Is Faster-than-Light Travel Possible with Constant Acceleration?

[JesseM]

The reason I bring up the speed of zero is because it is a "resting speed" simply by definition. If the speed of light was considered the "resting speed", then there would be reason to imagine a speed less than zero.

Again, speed cannot be less than zero by the definition of speed itself, so what you're saying doesn't make sense. Speed is always measured in the context of some coordinate system--for example, if we're using a coordinate system where an object is at position x=5 meters at time t=1 second, and then at position x=15 meters at time t=2 seconds, we'd say its speed was 10 meters/second in that coordinate system. If you want to, you can define a coordinate system where a given light wave is at rest, meaning its coordinate position doesn't change over time (though the equations of relativity are only supposed to work in a special class of coordinate systems called 'inertial reference frames', and this wouldn't be one of them so you'd have to write the laws of physics differently in such a coordinate system, and normal rules of relativity like 'nothing moves faster than light' wouldn't apply), but in this coordinate system objects which aren't moving at the speed of light would be moving faster than the light wave, not slower.

 

[peter0302]

Something that will come in handy for exploring Gliese 581. At a comfortable 1G, the trip takes a mere 6.1 years ship's time while the Earth time is a tolerable 22.6 years. I made note of this in my http://www.davesbrain.ca/science/gliese/gliese_primer.html" .

How much fuel would that take?

 

[DaveC426913]

How much fuel would that take?

It would take as many gallons of fuel as there are faeries that can dance on the head of a pin.

 

[peter0302]

I actually was serious. Fission fuel lasts a very long time.

 

[JesseM]

There are some fuel calculations at the bottom of the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html page. It turns out that even if you had a perfectly efficient way of converting fuel mass into forward kinetic energy (even a hypothetical matter/antimatter drive wouldn't do this 100% efficiently), you'd still need 10,000 times more fuel mass than the mass of the payload just to get to the nearest star 4.3 light years away at constant 1g acceleration (and if you wanted to accelerate at 1g for the first half of the trip and then decelerate at 1g for the second half so you could actually stop at your destination rather than zipping by it, you'd need 38,000 times more fuel mass). The basic problem is that if you carry your fuel with you, then at any given point in the trip you have to be burning enough fuel not just to accelerate the payload at 1G, but also to accelerate all the fuel you'll need for the remainder of the trip. So, the most "realistic" proposals for interstellar travel usually sidestep the problem of carrying your fuel with you by having the payload attached to something like a giant sail which is pushed along by a beam sent out from our solar system, either an enormous laser or perhaps something like a beam of tiny pellets (the advantage of the latter is that the pellets could be self-steering to some degree which would cut down on the problem of the beam becoming decollimated and missing the sail). In this case you wouldn't be accelerating at a constant rate though, since the force on the sail as seen in the solar system's frame would be at best constant and more likely decreasing due to decollimation, whereas the force needed to produce a given acceleration on a moving object is greater the larger its fraction of light speed.

 

[peter0302]

Yeah, your last point is exactly the biggest problem, I think. You don't want to carry your fuel with you, because it's heavy, but if you don't carry your fuel with you, then your relativistic mass increases requiring more fuel anyway. Bummer!

We need to learn to harness vacuum energy...

 

[JesseM]

I actually was serious. Fission fuel lasts a very long time.

A fission rocket would require a vastly larger fuel-to-payload ratio for a 1G trip to the nearest star than the one I quoted above for a 100% efficient rocket. Quoting something I wrote on this in another post a while ago:

This page gives Tsiolkovsky’s equation for the relation between change in velocity, payload mass and initial fuel mass:

Mpayload/mrocket = exp(-delta v/exhaust velocity)

This equation is a classical one which would need to be modified if delta v were close to the speed of light, but it can give you a sense of the huge amount of fuel needed if you just figure out the mass needed to get to some small fraction of light speed, like 0.01c, where the relativistic correction shouldn't be too big. They give the exhaust velocity for a chemical rocket as 4000 m/sec, and the exhaust velocity for a fission rocket as "12,000 m/sec (for solid-core nuclear thermal with oxygen augmentation), 40,000 m/sec (for nuclear electric propulsion), 100,000 m/sec (for more exotic and theoretical forms)". Using the 40,000 m/sec figure, to accelerate from being at rest wrt Earth to traveling at 0.01c relative to Earth (again, just calculating the answer using Newtonian physics without taking into account relativity, since the time dilation factor is very small at this speed), the equation tells us the mass of the rocket would have to be about e^75 times greater than the mass of the payload, which is about 3.5 * 10^32. If you want the answer in terms of acceleration, this thread gives the equation:

acceleration* time = specific impulse * ln(mass ratio)

with each type of rocket having its own specific impulse (wikipedia's relativistic rocket page mentions that specific impulse is the same as exhaust velocity)...rearranging, this should mean the mass ratio needed to accelerate at 1G for some time t would be:

e^(9.8 m/s^2 * t / specific impulse)

If we again use 40,000 m/s for the specific impulse, this becomes:

e^(t * 0.000245/s)

So, to accelerate at 1G for 3 days (259200 seconds) would require a mass ratio of e^63.5, or a total initial rocket mass about 3.8 * 10^27 greater than the payload mass. This page mentions that for an antimatter rocket you might have an exhaust velocity of 10,000,000 m/s, so plugging that into the equation would give the mass ratio as:e^(t * 0.00000098/s)

This would make 1G acceleration for a few days much more manageable, but to accelerate for 1 year (31536000 seconds) you'd need a mass ratio of e^(30.9), so the rocket would have to be about 26 trillion times more massive than the payload--that's a lot of antimatter!

 

[peter0302]

Wow. Thanks. I mean, you hear about air force fighters accelerating to many-Gs, but you don't really realize how much energy that actually requires until you consider what it would cost to reach such accelerations for a very long time.

Maybe a slingshot around the sun is the answer. :)

 

[DaveC426913]

I actually was serious. Fission fuel lasts a very long time.

Well, so was I. Your next question would ostensibly have been: 'for what size faery?' To which I would have responded 'for what type of fuel?' And there you would have had the missing element in your question.

 

[peter0302]

Every time I think the anal-retentiveness of these forums cannot get any higher, I am proven wrong.

 

[DaveC426913]

Every time I think the anal-retentiveness of these forums cannot get any higher, I am proven wrong.

"I do not sink zat means what you sink it means."

 

[JesseM]

"I do not sink zat means what you sink it means."

Well, your response was humorously pedantic, and to be pedantic myself I'll note that anal-retentive is listed as a synonym.

 

[peter0302]

HAHAH. Thanks Dr. Freud.