Ritz's Equation

In 1908, Walther Ritz published Recherches critiques sur l'Électrodynamique générale,^[1] (See English translation^[2]) a lengthy criticism of Maxwell-Lorentz electromagnetic theory, in which he contended that the theory's connection with the luminiferous aether (see Lorentz ether theory) made it "essentially inappropriate to express the comprehensive laws for the propagation of electrodynamic actions." Ritz proposed a new equation, derived from the principles of the ballistic theory of electromagnetic waves. This equation relates the force between two charged particles with a radial separation r relative velocity v and relative acceleration a, where k is an undetermined parameter from the general form of Ampere's force law as proposed by Maxwell. This equation obeys Newton's third law and forms the basis of Ritz's electrodynamics.

$\mathbf{F} = \frac{q_1 q_2} {4\pi \epsilon_0 r^2} \left[\left[1+\frac{3-k}{4} \left(\frac{v}{c}\right)^2 - \frac{3(1-k)}{4} \left(\frac{\mathbf{v\cdot r}}{c^2}\right)^2 - \frac{r}{2c^2} (\mathbf{a\cdot r}) \right] \frac{\mathbf{r}}{r} - \frac{k+1}{2c^2} (\mathbf{v\cdot r})\mathbf{v} - \frac{r}{c^2} (\mathbf{a})\right]$

Derivation of Ritz's Equation

On the assumption of an emission theory, the force acting between two moving charges should depend on the density of the messenger particles emitted by the charges ($D$), the radial distance between the charges (ρ), the velocity of the emission relative to the receiver, ($U_x$ and $U_r$ for the x and r components, respectively), and the acceleration of the particles relative to each other ($a_x$). This gives us an equation of the form:^[3]

$F_x = eD \left[ A_1 cos(\rho x)+ B_1 \frac{U_x U_r} {c^2} + C_1 \frac{\rho a_x} {c^2} \right]$.

where the coefficients $A_1$, $B_1$ and $C_1$ are independent of the coordinate system and are functions of $u^2/c^2$ and $u^2_\rho /c^2$. The stationary coordinates of the observer relate to the moving frame of the charge as follows

$X+x(t') = X'+x'(t')-(t-t')v'_x$

Developing the terms in the force equation, we find that the density of particles is given by

$D \alpha \frac {dt' e' dS} {\rho^2} = - \frac {e' \partial \rho} {c \rho^2 \partial n} dS dn$

The tangent plane of the shell of emitted particles in the stationary coordinate is given by the Jacobian of the transformation from $X'$ to $X$:

$\frac{\partial \rho} {\partial n} = \frac{\partial (XYZ)} {\partial (X'Y'Z')} = \frac{a e'}{\rho^2} \left(1 +\frac{\rho a'_\rho}{c^2} \right)$

We can also develop expressions for the retarded radius $\rho$ and velocity $U_\rho <\rho>$ using Taylor series expansions

$\rho = r \left(1+\frac{ra'_r}{c^2}\right)^{1/2}$

$\rho_x = r_x + \frac{r^2 a'_x}{2c^2}$

$U_\rho = v_r - v'_r +\frac{ra'_r}{c}$

With this substitutions, we find that the force equation is now

$F_x = \frac{ee'}{r^2} \left(1+\frac{ra'_r}{c^2}\right) \left[A cos(rx) \left(1-\frac{3ra'_r}{2c^2}\right) + A\left(\frac{ra'_x}{2c^2}\right)-B\left(\frac{u_x u_r}{c^2}\right)-C\left(\frac{ra'_x}{c^2}\right)\right]$

Next we develop the series representations of the coefficients

$A = \alpha_0 + \alpha_1 \frac{u^2} {c^2} + \alpha_2 \frac{u^2_r}{c^2}+...$

$B = \beta_0 + \beta_1 \frac{u^2} {c^2} + \beta_2 \frac{u^2_r}{c^2}+...$

$C = \gamma_0 + \gamma_1 \frac{u^2} {c^2} + \gamma_2 \frac{u^2_r}{c^2}+...$

With these substitutions, the force equation is now

$F_x = \frac{ee'}{r^2} \left[\left(\alpha_0+\alpha_1 \frac{u_x^2}{c^2}+\alpha_2 \frac{u^2_r}{c^2}\right) cos(rx) - \beta_0 \frac{u_x u_r}{c^2}-\alpha_0 \frac{ra'_r}{2c^2} + \left(\frac{ra'_x}{2c^2}\right)(\alpha_0-2\gamma_0) \right]$

Since the equation must reduce to the Coulomb force law when the relative velocities are zero, we immediately know that $\alpha_0 = 1$. Furthermore, to obtain the correct expression for electromagnetic mass, we may deduce that $2\gamma_0 -1 = 1$ or $\gamma_0 =1$.

To determine the other coefficients, we consider the force on a linear circuit using Ritz's expression, and compare the terms with the general form of Ampere's law. The second derivative of Ritz's equation is

$d^2 F_x = \sum_{i,j}\frac{de_i de_j'}{r^2} \left[\left(1+\alpha_1 \frac{u_x^2}{c^2}+\alpha_2 \frac{u_r^2}{c^2}\right) cos(rx) - \beta_0 \frac{u_x u_r}{c^2}-\alpha_0 \frac{ra'_r}{2c^2} + \frac{ra'_x}{2c^2}\right]$

Diagram of elements of linear circuits

Consider the diagram on the right, and note that $dq v = I dl$,

$\sum_{i,j} de_i de_j' = 0$

$\sum_{i,j} de_i de_j' u^2_x = -2dq dq' w_x w'_x$

$= -2 I I' ds ds' cos\epsilon$

$\sum_{i,j} de_i de_j' u^2_r = -2dq dq' w_r w'_r$

$= -2 I I' ds ds' cos(rds)cos(rds)$

$\sum_{i,j} de_i de_j' u_x u_r = -dq dq' (w_x w'_r+w'_x w_r)$

$= -I I' ds ds' \left[cos(xds)cos(rds)+cos(rds)cos(xds')\right]$

$\sum_{i,j} de_i de_j' a'_r = 0$

$\sum_{i,j} de_i de_j' a'_x = 0$

Plugging these expressions into Ritz's equation, we obtain the following

$d^2 F_x = \frac{I I' ds ds'}{r^2} \left[\left[2\alpha_1 cos\epsilon + 2\alpha_2 cos(rds) cos(rds')\right]cos(rx) - \beta_0 cos(rds')cos(xds)-\beta_0 cos(rds)cos(xds')\right]$

Comparing to the original expression for Ampere's force law

$d^2 F_x = - \frac{I I' ds ds'}{2r^2} \left[\left[(3-k)cos\epsilon - 3(1-k) cos(rds) cos(rds')\right]cos(rx)-(1+k)cos(rds')cos(xds)-(1+k)cos(rds)cos(xds')\right]$

we obtain the coefficients in Ritz's equation

$\alpha_1 = \frac{3-k}{4}$

$\alpha_2 = -\frac{3(1-k)}{4}$

$\beta_0 = \frac{1+k}{2}$

From this we obtain the full expression of Ritz's electrodynamic equation with one unknown

$\mathbf{F} = \frac{q_1 q_2} {4\pi \epsilon_0 r^2} \left[\left[1+\frac{3-k}{4} \left(\frac{v}{c}\right)^2 - \frac{3(1-k)}{4} \left(\frac{\mathbf{v\cdot r}}{c^2}\right)^2 - \frac{r}{2c^2} (\mathbf{a\cdot r}) \right] \frac{\mathbf{r}}{r} - \frac{k+1}{2c^2} (\mathbf{v\cdot r})\mathbf{v} - \frac{r}{c^2} (\mathbf{a})\right]$

In a footnote at the end of Ritz's section on Gravitation (English translation) the editor says, "Ritz used k = 6.4 to reconcile his formula (to calculate the angle of advancement of perihelion of planets per century) with the observed anomaly for Mercury (41") however recent data give 43.1", which leads to k = 7. Substituting this result into Ritz’s formula yields exactly the general relativity formula." Using this same integer value for k in Ritz's electrodynamic equation we get:

$\mathbf{F} = \frac{q_1 q_2} {4\pi \epsilon_0 r^2} \left[\left[1 - \left(\frac{v}{c}\right)^2 + 4.5 \left(\frac{\mathbf{v\cdot r}}{c^2}\right)^2 - \frac{r}{2c^2} (\mathbf{a\cdot r}) \right] \frac{\mathbf{r}}{r} - \frac{4}{c^2} (\mathbf{v\cdot r})\mathbf{v} - \frac{r}{c^2} (\mathbf{a})\right]$

References and notes

1. Ritz, Walther (1908), Annales de Chimie et de Physique, Vol. 13, pp. 145-275
2. Critical Researches on General Electrodynamics, Introduction and First Part (1980) Robert Fritzius, editor; Second Part (2005) Yefim Bakman, Editor.
3. O'Rahilly, Alfred (1965). Electromagnetic Theory. Dover Books. pp. 503–509.