# The Speed of Light - Nothing can go faster?

#### Korlus

Hello there, and I apologise if I do something wrong, I'm new to the forum and have a general interest in physics. I was thinking back to something earlier (as sparked by a post I happened on in the 5 light year long stick thread) and got to wondering about whether how I understood physics and relativity to work was true.

The way I understand it is that no object with mass can accelerate to the speed of light, as its mass will continue to increase, meaning that the energy required to get there via acceleration would be infinite (and thus impossible). As far as I can remember then, based upon that, any object without mass must then go at the speed of light, and that theoretically, if an object had negative or imaginary mass, it would go faster than the speed of light.

Is that correct? Does that also mean that if, through some means, you could impart velocity/speed onto an object without causing it to accelerate (such as the ability to simply cause an object to come into existence with a velocity), would that then mean that that object could be created at, or even beyond the speed of light?

Furthermore, if that object that you created was moving at the speed of light, would it them have infinite mass, and so would have infinite momentum and never stop moving at the speed of light? There's something there that appears to be wrong to me, and so I would greatly appreciate some sorting out of my understanding of relativity, if no one minds?

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#### Calimero

The way I understand it is that no object with mass can accelerate to the speed of light, as its mass will continue to increase, meaning that the energy required to get there via acceleration would be infinite (and thus impossible). As far as I can remember then, based upon that, any object with mass must then go at the speed of light...

Hi Korlus, welcome to PF. You mean object with mass must go below the speed of light?

...and that theoretically, if an object had negative or imaginary mass, it would go faster than the speed of light.
You are probably referring to hypothetical particles - tachyons. Tachyons also can't break light speed barrier, just from opposite direction, i.e. they can't go slower than c. Interesting thing about them is slower they go - more energy they have. So, much the same as ordinary massive particle would require infinite energy to speed up to c, tachyon would require infinite energy to slow down to c. Although we can describe tachyon with real mass, it is often said that they have imaginary mass because in

$$E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

you would have imaginary denominator, since v>c. In order to have real energy, you must introduce imaginary nominator as two divided pure imaginary numbers result in real number.

Does that also mean that if, through some means, you could impart velocity/speed onto an object without causing it to accelerate (such as the ability to simply cause an object to come into existence with a velocity), would that then mean that that object could be created at, or even beyond the speed of light?

No, that is unphysical question, something like: what would be the laws of nature, if we were able to suspend them for a while, and then restore them?

#### Korlus

Hi Korlus, welcome to PF. You mean object with mass must go below the speed of light?
Yes, I'm sorry for the typo.

You are probably referring to hypothetical particles - tachyons. Tachyons also can't break light speed barrier, just from opposite direction, i.e. they can't go slower than c. Interesting thing about them is slower they go - more energy they have. So, much the same as ordinary massive particle would require infinite energy to speed up to c, tachyon would require infinite energy to slow down to c. Although we can describe tachyon with real mass, it is often said that they have imaginary mass because in

$$E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

you would have imaginary denominator, since v>c. In order to have real energy, you must introduce imaginary nominator as two divided pure imaginary numbers result in real number.
Many thanks. I imagine that I was, although I was trying to look at the theory in terms of the theory and only the theory, outside of (most) physical terms to understand it better.

No, that is unphysical question, something like: what would be the laws of nature, if we were able to suspend them for a while, and then restore them?
I understand and apologise (as an aside: With quantum teleportation, do you know if it requires an input of energy on the other side to transfer the velocity of the initial particle - particularly in the case of between ions where such could have a drastic change in inherent energy? I assume it does, but have never read anything about it, other than noting that the experiments on ytterbium were done at near-zero-kelvin temperatures, I have no real idea).

My thought was simply that the theory of relativity is about acceleration and the energy required to create that acceleration (and the mass relationship etc) and not about the objects actually being at that velocity? Therefore the only barrier to an object attaining such a speed is that it cannot attain it, it could (and would) simply maintain it?

I suppose the logical next step from there would be that to get an object with mass over the speed of light, you would require "more than infinite" energy - which obviously, is also very impossible? Thereby meaning that if an object with mass were to somehow end up in travelling at the speed of light, it could no longer accelerate... Or decelerate (since any number plus or minus infinity is still infinity, resulting in a further proof that this would be impossible)?

I'm sorry for the "unphysical" questions - I'm just trying to get my head fully around the theory to put it to more practical uses later.

#### bcrowell

Staff Emeritus
Gold Member
The relationship between energy and momentum in relativity (in units where c=1) is, E2-p2=m2. The left-hand side goes from real to imaginary as v surpasses c, so a particle with a given rest mass m can never exist in both v<c and v>c states. The following FAQ entry may also be useful.

FAQ: Why can't anything go faster than the speed of light?

In flat spacetime, velocities greater than c lead to violations of causality: observer 1 says that event A caused event B, but observer 2, in a different state of motion, says that B caused A. Since violation of causality can produce paradoxes, we suspect that cause and effect can't be propagated at velocities greater than c in flat spacetime. Special relativity is one of the most precisely and extensively verified theories in physics, and in particular no violation of this speed limit for cause and effect has ever been detected -- not by radiation, material particles, or any other method of transmitting information, such as quantum entanglement. Particle accelerators routinely accelerate protons to energies of 1 TeV, where their velocity is 0.9999996c, and the results are exactly as predicted by general relativity: as the velocity approaches c, a given force produces less and less acceleration, so that the protons never exceed c.

The corresponding speed limit in curved spacetime is far from being established. The argument from causality is not watertight. General relativity has spacetimes, such as the Godel solution, that are valid solutions of the field equations, and that violate causality. Hawking's chronology protection conjecture says that this kind of causality violation can't arise from realistic conditions in our universe -- but that's all it is, a conjecture. Nobody has proved it. In fact, there is a major current research program that consists of nothing more than trying to *define* rigorously what the chronology protection conjecture means.

There are certain things we *can* say about faster-than-light (FTL) motion, based on the fundamental structure of general relativity. It would definitely be equivalent to time travel, so any science fiction that has routine FTL without routine time travel is just plain wrong. It would probably require the existence of exotic matter, which probably doesn't exist. If it were possible to produce FTL artificially, it would certainly require the manipulation of godlike amounts of matter and energy -- so great that it is unlikely that beings able to carry it out would have anything like ordinary human concerns.

There are many ways that velocities greater than c can appear in relativity without violating any of the above considerations. For example, one can point a laser at the moon and sweep it across, so that the spot moves at a speed greater than c, but that doesn't mean that cause and effect are being propagated at greater than c. Other examples of this kind include a pair of cosmic-sized scissors cutting through a gigantic piece of paper at greater than c; phase velocities greater than c; and distant, observable galaxies receding from us at greater than c, which is interpreted as an effect in which space itself is expanding in the space in between.

#### Calimero

With quantum teleportation, do you know if it requires an input of energy on the other side to transfer the velocity of the initial particle - particularly in the case of between ions where such could have a drastic change in inherent energy?
Quantum teleportation is way of sending quantum information, not particles or energy. And it does obey simple rule - no faster then light communication is possible.

My thought was simply that the theory of relativity is about acceleration and the energy required to create that acceleration (and the mass relationship etc) and not about the objects actually being at that velocity? Therefore the only barrier to an object attaining such a speed is that it cannot attain it, it could (and would) simply maintain it?

I suppose the logical next step from there would be that to get an object with mass over the speed of light, you would require "more than infinite" energy - which obviously, is also very impossible? Thereby meaning that if an object with mass were to somehow end up in travelling at the speed of light, it could no longer accelerate... Or decelerate (since any number plus or minus infinity is still infinity, resulting in a further proof that this would be impossible)?
There are probably dozens of explanations why it is impossible for massive particle to attain speed of light. You can search this forum and find many many similar questions. For starters, I would suggest you to take it as axiom, and deal with reasons as you go.

Edit: Nevermind, bcrowell was kind enough to find and post this FAQ entry.

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#### kamenjar

I don't understand nor "believe" in using equations to prove something, so please explain with examples.

Case:
A spacecraft with a person in it is sent in some direction. the spacecraft has the latest propulsion drive that is able to accelerate at gazillion G per second and require only tiny bit of mass to propell (assume the passenger can withstand this).

Assumptions/Observations:
2a. To the outside observer, the clock of the craft is slowing and it seems to the outside observer that the craft is pumping less and less G (energy). The effect of "mass increase" is apparent to the outside observer.
2b. To the outside observer, in the last billion years of observing, it looks like the craft is approaching speed of light but slowing. It seems to never reach the speed of light.

1a. The craft is constantly pumping gazillion Gs and is accelerating the craft in the craft's reference frames. The clock of the craft "looks normal" to the observer inside the craft. It is constantly pumping gazillion Gs every second and accelerating.
1b. The universe around the observer inside the craft is "spinning" faster and faster. Galaxies collide and merge in matter of seconds. Things are blue-shifted, but inside the craft everything is normal. In the next second the observer will reach the speed of light.

Conclusion/Outcome for the Observer inside the craft:
3 - As thngs were speeding up around them, the universe has collapsed on itself and the observer saw "The End"?
4 - The observer inside the craft realizes that they have reached the speed of light, but that the speed of light has increased due to expansion of the universe?

If 4 is correct, can we use 4 as a benchmark for calculating the age of the universe? Can we use it to predict that the universe will expand forever? Dos the increase of speed of light mean that the universe is not blue-shifted to the observer, but just plain white?

#### Ich

In the next second the observer will reach the speed of light.
In the next second, as in the seconds before and after, this observer is definitely at rest wrt himself. No speed.
He will observer galaxies moving closer and closer to the speed of light, however. Just like the galaxies see him moving closer and closer to the speed of light. Speed is the same in both directions, of course.

#### G037H3

tachyons go faster :surprised

#### kamenjar

In the next second, as in the seconds before and after, this observer is definitely at rest wrt himself. No speed.
He will observer galaxies moving closer and closer to the speed of light, however. Just like the galaxies see him moving closer and closer to the speed of light. Speed is the same in both directions, of course.
I don't get your point. He is at rest with himself. I don't claim otherwise.
He is moving at C with respect to the rest of the universe (original reference frame), and yes the standing observer (that died long time ago) is moving away from the craft at -C. So what?
Edit: And in fact at t + 1s (spacecraft clock) he is moving at a speed larger than C, I don't think you are realizing that. What is preventing him in his reference frame to reach C+1?

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#### Korlus

Sorry, I posted a little early before going on to explain the rest. I've left what I was going to write below. Anyway...

What I recall being told is that space-time is finite, but effectively 4D - so an object moving in 3D cannot simply escape it (much as if you extrapolate the idea of the mobius strip), and following the idea of the big crunch (if you partake in such theories) is that as the big bang happened, space-time was created, and space-time, in effect, pushed outwards. If/when the/a big crunch occurs, it'll pull space-time inwards, effectively meaning that anything watching the Universe end will, itself, also end?

I'm afraid that I don't have the sources right now to cite any of that, and my understanding is a bit wavy at best, but...

Stuff I was going to post as would be seen from the outside of the craft said:
My understanding is simply that as you go faster, you get heavier (more massive). While that's not an issue when moving at slow speeds, it starts to become problematic at higher ones.

Even if you were the weight of just a single "average" man when stationary, moving at 99% of the speed of light suddenly makes you absolutely massive. When you hit 99.999999999% (or whatever) of the speed of light, you would be so absolutely massive (in mass, not necessarily size) that suddenly the force providing your gazillion g is increasing your speed (velocity) very very slowly. As you go faster, the rate of speed increases. Soon you're going 99.9999999999%... and still technically increasing, but the amount you're speeding up will slow down and down and down...

It's like if you add 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....
You should get 2 - except that you'll never make it to 2 because you would have to halve the number previously an infinite number of times, you'll probably end up with 1.99-something when you stop. It's the same here.

... Or so I believe.

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#### DaveC426913

Gold Member
1b. The universe around the observer inside the craft is "spinning" faster and faster. Galaxies collide and merge in matter of seconds. Things are blue-shifted, but inside the craft everything is normal.
Yes.

In the next second the observer will reach the speed of light.
No.

If it took him 5 minutes to get from zero to .99c, then ni the next five minutes, he will get to (say) .9999c. In the 5 minutes after that, he will get to .999999c, etc.

Conclusion/Outcome for the Observer inside the craft:
3 - As thngs were speeding up around them, the universe has collapsed on itself and the observer saw "The End"?
Essentially yes. The universe will be compressed in the direction of his travel into a thin disc, and will age faster and faster.

#### kamenjar

...
If it took him 5 minutes to get from zero to .99c, then ni the next five minutes, he will get to (say) .9999c. In the 5 minutes after that, he will get to .999999c, etc...
By spacecraft's clock? Why? He is pushing constant G stilll.. Unless the universe expands and C is increasing (as predicted), there's nothing that I am aware of preventing the the guy in the spacecraft to reach C in the next second.

To the outside observer, yes. According to an external clock the speed will be .99999 then .9999999 etc etc., but I don't see the internal clock doing that.

Edit:
"here's nothing that I am aware of preventing the the guy in the spacecraft to reach C in the next second." .. unless the space flattening, as seen by the observer inside the ship is causing him to travel shorter distance with each second of his tick.

#### Ich

kamenjar said:
I don't get your point. He is at rest with himself. I don't claim otherwise.
You do claim otherwise. Problem is, you don't realize it. See:
What is preventing him in his reference frame to reach C+1?
Look up what "in sb's reference frame" usually means. It means at rest wrt sb.

I'm quite sure that I know what you want to say, you have something like rapidity in mind. But I can't help you sort it out unless you try to get a clear idea of what you have in mind.
Who, relative to whom, are you claiming is moving faster than c?
You said
To the outside observer, in the last billion years of observing, it looks like the craft is approaching speed of light but slowing. It seems to never reach the speed of light.
which is correct.
Now you say that for the galaxies, his speed is always smaller than c, while for him the galaxies move faster that c. That's nonsense.

#### Mentz114

Gold Member
By spacecraft's clock? Why? He is pushing constant G stilll.. Unless the universe expands and C is increasing (as predicted), there's nothing that I am aware of preventing the the guy in the spacecraft to reach C in the next second.
When he gets very close to c with respect to something, it appears to be receding from him at a decreasing rate as DaveC has said. It's explained in these refs

http://en.wikipedia.org/wiki/Rindler_coordinates
http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

#### kamenjar

... wrt sb....
Wrt sb?
I'm quite sure that I know what you want to say, you have something like rapidity in mind. But I can't help you sort it out unless you try to get a clear idea of what you have in mind.
Who, relative to whom, are you claiming is moving faster than c?
If the person inside the craft has a drive that can push 3333G per second (that is, 30km/s^2). That means that in 10 000 seconds of HIS TIME he will reach speed of light. Whatever THAT means. It says on the darn user manual :) And my problem is, I see nothing preventing him from doing that except flattening of space. In some explanations of "fast movement" it is noted that the observer inside the craft does not see much out of the ordinary it his reference frame.
which is correct.
Now you say that for the galaxies, his speed is always smaller than c, while for him the galaxies move faster that c. That's nonsense.
"maybe you misunderstood the "spinning faster" thing. I meant aging faster. But if you understood, then that brings the point the thought - your paradox - If to outside observer the craft can never reach C, that means that C is increasing at an increasing rate. That's what I concluded, and that seems to correspond expanding universe theories. That is the only answer to observer inside the craft, because he doesn't seem to reach C.

When he gets very close to c with respect to something, it appears to be receding from him at a decreasing rate as DaveC has said. It's explained in these refs

http://en.wikipedia.org/wiki/Rindler_coordinates
http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html
Sorry can't read formulas. Maybe someone could explain what happens and why.

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#### DaveC426913

Gold Member
If the person inside the craft has a drive that can push 3333G per second (that is, 30km/s^2). That means that in 10 000 seconds of HIS TIME he will reach speed of light. Whatever THAT means.
Precisely. What does it mean?

It means that, after x seconds, he will measure planets and stars going by him at .9c. After 2x seconds, he will measure stars going by at .99c, and after 3x seconds, he will measure stars going by at .999c etc.

"maybe you misunderstood the "spinning faster" thing. I meant aging faster. But if you understood, then that brings the point the thought - your paradox - If to outside observer the craft can never reach C, that means that C is increasing at an increasing rate.
No, it means that constant acceleration does not result in a constantly increasing velocity.

#### DrGreg

Gold Member
If the person inside the craft has a drive that can push 3333G per second (that is, 30km/s^2). That means that in 10 000 seconds of HIS TIME he will reach speed of light. Whatever THAT means.
You might think that a G-force equivalent to 30 km/s2 ("proper acceleration") means that the velocity relative to the starting point increases second-by-second as 0, 30 km/s, 60 km/s, 90 km/s etc. But as you change speed, your notion of distance and time keeps changing, so in fact the sequence is
• 0
• 29.9999999 km/s
• 59.9999992 km/s
• 89.9999973 km/s
• ...

The sequence never reaches c, no matter how long you wait. Although you in the rocket will experience a constant G-force equivalent to 30 km/s2, the acceleration measured by an external inertial observer will steadily decrease.

For anyone reading this who likes formulas, the numbers above were calculated using
$v=c \tanh \frac {\alpha \tau}{c}$​

#### kamenjar

Precisely. What does it mean?
...No, it means that constant acceleration does not result in a constantly increasing velocity.
Since when and WHY? That is what I am asking. What is measured to show that? Which theory says that?

His clock is constant. Why does he not accelerate at constant speed? What is the reasoning behind this? Relativity says that his reference frame is "normal" as far as he is concerned. As far as I am concerned Earth and Milky way could be moving at 0.1 C with respect to something. So does that mean that 1m/s is no longer 1m/s on earth?

If his ship is 1 meter in length had ship acceleration is 3333G, his velocity should increase by 30 000 ship lengths per second for every second he has his drive on. Isn't that what relativity proposes? In his frame of reference everything is normal?

#### DaveC426913

Gold Member
Since when and WHY? That is what I am asking. What is measured to show that? Which theory says that?
Relativistic velocity addition follows the http://en.wikipedia.org/wiki/Lorentz_factor" [Broken].

No matter how much you increase the velocity, it will never reach c.

If his ship is 1 meter in length had ship acceleration is 3333G, his velocity should increase by 30 000 ship lengths per second for every second he has his drive on. Isn't that what relativity proposes? In his frame of reference everything is normal?
No it shouldn't. Velocities add simply only when they are very small - the dilation factor is still there, it's just vanishingy small.

Every time you walk to the store, you are undergoing time dilation wrt the rest of the universe. Your mass would be measured by a passerby as slightly greater than if you were still wrt to him. His mass would be greater as measured by you. He would also be flattened in the plane perpendicular to your direction of movement. This is how real life really works, we just don't encounter it in day-to-day life.

But as velocities approach c, this dilatory factor becomes significant enough that we can no longer use the Newtonian approximation.

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#### kamenjar

Relativistic velocity addition follows the http://en.wikipedia.org/wiki/Lorentz_factor" [Broken].
No matter how much you increase the velocity, it will never reach c.
Why?
No it shouldn't. Velocities add simply only when they are very small - the dilation factor is still there, it's just vanishingy small.
I never said time dilation does not exist. Time dilation is proven to exist. At higher G and (probably) at higher velocity.
Every time you walk to the store, you are undergoing time dilation wrt the rest of the universe. Your mass would be measured by a passerby as slightly greater than if you were still wrt to him.
But not by myself. I see my OWN mass wrt myself to be constant regardless of where I am or at what speed I move. If I could take earth and my scale with me on a 0.9 C trip, I would weigh same on that scale.
His mass would be greater as measured by you.
His mass wrt me nor my own mass wrt him does not concern me. I have my little world with me on a 0.9 C trip.
He woudl also be flattending in the palne prependiclaur to your direction of movement.
Again, anything relative to myself in my ship doesn't really concern me. I am on a speed record trip, and I don't care if there's no earth to return to when I come back.

As velocities approach c, this dilatory factor becomes significant.
Yes, my time slows wrt the rest. But wrt myself, it is all the same. Electrons don't spin slower. Light is white. when I point a flashlight from the back towards the end of the ship it still takes 1/300 000 000th of MY second to reach it, and it is white. So there's no reason for my propulsion drive to function slower or less efficient wrt MY OWN clock.

I saw the Lorentz transform but that is math. I also could not find experimental confirmation for Lorentz contraction. I see no empirical proof, and I still see no explanation (reason) behind anything.

I really hate to sound like a nuisance, but I'm not understanding this :(

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#### kamenjar

... measured by an external inertial observer will steadily decrease....
You said EXTERNAL. What about INTERNAL measurement? I claim that INTERNALLY, I can reach C (I doubt this is possible, but it sounds reasonable).

I never contradicted your claim. Because my clock is slowing, to an external observer, my drive is functioning less and less efficiently. he sees that my clock is slowing and my drive spits out less FOOBAR per his second (FOOBAR propels me). So I am accelerating less and less, as far as his clock goes. but INSIDE my ship I am traveling the same distance and increasing my speed constantly on MY clock in MY universe.

#### yuiop

If his ship is 1 meter in length had ship acceleration is 3333G, his velocity should increase by 30 000 ship lengths per second for every second he has his drive on. Isn't that what relativity proposes? In his frame of reference everything is normal?
Well let's say the accelerating ship's velocity relative to some inertial observer is 0.8c then the ship length is 0.6 meters so he the ships velocity is only increasing by 30000*0.6= 18,000 meters per second at that point and the second measured by the ship's clock is 1.6666 seconds according to the inertial observer so the acceleration relative to the inertial observer is actually 18000/1.66666^2 = 6480 m/s^2 when the ship thinks it is accelerating 30000 m/s^2. The faster the ship goes, the shorter it gets due to length contraction and the slower its onboard clock rate gets and the slower its actual acceleration relative to an inertial observer.

You could look at another way. Let us say ejecting 1kg of some fuel out the back every second produces a proper acceleration of 30000 m/s^2. The ship thinks is it burning fuel at a constant rate, but to the inertial observer when the ship gets to 0.8c it is actually taking 1.66666 seconds to eject 1kg of fuel, so its thrust according to the inertial observer is not as great as the rocket observer thinks. If it was a chemical rocket, then the faster the rocket goes the slower the chemical reactions become and the less efficient the rocket. Another way to look at it is the inertial mass of the rocket is effectively increasing or getting harder to accelerate at any rate. Basically everything conspires to make it impossible to reach the speed of light relative to anything.

Now you might think "who cares what the inertial observer thinks" but if the rocket observer is going to claim he is going faster than the speed of light than he it has to be relative to something and that something is the inertial observer.

You might also try accelerating at a constant rate relative to the inertial observer and in this case the proper acceleration measured onboard the rocket appears to be increasing. Now let us say you have x amount of fuel and you work out that by accelerating the rocket at a constant rate gets you to 0.9c when your fuel runs out. So you increase the fuel but this increases the payload and you need extra fuel to carry the extra fuel and so the fuel requirements increase exponentially and a fuel load of 10 x might get you to 0.91c and it only gets worse. (I have not done the exact calculations but you get the idea.) When you work out how much fuel you need to get lightspeed, it turns out to be infinite.

Then again, the rocket captain could decide to ignore inertial observers altogether and calculate his theoretical speed based on how much his proper acceleration is and how long he has been accelerating and proudly declare his ship is travelling at 2x the speed of light! His proud claim is soon demolished, when someone* turns the rocket headlights on and the light races ahead of the rocket.

*The crew member that turned the headlights on is later sacked for embarrassing the captain.

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#### Mentz114

Gold Member
You said EXTERNAL. What about INTERNAL measurement? I claim that INTERNALLY, I can reach C (I doubt this is possible, but it sounds reasonable).

I never contradicted your claim. Because my clock is slowing, to an external observer, my drive is functioning less and less efficiently. he sees that my clock is slowing and my drive spits out less FOOBAR per his second (FOOBAR propels me). So I am accelerating less and less, as far as his clock goes. but INSIDE my ship I am traveling the same distance and increasing my speed constantly on MY clock in MY universe.
In your own frame your velocity is undefined. Only relative velocity can be measured, and those observers measuring your velocity will never see you reach c. They will disappear behind the Rindler horizon before then. Whatever happens, you will never be seen to reach the speed of light, nor will you see any other object reach the speed of light.

#### jtbell

Mentor
For mathematical details, see the following article about the "relativistic rocket":

http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html [Broken]

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#### yuiop

For mathematical details, see the following article about the "relativistic rocket":

http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html [Broken]
Especially the part about the amount of fuel required (4.2 thousand million tonnes for each Kg of payload) to maintain constant proper acceleration of 1g = 9.8 ms^2 half way to Andromeda and then decelerate at 1g to arrive at a stop at Andromeda. This journey will take about 2,000,000 years of Earth time and the astronauts will age by about 28 years. Not much chance of returning to Earth and finding anyone they knew still alive.

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