G5 Different Speeds of Light ?!
He who can measure lengths and times can also measure speed. Let’s consider how to transform the measurements of an observer at rest at a distance r from the center of our spherical mass into those of an observer at the OFF position.
As in STR we must distinguish whether a speed is observed in the x-direction, i.e., along the field lines (and thus parallel to the velocity of our free falling laboratory), or in the y-direction, i.e., perpendicular to the field lines. The measured time intervals are transformed independent of the direction, while distances are dependent. With the notation vy (r;r) we denote the velocity of an object at distance r from the center of M in the y-direction, as it is measured from a laboratory at rest at position r. vy (r;∞) denotes the velocity, that an observer at the OFF position attaches to the same object with the same motion. We calculate:
vy(r;r) = ∆y(r)/∆t(r) = ∆y(∞) / (∆t(∞)•√(1 - 2•α/r)) = (∆y(∞)/∆t(∞)) / √(1 - 2•α/r ) = vy(r;∞) / √(1 - 2•α/r) mm and thus
vy(r;∞) = vy(r;r) • √(1 - 2•α/r) = vy(r;r) • √(1 - Rs/r) = vy(r;r) • √(1 + 2• Φ(r) / c2)
It is exactly as in STR: A lateral velocity uy' measured locally in the red frame presents itself in the black frame slowed by our radical term: uy = uy' • √ mm (see D5)
We use the approximations of G3 for √ (1-2 • α / r) to obtain the simpler expression
vy(r;∞) ≈ vy(r;r) • (1 - α/r) = vy(r;r) • (1 + Φ(r) / c2)
Applying the same for velocities in the x-direction produces a different result:
vx(r;r) = ∆x(r)/∆t(r) = (∆x(∞)/√(1-Rs/r)) / (∆t(∞)•√(1-Rs/r)) = (∆y(∞)/∆t(∞)) / (1-Rs/r) = vx(r;∞) / (1-Rs/r) mm and thus
vx(r;∞) = vx(r;r)•(1 - Rs/r) = vx(r;r)•(1 - 2•α/r) = vx(r;r)•(1 + 2•Φ(r)/c2)
Here our radical term is squared and thus the root disappears! In this case we do not need the approximations.
Elementary, but somewhat cumbersome calculations [27-135f] result in an approximation which is good for all directions. Let δ denote the angle between v and the field lines:
v(r;∞;δ) ≈ v(r;r;δ) • (1 – α • (1 + cos2(δ)) / r)
The term 1 + cos2(δ) is equal to one, when δ = 90 º, that is, when the motion is in the y-direction, and it is equal to 2 when δ = 0 ° and δ = 180 º, that is, when it is in the x-direction. We could have actually guessed this formula ...
Thus, as seen from a distance, speeds slow in the vicinity of large masses. That is not surprising, since time itself seems to slow down. But now the hammer: This holds for the speed of light, too !!
If we measure the speed of light in a vacuum in any direction in a laboratory resting at position r, we obtain the default value of c(r;r;δ) = c0 ≈ 3 • 108 m / s. From the viewpoint of an observer at position OFF, however, we obtained this value with our possibly-shortened yardsticks and our certainly-slowed clocks. Thus for an observer from OFF, the light at the location r in the gravitational field of mass M must spread out with the slower and direction-dependent speed c, as follows:
c(r, ∞, δ) ≈ c(r, r, δ) • (1 - α • (1 + cos2(δ)) / r) = c0 • (1 - α • (1 + cos2(δ)) / r) !!
This means that for an observer at OFF c0 represents only the upper limit of the speed of light for gravity-free space. In the vicinity of masses, light (as observed from OFF) is slower. Meanwhile, local observers always measure at their position the known value of c0. Locally the world is everywhere Lorentzian ...
We make a note of the speed which light has in the vicinity of large masses for an observer in the OFF position:
Gravity acts (observed from a distance) on light like an index of refraction! The stronger the field is the slower light advances. The index of refraction depends not only on the position r, but also on the direction ∂. If the index of refraction changes then light no longer spreads in a straight line – except when the light beam runs exactly along a field line from or towards the center of M. In all other cases, the phenomenon of refraction occurs. We are now prepared for the calculation of the experimental result which made Einstein a celebrity of the first rank over night: the deflection of light at the solar limb, which can be photographed only during a solar eclipse (I2).
All velocities indeed transform themselves according to our formulas above, we need only replace c0 by the locally measured velocity v(r;r). This means that the quarter of a satellite, which is closest to the earth moves slightly slower than the part that lies further away! If, in a row of four people, the ones on the right are always a bit faster than those on the left, then the direction in which they march will gradually turn to the left. Instead of the free falling (or flying) satellite moving ‘straight’, it will therefore run in an orbit around the earth! In just this way Einstein explains the effect of gravity – without using the concept of force. The ‘distortion’ of the metric of space and of space-time itself produce “inertial trajectories” just as we know them from Newtonian mechanics.
If we describe everything from the perspective of the observer in OFF, then we can do away with ‘shortened yardsticks’ and ‘slow clocks’ and infer all phenomena just from this refraction in the vicinity of masses. In fact, however, we are sitting in the midst of such gravitational fields, and our atomic clocks today show directly the effects demanded by Einstein (I5, I6 and I7). It would be in any case somewhat elitist or aloof, just to deal with the world only from the OFF position ...
Epstein devotes four pages to the theme of "gravitation by refraction" [15-160ff] - recommended to the interested reader. A very nice relevant illustration can be found at the end of the next section.
We can now perhaps even understand the pithy comment of Misner, Thorne and Wheeler which summarizes general relativity theory in an unsurpassed manner:
“Matter tells space how to curve, and space tells matter how to move.” ‘The phone book’ [29-0005]
With 'space' designating 4D space-time, of course ...