It is said that a speeding clock shows less elapsed time than
the stay-at-home clock, because (if already speeding) it is
running at a slower rate. Or else, because it had run at a
slower rate when it was speeding, assuming now it is
stopped near the stay-at-home clock.

The funny thing about this is that time and length change
at the same time, but not the ratio between both (velocity).

If we keep length constant, the only possible solution is
uniform circular motion. That is a twin travelling in circles,
of constant radius r, around the first twin assumed to be
stopped at the center of rotation.

Let's say the radius r is a constant value of 100 light-seconds
(r = 100c).
The speeding twin goes on a spaceship at 0.999c.
Therefore, the angular speed 'omega' is v/r = 0.999c/100c
= 0.01 rad/s.
The speeding twin takes 628 seconds to have a complete turn
of 360 degrees.

For small values of t, the speeding twin is almost going in
a straight line, but is fact it has a centripetal force f = m c^2/r
= m c/100, being the centripetal acceleration c/100, towards
the first stopped twin in the center of rotation.

Both twins have powerful antennas that broadcast radio
spherically around the entire space. Both twins are tuned
to each other frequency/radio-station.

Since the distance r = 100c between the emitter and the
receptor is constant, obviously that both twin will hear
each other radio (music) in perfect conditions.

Nevertheless, relativity says that the clock synchronising
the emission of the speeding twin must be running at
a clock rate close to zero. Theoretically, the speeding
twin won't have any trouble receiving the stay-at-home
radio emission, but the stay-at-home twin cannot
receive the speeding twin radio emission, because
the speeding clock is running near zero.
The speeding twin radio emission will take infinite
time to broadcast one single spoken word. The
stay-at-home will be dead by the time the speeding
twin could say a single word.

The trouble seams to be the acceleration:
a = (0.999)^2 c/100 which is about c/100.
(Here the number 100 means 100 seconds).
That's a huge gravity field of 300,000g at
a radius of 100 light-seconds, just imagine
the value it will be at Earth radius based
on the inverse-square Law.)

I presume that such acceleration of 300,000g will
be responsible for a clock speed up rate that
should keep time unchanged after all.

Any comments welcome.