Theoretical problems with relativistic space flight

[Nabeshin]

The following is a paper I wrote mainly to collect my thoughts and document some research I did about interstellar space flight. The main ideas for this paper grew out of the things I always hear people talk about (blueshifted radiation) but I never see the math behind it, so I developed a few models of my own. I wanted to keep the paper as theoretical (i.e away from any specific model of space flight. I don't care what method of propulsion is used, for example). For this reason the first section is a little more ambiguous than, say, the relativistic rocket page. I aim just to either a) get order of magnitude estimates or b) upper/lower bounds.

That said, feel free to criticize and comment, but keep in mind my purpose isn't to be completely rigorous!

http://docs.google.com/fileview?id=...jYtODE2NC00ZmRlLWI1ZDctOTRiZTRmZWNjMzll&hl=en

Note: Admins if this is better suited for the independent research form I'm not sure, but I'll leave that up to you.

 

[phyzguy]

Interesting paper. The issues you present are formidable. Because of these issues, I believe that the most likely scenario for us traveling to the stars is on large space habitats which move much more slowly than you posit, and require generations to travel between stars. If we do begin in the coming centuries to colonize nearby space in large habitats as envisioned by Gerard O'Neill, it is merely an evolutionary step to make these habitats large enough and self-sustaining enough that they could travel to the nearest stars. Interstellar colonization then becomes more of a diffusion mechanism, where we slowly move from one star to the next, building up resources and population at each star before taking the leap to the next star.

 

[Lsos]

I think we'll be able to digitize our minds before we make attempts with interstellar arcs.

Once we do that, we we'll be able to travel to other worlds without spending eons inside spaceships, and consequently without having to worry about the effects of relativistic travel.

 

[Nabeshin]

Interesting paper. The issues you present are formidable. Because of these issues, I believe that the most likely scenario for us traveling to the stars is on large space habitats which move much more slowly than you posit, and require generations to travel between stars. If we do begin in the coming centuries to colonize nearby space in large habitats as envisioned by Gerard O'Neill, it is merely an evolutionary step to make these habitats large enough and self-sustaining enough that they could travel to the nearest stars. Interstellar colonization then becomes more of a diffusion mechanism, where we slowly move from one star to the next, building up resources and population at each star before taking the leap to the next star.

Perhaps. Of course, building self-sustaining habitats presents its own myriad of challenges, but most of these would be engineering concerns. Plus it's just much less interesting. Without relativistic space travel, for all intents and purposes for the individual human the stars are forever disconnected. (Perhaps, actually, communication would be possible. So long as the target is within, oh, 30 light years, one could imagine one exchange of messages in a lifetime!) I think most people who believe the challenges of relativistic spaceflight are too vast turn to cryogenics for this reason. It's better to go to sleep during the voyage than condemn (several) generations of people to know only life aboard a space craft.

 

[phyzguy]

It's better to go to sleep during the voyage than condemn (several) generations of people to know only life aboard a space craft.

You should read O'Neill's "The High Frontier". If the space habitat is cubic miles in extent with several million people on board, the quality of life would be as good or better than what you have on Earth today, so you would hardly be "condemned". It would be like being "condemned" to spending your life on an island - imagine it was Maui - or Manhattan - or Singapore - hardly an onerous existence. And, as you say, you could continue to communicate with the folks back home (although with an ever growing time delay). And you would have the added benefit of knowing that when your descendents arrived at the next star they would have a whole new solar system to explore and populate. It would not be a bad life. I'd go.

I think, given the laws of physics, this is the most likely scenario.

 

[Filip Larsen]

Interesting paper. It sure detracts from the possibility of traveling to the Andromeda galaxy in a lifetime that is otherwise offered by the relativistic rocket equation using mass annihilation.

Have you considered including solar sail and similar energy transfer setups in your analysis? There are several proposed designs in the literature, but I don't recall any analysis of the heating from ISM on those.

I also have a few comments, not specifically to your paper, but more related to the overall concept of human interstellar travel.

As an engineer I have long had the belief, that any humans presence outside low Earth orbit will be pioneered well in advance by automatic probes. No one in their right mind would travel to a solar or extra-solar planet without having it probed out first, even if this means waiting 30, 50 or even 100 years for the results. Add to this the complexity of engineering manned spaceflights systems compared to autonomous unmanned probes, which I can imagine only gets even worse for interstellar travel, and you are almost certain that probes will go in first.

So, assuming that probes have been sent out to nearby star systems, what should then be the reason to send humans to one or more of such selected systems? Even for the closest systems it is hard to imagine the possibility of ever returning to Earth again, so it seems that who ever go, will go to stay and this seems to imply they will have to build some form of colony, even if this (for some reason) is not be their primary goal. Of course, it may be quite possible to find people that would be willing to accept a "suicide mission" to a system that do not involve colony building, but I simply cannot imagine that anyone would design and implement such an extremely complex mission and then miss the chance to "keep the foothold" so to speak if it were at all possible.

So, you would have probes going to systems and you would want colonies in (some of) those systems. These two things together seems to hint that it is far more likely that we send a small probe with the capability to autonomously "bootstrap" a colony. It could for instance build a small incubator station around or on a selected planet from asteroid or comet like raw material (including raw materials for building organic matter like proteins) and then "simply" use bio-engineering technology to establish an initial human presence on that station, who could then migrate planetside once ready for it. The bio-engineering could range from "cloning" of single-cell "embryos" for each individual (which could also include animals) to some sort of cryo-sleep that would allow a selected few "earth humans" to be there to supervise the process.

While such a scenario may seem as far fetched as group of humans zipping back and forth in highly relativistic spaceships, I think it is more likely that the "needed" parts of such a scenario will be "technologies we have anyway" at some point in the future. While AI of today still does not quite live up to the term intelligent, it seems fairly likely that autonomous probes in the no so distant future could be capable of, say, navigating and rendezvous with asteroids and comets in the solar system and process their materials into raw materials which is send back to Earth or otherwise collected. If we can combine such autonomous operation to also include manufacturing of a range of "finished" structures we almost have what we need. Regarding bio-engineering, it also seems likely, with the present research into growing human "parts" from stem cells for instance, that we may end up with bio-engineering technologies that we have anyway and which will be useful for a colonization mission.

So, if I had a huge amount of patience and life-expectancy way longer than normal, I would definitely bet my money on the first human being delivered to a nearby exo-planet by a slow-boating dormant arkship rather than by a fast ship carrying "live" humans. That said, I sadly think that the most likely scenario is that we never get out of the solar system. While humans as individuals can show great persistence and dedication to a goal, current human society do not seem to be able to show consistent dedication to long term goals, even more so when such goals would be of insignificant importance to the average voting citizen. Only chance I see, is that one day we realize we have most of the needed technology anyway and that we with a minor effort on the scale of, say, 10-20 years and correspondingly small funding, could launch (and forget) such a mission. That, or some multi-billionaire nut-case who one day decides he wants to send himself or his heir along with a staff of thousands to a nice exo-planet and build a new galactic empire there.

 

[Nabeshin]

Interesting paper. It sure detracts from the possibility of traveling to the Andromeda galaxy in a lifetime that is otherwise offered by the relativistic rocket equation using mass annihilation.

Have you considered including solar sail and similar energy transfer setups in your analysis? There are several proposed designs in the literature, but I don't recall any analysis of the heating from ISM on those.

No, I didn't consider these explicitly. My point was to give a somewhat propulsion-independent lower limit on the power requirements of a relativistic journey. I don't think a solar sail falls into any of the acceleration categories I considered -- acceleration would probably be significantly less than .01g (of course, we could make the sails arbitrarily large, but I get the feeling the answer would be analogous as to the size of the ISM scoops I derived in the paper). So in this case, it's not relativistic space flight, and you need to turn your attention to sustaining the population for such a long period of time.
I also have a few comments, not specifically to your paper, but more related to the overall concept of human interstellar travel. Bottom line: you want to go fast, you need a TON of power. Nothing can deliver this more efficiently than matter-antimatter, and even this is not so good.

As an engineer I have long had the belief, that any humans presence outside low Earth orbit will be pioneered well in advance by automatic probes. No one in their right mind would travel to a solar or extra-solar planet without having it probed out first, even if this means waiting 30, 50 or even 100 years for the results. Add to this the complexity of engineering manned spaceflights systems compared to autonomous unmanned probes, which I can imagine only gets even worse for interstellar travel, and you are almost certain that probes will go in first.

Certainly, as is all the cases for human exploration nowadays.

So, assuming that probes have been sent out to nearby star systems, what should then be the reason to send humans to one or more of such selected systems? Even for the closest systems it is hard to imagine the possibility of ever returning to Earth again, so it seems that who ever go, will go to stay and this seems to imply they will have to build some form of colony, even if this (for some reason) is not be their primary goal. Of course, it may be quite possible to find people that would be willing to accept a "suicide mission" to a system that do not involve colony building, but I simply cannot imagine that anyone would design and implement such an extremely complex mission and then miss the chance to "keep the foothold" so to speak if it were at all possible.

It's reminiscent of the argument against sending humans anywhere even in our own solar system. People will give you all sorts of reasons for the limits of robotic exploration and the need for flesh and blood humans. The discussion is rather besides the point here, and could certainly be a thread of its own (and likely has!).

So, you would have probes going to systems and you would want colonies in (some of) those systems. These two things together seems to hint that it is far more likely that we send a small probe with the capability to autonomously "bootstrap" a colony. It could for instance build a small incubator station around or on a selected planet from asteroid or comet like raw material (including raw materials for building organic matter like proteins) and then "simply" use bio-engineering technology to establish an initial human presence on that station, who could then migrate planetside once ready for it. The bio-engineering could range from "cloning" of single-cell "embryos" for each individual (which could also include animals) to some sort of cryo-sleep that would allow a selected few "earth humans" to be there to supervise the process.

Interesting, I must honestly say I never considered this possibility! I mean surely everyone knows of the von neumann machines, but I never thought to make one create human life on the habitable planets they find!

So, if I had a huge amount of patience and life-expectancy way longer than normal, I would definitely bet my money on the first human being delivered to a nearby exo-planet by a slow-boating dormant arkship rather than by a fast ship carrying "live" humans. That said, I sadly think that the most likely scenario is that we never get out of the solar system. While humans as individuals can show great persistence and dedication to a goal, current human society do not seem to be able to show consistent dedication to long term goals, even more so when such goals would be of insignificant importance to the average voting citizen. Only chance I see, is that one day we realize we have most of the needed technology anyway and that we with a minor effort on the scale of, say, 10-20 years and correspondingly small funding, could launch (and forget) such a mission. That, or some multi-billionaire nut-case who one day decides he wants to send himself or his heir along with a staff of thousands to a nice exo-planet and build a new galactic empire there.

Sadly agree!

 

[Nabeshin]

Anyone else want to help me out?

 

[TimGrebin]

Interesting read!

However there's a point which confuses me, are you proposing traveling 4.2 lys in 3.5 years? (Top of page 3)

 

[Filip Larsen]

However there's a point which confuses me, are you proposing traveling 4.2 lys in 3.5 years? (Top of page 3)

Allow me to answer. The time in equation 1.2 (and in all of section 1.1 for that matter) is the proper time for the rocket and not the time for an earth-bound observer. If you can somehow manage to keep an acceleration of, say, 1 G for years on end you would be able to travel exponentially farther away as time goes. You could go, say, 1 million light years in little under 27 years on the ship clock (of course, as measured from the Earth the journey will take a bit more than one million year to complete).

 

[TimGrebin]

How interesting! Thank you.

 

[Filip Larsen]

Now I was there, I noticed that it is a bit peculiar to apply the relativistic proper speed from equation (1.3) to equation (1.1) for required energy. I would think that to make sense you would either have to integrate vehicle power up in the Earth frame (to calculate how much energy you must transfer from the Earth frame to move the vehicle) or you would have integrate jet power in the rocket frame (to get the amount work the engine would have to provide to obtain the given acceleration for the whole trip). Applying (1.3) to (1.1) seems to be doing a bit both, but I cannot see the rationale behind it.

Just to illustrate what I mean with jet power, assume for a minute that the rocket is self-contained and does not receive energy or momentum from outside, then I find that the integral of jet power in the local rocket frame for the whole trip is

(1) $$W_{jet} = \frac{1}{2} m_s v_e^2 [ \exp \left( \frac{2 a_0 t_1}{v_e} \right) - 1 ]$$

and the total and specific jet power at any time during the trip is

(2) $$P_{jet} = \frac{1}{2} v_e a_0 m_s \exp \left( \frac{a_0}{v_e}(2t_1-t)\right)$$

and

(3) $$P_{jet} / m_t = \frac{1}{2}v_e a_0$$

respectively, where $v_e$ is the reaction mass ejection speed (with maximum possible value $v_e = c$ corresponding to a mass-annihilation drive), $m_t$ is the instantaneous rocket mass at time $t$, and the remaining symbols are from the paper. Inserting the values from the paper into (1) gives a value of around 1.7*10²³ J and (3) gives 1.5*10⁹ J/kg. This should be compared to the values in the paper of 8.0*10²³ J and 7.2*10¹⁰ J/kg (where I have converted the last power to specific power using 10⁵ kg).

While the above numbers are within an order of magnitude or so of each other, I suspect that the result of equation (1.3) applied to (1.1) will diverge heavily from (1) for longer trips, like a trip for 10⁶ ly and 26.9 years. For that trip, equation (1) gives around 4.8*10³³ J (specific power remains at 1.5*10⁹ J/kg no matter how long the trip is).

 

[Lsos]

Interesting, I must honestly say I never considered this possibility! I mean surely everyone knows of the von neumann machines, but I never thought to make one create human life on the habitable planets they find!

That's the point I was trying to make. I'm surprised you skipped right over it.

We can already clone mammals, so half the problem is already solved.

For the other half, at the rate computing power is growing, they say we'll have computers as powerful as human brains in 20 years. Not long after that, we'll be able to download our minds onto them...and vice versa.

At this point, it's just a matter of sending a spaceship that can clone a human to another planet. We can probably already do that part now if we tried. Once you wait the initial 10-20-whatever years for the spaceship to get there, you can beam the contents of your mind to it, and effectively teleport there. Since it will be at the speed of light, for you it will be instantaneous...but of course on Earth years will have passed. I don't think that will be a big deal by then...

 

[Nabeshin]

Now I was there, I noticed that it is a bit peculiar to apply the relativistic proper speed from equation (1.3) to equation (1.1) for required energy. I would think that to make sense you would either have to integrate vehicle power up in the Earth frame (to calculate how much energy you must transfer from the Earth frame to move the vehicle) or you would have integrate jet power in the rocket frame (to get the amount work the engine would have to provide to obtain the given acceleration for the whole trip). Applying (1.3) to (1.1) seems to be doing a bit both, but I cannot see the rationale behind it.

Just to illustrate what I mean with jet power, assume for a minute that the rocket is self-contained and does not receive energy or momentum from outside, then I find that the integral of jet power in the local rocket frame for the whole trip is

(1) $$W_{jet} = \frac{1}{2} m_s v_e^2 [ \exp \left( \frac{2 a_0 t_1}{v_e} \right) - 1 ]$$

and the total and specific jet power at any time during the trip is

(2) $$P_{jet} = \frac{1}{2} v_e a_0 m_s \exp \left( \frac{a_0}{v_e}(2t_1-t)\right)$$

and

(3) $$P_{jet} / m_t = \frac{1}{2}v_e a_0$$

respectively, where $v_e$ is the reaction mass ejection speed (with maximum possible value $v_e = c$ corresponding to a mass-annihilation drive), $m_t$ is the instantaneous rocket mass at time $t$, and the remaining symbols are from the paper. Inserting the values from the paper into (1) gives a value of around 1.7*10²³ J and (3) gives 1.5*10⁹ J/kg. This should be compared to the values in the paper of 8.0*10²³ J and 7.2*10¹⁰ J/kg (where I have converted the last power to specific power using 10⁵ kg).

While the above numbers are within an order of magnitude or so of each other, I suspect that the result of equation (1.3) applied to (1.1) will diverge heavily from (1) for longer trips, like a trip for 10⁶ ly and 26.9 years. For that trip, equation (1) gives around 4.8*10³³ J (specific power remains at 1.5*10⁹ J/kg no matter how long the trip is).

Alright excellent this is the kind of analysis I was looking for! So let me see if I understand your concern... You think I'm accidentally mixing equations from two different reference frames in my work equation? Here was my thought process in deriving it:
Start with the most basic, everyone knows that
$$W=\int F dx$$
So initially the Earth will observe an acceleration a₀, which by Newton's good ol' 2nd is a force of ma₀. Seeing as we run the engine at the same capacity throughout the entire voyage, I call this force a constant and remove it from the integral.
$$W=ma_0 \int dx$$
Are we in agreement up until this point? Or do you disagree with this argument so far? I'll assume we're okay...

Now the question is, which path length is this: the one observed from the Earth or the one observed from the ship's point of view? Owing to length contraction, the ship necessarily traverses a shorter path than the 4.2 ly, so the two numbers do not coincide. I think this is where your disagreement actually lays, no? In the paper, the original version had just the Earth distance, but I dropped this in favor of the ship distance in about September. Now that you got me thinking about it again, I'm not entirely sure...

We can imagine some giant hand from the Earth physically pushing the ship along its voyage. No doubt the hand pushes through the full distance d. But is the force it pushes with always necessarily ma₀?

I think a (somewhat) simpler example might be the following: The spacecraft consumes power at a rate P, for simplicity, say 1MW. Say they go on a quick trip. Earth says it took you 100 days to complete the trip, you must have used 100days*1MW of power. The ship says no, it took us 10 days and we used 10days*1MW of power. I think in this circumstance my intuition is much stronger to suggest that the ship's bookkeeping is the correct one. By analogy, then, the formula given in the paper should be correct.

What do you think, or have I completely missed your argument?

 

[Filip Larsen]

You think I'm accidentally mixing equations from two different reference frames in my work equation?

Yes, and I guess I am also a bit puzzled about what energy it really makes sense to analyse. I assume that since the paper strives to establish limits and constraints to "practical" relativistic flight the energy used must be something that would be relevant for the design of a propulsive system for such flights.

I can only see to options (or extreme end-points, if you will) for the propulsive system: either the mass of the vehicle is unchanged by propulsion and all change in vehicle momentum and energy comes from the Earth frame, or the vehicle is a self-contained rocket delivering its "own" change in momentum and energy with no particular tie to the Earth frame once the trip has started. In the first case I gather it makes sense to analyse the power transferred to the vehicle, like if we consider the vehicle as a big particle in a linear relativistic accelerator. In the second case it seems to make more sense to analyse the engine energy (or jet power) required in the rocket frame since this is where all the action takes place.

Here was my thought process in deriving it:
Start with the most basic, everyone knows that
$$W=\int F dx$$
So initially the Earth will observe an acceleration a₀, which by Newton's good ol' 2nd is a force of ma₀. Seeing as we run the engine at the same capacity throughout the entire voyage, I call this force a constant and remove it from the integral.
$$W=ma_0 \int dx$$
Are we in agreement up until this point? Or do you disagree with this argument so far? I'll assume we're okay...

I'm not sure what frame you are using here.

If you select the Earth frame, then the acceleration is decreasing with a factor $1/\gamma^3$ and (relativistic) mass is increasing with a factor $\gamma$, so force is overall decreasing with factor $1/\gamma^2$ and can hardly be considered constant except for short trip, right? And if you do want to evaluate this integral in the Earth frame, wouldn't it then make more sense to use relativistic energy-momentum equation like for a particle accelerated to relativistic speeds in a lab?

Alternatively, if you select to integrate in the rocket frame but use the contracted path length "back" to Earth as distance, then I'm puzzled if this Newtonian integral is a "legal" integral for work in a relativistic context. I mean, the path is obviously being contracted which means that if the propulsive system delivers more energy (e.g. operates for a longer time) it will have done its work over a shorter path and therefore delivered less energy? (Edit: there is a bit too much hand-waving in that last conclusion - please just read this as an indication that I am confused) That does not make sense to me, so either I missed something or the integral is "dubious". It may be that there are some "trick" where you perhaps can transform a Newtonian work integral from the rocket frame to the Earth frame using hyperbolic trigonometry (like Newtonian speeds that are added algebraically can be transformed using hyperbolic tangent), but it seem you then ought to arrive with something similar to the energy-momentum equation if conservation of energy and momentum still are to hold.

Finally, if you select to integrate in the rocket frame, but use the uncontracted "Newtonian" path length the rocket itself "sees", then we are back at a work integral that looks valid, but which are then expressing the energy required solely from the vehicles frame, i.e. its the jet power. Using the specific jet power (like equation (3)) has the benefit that its value seems to be constant and equal both for constant mass vehicles getting propulsion energy from the Earth frame and for self-contained rockets, thus charactering the lower bound of energy usage not matter the design of the propulsive system.

So, as you can probably gather, I have talked myself pretty warm on giving the last option the most sense, ie. that the energy considered are the integral of the jet power.

Now the question is, which path length is this: the one observed from the Earth or the one observed from the ship's point of view? Owing to length contraction, the ship necessarily traverses a shorter path than the 4.2 ly, so the two numbers do not coincide. I think this is where your disagreement actually lays, no? In the paper, the original version had just the Earth distance, but I dropped this in favor of the ship distance in about September. Now that you got me thinking about it again, I'm not entirely sure...

We can imagine some giant hand from the Earth physically pushing the ship along its voyage. No doubt the hand pushes through the full distance d. But is the force it pushes with always necessarily ma₀?

According to my argument earlier I would say no. As seen from the Earth frame the force cannot in general be constant if the local acceleration of the rocket is constant.

I think a (somewhat) simpler example might be the following: The spacecraft consumes power at a rate P, for simplicity, say 1MW. Say they go on a quick trip. Earth says it took you 100 days to complete the trip, you must have used 100days*1MW of power. The ship says no, it took us 10 days and we used 10days*1MW of power. I think in this circumstance my intuition is much stronger to suggest that the ship's bookkeeping is the correct one. By analogy, then, the formula given in the paper should be correct.

While I do agree that the rocket frame is the "interesting" frame here, I also think the problem with this particular argument is that its assumptions do not hold. If you select the Earth frame the power will decrease over time, and I would in fact boldly claim, that if we invoke the principle of conservation of energy, then all types of work integrals should end up giving the same value for the whole system (e.g. possibly including any ejected fuel). So, selection of frame or path really should not affect the final value of the energy, only the ease of which the integral may be calculated with.

What do you think, or have I completely missed your argument?

Not completely missed, no, but it is not right on target either as I have tried to argument for above

If you are interested in resolving this issue, then perhaps you can make some numerical experiments and test the various ways to calculate energy for short and long trips. While a trip for, say, one million light years are pure engineering science fiction, it may provide a hint to which energy values that are sensible considering the vehicle only travels for a short time.