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Let's say I build a ship, get in it and take off into the dark. When I leave Earth, I measure my acceleration at 1g and my ship is capable of producing that level of power indefinitely. There will be a point when the amount of energy I am using will not be enough to continue accelerating me.

As I reach the point of diminishing returns, do I still feel 1g acceleration? At some point I will stop accelerating - how fast will I be going? What would be the time dilation? What would I, as a passenger on that ship, experience during my travel?

I'm just thinking if we could find a way to accelerate indefinitely (I'm looking, it's around here somewhere), we wouldn't actually have to achieve c in order to make the trip to wherever seem short to the people on the spacecraft.

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Right I'm going to assume the constant power your ship is capable of producing is actually a constant force your ship is under. (i.e. your ship is using some purely electrical form of acceleration - no loss of mass due to propellant).

The basic law of physics behind this is that as you approach the speed of light you mass increases. Therefore to maintain a constant acceleration you need an increasing force. However, if your force is not increasing you acceleration decreases. The rate of decrease is such that you will approach the speed of light but never reach it, or to put it another way you would need an infinite length of time to reach the speed of light.

This being said, you never actually stop accelerating. Your acceleration may get so low that you can't measure it. As for the observer on the spacecraft, they will still experience the 1g of acceleration - as in their frame of reference they aren't going at/near the speed of light. The time dilation experienced will be dependent on the length of the journey. As a passenger on the ship you will experience the universe around you accelerating in the opposite direction to your travel. You will also experience the blue/red shift of object outside the spacecraft infront/behind you.

Also to answer your title question you could get infinitely close.

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  • $\begingroup$ If I could have limitless energy to propel me at a constant rate indefinitely I could get to c in just under a year. But without that energy I can't get to c at all. Would I still get to "very near" c in a year, or would my acceleration slow to the point that it takes significantly longer? Maybe I can only get to 1/2 c in a year, for example. Then after to 3/4 c over the next year and so on. $\endgroup$ – Bill Apr 17 '14 at 16:51
  • $\begingroup$ I suppose if you picked some set percentage of the speed of light, like 99.999%, you could calculated how long it would take to get to that speed, and that will give you some idea of how long it would take to get to some practical percentage of the speed of light. $\endgroup$ – Nickolai Apr 17 '14 at 16:53
  • $\begingroup$ Why would it take more energy to maintain an apparent 1g as you go faster? That violates relativity. The outside observer will see the acceleration drop, the person in the starship won't. $\endgroup$ – Loren Pechtel Apr 18 '14 at 0:17
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    $\begingroup$ @FraserOfSmeg Oops, misread. $\endgroup$ – Loren Pechtel Apr 18 '14 at 18:23
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    $\begingroup$ Actually, your mass doesn't increase, your local time slows. Net effect similar, but it does matter for fuel use calculations. $\endgroup$ – aramis Apr 19 '14 at 4:41
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Assuming constant acceleration you'd be sailing across a Poincare disk. See M. C. Escher's Circle Limits for nice illustrations of Poincare Disks.

From the point of view of each angel or devil in Escher's circle limit, he/she is in the middle of the disk. Near neighbors are somewhat scrunched. Distant neighbors closer to the edge are a lot scrunched.

Each angel or devil could be thought of as an inertial frame. With constant acceleration you'd be sailing across the disk. After you sailed over a devil you could watch him receding. He'd get closer and closer to the edge but never reaching it. From the devil's point of view he'd see you receding. He'd watch you getting closer and closer to the speed of light but never reaching it.

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You can get as close as you want. The usual E=1/2 mv2 breaks down at relativistic speeds. Instead, you get the famous formula E=mc2. And as you might know, your mass increases as you approach light speed, by a factor (1-v2/c2)-1/2. In other words, as you approach light speed your mass goes to infinity. That in turn means with a constant force your acceleration decreases to zero, but it never becomes zero.

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  • $\begingroup$ Although relativity is often traditionally taught as if mass increases at relativistic velocity (a concept called relativistic mass) this is not actually what happens, and many including Einstein, advocate that, to avoid confusion, the term mass should only apply to the invariant rest mass. en.wikipedia.org/wiki/Mass_in_special_relativity#Controversy $\endgroup$ – Caleb Hines Nov 11 '14 at 23:29
  • $\begingroup$ "It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M (relativistic mass) it is better to mention the expression for the momentum and energy of a body in motion." - Einstein $\endgroup$ – Caleb Hines Nov 11 '14 at 23:30

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