Janus

in terms of time dilation' date=' yes. According to your post, Zach is sitting at the North pole of Earth 3, so he wouldn't be moving at all wrt Bob, who is sitting at the North pole of Earth 1.

How do they end up feeling the same? Bob is traveling in a circle of Earth radius such that he feels 1g. Dave is traveling in the same circle around a planet of Earth mass, thus is essentially traveling at orbital speed and feels weightless. Perhaps you meant Bob and Frank? But even then, the local acceleration due to gravity or what you 'feel' locally does not determine what time dilation you will show. The interesting part is that Zach and Chuck will show equal time dilation.(Though Zach will feel half the g-force that Chuck does.)

Due to the way the problem was set up, I also assumed no equatorial bulge. (so that Bob, Dave and Frank all traveled in circles of equal radius. )

Archie shows no SR or Gravitional dilation Bob only shows SR dilation (even though he is accelerating, this acceleration adds no additional time dilation. Chuck shows only gravitational dilation, but it will be greater than the SR dilation shown by Bob. Dave shows the same gravitational dilation as Chuck and in addition the SR dilation of Bob. Edgar shows the only gravitational dilation but it is greater than the combined gravitational and SR dilation's shown by Dave. Frank shows the gravitaional dilation of Edgar in addition to the SR dilation shown by Bob and Dave. Gus shows only SR dilation due to his instantaneous speed at any given moment, but his average velocity will be high enough to show a greater time dilation than any of the others. On a side note, an interesting result occurs if Earth 2 has both twice the mass and twice the Radius as Earth 1

No, it is not. The equivalence of gravity and acceleration is due to the equality of 'gravitational' mass and 'inertial' mass(if I increase the gravitational mass of an object so that it attracts another mass more, I also increase its inertial mass and thus its resistance to acceleration, by the same factor). There is no such equality with magnetism. I can increase or decrease the magnetic field of an object without causing the same increase or decrease in its inertial mass and vice-versa. (I can increase the force of the magnetism of the object without increasing its resistance to acceleration.)

No more than I would count the fact that the movie Titanic accurately described many real events that took place on that ship as evidence that the characters of Rose and Jack actually existed in real life.

As pointed out time runs slow for you only as measured by someone who measures you as moving with respect to them. Now, from their standpoint it means that your clock will accumulate 1/5 of a year while you travel the distance from earth to Alpha Centauri (4.3 ly), while their clock accumulates 4.3 years. From your viewpoint, your clock will also accumulate 1/5 y, but not because your clock ran slow, but because, due to length contraction, you will measure the distance between Earth and Alpha Centauri as only being 1/5 ly, which you traveled at .999c

The Hubble has a resolution of .1 arc-secs.

It would take a telescope with an aperture of about 32,000 in. to resolve objects as small as 5 km on the surface of Pluto.

But you don't need to run into faster light waves to see a doppler effect any more that you need to run into faster sound waves to hear a doppler shift. If I'm standing on the train platform on a windless day and someone on the platform yells at me, the sound travels at the speed of sound in air towards me. If a train approaches and blows its whistle the instant it passes that person, the sound from it will travel at the same speed as the yell (they will both reach me at the same time). The train whistle will be doppler shifted however.

To say that Ion engines "go faster" is a bit of a misnomer. They are just more efficient due to the fact that they have higher exhaust velocities than chemical rockets, which means that they can reach higher velocities with the same reaction mass to payload ratio. Oh, and by the way, Ion engines are still rockets as they use the action/reaction principle for propulsion. There is a better system on the drawing board, called the VASIMR (VArible Specific Impulse Magnetohydrodynamic Rocket) It could produce even greater exhaust velocities than the Ion engine and more thrust (producing better acceleration).

Um no what? Have you even done the math?

To any observer traveling with the ship, this is what happens: No matter what direction he is facing.

The stars are there, it is just the they are too dim to be seen with the exposure used. The main objects in the picture (planet etc) are so bright compared to the stars, that the camera setting is such that the stars don't show up. You see the same effect if you go out and look at the night sky during a moonless night compared to a night with a full moon. During a full moon you'll see a lot fewer stars than on a moonless night. If the moon were a little cloeser to the Earth, it would be bright enough that when you looked at it, you wouldn't see any stars at all.

Here is an animation that shows what happens according to an observer in S for which the ship is moving. The white dots are the light pulses, the yellow lines are the path that the pulses follow with respect to S' the green lines show the paths the pulse's take with respect to S'(the ship}

It does not matter whether you consider the light as being shot from the S or S', you get the same results either way. That is the whole point of Relativity. As they would be. We don't live in a Newtonian Universe, we live in a Relativistic Universe. And in a Relativistic Universe the speed of light is invariant; Meaning that all observers will measure light in a vacuum as moving at c with respect to themselves regardless of any relative motion thye may have to the source. This is the second postulate of Relativity. there is no problem here, it is just a different aspect of the fact that time is relative. In this case it is the idea of simultaneity that is relative. In S', the three pulses strike the walls at the same time

You said: ct1 and ct2 are distances the light travels in S before it hits the sphere. because that is not what relativity predicts. If you wish to discuss Relativity you must first learn it, something you've obviously have neglected to do. I stand by my images, as they are accurate. If you cannot see that, then that is your problem. I start out with the assumption that the light is emitted at angle Alpha as seen by the inside observer. I then can calculate the path the light will take with respect to the outside obserever, based on that assumption and the velocity v and knowing the speed of light. one does not have to see something to know that it is true. I didn't have to be around 2,000,000 yrs ago to see the light leaving the Andromeda galaxy to know that the light from it reaching the earth now left it 2,000,000 yrs ago. Thus just because the Outside observer doesn't see the light traveling at angle ALPHA does not mean he cannot know that the light travels at angle alpha as seen by the inside observer. OF course the outside observer in S can see ct1 and ct2 As these are the distances the light travels as seen by him by definition!Sure, I could have determined angle BETA and worked from that, but to determine angle BETA, I have to start from angle ALPHA and determine it from that. Or conversely, I could start with Angle BETA and derive angle ALPHA from it. Either way you go,the two angle are co-dependent. [math]Y \sin \alpha[/math] is measured along a line perpendicular to the line of relative motion, Length contraction only applies to distances that are parallel to the line of relative motion. Again, if you had actually tried to learn Relativity, you would know this. [math]\arccos \left( \frac{v}{c} \right)[/math] Sure, just imagine that the spaceship is open and the light is passing through a series of glass plates. The light will slightly illuminate each plate as it passes though. If the outside observer draws a line through these spot of illumination they will follow this angle. If you can't follow the example just say so, but your inability to understand it speaks more about you than it does the example.

Why? What prevents him from knowing it? Unless you are claiming that the sphere is opaque and that the outside observer cannot see what happens in the spaceship, in which case this whole exercise is pointless. What speed [math]\gamma Y \cos \alpha[/math]? The only speeds in this example are v which can have any value <c and c, the speed of light in a vacuum.