[paul cotter (OP)]

Following on an error I made on another topic this question arose. Supposing one had a space vehicle with unlimited power and one wished to see the earth disappear from view at the fastest possible rate, what would be the optimum speed? Go too fast and the earth might never disappear from view as Halc has kindly explained in the other topic. I may or may not  be able to calculate this myself. Any ideas?

[alancalverd]

Presumably you would have to depart at  v > c and maintain your speed for (longer than the object has existed) x c/v. However it is not clear how you could achieve the first condition since you could not accelerate to v > c if you could see the object in the first instance! 

By adding "unlimited power" to your specification, you have fallen into the immovable object/unstoppable force black hole of philosophy. Fortunately physics tells us that you can't do that because the only way you could generate unlimited power would be to convert all the mass of the universe to energy, so the question resolves to which would disintegrate first - the planet or your ship? Either way, 'twould be a glorious finale.

[paul cotter (OP)]

Okay, drop the unlimited power as it is not needed anyway. Obviously I cannot reach c even with continuous acceleration. Going slow the earth will recede from view slowly, going faster it will recede faster but if I go too fast the earth will remain in view as all distances have been shortened by the Lorentz factor. So what is the optimum speed to make my view of the earth disappear the fastest?

[Halc]

Go too fast and the earth might never disappear from view as Halc has kindly explained in the other topic.

Maybe you should quote what I said instead of making up something that I didn't.

Presumably you would have to depart at  v > c and maintain your speed

Positing impossibilities opens things up for making up any fiction you want.  No amount of power will do this, and you know that.
If you balk at unlimited power, just make yourself very small. It takes terribly little energy to make a neutrino go fast.

My solution?  It can be done in under a minute.  Say you wanted to visit this cool place where two massive galaxy clusters are colliding (about 2.4 GLY away).  You embark on a circular path that takes you there and back to where you started.  If you do that fast enough, you'll arrive back home after less than a minute of your time, and Earth will have disappeared from view, but only very near the end of your trip.

OK, there are easier ways to do it.  Take your ship to the far side of the moon.  Or for that matter, close your eyes. Both will cause Earth to disappear from view.

[Bored chemist]

If you crank up a big enough engine, and point the exhaust at the earth, the earth will disappear from view.

[paul cotter (OP)]

Hi Halc, I apologise if you feel I "put words in your mouth" . In the example in the other thread where the spacecraft was travelling at 2c relative to the tape measure you said the earth would be hardly receding at all and that it would not reach 1 light year with unlimited time of travel of the craft. To me that suggests at these speeds and above the earth would remain distant but cease to recede significantly. So there must be a speed at which the earth appears to recede the fastest as viewed from the craft and above or below that speed the recession rate will appear less. Circling around to come back >2 billion years in the future is not what i'm looking for. Linear travel and apparent maximum recession rate is what interests me. 

[Halc]

I read-read that and I did come across kind of snarky. Apologies for that.

In the example in the other thread where the spacecraft was travelling at 2c relative to the tape measure you said the earth would be hardly receding at all and that it would not reach 1 light year with unlimited time of travel of the craft.

Right, but it has nothing to do with speed, but rather continuous proper acceleration.  The greater the proper acceleration, the closer the event horizon is to the ship, and Earth will never fall through the event horizon, but will instead remain close (in the accelerating frame of the ship) and forever visible, just like a lantern dropped into a black hole.
Of course quantum mechanics says that at some point the last photon to reach you will leave Earth, so looking at it from a quantum viewpoint, Earth will remain visible only until that last photon reaches you, but the 'never disappear' statement presumes continuous light, not quantum.

So there must be a speed at which the earth appears to recede the fastest

No, there's no such speed, especially since ship speed reaches any arbitrary value regardless of rate of acceleration. First of all, the appearance of velocity is dependent on one's choice of frame type.  Secondly, for all constant levels of acceleration and relative to said accelerating frame, Earth will 'appear' to drop away slowly at first, pick up speed as the ship velocity relative to Earth grows, and then slow as Earth approaches the ship's event horizon.  It will never appear to cross that horizon.

For linear non-accelerating travel, that is limited to c, and in flat spacetime, Earth will always be visible.  In real spacetime, at any ship speed, the ship will eventually cross Earth's event horizon, but both Earth and ship will always be able to see each other.  There is no way to exit the visible universe.  Despite recession velocities > 3c relative to the cosmic frame, new stars become visible (enter the visible universe) every day, and none of the ones visible ever vanish from sight (unless they cease emitting light).

[alancalverd]

Or for that matter, close your eyes.

This works very well.
One night flying training exercise involves having the lights fail whilst the student is making an approach to an airfield. On my first attempt, the Tower apologised that they couldn't douse the lights as there was a commercial flight lining up a few miles behind me. I responded "No problem, I'll shut my eyes". The only time I ever scared an instructor.

[paul cotter (OP)]

Thanks Halc, it looks like I have been in error, on both threads. I will avoid making comments on relativity matters until I get on a more sure footing.