# Gradual acceleration with converging series (a thought experiment)

Let's imagine a rocket that travels straight path through space, accelerating gradually (or being boosted by outside force). The speed of rocket is such that

1st light year takes 1/2 hours in rocket's own frame
2nd light year takes 1/4 hours
3rd light year takes 1/8 hours
4th light year takes 1/16 hours
5th light year takes 1/32 hours
and generally, nth light year takes (1/2)^n hours in rocket's own frame

As it's seen, the sum of travel times form converging series that has limit 1. In the outside rest frame (the path frame), the rocket's speed approaches light speed and kinetic energy grows without bound, but light speed or infinite energy is never actually reached.

However, in rocket's own frame, the passengers (strong enough to survive the acceleration) expect that they reach infinity at 1 hour and eagerly wait to see what happens and where they end up to. Will they be disappointed? Is there something in relativity why, at 1 hour, the passengers see that they still are far away from infinity, and actually will never reach it? Thanks.

DrGreg
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What will happen is that before the hour is over, the rocket will run out of fuel, as its supply cannot be infinite.

Dale
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Also, the thrust cannot be infinite.

In the outside frame, rocket never has infinite energy or reaches light speed. This sounds correct, but this would also mean that rocket's observation is somehow wrong. I thought that in relativity every viewpoint is equally valid.

Dale
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The first postulate asserts the equivalence of inertial frames. The rocket's frame is non inertial.

PAllen
In the outside frame, rocket never has infinite energy or reaches light speed. This sounds correct, but this would also mean that rocket's observation is somehow wrong. I thought that in relativity every viewpoint is equally valid.

1) In the inertial frame, the rocket's KE approaches infinite, as t goes to infinity.

2) In the rocket, the mapping between the rocket's proper time and inertial time is such that [0,∞) is mapped to [0,1). Just as ∞ is never reached in the inertial frame, proper time of 1 is never reached for the rocket. Another feature of the rocket experience is that any feature of the universe, no matter how big the universe, is reached before t=1. Thus there isn't enough universe to reach 1.

3) Relativity does not say inertial frames are equivalent to non-inertial frames any more than Newtonian mechanics does, and for fundamentally the same reason. This is a surprisingly common misconception, even though there is no basis for it in scientific literature. The closest you can say is: a non-inertial frame far from any massive bodies may be considered, for most purposes, locally equivalent to a static frame near a massive body.

1) In the inertial frame, the rocket's KE approaches infinite, as t goes to infinity.

2) In the rocket, the mapping between the rocket's proper time and inertial time is such that [0,∞) is mapped to [0,1). Just as ∞ is never reached in the inertial frame, proper time of 1 is never reached for the rocket. Another feature of the rocket experience is that any feature of the universe, no matter how big the universe, is reached before t=1. Thus there isn't enough universe to reach 1.

3) Relativity does not say inertial frames are equivalent to non-inertial frames any more than Newtonian mechanics does, and for fundamentally the same reason. This is a surprisingly common misconception, even though there is no basis for it in scientific literature. The closest you can say is: a non-inertial frame far from any massive bodies may be considered, for most purposes, locally equivalent to a static frame near a massive body.

Ookke never mentioned the word "equivalent". He was just talking about the mapping [0,∞) to [0,1). What he actually said was "I thought that in relativity every viewpoint is equally valid." and not that every viewpoint is equivalent. The point he was trying to make is that if ∞ is NOT reached in the inertial frame then 1 hour should NOT be reached in the rocket frame, which is correct. However, I suspect his intuition is that it is inevitable that the rocket observers will see 1 hour ion their clocks, unless time stops for the rocket observers. What he is missing is that when the rocket observers do see 1 hour on their clocks they will have stopped accelerating (because there is not enough fuel in the universe) and the mapping [0,∞) to [0,1) no longer applies.

His statement that "I thought that in relativity every viewpoint is equally valid." is correct for the inertial case, but not completely restricted to inertial cases only. For example the viewpoint that only 1/2 hour has elapsed for the accelerating rocket observers is as valid as the opinion of the inertial observers that 1 year has elapsed. However, it is correct to point out that in some non-inertial situations not all viewpoints are equally valid. For example the viewpoint of an observer that is riding on the edge of a rotating disc, that the rest of universe is rotating while they are stationary, is not a valid viewpoint, because rotation is absolute. The viewpoint the rocket is stationary and the rest of the universe is accelerating backwards is also not a valid view because acceleration is absolute.

The point he was trying to make is that if ∞ is NOT reached in the inertial frame then 1 hour should NOT be reached in the rocket frame, which is correct. However, I suspect his intuition is that it is inevitable that the rocket observers will see 1 hour ion their clocks, unless time stops for the rocket observers. What he is missing is that when the rocket observers do see 1 hour on their clocks they will have stopped accelerating (because there is not enough fuel in the universe) and the mapping [0,∞) to [0,1) no longer applies.

That's pretty much it, thank you and others. I could accept the solution that rocket runs out of fuel, but let's not give up just yet.

We can imagine an infinite chain of fueling stations, one light year apart. Station 1 gives rocket enough fuel to travel light year in 1/2 hour, station 2 gives fuel to travel next light year in 1/4 hour, and so on. The total amount of fuel in stations is infinite, but each single station needs to hold only finite amount of fuel. So infinite mass of the universe is assumed here (sounds possible to me), but no single point with infinite mass (which sounds impossible).

Of course, if the universe must have finite mass, this wouldn't work. Actually this involves two questions: does our universe have infinite mass, and is it even logically possible to have an universe with infinite mass.

Dale
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The total amount of fuel in stations is infinite, but each single station needs to hold only finite amount of fuel.
No, the fuel for a single station is infinite. The acceleration is infinite and an infinite acceleration on a finite mass requires an infinite force. Since the distance between stations is finite it requires an infinite energy to get to the next station.

PAllen
That's pretty much it, thank you and others. I could accept the solution that rocket runs out of fuel, but let's not give up just yet.

.

That's why I tried to give another explanation that has nothing to do with fuel (though it really is equivalent). That is that proper time on the rocket reaching 1 means that actual infinite distance in any inertial frame has been traveled. Infinite distance can never be reached, only approached.

Note that because acceleration is absolute, not relative, you feel exponentially increasing proper acceleration. You know, from SR or equivalence principle of GR, that your clock is getting slower, moment to moment compared to an inertial clock you release a moment ago. Thus, this really amounts to just a disguised form of the following 'paradox': Suppose I program a clock with infinite battery life to half its rate every second. Then, when this clock reaches 2 seconds you will have passed actual infinite time on a normal clock. Obviously, this programmed clock will simply never reach 2 seconds. Neither will your rocket reach one hour.

I brought up equivalence for a reason. I see that you are thinking of 'valid' more like equivalent. The rocket has a 'valid' experience. It is not remotely similar to inertial experience, and SR and GR both predict that this should be so. Exponentially increasing proper accelerations is an invariant feature, not something that is relative.

PAllen
No, the fuel for a single station is infinite. The acceleration is infinite and an infinite acceleration on a finite mass requires an infinite force. Since the distance between stations is finite it requires an infinite energy to get to the next station.

The acceleration is exponentially increasing not, infinite. The fuel required at each successive station is exponentially increasing, but not infinte at any one station.

Dale
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The acceleration is exponentially increasing not, infinite. The fuel required at each successive station is exponentially increasing, but not infinte at any one station.
Exponentially increasing is infinite in an infinite universe. Even linearly increasing would be infinite.

Pick any finite value of acceleration or fuel, and there is a station which requires more. That is infinite, by definition.

That's why I tried to give another explanation that has nothing to do with fuel (though it really is equivalent). That is that proper time on the rocket reaching 1 means that actual infinite distance in any inertial frame has been traveled. Infinite distance can never be reached, only approached.

Note that because acceleration is absolute, not relative, you feel exponentially increasing proper acceleration. You know, from SR or equivalence principle of GR, that your clock is getting slower, moment to moment compared to an inertial clock you release a moment ago. Thus, this really amounts to just a disguised form of the following 'paradox': Suppose I program a clock with infinite battery life to half its rate every second. Then, when this clock reaches 2 seconds you will have passed actual infinite time on a normal clock. Obviously, this programmed clock will simply never reach 2 seconds. Neither will your rocket reach one hour.

I brought up equivalence for a reason. I see that you are thinking of 'valid' more like equivalent. The rocket has a 'valid' experience. It is not remotely similar to inertial experience, and SR and GR both predict that this should be so. Exponentially increasing proper accelerations is an invariant feature, not something that is relative.

Ok, I'm trying to get it. We can forget about the fuel, but then there must be something that prevents the rocket to reach infinity in its own frame (since if it doesn't reach infinity in outside inertial frame, it cannot do that in any other frame either).

I suppose this preventing physical thing cannot be that the clock slows down, because each observer's (inertial or not) own clock goes at normal rate according to his perception. It could be that the acceleration, which approaches infinity, makes my assumption about "strong enough passengers to survive acceleration" inevitably fail, and that would sync the different point of views.

With "valid" I ment something like that all observers, inertial or not, must agree certain basic things, for example local events: if two cars collide on a highway, they must do that in any observer's frame. If acceleration is felt, this does not invalidate all observations. Non-inertiality can have effect on observations about light speed, distances, times etc, but not on anything that is absolute.

Suppose that at some point in its journey the ship emits a flash of light in the forward direction. Will that flash of light ever reach infinity, according to the ship-board observer? No. According to the at-home observer? No. Will the ship ever catch it?

Basically no to both, but if the ship itself was to reach infinity, the light must do that too: it would be "squeezed" between the ship and infinity, so to speak. Maybe if light had a reference frame (I know, it doesn't) it would end up to infinity with finite (or zero) time.

Dale
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We can forget about the fuel, but then there must be something that prevents the rocket to reach infinity in its own frame (since if it doesn't reach infinity in outside inertial frame, it cannot do that in any other frame either).
OK, let's be clear about something. It is possible to make a valid coordinate system which maps time and space in the way which you suggest. In that coordinate system an "at rest" worldline would be always timelike, but it would also have infinite proper acceleration.

You could consider it like a time varying gravitational field in this frame, which becomes infinite. Regardless of how strong or powerful you make the rocket, eventually it will succumb to the gravitational field.

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PAllen
Exponentially increasing is infinite in an infinite universe. Even linearly increasing would be infinite.

Pick any finite value of acceleration or fuel, and there is a station which requires more. That is infinite, by definition.

Well, I look at these terms differently. Every station requires finite fuel (in principle). The limit of the amount of fuel in a station along the path is infinite. But no station requires infinite fuel.

Similarly, I would never say the proper acceleration is infinite. Proper acceleration is defined at some event on a world line. Every event on the proposed world line has finite proper acceleration. It approaches infinity as proper time approaches 2. The point on the world line with proper time 2 does not exist, any more than points at infinity exist in Minkowski space (you can add them, getting a different topology altogether).

Dale
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You are using the terms wrong. Infinite isn't a number. It means that you have some sequence that increases without bound. As the time approaches 1 the fuel at each station and the acceleration increase without bound. They are therefore infinite.

If you disagree then what is the finite amount of fuel or acceleration which exceeds the fuel and acceleration at all stations?

PAllen
You are using the terms wrong. Infinite isn't a number. It means that you have some sequence that increases without bound. As the time approaches 1 the fuel at each station and the acceleration increase without bound. They are therefore infinite.

If you disagree then what is the finite amount of fuel or acceleration which exceeds the fuel and acceleration at all stations?

There isn't one, but that is describe, according to what I learned, as absence of an upper bound. Infinity exists as a limit or as a special point in projective or conformal geometry, or in formalisms of transfinite arithmetic. What you are calling infinite, I call (and learned) as without upper bound or approaching infinite.

Dale
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Increasing without bound, without upper bound, absence of an upper bound, those are all just different ways of saying infinite.

##\lim_{t\to 1} \, a = \infty##

It doesn't matter how strong the rocket is nor how much fuel your fuel depots can store, more is required at some point. The required amount is larger than any finite number, i.e. it is infinite.

The amount of fuel in a depot is a real number, and infinity is not a real number. When I say that the amount of fuel required is infinite that means that there is no real number which is big enough to meet the requirement.

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About mass of the universe: Is our universe known to have finite mass? And could relativity hold in an universe that has infinite mass, or is there some logical obstacle for that? Sorry, difficult questions, but just in case someone has insight on these...

Dale
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I don't think that the mass of the universe is known, and yes, some of the cosmological solutions of GR have an infinite universe with non-zero mass/energy density everywhere.