Are Space Contraction and Time Dilation for Real?

We know that light aberrates due to its finite speed of propagation so that if gravity propagates at a finite speed it too must aberrate. Conventional wisdom claims that there is an apparent cancellation of retarded effects. This is nonsense.

Reception of a radiating source at the past position s' and its present position, s.

The usual example given is a radiating electromagnetic source moving at constant velocity $\vec{v}$. The electric field created by a moving charge $q$ is


where $\vec{s}'$ is the vector from the past position to the receiver, $\vec{s}$ that of the present position, and $\vec{\beta}=\vec{v}/c$. Written in this form, it will be apparent that the observer sees an electric field is pointing from the present position. To see this take the inner product of the numerator with $\vec{s}'$ to obtain






and the story is over. There is nothing to show, and there is no cancellation of retarded effects. But does that mean there is no aberration? Absolutely not!

By computing $s'^2=(\vec{s}+s'\vec{\beta})^2$, one finds


and introducing this into


one gets


What was not done was to equate the two expressions for the ratio $s'/s$. For then they would have obtained


For $K=1$ this reduces to the usual expression for aberration [cf. (8.3.9) in my book A New Perspective on Relativity.] So aberration is present; it is just that the peculiar form of the electric field used does not pick it up.

Now, the relativists in general, initiated by Einstein in particular choose a particular form of the aberration formula which does not involve $K$ which turns out to be the contraction factor $\surd(1-\beta^2)$. They set the numerator equal to zero which implies $\beta=\cos\omega'$ and $\omega=\pi/2$, a right angle!

Using Pythagoras' theorem $s$ undergoes a contraction from its past position by an amount $s=s'\surd(1-\beta^2)$, and the time difference, $t-t'$ becomes dilated by an amount $s/c\surd(1-\beta^2)$. These effects can hardly be considered as general results since they depend on a specific configuration, a radiating source that is traveling perpendicular to the observer at the receiving end.

Since we have come this far, we can obtain some general results that go beyond what has been called special relativity. The law of sines is


which looks like Snell's only--except for one particular. If the wavelengths $\lambda$ and $\lambda'$ are proportional to $s$ and $s'$, respectively, we would have


and not


as Snell's law requires. Obviously not, since we are dealing with a Doppler shift, and not a change in wavelength as light passes from one medium to another with a different index of refraction.

Letting $\ell=s\sin\omega$, and $\ell'=s'\sin\omega'$, Doppler's principle requires $\ell=\ell'$, while Snell's law requires $s=s'$, i.e., no contraction. If we take a cone and cut it obliquely, the eccentricity is $e=\cos u$. The ratio of the semi-minor, $b$ to semi-major, $a$, axes of the ellipse is $\sin u=b/a$. $b$ can be thought of as the radius of the corresponding circle. Then if we have two ellipses with semi-minor axes, $b$ and $b'$ and semi-major axes $a$ and $a'$, the condition that they correspond to the same circle, $b=b'$, but with different inclinations, $u$ and $u'$, is 

$$a\sin u=a'\sin u'.$$           

Finally, by differentiating the generalized formula for aberration with respect to $t$, and defining the frequencies as $d\omega/dt=\nu$ and $d\omega'/dt=\nu'$, Doppler's principle can be expressed as


For $K=1$, purely rectilinear motion, there is no change in the speed of light.  For radiation normal to the direction of propagation, $K=\surd(1-\beta^2)$, and there is contraction. Yet space contraction and time dilation cannot be considered as  general principles since they depends upon a very special configuration that results by setting $\cos\omega'=\beta$.