antimatter drive / relativity question

[quadraphonics]

Here's a toy problem that's always puzzled me. I've put it past a couple of physics teachers and TAs, but never received a satisfactory explanation.

Suppose we have a space ship powered by an anti-matter drive. That is, it has a stock of fuel in the form of some material (say, hydrogen ions) and an equal quantity of anti-fuel. The drive then extracts energy by performing annihilation reactions. We know from special relativity that it's impossible for the ship to accelerate past the speed of light, but the usual explanation given for this effect (at least in introductory relativity courses) is that the mass increases with speed, and so it takes an ever-increasing amount of energy to accelerate the space ship. However, it would seem that the mass of the fuel, and so energy available from the annihilations, would increase at precisely the same rate as the space ship, thus providing the additional energy, allowing the ship to accelerate without bound.

I know there must be some flaw in this thinking, but I can't see it. I'm guessing it would be instructive to consider what the situation looks like from an earth frame vs. the spaceship frame?

 

[Pete]

Hi quadraphonics,
You're thinking goes beyond the beginner stuff. Good work!

Very briefly, the key is that the energy released on annihilation depends on the rest mass of the annihilated particles, not their relativistic mass.

Something to think about:
Here's the full version of Einstein's most famous equation:
E² = (mc²)² + (pc)²

m in this equation is rest mass.
When p (momentum) is zero, it reduces to E = mc²

 

[Farsight]

I'd like to back Pete up on that. Check out invariant mass:

http://en.wikipedia.org/wiki/Rest_mass

One way to think about this is to think about matter in terms of spring steel hoops. Let's say the spaceship is made out electrons, and every electron is a hoop like this O. The hoops are springy, if you could cut one in half, you release the energy, and the two sections would fly off in opposite directions like this <~ boing ~>. The trouble is, if you go real fast, you suffer length contraction. So that's like having to push your hoop into an oval, until eventually it's got zero length and looks like this |. There just isn't enough tension in the hoops you annihilate to do this to the other hoops. There isn't enough energy.

 

[Trilairian]

Here's a toy problem that's always puzzled me. I've put it past a couple of physics teachers and TAs, but never received a satisfactory explanation.

Suppose we have a space ship powered by an anti-matter drive. That is, it has a stock of fuel in the form of some material (say, hydrogen ions) and an equal quantity of anti-fuel. The drive then extracts energy by performing annihilation reactions. We know from special relativity that it's impossible for the ship to accelerate past the speed of light, but the usual explanation given for this effect (at least in introductory relativity courses) is that the mass increases with speed, and so it takes an ever-increasing amount of energy to accelerate the space ship. However, it would seem that the mass of the fuel, and so energy available from the annihilations, would increase at precisely the same rate as the space ship, thus providing the additional energy, allowing the ship to accelerate without bound.

I know there must be some flaw in this thinking, but I can't see it. I'm guessing it would be instructive to consider what the situation looks like from an earth frame vs. the spaceship frame?

Mass doesn't change with speed as we've had that thread, and also in a recent thread I posted the equation for the rocket. Fine, for the second time in about two days, from conservation of relativistic momentum and energy the equation is
$$v = ctanh[\frac{v_{ex}}{c}ln(\frac{m_{i}}{m})]$$
where $$v_{ex}$$ is the velocity of the exhaust.
The tanh function has an asymtotic behavior that limits its speed to c no matter what exhaust velocity you have, even if it were tachyon exhaust.
Since you probably won't believe the math anyway think of it this way. No matter how miniscule you make the cabin it must have some finite amount of mass. It would take an infinite amount of energy to take something that has nonzero mass and push it up to the speed of light, so no matter how much of the rest of the ship you burn off as fuel even as matter/antimatter, whatever remains would never have reached the speed of light.

 

[weed_eater_guy]

I'm very much a noob with this sort of thing, but that formula looks suprisingly similar to the one I used to determine launch velotity for an exhaust velocity and a given mass-fraction. I'm guessing it's a modification? Very cool though, nice to know this formula, thanks!

 

[Trilairian]

I'm very much a noob with this sort of thing, but that formula looks suprisingly similar to the one I used to determine launch velotity for an exhaust velocity and a given mass-fraction. I'm guessing it's a modification? Very cool though, nice to know this formula, thanks!

Your welcome. It's not a modification , but is the exact relativistic solution for that same problem. It is relativity's version of the equation that you are thinking of.

 

[Positron]

yes I appreciate that formula as well. As for antimatter engines/reactors whatever, how can you contain the energy released? I mean if it is released all at the same time like it should in an elimination wouldn't it just destroy the ship? I imagine you would be using positrons because the mass of an antiproton is just to large, but still, is there any way to contain the energy at present or even theoretically?

 

[zanket]

However, it would seem that the mass of the fuel, and so energy available from the annihilations, would increase at precisely the same rate as the space ship, thus providing the additional energy, allowing the ship to accelerate without bound.

The easy way to think about this, is to realize that we are now moving at almost c, the speed of light, relative to something else, yet we obviously don't get additional energy because of that.

Also a ship can accelerate indefinitely without ever attaining c. The speed would asymptote to c.

 

[quadraphonics]

Very briefly, the key is that the energy released on annihilation depends on the rest mass of the annihilated particles, not their relativistic mass.

Yeah, I suspected that was the problem. The situation makes sense to me from the perspective of the earth frame: the ship is moving really fast, and so its inertia is increasing with the relativistic mass, but the propulsion energy, being proportional to the rest mass of the fuel, is not keeping up. It seems that the bit about antimatter fuel was just a red herring then.

I still have trouble with what the physics look like from the frame of the space ship, though. In that frame, the inertia of the space ship doesn't change, so why does the same propulsion energy give decreasing returns? Obviously the spaceship frame isn't inertial; presumably this gets accounted for by a frame force?

 

[Montec]

Hi quadraphonics

I believe the slowing of time may be the a key to the decreasing energy returns.

 

[zanket]

I still have trouble with what the physics look like from the frame of the space ship, though. In that frame, the inertia of the space ship doesn't change, so why does the same propulsion energy give decreasing returns?

The "decreasing returns" exist only from the misleading perspective of velocity. The longer the ship accelerates in anyone's frame, the shorter the time it takes on the crew's clock for them to travel between any given two points in the earth frame. That time can be arbitrarily short.

 

[Pete]

I still have trouble with what the physics look like from the frame of the space ship, though. In that frame, the inertia of the space ship doesn't change, so why does the same propulsion energy give decreasing returns? Obviously the spaceship frame isn't inertial; presumably this gets accounted for by a frame force?

Hi quadraphonics,
I find that the best way to get a good intuitive grasp is to immerse yourself in the maths. Burrow in deep, and discover the real relationships. The mathematical description is the concise and authoratative description of relativity, so it should be the benchmark for all intuitive interpretations.

I don't have a great grasp of the accelerating spaceship frame. Check out Trilairian's posts for that. He's a bit abrasive and often reads more stupidity into posts than is warranted, but he knows his stuff.


Here's my best approach. Take with salt :
It might be instructive to consider an inertial frame in which the spaceship is at rest for an instant. Perhaps one in which the Earth is moving at 0.999c.
Imagine an observer in that reference frame.

They will measure the Earth to be moving at 0.999c.
They will measure the rocket to first be going backward, decellerate to rest, then accelerate forward. During the time that the rocket has a low speed in that frame, the observer measures the rocket to be accelerating as expected - the propulsion energy doesn't give decreasing returns, it works precisely as expected.


So what's really going on? Does the motion of rocket dictate the decreasing returns, or the motion of the frame that we measure it in? Or is it the relationship between the two?


You might also think about what the "returns" are. In terms of kinetic energy and momentum, do the returns really decrease?

 

[quadraphonics]

I don't have a great grasp of the accelerating spaceship frame. Check out Trilairian's posts for that. He's a bit abrasive and often reads more stupidity into posts than is warranted, but he knows his stuff.

Do you have a specific post in mind? I've seen him post the final result for the rocket velocity in a couple of places, but I haven't seen the actual derivation. Perhaps it's in one of the name-calling threads, which are very difficult to glean any useful information from...

You might also think about what the "returns" are. In terms of kinetic energy and momentum, do the returns really decrease?

Well, I was thinking of "returns" as change in the ship's speed due to, say, a single annihilation reaction. It seems to me that in the (instantaneous) rest frame of the ship, the usual Newtonian relationships should still hold, and so a constant propulsion energy should result in a constant change in speed (neglecting the decrease in the ship's mass due to fuel being expended). Perhaps the problem lies in computing the new speed of the earth in space ship's rest frame; since, eventually, the earth is moving at relativistic speeds in the (pre-acceleration) spaceship frame, one can't simply add the change in ship speed to the old earth speed to get the new earth speed... That is, even though the Newtonian relations hold for the space-ship in the instantaneous rest frame, at the end of the process we're interested in the relative velocity with an object (the earth) that is moving at relativistic speeds in both the pre- and post- acceleration ship frames, which then accounts for the difference.

 

[Trilairian]

Yeah, I suspected that was the problem. The situation makes sense to me from the perspective of the earth frame: the ship is moving really fast, and so its inertia is increasing with the relativistic mass, but the propulsion energy, being proportional to the rest mass of the fuel, is not keeping up. It seems that the bit about antimatter fuel was just a red herring then.

I still have trouble with what the physics look like from the frame of the space ship, though. In that frame, the inertia of the space ship doesn't change, so why does the same propulsion energy give decreasing returns? Obviously the spaceship frame isn't inertial; presumably this gets accounted for by a frame force?

You need to get a new teacher. Mass doesn't change with speed. Confront him about this:

http://www.geocities.com/zcphysicsms/chap3.htm

 

[Trilairian]

Do you have a specific post in mind? I've seen him post the final result for the rocket velocity in a couple of places, but I haven't seen the actual derivation. Perhaps it's in one of the name-calling threads, which are very difficult to glean any useful information from...

You never asked me for it. I shouldn't give it to you because with your attitude you don't deserve to know, but for the interest of others that do I have posted it in a new thread.

 

[Zephyr]

<p class=MsoNormal>The tanh function has an asymtotic behavior that limits its speed to c no matter what exhaust velocity you have, <b>even if it were tachyon exhaust</b>.</p>

Although it's possible to get big numbers out of tanh if you apply it to a complex number.

All we need is imaginary mass...

 

[orcot]

Yust wondering but if you travel 1 lightyear at:

0.5C you would travel 730 days from a earth point of vieuw however due to relative effects you would only experience 632 days in the space ship

if you travel at

0.8C you would travel for 456.25 days from earths observation point but in reality you would only experience 273.75 days

If you simply assume that your speed = distance / time it takes you

then it will take you less then 366 days to reach your destination.

If you would travel at C then you would get a devided by 0 error in your math because of the relative effects (time=0). So at C you would move at a infinite speed because it would take no time at all to reach any distination.
The fact that you would be seen moving for a year on earth is nothing more then a optical illusion.

Now if it was only possible to construct a engine that doesn't shorten the time to get there but shortens the distance

Annyway that's my opinion why we can't move fasther then C

 

[Trilairian]

Although it's possible to get big numbers out of tanh if you apply it to a complex number.

All we need is imaginary mass...

No complex numbers enter in, only the exaust velocity and the initial and final masses of the real mass ship and fuel and the speed of light which are all real.

 

[Trilairian]

Yust wondering but if you travel 1 lightyear at:

0.5C you would travel 730 days from a earth point of vieuw however due to relative effects you would only experience 632 days in the space ship

if you travel at

0.8C you would travel for 456.25 days from earths observation point but in reality you would only experience 273.75 days

If you simply assume that your speed = distance / time it takes you

then it will take you less then 366 days to reach your destination.

If you would travel at C then you would get a devided by 0 error in your math because of the relative effects (time=0). So at C you would move at a infinite speed because it would take no time at all to reach any distination.
The fact that you would be seen moving for a year on earth is nothing more then a optical illusion.

Now if it was only possible to construct a engine that doesn't shorten the time to get there but shortens the distance

Annyway that's my opinion why we can't move fasther then C

I suppose if we had something like an Alcubierre warp drive then you could do that, but there are no illusions involved in the transformations.

 

[CANGAS]

I suppose if we had something like an Alcubierre warp drive then you could do that, but there are no illusions involved in the transformations.

Now that is a really clever response in defense of your theory, that will impress the hell out of the guys in Stockholm.