accelerated the rocket, we gave more energy to each box which represents a particle. The subparticles inside the box gained energy, were "heated". More energy means more inertia, and inertia slows down clocks.
When an electron descends down toward a positive charge, the subparticles inside the electron are heated in a similar fashion. The electron gains more energy, and consequently, more inertia.
Question. Does newtonian mechanics explain the gain of inertia? If we have fast particles inside a box, accelerating the box takes more energy in newtonian mechanics because
1/2 m (V + v)² + 1/2 m (V - v)²
> 2 * 1/2 m V².
The box has more inertia!
Our question may offer a way to reduce special relativity to newtonian mechanics. Friends of the aether hypothesis tried this in the 19th century, but were not able to produce a beautiful model.
Let us analyze why the x acceleration and y acceleration calculated above match a magnetic field B.
1. Both effects are linear in the strength of the interaction.
2. The x acceleration is linear in v and V_y. The y acceleration, through a happy coincidence, is linear in v and V_x.
3. The inertia added to the test charge q is proportional to 1 / c². Length contraction, similarly, is proportional to 1 / c².
There is no simple reason why these two effects have a similar magnitude. It looks like a coincidence.
How can the magnetic field B function in an electromagnetic wave?
In electromagnetic waves, the magnetic field B and the electric field E are interwoven. In that context, B seems to be one unified field. Why is that?
If an electromagnetic wave propagates in a polarizable material, then the wave can be regarded as an interaction between actual charges.
But there are no charges in empty space. What is interacting with what?
Fields store energy, and a wave is a phenomenon where energy cyclically changes its form. In a harmonic oscillator, or in a violin string, kinetic energy and elastic energy alternate.
What is the cyclic process in an electromagnetic wave?
^
|
• Q dipole radio • q
| transmitter
v
^ y
|
-------> x
If we take for granted that the electric field is sourceless in empty space, then we may be able to prove that the electric field far a away from a dipole radio transmitter oscillates in the y direction in the familiar way:
E = E₀ sin(ω t).
In the x and z direction the electric field is zero.
In electromagnetism, in vacuum,
∇ × B = 1 / c * dE / dt.
From this we can probably derive that the magnetic field B oscillates in the z direction if the wave propagates to the x direction. Then B is just a byproduct of an oscillating field E.
Is that field B born from Lorentz transformations or from changes in the inertia of a test charge q?
It has to be both. If q in the diagram moves toward Q, or to the direction of the y axis, it feels the magnetic field B which points out from the screen.
We conclude that the magnetic field of an electromagnetic wave has to be a result of some kind of a hidden electric current in empty space (= electric polarization). It cannot be born from the visible electric field E itself.
Space is polarized in a fashion similar to any polarizable material. Polarization changes with time, which means hidden electric currents.
Polarization creates both the electric field E and the magnetic field B of the electromagnetic wave, just like it would in a polarizable material.
Since an electromagnetic wave propagates through a polarizable material through polarization, it would actually be surprising if the propagation method would be completely different in empty space.
Thus, what is the magnetic field B in an electromagnetic wave? It is an effect of the Coulomb fields of hidden electric current carriers. We do not need to assume the existence of any "physically real field" B at all, but we have to assume the existence of hidden currents.
If there is polarization in space, is there an excess charge somewhere? An observer measures an electric field E. Does that mean that there is more positive charge in some direction?
In the radio transmitter above, the excess charge is Q. It might be that we can order polarization in such a fashion that Q is the only excess charge. We might add another charge -Q to make the radio transmitter electrically neutral. Then all electric field lines start from Q and end at -Q. There are no sources of electric field anywhere else. Then empty space stays electrically neutral at a macroscopic level.
In a polarizable material, there are charges at the microscopic level. Polarization is a displacement of those microscopic charges.
Conclusions
Let us close this lengthy discussion. We will study the other component of our "unified field theory" – gravity – in subsequent blog posts.
We were able to derive the magnetic field B from Lorentz transformations and changes in inertia inside an electric field. This is a new result in physics and may tell us what a magnetic field "really" is. It is an effect of the Coulomb electric field.
There cannot exist magnetic monopoles because there is no "independent physical field" B.
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