Half of Special Relativity Is Not Needed!

“Apparent c/D” completely removes length contraction and space curvature❗❗

On several occasions we discussed real meaning of c = 299,792,458 m/sec, which is just a definition of local time unit:

• SECOND is the time interval during which a light beam travels 299,792,458 meters.
• Or to make it more local to an area: NANOSECOND is the time interval during which a light beam travels about 30 cm (about a foot).

c is the local speed of light, and there could be areas with different speeds of time, like in this drawing:

While 1 second passes in “normal time” (on the left side), only half a second passes in “twice slower time” (on the right side) and the beam of light on the right side travels half the distance of the beam on the left side. From the “normal time” perspective, the speed of the light on the right side is c/2, or c/D, with D = 2 as the time dilation factor in the pink area relative to the time in the blue area. That is where “apparent” speed of light c/D comes from (vs. “local” speed of light, which is always c). How does c/D concur with c? When an observer is local, there is no time dilation, and no time dilation means D = 1, therefore, apparent speed of light is c/D = c when observed locally. Now, an interesting question arises: “Can apparent speed of light be greater than c?” The answer lies in the same drawing:

Blue-area time is twice faster than pink-area time, thus, D = 1/2 in the blue area from the pink-area perspective, and a light beam in the blue area covers 299,792,458 m while half a second passes in the pink area, thus “apparent” (for a pink-area observer) speed of light in the blue area is c/(1/2) = c/D = 2c. Formula c/D remains, but D < 1 now.

And since apparent speed limit can be above c, like 2c or higher, that raises another valid question: “Why have we not heard about objects moving at a speed higher than c?” Several reasons for that are:

1) When we observe distant parts of the Universe, which we see as they were in the past (it takes time for the light from there to reach the Earth), time in that past/younger Universe runs slower than time runs for us now, thus, D > 1 for observed remote regions of the Universe. That is because time in the Universe speeds up (which is the reason behind the Cosmological redshift, check chapter 1 and chapter 11 in Time Matters). Therefore, “apparent” speed limit for objects observed in the younger Universe parts is c/D ≤ c. And we do not know areas in the Universe, where time is faster than our time. That is because time in Universe accelerates, the later in time the faster it is, and the latest (and, therefore, fastest) time is here and now.

2) Time dilation causes refraction (literally by Snell’s law, check chapter 1). Such refraction messes up assessment of an object’s location and its velocity.

3) Objects with extremely high velocities can be observed only if they move almost directly toward us (as explained in chapter 61) and not across the sky. That complicates determining their speed (determining Doppler blueshift is not enough, because it shows v < c always).

We explained how “apparent” speed of light c/D concurs with the “local” speed of light constant c. Let’s show now that c/D rejects “length contraction” of moving objects. Einstein’s justification for length contraction was based on “apparent” speed of light being the same as c. But with “apparent” speed of light c/D we don’t need contraction:

• “Apparent” distance covered by a light beam is (c/D)×t, where t is time elapsed for an external observer;
• “Local” distance covered by the same beam is c×(t/D), because only t/D seconds passed in D-times-slower time.

No disparity between “apparent” and “local” distances, thus, no length contraction is needed! (The same applies to an object moving at local speed v: its apparent to an external observer speed is v/D, check chapter 83).

Having apparent speed of light c/D and no length contraction, we need to confirm (or reject) a well–known formula for relativistic time dilation:

Having apparent speed of light c/D, in chapter 92 we derived formula for gravitational Potential = 0.5×c²/D², which solved major problems in physics of gravity, removed space curvature, and even explained UFO gravitational propulsion (check chapter 84). Gravitational potential 0.5×c²/D² means that gravitational acceleration g = – (0.5×c²/D²)', where apostrophe stands for derivative by location, often denoted as gradient ∇:

g = – (0.5×c²/D²)' = – ∇ (0.5×c²/D²).

Let’s consider an object that starts at speed of zero and uniformly (at a constant acceleration g) accelerates to speed v in t seconds. Acceleration, by definition, is change in speed per second: g = v/t. By the formula for uniform acceleration S(t) = S₀ + v₀t + gt²/2 (with initial S₀=0, v₀=0), distance covered by this object in t seconds is L = g×t²/2 = (v/t)×t²/2 = v×t/2. Integral of g by location (in this case by variable x from zero to L) is:

(We used the fact that g is constant). And since g = – (0.5×c²/D²)', this integral is the same as

Initial time dilation D(0) = 1, and we are focusing on the final time dilation D(L). Let’s find the value of D(L), denoting it as D (intermediate D(x) values are not interesting to us):

Since all integrals above are the same, we have:

We got the same formula for time dilation D as Einstein’s Lorentz factor

But this time we ditched the length contraction (by the same Lorentz factor).

Time dilation and local speed of light constant c stay, but length and space metric manipulations go away.

P.S. With acceleration g > 0, D value increases; with deceleration g <0, D value decreases; with g = 0 (without acceleration or deceleration), D value does not change. That is enough to explain Twin Paradox, since acceleration and deceleration are needed for twins to meet eventually and compare their clocks.