When Ampere Meets Doppler

It has often been claimed that Ampere's force is equivalent to Grassmann's force, which is a precursor of the Lorentz force, when the force is integrated round a circuit, in which the current element forms a part, vanishes. This is taken to mean that a force exerted by a complex circuit element is a right angles to the element. As such it was taken to be equivalent to the earlier discovered Biot-Savart law. The Biot-Savart law,

$$\vec{B}=\frac{\mu}{4\pi r^2}(\vec{v}\wedge\hat{r}),$$

is just the definition of the magnetic field. Obviously, it does not act along the unit vector $\hat{r}$ from one charge to the other. Neither does the Grassmann force,

$$\vec{F}^G=\frac{\mu}{4\pi r^2}[\vec{v}'\wedge(\vec{v}\wedge\hat{r})],$$

which acts also in the direction $\vec{v}$, from the charge $e$ to the charge $e'$.

J J Thomson, and other illuminaries of late nineteenth century could not digest Lorentz's law because it violated Newton's law of action-and-reaction, i.e, something is being picked-up by its own bootstraps. Now Ampere's law acts along the line connecting the two circuit elements, which Grassmann's law has a component in the direction of one of the two circuit elements which is acting on the other. Hence, they do not in general point in the same direction. How then can they be considered equivalent in any respect?

Moon and Spencer [J Franklin Inst. 257 (1954) 305] go even farther and claim that Grassmann's force "is not derivable from the force between charges. Thus the equation is untenable, and the whole process of defining a magnetic field from this equation is unsound. We wouldn't go so far as to agree with them, because it is backed-up experimentally, but it is interesting to note that in the same paper, these authors claim "Einstein's famous paper of 1905,$\ldots$, which enunciated the theory of special relativity, was an attempt to remedy this non-relativistic aspect of electromagnetic theory. Looking back from the vantage point of present knowledge [1954], we may wonder if Einstein's work was not directed up a blind alley." This statement we find more sympathy for, but wonder how it could have come from a tenured professor at MIT?

Ampere's law, which was an addendum to the static Coulomb law, depends on the acceleration as well as the relative velocities of the two current elements. Shouldn't it, therefore, be amenable to the longitudinal Doppler law? 

The Weber force acting from charge $e$ to $e'$ along $r$ is

$$\vec{F}^W=\hat{r}\frac{ee'}{4\pi\epsilon r^2}\left\{1+\frac{1}{c^2}\left(\vec{v}\cdot\vec{v}-\frac{3}{2}(\hat{r}\cdot\vec{v})^2+\vec{r}\cdot\vec{a}\right)\right\},$$

where $c=1/\surd(\epsilon\mu)$, which is Maxwell's constant, not necessarily coinciding with the speed of light.

If $\theta$ is the angle between $\hat{r}$ and $\vec{v}$ as shown in the figure below, then if $e$ is moving at constant velocity $\vec{v}$ with respect to $e'$, the Moon and Spencer claim that the Ampere force resulting from this motion is

$$\vec{F}'=\hat{r}\frac{ee'}{4\pi\epsilon r^2}\beta^2\left(1-\frac{3}{2}\cos^2\theta\right),$$

The force acting on charge $e'$ due to $e$ at a distance $r$ moving with uniform velocity.
The force acting on charge $e'$ by a charge $e$ moving with velocity $\vec{v}$ and acceleration $d\vec{v}/dt$.

where $\beta=v/c$, for $d\vec{v}/dt=\vec{a}=0$.If the motion is in the direction toward $e'$, the combination of the force with Coulomb's force is

$$F'=\hat{r}\frac{ee'}{4\pi r^2}\left(1-\frac{1}{2}\beta^2\right),$$

which looks like an approximation to $\surd(1-\beta^2)$. But, that makes no sense since the force is acting longitudinally, and $\surd(1-\beta^2)$ would be a second-order Doppler shift in the direction normal to the motion. So it would appear that Ampere is at odds with Doppler!

From an historical perspective, Biot and Savart experimented with forces between currents and magnets. Ampere was the first to experiment on forces between two current elements $ids$ and $i'ds'$. He first came up with the force

$$F^A=\frac{ids\cdot i'ds'}{r^2}\left\{\sin\theta\sin\theta'\cos\eta+k\cos\theta\cos\theta'\right\},$$

somewhere around 1820, but he was not able to determine the unknown constant $k$ until 1823 when he set it equal to $-1/2$. $\theta$ and $\theta'$ are the angles that the velocities of the two current elements make with $\hat{r}$, and $\eta$ is the angle between the two planes that the current elements find themselves in. 

Now, in terms of the radial velocity $\dot{\vec{r}}$ and radial acceleration $\ddot{\vec{r}}$, the combined Coulomb and Ampere laws is

$$F^W=\frac{ee'}{r^2}\left\{1+\frac{1}{c^2}\left(k\dot{r}^2+r\ddot{r}\right)\right\},$$

in the radial direction uniting the two charges. In terms of  the relative velocity, $\vec{v}$ and relative accelerations $\vec{a}$, the force acting on $e'$ has magnitude

$$F^W=\frac{ee'}{r^2}\left\{1+\frac{1}{c^2}\left(\vec{v}\cdot\vec{v}+(k-1)(\hat{r}\cdot\vec{v})^2+\vec{r}\cdot\vec{a}\right)\right\}.$$

Now, having traced the origin of $k$ this far, insteading of choosing $k=-1/2$, we set $k=-1$, we get the numerical coefficient in the numerator of the Moon and Spencer expression as $(1-2\cos^2\theta)$, so that it becomes

$$F=\frac{ee'}{r^2}\left(1-\beta^2\right),$$

when combined with Coulomb's law and the motion is along $\hat{r}$ so that $\theta=0$. Now things are looking a bit brighter for if we replace $r$ by its retarded value $r'$, according to $r=(1-\hat{r}\cdot\vec{\beta})r'$, we get

$$\vec{F}=\hat{r}\frac{ee'}{r'^2}\frac{1-\beta^2}{(1-\beta)^2}=\frac{ee'}{r'^2}\left(\frac{1+\beta}{1-\beta}\right).$$

This is none other than (the square) of the longitudinal Doppler shift! This says that the Weber force is the result of a Doppler shift of the Coulomb force when Ampere's constant is chosen $k=-1$.

This is corroborated by the Lienard-Wiechert expression for the electric field, found in all texts on electrodynamics,

$$\vec{E}=e\frac{(\vec{r}'-r'\vec{\beta})(1-\beta^2)}{(r'-\vec{r}'\cdot\vec{\beta})^3},$$

when $\vec{\beta}\parallel\vec{r}'$. When they are not parallel to one another we have aberration; it's as simple as that.

This brings to mind the conclusion arrived at by Ibison, Puthoff, and Little in the paper "The speed of gravity revisited" in their attempt to refute Van Flandern that gravity propagates much faster than the speed of light, by explaining "the origin of apparently instantaneous connections, first within EM, and then within strong-field GR."

For EM, they express the electric field as

$$\vec{E}=e\frac{\hat{r}}{r^2}\times\left\{\frac{1-\beta^2}{K^3}\right\},$$

where $\theta$ is the angle between $\vec{r}$ and $\vec{v}$ and $K=\surd(1-\beta^2\sin^2\theta)$. Into this expression we can use their relation

$$r'/r=K/(1-\beta\cos\theta'),$$

to obtain

$$\vec{E}=e\frac{\hat{r}}{(r'-\vec{r}'\cdot\vec{\beta})^2}\frac{1-\beta^2}{K}=\vec{E}'\frac{c_0}{c},$$

where $c_0$ is one-way speed of light. This would have been the ratio of frequencies shifted by a Doppler effect had considered only one way propagation. In terms of the Kennedy Thorndike interferometer, the time taken to run along an optical path of length $\ell$ is

$$t_1=\frac{\ell}{c}\left\{\frac{K+\beta\cos\theta}{1-\beta^2}\right\}.$$

On the return leg, the only thing that changes is the sign of $\beta$ so that the time to complete both legs is

$$\frac{2\ell}{c}\frac{K}{1-\beta^2}.$$

But, this time should be equal to the ratio of the distance covered, $2\ell$, to the speed, $c_0$. This speed will equal $c$ only when $\beta=0$. The factor in the braces of the time for the outward journey is Doppler shift, and since the electric field cannot depend on the sign of $\beta$, it is suppressed in the Ibison et al. expression, just like considering the round-trip in the Kennedy-Thorndike interferometer. In other words, the factor $\gamma^2K$, where $\gamma=1/\surd(1-\beta^2)$, can be considered as the two-way Doppler shift, obtained by reflecting the light beam by a mirror placed at a distance $\ell$ from the source.

 

The fact that electric field is pointing in the present direction was loaded from the start,

$$\vec{r}=\vec{r}'-r'\vec{\beta},$$

so that should raise no eyebrows. But listen to their conclusion: "This result proves the claim that the electric field from a uniformly moving source is not aberrated. The force on a test charge is directed towards the instantaneous--not the retarded--position of the source. The factor in braces affects the magnitude, but not the direction, of the field." How can it when it was put in from the very start?! Moreover, the factor 

$$\frac{c_0}{c}=\frac{1-\beta^2}{K(\theta)},$$

is the Doppler shift for "there and back". Moreover, it can be generalized to aberration from the relation

$$\cos\theta=\frac{K(\cos\theta'-\beta)}{1-\beta\cos\theta'},$$

when $\cos\theta'\neq\beta$, as in the Einstein convention.  

We can even generalize further. Any relation between past and present space coordinates, $r'$ and $r$, past and present times, $t'$ and $t$, or frequencies, or time and space increments, are related by the same Doppler shifts. This means that the Einstein metric,

$$ds^2=g_{00}dt^2+g_{ij}dx^idx^j,$$

has only meaning for $ds^2=0$, along a null, or  light, cone. As such it cannot accommodate accelerations since it would no longer have a quadratic form. As we have always said, accelerations have nothing to do with Doppler shifts and aberration. When included they would raise the metric to a cubic!

This brings to mind Carlip's claims, in his manuscript "Aberration and the speed of gravity", where he tries to extend the preceding electromagnetic argument to general relativity. According to him, "we cannot simply require by fiat that a massive source accelerate. The Einstein field equations are consistent only when all gravitational sources move along trajectories determined by their equations of motion, and in particular, we can consistently represent an accelerated source only if we include the energy responsible for its acceleration." Masses, according to Einstein's equations move along geodesics--trajectories of constant velocity. Energy can't be localized in general relativity, so it is an enigma how energy is to added to make a mass accelerated. Newton had the answer to that! Admitting to the fact that a "test particle in spacetime will travel along a geodesic, $\ldots$, the 'acceleration' of such a particle, in Newtonian language, is determined by the connection $-\Gamma^i_{00}$", if the particle was initially at rest. How can this be if the "test" particle follows a geodesic?

The answer, Carlip contends, is to incorporate 'acceleration", $Gm/r^2$, in the $g_{00}$ term. However, this is inconsistent with the structure of the metric, as we now show. Neglecting terms quadratic in the acceleration term, $\alpha dt$, where $\alpha=a/c$, the KT metric can be written as

$$c^2dt^2(1-\beta^2)=dr^2+(2\beta+\alpha dt)\cos\theta drdt.$$

Completing the square, we find that to lowest order

$$\frac{ds}{dt}=\frac{1-\beta^2}{\sqrt{K^2+\alpha\beta\cos\theta dt}+\beta\cos\theta}.$$

But, this makes no sense since the rhs depends on the time interval $dt$ on account of the fact that the metric contained a cubic term, $dt^2dr$. Geodesics, metrics, Doppler, and aberration abhor terms that depend on the acceleration. The key to incorporating accelerations is found in Ampere's law, and it will be the subject of our next blog.

However, the point to emphasize is that space $ds$ and $dt$ behave in the same way as past and present times $t'$ and $t$, past and present positions, $r'$ and $r$, and Coulomb and Ampere forces for $k=-1$. They are all related through Doppler shifts caused by the inclusion of velocities.