The relativistic consistency of gravity
The relativistic consistency problem of Maxwellian gravity has been resolved.
Initially, the relativistic consistency problem of gravity was a problem I had set aside, presuming it would be difficult to solve immediately. It was a problem that first required solving how the potential appears differently in each inertial frame. However, after a time, I decided to examine such correlations numerically, and unexpectedly discovered that the relationship between potentials in different inertial frames is remarkably simple.
Specifically, it was found that \(V_v = \frac{1}{\gamma_v} V_r = \frac{1}{\gamma_{\text{id}}} V_u\). In other words, the gravitational potential in each inertial frame appears larger by the γ factor of the gravitational source as observed in that frame.
The gravitational time dilation effect due to this potential is \(1 – \frac{1}{c^2} V\) in the v-inertial frame. The concept of how this time dilation effect is applied is simple. The v-inertial frame is the rest frame of the gravitational source. The idea is that an object experiencing slowed time within this frame’s gravitational field will, when observed from another inertial frame, exhibit a compounded effect: the product of the gravitational time dilation and the time dilation due to the velocity v of the v-inertial frame.
If this object’s motion is viewed from another inertial frame, it would appear as a corrected motion where the gravitational time dilation is applied to the motion induced by the field in that frame. In this case, following the analogy of electromagnetism (the basis for Maxwellian gravity), we can infer that we only need to consider the gravitational time dilation from the v-inertial frame. This is because the effect of the field as observed from another inertial frame already includes the relativistic time dilation caused by the velocity difference between the frames.
The effect of gravitational time dilation on motion, as discussed in the context of the perihelion precession of Mercury, is proportional to \(\left( 1 – \frac{1}{c^2} V \right)^3\) from the perspective of the object experiencing gravity. However, when observing the object from outside the gravitational field, we only need to consider the effect of its decreasing inertial mass within the gravitational potential, making the effect proportional to the \(\left( 1 – \frac{1}{c^2} V \right)\) term.
When the acceleration of an object, taking this effect into account, is expressed by the fields observed in each inertial frame:
The acceleration in the most general inertial frame (a rest frame), where both the object and the gravitational source are observed to be moving, is \(\vec{a}_r = \frac{1}{\gamma_u} \left( 1 – \frac{1}{c^2 \gamma_v} V_p \right) \left( \vec{G}_p + \vec{u} \times \vec{W}_p – \frac{1}{c^2} \vec{u} (\vec{G}_p \cdot \vec{u}) \right)\).
The acceleration in the v-inertial frame, where the gravitational source appears to be at rest, is \(\vec{a}_v = \frac{1}{\gamma_d} \left( 1 – \frac{1}{c^2} V_v \right) \left( \vec{G}_v + \vec{d} \times \vec{W}_v – \frac{1}{c^2} \vec{d} (\vec{G}_v \cdot \vec{d}) \right)\).
And the acceleration in the u-inertial frame, where the object appears to be at rest, is \(\vec{a}_u = \left( 1 – \frac{1}{c^2 \gamma_{\text{id}}} V_d \right) \vec{G}_d\).
I have confirmed that these accelerations perfectly match one another when they are interconverted using the relativistic acceleration transformation formulas. In other words, the relativistic consistency of accelerated motion in Maxwell’s gravity has been verified.
As a side note, the reason the magnetic field effect must be considered for motion in the v-inertial frame is that a gravitomagnetic field exists due to the accelerated motion of the gravitational source.
An excerpt from the relevant section of the book can be found below, and the friCAS file can be downloaded from the GitHub link below.
rcoghttps://github.com/kycgit/gsimm
The field generated by an arbitrarily moving charge
I have completely removed the ambiguity in the method for deriving the field of an arbitrarily moving charge.
Several formulas describing the field of a moving charge are already known, including the Heaviside-Feynman formula, the formula in Griffiths’ electrodynamics textbook, and the one derived from the Liénard-Wiechert potentials. In my book, I showed that these are all ultimately the same formula and presented a simplified, practical version that clarifies their complex form and meaning. However, I had considered their derivation processes to be incomplete. Feynman left no explanation for his derivation, and Griffiths’ formula appears to hide the ambiguity of its derivation with mathematical tricks. Given that the well-known formula for the Liénard-Wiechert field is overly complex for its actual physical meaning, its derivation also seems unlikely to be complete.
In the previous edition of my book, I acknowledged a mathematical incompleteness in the derivation process and presented a method that filled this gap with physical intuition. Now, I have discovered a mathematically clean solution for that part and added it to the revised edition. I believe this is noteworthy as one of the book’s key achievements.
The part in my previous derivation that was mathematically incomplete, which I had resolved using physical intuition, concerned the definitions of ∇r and \(\nabla \times \vec{r}\). I have since found a complete derivation method for this section, added it to the book, and will introduce it here as well.
First, regarding the definition of ∇r, a somewhat plausible explanation was already known even before my work.
The diagram of ‘the definition of ∇r’ shows that if the information about the distance between Q and P is assumed to be a value of some physical quantity traveling from Q to P at the speed of light, then the gradient of that physical quantity with respect to distance is \(\frac{dy}{dx} = \frac{1}{1 \pm \frac{v}{c}}\). Replacing y with r and expressing this as a vector derivative in three dimensions yields \(\nabla r = \frac{1}{1 + \frac{\dot{r}}{c}} \frac{\vec{r}}{r} = \frac{\dot{r}’}{\dot{r}} \frac{\vec{r}}{r}\), providing justification for the value of ∇r previously obtained by conjecture. However, the case for \(\nabla \times \vec{r}\) is a bit more complex.
For \(\nabla \times \vec{r}\), the description must be based on \(\vec{v}’\) rather than \(\vec{v}\). At this point, it is not easy to depict the relationship between \(\vec{v}\) and \(\vec{v}’\) in the same diagram because \(\vec{v}’\), which is Q’s apparent velocity as observed from the ′ frame of P, can reach infinite. Therefore, this relationship must be understood through the algebraic methods derived earlier: \(\vec{v}’ = \frac{\vec{v}}{1 + \frac{\dot{r}}{c}}\) and \(\vec{v} = \frac{\vec{v}’}{1 – \frac{\dot{r}’}{c}}\), where \(-c < \dot{r} < c\) and \(-\infty < \dot{r}’ < \frac{c}{2}\).
The ‘definition of \(\nabla \times \vec{r}\)’ diagram defines ∇r and \(\nabla \times \vec{r}\) from a perspective looking perpendicularly at the r-v plane, given arbitrary \(\vec{r}\) and \(\vec{v}’\). Since vector expressions are inherently coordinate-independent, the result remains the same regardless of the chosen perspective for calculation. Therefore, one can freely select a conceptually easier perspective without issue. First, an explanation of the diagram will be given by demonstrating how to calculate ∇r using this diagram.
The familiar definition of ∇r is \(\nabla r = \tfrac{\partial r}{\partial x} \hat{x} + \tfrac{\partial r}{\partial y} \hat{y} + \tfrac{\partial r}{\partial z} \hat{z}\). In this diagram, the direction corresponding to the z-axis is not depicted, and \(\tfrac{\partial r}{\partial z} = 0\). The direction corresponding to the y-axis is the P+dr⊥ direction, and its magnitude is infinitesimal. Therefore, the distance information originating from Qt, which is Q’s position observed at time t from P, reaches P+dr⊥ and P simultaneously and is equal. Thus, \(\vec{r}_t + d \vec{r}_{\perp}\) is parallel to and equal to \(\vec{r}_t\), so \(\tfrac{\partial r}{\partial y} = \frac{\partial r}{\partial \vec{r}_{\perp}} = 0\). The direction corresponding to the x-axis is the P+dr∥ direction, and the magnitude corresponding to dx is \(d r_{\parallel} = cd t\). At this point, the information about Q observed from P+dr∥ is information from Qt-dt, as it passed through P at time t-dt, and is represented as \(\vec{r}_{t – d t} + d \vec{r}_{\parallel}\). Since this can also be considered parallel to \(\vec{r}\), the length of \(\vec{r}_{t – d t} + d \vec{r}_{\parallel}\) is \(| \vec{v}’_{\parallel} d t | + r + d r_{\parallel} = v_{\parallel}’ d t + r + cd t\). Therefore, \(d r = v_{\parallel}’ d t + r + cd t – r = v_{\parallel}’ d t + cd t\). Combining these, it can be confirmed that:
\[\begin{array}{lll} \nabla r & = & \tfrac{\partial r}{\partial x} \hat{x} + \tfrac{\partial r}{\partial y} \hat{y} + \tfrac{\partial r}{\partial z} \hat{z}\\ & = & \tfrac{\partial r}{\partial r_{\parallel}} \hat{r} + 0 + 0\\ & = & \frac{v_{\parallel}’ d t + cd t}{cd t} \hat{r}\\ & = & \left( 1 + \frac{v_{\parallel}’}{c} \right) \hat{r}\\ & = & \left( 1 – \frac{\dot{r}’}{c} \right) \hat{r} \end{array}\]
This has provided a mathematically sound derivation for \(\nabla r = \left( 1 – \frac{\dot{r}’}{c} \right) \hat{r} = \frac{1}{1 + \frac{\dot{r}}{c}} \hat{r}\), which was previously obtained by intuitive conjecture. Although ∇r could also be derived in a mathematically sound manner, using the ‘Definition of ∇r’ diagram, I derived it first to explain the meaning of the ‘Definition of \(\nabla \times \vec{r}\)’ diagram and to verify its content. Additionally, examining this spatial differentiation through a concrete diagram confirmed that spatial differentiation is defined by analyzing the space around point P. This provided some persuading as to why the reordering of differential operators functioned correctly only in the ′ frame, where physical quantities are defined based on observations from point P. Now, the actual purpose for devising this diagram, which is \(\nabla \times \vec{r}\), will be explored.
\newpage First, introducing the definition as in the explanation of ∇r:
\[ \nabla \times r = \det \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ r_x & r_y & r_z \end{vmatrix} = \left( \frac{\partial r_z}{\partial y} – \frac{\partial r_y}{\partial z} \right) \hat{x} + \left( \frac{\partial r_x}{\partial z} – \frac{\partial r_z}{\partial x} \right) \hat{y} + \left( \frac{\partial r_y}{\partial x} – \frac{\partial r_x}{\partial y} \right) \hat{z} \]
Rewriting this using the representation of the ‘Definition of \(\nabla \times \vec{r}\)’ diagram:
\[ \nabla \times r = \det \begin{vmatrix} \hat{r}_{\parallel} & \hat{r}_{\perp} & \hat{r}_{\parallel \times \perp} \\ \frac{\partial}{\partial r_{\parallel}} & \frac{\partial}{\partial r_{\perp}} & \frac{\partial}{\partial r_{\parallel \times \perp}} \\ r_{\parallel} & r_{\perp} & r_{\parallel \times \perp} \end{vmatrix} = \left( \frac{\partial r_{\parallel \times \perp}}{\partial r_{\perp}} – \frac{\partial r_{\perp}}{\partial r_{\parallel \times \perp}} \right) \hat{r}_{\parallel} + \left( \frac{\partial r_{\parallel}}{\partial r_{\parallel \times \perp}} – \frac{\partial r_{\parallel \times \perp}}{\partial r_{\parallel}} \right) \hat{r}_{\perp} + \left( \frac{\partial r_{\perp}}{\partial r_{\parallel}} – \frac{\partial r_{\parallel}}{\partial r_{\perp}} \right) \left( \hat{r}_{\parallel} \times \hat{r}_{\perp} \right) \]
Here, since this diagram is viewed from the \(\hat{r}_{\parallel} \times \hat{r}_{\perp}\) direction, perpendicular to the r-v plane, all terms in the numerator with \(\partial r_{\parallel \times \perp}\) are 0. Also, for terms where the denominator is \(\partial r_{\perp}\), the observation of Q is the same as position P, that is Qt, so there is no change in the r vector perpendicular to that direction. Therefore, all terms with \(\partial r_{\perp}\) in the denominator are also 0. The same applies to terms where the denominator is \(\partial r_{\parallel \times \perp}\). Consequently, it can be seen that:
\[\nabla \times r = (0 – 0) \hat{r}_{\parallel} + (0 – 0) \hat{r}_{\perp} + \left( \tfrac{\partial r_{\perp}}{\partial r_{\parallel}} – 0 \right) (\hat{r}_{\parallel} \times \hat{r}_{\perp}) = \tfrac{\partial r_{\perp}}{\partial r_{\parallel}} (\hat{r} \times \hat{r}_{\perp})\]
According to the ‘Definition of \(\nabla \times \vec{r}\)’ diagram, \(\tfrac{\partial r_{\perp}}{\partial r_{\parallel}}\) is the value obtained by dividing the r⊥ component of the difference between \(\vec{r}_{t – d t} + d \vec{r}_{\parallel}\) (viewing Qt-dt from point P+dr∥) and \(\vec{r}_t\) (viewing Qt from point P) by dr∥. Thus, it is known that \(\tfrac{\partial r_{\perp}}{\partial r_{\parallel}} = \frac{- v_{\perp}’ d t}{d r_{\parallel}}\). Therefore,
\[\nabla \times \vec{r} = \tfrac{\partial r_{\perp}}{\partial r_{\parallel}} (\hat{r} \times \hat{r}_{\perp}) = \frac{- v_{\perp}’ d t}{c d t} (\hat{r} \times \hat{r}_{\perp}) = \frac{\vec{v}’}{c} \times \hat{r}\]
This result is the same as \(\nabla \times \vec{r} = \frac{\vec{v}’}{c} \times \frac{\vec{r}}{r}\), which was previously obtained through physical intuition. With this, the mathematical incompleteness and physical ambiguity in the derivation of Feynman’s formula for the electromagnetic field produced by a moving charge have been completely resolved. Consequently, it has been confirmed that the Heaviside-Feynman formula and its practical form, as I modified, are not mere hypotheses but rather the inevitable result of Maxwell’s equations and the principle of the constancy of the speed of light.
Regarding the Wigner Deflection Force
I interpreted the effect of the Wigner rotation as a ‘Wigner deflection force’ that is directly included in the expression for the force, but I could not determine its exact form. There were two candidates, and since I could not decide which was the correct formula, I simply added the section and left it as an unsolved problem.
For this reason, I did not address gravitational time dilation that includes the effect of Wigner rotation. This is also because, for the practical purpose of calculating planetary motion, this term would represent a negligible difference, even compared to the other correction terms added in this book.
For those interested in this unresolved section, I encourage you to read that part of the book for yourselves. Here, I will limit myself to simply stating the fact that this section has been added.