Calculating Ship Acceleration with Gauss Gun Propulsion

[BitWiz]

Hi,

Say I have a spaceship in ideal gravity-free, friction-free space. I have a source of power capable of producing a maximum of E joules per second, and I want to use some form of continuous electric propulsion, such as a Gauss gun or ion thruster, to get around.

I have these questions:

1) From the spaceship's frame, how do I calculate the ship's acceleration from E, given the ship's Lorentz-adjusted mass, the mass of projectiles fired per second, and any other required parameters? An equation would be very helpful, along with an example using simple values such as 1 second, 1 kg, 1 meter, etc.

2) Does the projectile's actual rest mass and final velocity really matter in calculating the reactive force F' that accelerates the ship? If the gluons will put up with it, can I theoretically accelerate a single proton in a Guass gun fast enough to produce an F' that accelerates a (very sturdy) ship to any Lorentz gamma in one ship second? In other words, given perfect engineering, can I directly translate any power to acceleration without regard to the speed-of-light c in the Gauss gun, simply because the projectile's effective mass will always adjust to balance momentum, relative to the ship?

3) As the Gauss field "pushes" the projectile, does the field itself change in a Lorentzian and/or Doppler way from either the projectile's frame or the ship's frame, that reduces the ability to continue to apply force to the projectile? For instance, if the projectile is traveling too fast to see changes in the field propagate from the source, does it still see the field at all?

4) How does the projectile appear to me (in the ship)? Does it seem to slow down as relativistic effects become conspicuous? If I manage to accelerate it to very close to c, does it ever leave the gun?

5) Finally, if I accelerate a proton to c minus one-zillionth (or so), does the proton become a black hole from my frame, ie does the Schwartzschild radius expand to encompass the boundary of the particle? Would it know it? How about for larger masses? To me, would the black hole appear virtually stationary?

Thanks very much.

Bit

 

[Dale]

There is too much to answer in depth, so I will answer briefly and then you can ask more detailed and specific questions.

1) From the spaceship's frame, how do I calculate the ship's acceleration from E, given the ship's Lorentz-adjusted mass, the mass of projectiles fired per second, and any other required parameters? An equation would be very helpful, along with an example using simple values such as 1 second, 1 kg, 1 meter, etc.

I would use the conservation of four-momentum.

2) Does the projectile's actual rest mass and final velocity really matter in calculating the reactive force F' that accelerates the ship? If the gluons will put up with it, can I theoretically accelerate a single proton in a Guass gun fast enough to produce an F' that accelerates a (very sturdy) ship to any Lorentz gamma in one ship second? In other words, given perfect engineering, can I directly translate any power to acceleration without regard to the speed-of-light c in the Gauss gun, simply because the projectile's effective mass will always adjust to balance momentum, relative to the ship?

The mass of your projectiles will matter. Remember, both energy and momentum are conserved, so the lighter the projectile the more of the E will go into the projectile.

3) As the Gauss field "pushes" the projectile, does the field itself change in a Lorentzian and/or Doppler way from either the projectile's frame or the ship's frame, that reduces the ability to continue to apply force to the projectile? For instance, if the projectile is traveling too fast to see changes in the field propagate from the source, does it still see the field at all?

Yes, but since the projectile is accelerating it is somewhat more complicated than simply a Lorentz transform.

4) How does the projectile appear to me (in the ship)? Does it seem to slow down as relativistic effects become conspicuous? If I manage to accelerate it to very close to c, does it ever leave the gun?

Yes, obviously if it accelerates close to c then it will leave the gun slightly later than a photon would.

5) Finally, if I accelerate a proton to c minus one-zillionth (or so), does the proton become a black hole from my frame, ie does the Schwartzschild radius expand to encompass the boundary of the particle? Would it know it? How about for larger masses? To me, would the black hole appear virtually stationary?

If it is not a black hole in its rest frame then it will not be a black hole in your frame.

 

[BitWiz]

Hi, Dale,

Thanks for the reply. I think I'm going to divide my response if you don't mind.

I would use the conservation of four-momentum.

Thanks, but can we collapse the three space vectors? I just need linear acceleration from E. My starting or ending coordinates don't matter (yet), just my (rocketry-style) delta-V between two points in time. Is there something simple enough to plug into Excel?

Thank you, sir.
Bit/Chris

 

[BitWiz]

The mass of your projectiles will matter. Remember, both energy and momentum are conserved, so the lighter the projectile the more of the E will go into the projectile.

Then I guess I don't understand Lorentz. If I use a force F to accelerate a mass for time t, I will end up with a delta-V of X and momentum M (from the object frame?). If I then repeat this process using the identical force and time, I will end up with a smaller X and 2M, will I not? The balance of the momentum change going into mass? Am I screwed up on this?

Thanks,
Chris

 

[Dale]

Thanks, but can we collapse the three space vectors? I just need linear acceleration from E. My starting or ending coordinates don't matter (yet), just my (rocketry-style) delta-V between two points in time. Is there something simple enough to plug into Excel?

The four-momentum won't give you position, just momentum and energy. It is the simplest mechanism I know for keeping track of both and is certainly appropriate for Excel.

 

[Dale]

Then I guess I don't understand Lorentz. If I use a force F to accelerate a mass for time t, I will end up with a delta-V of X and momentum M (from the object frame?). If I then repeat this process using the identical force and time, I will end up with a smaller X and 2M, will I not?

Yes, this is correct. However, you need to remember that in this type of propulsion system the force is not just on the gun, but also on the projectile. You cannot ignore the momentum and energy that goes into the projectile.

 

[BitWiz]

If it is not a black hole in its rest frame then it will not be a black hole in your frame.

Hi, Dale,

Let me try this another way. If I accelerate a mass, do its gravitational effects on space-time deepen from any frame along with increases in velocity, or is the "Lorentz mass effect" some other artifact that acts like an increase in mass?

Thanks!
Chris

 

[Dale]

Here is a FAQ link that I like. It explains how black holes do not form by going too fast.

http://www.edu-observatory.org/physics-faq/Relativity/BlackHoles/black_fast.html

The most important point in the FAQ entry is that in GR gravity couples not only to mass but also to momentum and momentum flow. While in Newtonian gravity the source of gravity is a single scalar field (mass density), in GR the source of gravity is the entire stress-energy tensor which has 10 independent components. While a relativistic mass does have a large energy component, it also has one or more large momentum components. This interaction prevents the formation of a black hole in other frames.

See:
http://en.wikipedia.org/wiki/Stress-energy_tensor
http://en.wikipedia.org/wiki/Einstein_field_equations