On 15/11/2013 12:44 AM, David Roberson wrote:
...Since we knew the original distance to the star was 10 light years,
it suggests that we should reach it within 1 year our time at our
calculated velocity. Is this what should actually occur?
Yes this is correct and this is the essence of the twins paradox. There
is no limit to your accumulated velocity as measured by yourself using
your clocks that are slowing down the faster you go. If you were a
photon (or had no rest mass) you could travel the 10 light years in an
instant of your time but still take 10 years of observers time.
I realize that an observer located near the star and stationary to it
would determine our velocity as less than light speed and thus take
longer than 10 years to reach his location. Also, the observer would
detect that time passes slower on our ship due to our relative
velocity. We of course would see his time as passing slower by the
same factor during our high velocity trip.
Yes.
I also understand that we can measure the distance to the star once we
reach our stable velocity by using radar signals for example. The
signal would leave our ship at a velocity of light relative to us and
head toward the star which appears to be significantly closer to us by
Lorentz contraction. Our high specification radar beam would reach
the star and some would reflect back toward us. The frequency of the
reflected beam would be shifted by the velocity of the star relative
to our velocity and we could thus accurately calculate the star's
relative velocity which would be the same as the velocity the observer
sees us moving toward him.
That is correct. However for us to measure how fast our signal leaves
our ship, we need 2 clocks - Say one at the back of the ship where the
signal is launched from and one at the front of our ship to time how
long it takes to travel the length of the ship. The signal *only*
leaves our ship at the speed of light *if* we have taken care to
re-synchronise our clocks (using the so-called Einstein method) after
reaching a steady speed. If instead we kept the same synchronisation
that we had before we started to accelerate, then we would measure the
same speed that any previously stationary observer (remote or otherwise)
measures (ie the light pulse would travel *much* slower than c
travelling from the back of our ship towards the front, and *much*
faster than c in the reverse direction! (this is not well known and is a
surprise even to many physicists).
The important thing here is that once we have reset our clocks to be
synchronous in our new high speed inertial frame, (or once we consider
ourselves to be at rest), then all distances with respect to our new
coordinate system have changed. In particular the 10 light year remote
star, has now instantly (with the synchronism or the consideration that
we are stationary) become only 1 light year away. That is why it will
only take us one light year to reach it. Distances in the reverse
direction (places behind us) are likewise increased (instead of
decreased) simply by the change of inertial reference frame.
However if we consider ourselves using our initial clock
synchronisation, then we know our true accumulated speed because we can
see that the light pulse is only just travelling a bit faster than us
(it takes the pulse a very long time to travel from the back of the ship
to the front) and so we are travelling just a shade slower than c. Also
since any clock tick rate is given by an oscillation time, if we use the
round trip time of a light pulse travelling from the back of the ship,
to the front and back again, as our oscillation tick time, then we know
that our time is ticking a lot slower than it was before we
accelerated. If we divide the known distance (10 light years) by our
speed measured this way (~0.99c or thereabouts) then we know how many
ticks of our (slowed down) clock will happen in that distance - and it
will be 1 years worth. Since our clock seems to us to be ticking at its
normal rate, we will get there in what feels to us like a year.
The observer near the star has his own radar which he directs towards
us. He also determines the same relative velocities by measuring the
reflected signal from our ship, so everyone is in agreement that the
space between us is closing at a velocity that is somewhat less than
light speed.
Yes.
Since velocity is relative,
This is an assumption which cannot be proved. Given a peek out the
window at the CMBR, we could determine our velocity with respect to the
rest of the matter in the universe - which conceptually at least is an
absolute value.
...the observer near the star concludes that he is the one moving
rapidly toward us and we are stationary. From his perception the
Lorentz contraction of the distance between both parties is the same
as we located upon the high velocity ship calculated earlier. He
therefore determines he and the nearby star will close the gap in much
less than 10 years of time passing.
This is not correct. This observer has had no reason to re-synchronise
his clocks to match the inertial reference of the space ship (because he
has experienced no acceleration). So nothing changes for him. It will
still take 10 years for the spaceship to travel the distance because it
is still 10 light years away.
