So if spaceman Spiff travels to the moon in one second and it takes one more second for the ligth to return to earth so that we see Spiff at the moon that is an elasped time of two seconds.

If Spiff used the gravity of the moon to make a full speed U turn he would make the return trip to earth in one second.

Would he not appear in two places at once?

if he were traveling the speed of light, then you would see the image of him on the moon, and the image of him next to you at the same time.

if he was traveling faster than the speed of light you would see him next to you before you saw him on the moon.
just like you hear the sonic boom of an aircraft after you see it, because the plane is traveling faster than the sound. If he was traveling faster than light, he could even turn around and see himself.

however it is impossible(so far) to travel anywhere near the speed of light, let alone the speed of light, or faster, so dont worry about seeing double of spaceman spiff anytime soon.

Yes, and not only would you see him on the moon and earth at the same time, you would see his entire trip at the same time, and when he's back on earth, you will see that his watch will not have moved one bit. He/she will have travelled thru space and not thru time. Of course, nothing with mass can ever travel at lightspeed, so there would be some tiny fraction of a second increase on his watch.

to be honest my relativity is pretty rusty so i'm not going to try to really answer this, but i would like to point out that the situation you described isn't covered by special relativity. because it involves acceleration (acceleration is any change in direction or speed, so the U-turn is acceleration). You need to use the theory of general relativity to analyze this situation, which is a bit more complex than special relativity. special relativity gets its name from the fact that it is a special case (the case where there is no aceleration) of general relativity.

Gerry and Bibc14 - are you sure nothing can travel at the speed of light? We can't make anything do it now but someday maybe. Wasn't there recently an article where in a certain gaseous enviroment the speed was calculated at 300 times the "normal" speed of light? Wasn't the speed of sound and even 60 mph considered a limit at one time?

I am 99.99999 percent certain that anything with mass cannot EVER reach light speed. I am 99.9 percent certain that electromagnetic waves (light , radio, microwave, etc) cannot ever exceed light speed. (I believe the experiments you are talking about were flawed, they may have been measuring the front waveform of the radiation as it left the tube).
However, I am not at all certain about the speed of gravity waves. I believe that certain gravity waves (through the postulated carrier called a graviton) may be able to exceed lightspeed into the higher order space-time dimensions, especially at the singularity of a black hole, or for that matter, the singularity at the Big Bang or Big Crunch. Much still needs to be done in this area.

just a comment on the exceeding the speed of light experiments. the experiments were in no way flawed, the popular press just made a bigger deal out of it than it should have. The issue is what velocity you are measuring. There are different velocities dealing with a pulse of light, the phase velocity and the group velocity.

before i can explain the difference well, you need to understand interference. As you may know, when two waves are superimposed on each other, the intensities of the two waves add up at each point in space. Keep in mind that they have both positive and negative parts, so the resulting waveform can be large in some places and small in others. If you have a very large number of these waves put together in just the right way, you can end up with what looks like a pulse in one place and nothign elsewhere, even though the individual waves do exist elsewhere they cancel each other out at all but one spot.

The phase velocity is the actually velocity of the individual waves of light. The group velocity is the velocity at which the pulse seems to travel. In this experiment, they were able to adjust things in such a way that the pulse, which is the group velocity, seemed to travel faster than light. The actual individual waves of light, however, traveled at the phase velocity, which was below the speed of light.

now, what may be harder to see is that this means that no information travelled faster than light. remember that, even though they cancel each other out everywhere except the pulse, the individual light waves cover a large area. They actually hit the cesium well ahead of the pulse and traveled through it at normal speed. These waves contained all the information in the pulse. The tricky thing was, passign through the cesium altered the phase relationships so that the waves moving ahead of the pulse, which had been cancelling each other out, now formed a pulse.

The waves, and the information they contained, didn't travel faster than the speed of light, even though the pulse seemed to.