Talk:Relativistic rocket

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[edit] Relativistic Rocket Problem

I will present a credible theoretical design for a relatavistic , or semi-relatavistic rocket here. For now I will use algebra & geometry based newtonian physics to describe such a rocket . A calculas based version of this description may also be presented here in a future improved version of this article.* Improvements to the article may be made by other editors here if the nature of their work is good, and correct. Let all editors feel free to correct what ever spelling errors, spacing errors, or puntuation errors may be found in this article. Tmayes1965


Please excuse my clumsy editing. This is my first contribution here, and I don't know much about the markup and conventions. I was trying to find the tag for attaching a signature with time and date, but I'll just type them in.

1) Please kill "relatavistic", "relavistic". relativistic please.
2) 2nd paragraph contains two conflicting definitions of "relativistic rocket"

I have now fixed this contradiction.*tim

3) Punctuation is messed up. Please put a space after commas and periods, not before.
4) "wickipedia"?! "cruising velocity"?!!

Article begun by OneRock on 7/23/04.

Somewhere, I have a numerical Fortran solution (or did I write that in Basic) of the general-relativistic rocket problem on a 5 1/4 floppy, which I can no longer access. It ran quite well on my old TRS-80 with no hard drive. I don't think I have the will to solve it again, but I am curious whether anyone else has such a program.

In case you want to duplicate my work, here's what it involves:

First, you must postulate the existence of a propulsion mechanism with specific parameters. My imaginary rocket was propelled by a laser which converts matter and antimatter to a light beam with 100% efficiency. (Onerock 19:40, 28 Jul 2004 (UTC)) I propose this engine as an absolute upper limit on what can plausibly be done without violating widely accepted physical principles.) The program user was asked to specify the mass and thrust of the engine. P.S.: Exploring this site, I see that Eugen Sänger proposed a nuclear photonic rocket[1] long before the discover of the laser.

Second, you must determine the rate at which fuel must be consumed to deliver the required thrust. In the case of the laser drive, thrust is equal to the power of the laser divided by the speed of light. Begin with thrust (force), solve for power, and covert to mass--using E= Mc^2.

Third, you must attach your laser drive engine to a vehicle, and set up the problem of incrimentally calculating velocity after a short time interval. If you do this in a general relativity format, the solution will be valid all the way to relativistic speeds. (See correction below.)

I originally attempted to obtain a classical formula for the result, but I was thwarted by an integral of a form I never encountered in three years of Integral Calculus. Does anyone know how to solve an integral of the following form?

Integral from A to [the integral from A to B of F(x) dx] of G(x) dx.

Sorry I can't be more specific about that integral without tackling the rocket problem anew.

Interestingly, it turns out that such a rocket could take you across the galaxy and back in a few years (on the traveler's clock)---if not for some rather daunting obstacles. First, you would need to start your journey with a matter-anitmatter fuel supply with about as much mass as Earth's Moon. All that mass would come in very handy as a shield against the inevitable collision with a grain of sand which would explode like an atomic bomb. The travelers would have to be frozen in a solid block of ice to withstand the high G-forces. And of course, the Earth might not be here when you return a few years older but hundreds of thousands of Earth-years later.

(Onerock 19:40, 24 Jul 2004 (UTC)) Yesterday, I wrote, impromptu, during my 1-hour daily internet slot (Tuesday thru Saturday at the public library). Today’s message has been written at a more leisurely pace at home.

I hypothesized the 100% efficient matter-antimatter laser-drive engine, not because it is feasible, but because I believe it places an absolute upper limit on what can plausibly be achieved without violating any widely accepted principles of physics. This is not Star Trek; it is real physics. If you want feasible results, you need only reduce the efficiency of the engine and its power to mass ratio. I’m afraid today’s lasers are many orders of magnitude too wimpy to get you to the corner drug store, let alone the far side of the galaxy.

Correction: It is not necessary to apply, or even know, the general relativity formulas. As the time interval, Delta-T, approaches zero, general relativity reduces to special relativity. Consequently, my numerical approach does not rely on such esoteric concepts as curved four-dimensional space.

Apply special-relativity formulas to calculate changes in position, velocity, mass and time for a short time interval, Delta-T. Plug the new values back into the algorithm, and the result after many short intervals should be exactly equal to that predicted by general relativity formulas for the total interval—provided that you chose a small enough Delta-T.

Start with a fairly large Delta-T (perhaps one hour); repeat with smaller values of Delta-T until successive calculations yield the same result within tolerable limits.

Relativistic effects are detectible at much lower velocities if you use double precision math; I’m not sure how much difference that makes. Lower precision with smaller Delta-T’s will probably yield the same result; I’m not sure which method takes more computing time. I used double precision, which yielded an accurate result in a few minutes of TRS-80 computer time for a trip across the galaxy. Today’s PC is a thousand times faster.

Calculations must be kept within a single reference frame. Mix your coordinate systems and the results will be invalid.

From the traveler’s point of view: My hypothetical engine’s thrust is constant; remaining fuel mass decreases at a constant rate; and acceleration increases in inverse proportion to the ship’s total mass with fuel. There is no relativistic mass to consider because the ship and fuel are stationary in the traveler’s coordinates. Due to this simplicity, it is probably easier to calculate the Earth’s position in the traveler’s coordinates, rather than the other way around.

From an Earthling’s point of view:

I’m not sure if thrust is increasing or decreasing. Since the relativistic mass of the engine is increasing, we might expect it to produce more light, but the thrust would be diminished in proportion to the red shift of the retreating light source. Conservation of mass-energy requires that the total of kinetic energy plus relativistic mass of the ship (including fuel) decreases by an amount equal to the total energy imparted to the laser beam since the beginning of the trip. I’m pretty sure I didn’t write my old program from this point of view; it’s too confusing.

I recommend that you calculate Earth’s parameters in traveler’s coordinates, but output spaceship parameters in Earth coordinates. With each millisecond on the traveler’s clock, you calculate the time on Earth; then for each hour on Earth, you find the corresponding time on the traveler’s clock and apply special relativity formulas to convert the other parameters. (At least I think that’s sort of the way it’s done. It’s been a very long time since I did it.)

Further comments:

How much acceleration does it take, and for how long, to reach relativistic speeds? Suppose you accelerate at one G in one direction for 355 Earth days. The Newtonian formulas put your speed at just over the speed of light. I’m too lazy to look up the formulas and do the math, but I’d guess general relativity puts your speed at roughly half of light speed.

In case you’re wondering, this laser-drive engine is no kid’s toy. A laser gun having a recoil force of 1000 Newton (i.e. 225 lb.) would consume fuel at a rate of 1000 Newton ¸ c = 1/3 microgram per second. The power of the beam would be 1000 Newton times c = 300 trillion Watt—approximately 100 times the most powerful nuclear power plant on Earth. (Hope I did that correctly in my head.) I wonder if the astronomical phenomenon of gamma-ray bursts might be our passage thru the exhaust of such a space ship in a distant galaxy.

In traveler’s coordinates, the distance across our galaxy decreases as the relative velocity increases. That is why, from the traveler’s point of view, it may take only a few years to cross a distance of 100,000 light-years without traveling faster than light. When the ship is stationary at each end of the journey, the width of the galaxy is 100,000 light-years; mid-way through the trip, the galaxy is only a few light-years across. When Einstein said you can’t travel faster than light, he meant relative to any object in your immediate vicinity. While you are accelerating, even at very modest rates, distant galaxies (both fore and aft) grow nearer or farther at many times the speed of light—without moving! Ain’t language funny!

(Onerock 19:40, 28 Jul 2004 (UTC))I moved my comments re. Twins Paradox to the appropriate forum.


199.250.57.32Onerock 7/29//04

It may be argued that fuel can be collected along the way. Unfortunately, any potential fuel that may be encountered along the way will be coming at you very close to light speed. Collecting fuel will be the furthest thing from your mind, since its momentum would slow you down—besides which, any collision with it would vaporize you. A more feasible way to conserve fuel is to chuck your excess engines into the matter-antimatter reactor when acceleration reaches the max allowable. In that case, most of the mass of the engines may be considered as fuel mass.

I may have found that old 5 1/4" floppy; the label just says “Batch Files”. I’ll see if I can find someone with an operational B-drive to read it. Some of the program is coming back to me, now.

The program should be initiated by user input specifying the following: Mass of the ship, Mass of cargo and crew, Mass of fuel, Mass of one engine, Thrust of one engine, Number of engines, Distance to destination. You may assume 100% efficiency or input a lower figure; calculate fuel consumption accordingly. You may also want to specify a maximum allowable deceleration so you won’t apply the brakes hard enough to crush the frozen bodies of the crew. You may set the initial velocity at zero (relative to the Sun) or get fancy and factor in the Earth’s solar-orbital velocity and the Sun’s galactic-orbital velocity, as well as an escape-velocity correction for the gravity of Earth and Sun. Similar corrections can be made for the gravity at your destination, if you already know what’s there—fat chance.

Use Newtonian formulas to calculate position, mass, time and velocity of the ship (s, m, t, v) after one δT in ship coordinates, which are initially the same as Earth (no-prime) coordinates.

a = thrust / (rest mass);  v = a·δT;  s = v·δT / 2;  t = t + δT.

That gives you position, time and velocity of the 1-prime coordinate system. Note that you are not using relativistic mass to calculate acceleration because you will always calculate acceleration in the ship’s coordinates, not Earth coordinates.

Next use Newtonian formulas to calculate the position, mass, time and velocity of the ship (s', m', t', v') after a second δT in 1-prime coordinates;

a' = thrust / (rest mass);  v' = a·δT;  s' = v·δT / 2;  t' = t + δT.

That gives you the position, time and velocity of the 2-prime (next) coordinate system in 1-prime (present) coordinates.

Using special-relativity, calculate the position, time and velocity of the two-prime (next) coordinate system in Earth (zero-prime) coordinates. I think the velocity formula is v (i.e., velocity of next coordinate system relative to Earth) = (v + v') / (1+ v×v' / c2). I haven’t yet found the position and time formulas. Anybody here know them? I think the answer might be found under the heading of Lorenz transformation http://en.wikipedia.org/wiki/Lorentz_transformation or Poincare transformations http://en.wikipedia.org/wiki/Poincar%E9_group.

You will loop back to calculate a new s', m', t', and v', and repeat the same steps n times to get the position, mass, time and velocity of the ship in Earth coordinates after n δT’s. Before looping back, however, you must determine whether it is time to begin coasting.

You will need to begin coasting and turn the ship around when the remaining fuel is just sufficient to stop the ship at its destination. To calculate the deceleration trajectory, you must run time backwards from touchdown at the destination. In forward time, the arrival fuel mass will decrease, and acceleration will increase until it reaches the max allowable value. From then on, you will convert one engine at a time to fuel, and adjust your thrust, acceleration and fuel consumption accordingly. You will run out of fuel upon arrival.

However, you will be doing these calculations in reverse time; so arrival fuel mass begins at zero and increases, acceleration decreases until one additional engine can be created from fuel mass without exceeding max acceleration. In reverse time, you have negative fuel consumption as you accelerate toward Earth. You begin with the minimum number of engines needed to produce maximum acceleration; you convert fuel mass to engines each time your acceleration decreases below maximum.

Coasting must begin at the point where both velocity and fuel mass accumulated in reverse time relative to the destination are equal to velocity and fuel remaining in forward time relative to Earth. The program should alternate between the departure trajectory from Earth and arrival trajectory at the destination—in leap-frog fashion. Each time the velocity in one direction increases beyond the other, you compare fuel masses; if the arrival fuel mass is greater, then it is past time to begin coasting.

That concludes one iteration of the program. You should loop back to the beginning and repeat the calculation using a smaller δT—perhaps half as large. When two consecutive iterations yield the same result, withing tolerance limits, you output the resulting trajectory. The output should, at least, describe the duration of the journey in both Earth coordinates and in ship coordinates. It might also tell the time, location and mass at the beginning and end of coasting in both Earth and ship coordinates and the number of engines remaining at touchdown.

The program I am describing is one dimensional. A sensible space traveler would probably go the long way—following an arc well outside the plane of the galaxy’s disk—to hopefully avoid colliding with dark matter.

The program also ignores the effects of gravitational bodies, such as Earth the Sun the galaxy, etc. If you hypothesize a really peppy spacecraft, say ten G’s at liftoff, and you’re only going to Alpha Centauri and back, those gravitational fields will not greatly alter your trajectory. A little more realistically, you will begin with a tremendous mass of fuel (perhaps equal to the Moon’s mass) and very small acceleration; it may take a significant portion of your fuel to overcome the gravity of the Earth and Sun. You may need to spiral once or twice around the Sun before heading into deep space. At the other end of the trip, your fuel mass is greatly reduced, so gravitational fields are insignificant by comparison to your maxxed-out deceleration.


If you wish to make a round trip, you will have to take with you the ability to replenish your matter-antimatter fuel supply at the turn-around point. You’ll probably want to carry only the most basic tools and the information needed to build from there to a functional antimatter production facility. Bear in mind, you’ll be a few years older when you return to Earth, but Earth will be about 200,000 years older, so what’s the point of returning? How many dark ages will have ensued? Will the master race be silicon-based descendants of our computers or will it be an intelligent cockroach? I doubt if either will be friendly to a 21st Century human.

Since any information obtained by such a journey would be unavailable even to the descendants of those who remain behind, who would be willing to pay the bill? Would those making the trip have to pick up the tab before leaving Earth. The more people going, the greater the expense. So a very few people would have to enslave the rest and leave them behind after milking them dry. The cost of this project would make the pyramids of Egypt look like an item on an employee’s expense voucher.

Has anyone read this yet?

I've read as much as I can :) Anyway, there was a science fiction story many years back about a space faring race that created highways by collapsing suitable stars into black holes, turning all the resulting energy into laser beams that were bent into loops by the created blackholes. The idea was that to travel through space one simply dipped a mirror into these beams and got an immediate thrust. The beam also cleared the path of any debris. The reason I'm posting this is the fact that your calculations assume that the reaction mass-energy is carried with the ship, but in fact beaming the energy out makes the limit somewhat less useful. njh 05:30, 31 December 2005 (UTC)

[edit] Cleanup

This needs lots of cleanup such as spelling, italic variables, consistent symbols for variables, superscripts, and capitalization for starters. Plus it needs to be broken up into sections and otherwise wikified. Gene Nygaard 01:59, 28 January 2006 (UTC)

Not just consistent variables but conventional ones, too, and other things like leading zeros. Gene Nygaard 02:01, 28 January 2006 (UTC)

Be more specific about what things you think need changing, and state justification for change"? If you see any actual spelling, puntuation, or grammar errors you are welcome to fix them.?

"? -> .
puntuation->punctuation
.? -> .
(there are a lot of errors in this article, I just haven't had the time to go through and fix them - perhaps you could run the whole thing through a spell chequer and fix the worst offenders) njh 08:54, 6 February 2006 (UTC)

[edit] Relativistic rocket formulas

I've put a bunch of formulas at User:Wwoods/Relativistic rocket formulas. I collected or worked them out several years ago, so I don't have the derivations at hand, but for most of them it shouldn't be to hard to verify them. —wwoods 22:36, 10 February 2006 (UTC)

[edit] Additions by Tmayes1965

A long series of paragraphs have been added to this article by the above user, describing the design of a theoretical relativistic rocket. While there are a number of useful points on design considerations that would need to be taken into account, as it stands the additions are much more a prescription for the design on an individual rocket. Sentences such as "It would have a rotating crew habitat" or "There would be an array of high energy lasers, particle beams and some missiles mounted on the starship" are nothing more than crystal ball gazing. Would it make more sense to try and selectively eliminate passages that refer to non-critical design features, move the whole section to a new article on this ""beam core" pion rocket" or just wipe the entire section and try to rebuild a few useful points of consideration, such as the need for sheilding at the bow of the craft? Icelight 01:18, 7 June 2006 (UTC)


[edit] Addition/Correction/Question - Andy Stangeland 02 Oct 2006

In the article it mentions that "Gamma rays can be reflected by some materials such as beryllium." This is counter to every other discussion about gamma rays that I've seen. Other sites have said that only theoretical, quantum engineered materials, could deflect a gamma ray. Additionally, I can't find any other resource that says that gamma rays can be deflected. High energy X-rays (lower than gamma rays) can be deflected or refracted at small angles, but the only information I've found on gamma rays said it could be stopped with 2 inches of lead or tungsten. (CRC) Never anything about deflection though.

[edit] Nonsense

This whole article is quite a nonsense. It is not useful to derive from the special theory of relativity, since it is obvious from conversation of momentum that a relativic velocity never can be reached by a rocket. From the law of conversation of momentum one easly derives

m \ \frac{dv}{dt} + dm \ v_e = 0

for the velocity v of the rocket of mass m and the velocity ve of the exhaust gases. This leads to

dv = v_{e} \frac{dm}{m}

and finally by integrating

v = v_{e} \ln\left(\frac{m(t=0)}{m(t)}\right).

Moreover the relativistic calculation is just wrong, since as an example, there is not a uniform velocity ve of the exhaust gases. The velecity ve is only constant relative to the rocket but not in any frame which is not accelerated.
—The preceding unsigned comment was added by 84.169.245.54 (talk • contribs) 25 November 2006 (UTC)

The relativistic rocket equation is
\tanh^{-1}\frac{v(t)}{c} = \frac{v_e}{c} \ln\left(\frac{m_0}{m(t)}\right).
For v \ll c, this reduces to the non-relativistic version:
v(t) = v_e \ln\left(\frac{m_0}{m(t)}\right).
The exhaust "velocity", ve, is a figure of merit for a rocket, not necessarily the speed of anything.
—wwoods 21:33, 26 November 2006 (UTC)
Ok, one may define an exhaust velocity just as the ratio of the disired total momentum the space vehicle finally gains to the mass of the propellant. This ratio can indeed be regarded as a figure of merit for a space vehicle and has the dimension of a velocity, but is not the speed of anything. This is in contrast to the classical derivation of the rocket equation, since here ve is in fact the velocity of the exhausted gases measued in the rest frame of the rocket.


[edit] Additions by 71.124.2.194

I reverted these additions:

It should be noted that any electrical source can be used to have an effective exhaust velocity of nearly c using a linear acelerator.

This is not true because you have to include the energy source's fuel ...

The implication of this is that even a c exhaust velocity needs a fuel ratio of 99.9997%; effectively making a 50 ton spaceship require 2 million tons of fuel to have 0.5c final velocity. Being able to decelerate also would mean 4x the fuel mass.

The calculation is obviously flawed.

Icek 05:46, 28 December 2006 (UTC)

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