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So when people say: 'I am approaching the speed of light, and to get to 100% light I would need infinite energy' they are essentially saying that this situation is impossible?

I read this in Hawking's book and confused me because I assume when he says 99.9% speed of light, he means 99.9% speed of light in relation to someone outside observing?

I just cannot understand this notion of needing more and more energy to get closer to light as absolute velocity does not exist? (in that it is a purely relative concept). Surely the ability to accelerate further cannot possible be impeded because speed is all relative, there should be no limit to acceleration? If I 'accelerate' a further 50MPH, will I get to the destination exactly 50 miles early?

From what I can gather you 'can' accelerate FTL (sort off) but instead space bends towards you so you will get to your destination 'ftl' but only due to the curvature in space? So in effect, you can go light years in seconds (lets forget the practicals for a second), but from anyone observing, this will ALWAYS take light years.

Also, if for me I am going 'FTL', does outside observers see me as going light speed, or is it 99.999%.. is there a specific number?

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  • $\begingroup$ I suspect (but am not sure) that part of your confusion comes from the false belief that the energy of a body appears the same to all observers. $\endgroup$
    – WillO
    Dec 24, 2013 at 21:22
  • $\begingroup$ The 99.9% speed of light is to be understood with respect to some chosen reference frame. Of course you can find another reference frame, in which you are stationary. But in such reference frame the relativistic effects would not appear. $\endgroup$
    – mpv
    Jan 19, 2015 at 20:16

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So when people say: 'I am approaching the speed of light, and to get to 100% light I would need infinite energy' they are essentially saying that this situation is impossible?

Yes.

I read this in Hawking's book and confused me because I assume when he says 99.9% speed of light, he means 99.9% speed of light in relation to someone outside observing?

Yes, but note that $c$ is a universal constant. If something is traveling at the speed of light, it is traveling at the speed of light to everyone (except other photons traveling parallel, see my answer to this question).

I just cannot understand this notion of needing more and more energy to get closer to light as absolute velocity does not exist? (in that it is a purely relative concept).

It boils to relativity. The energy of a particle is related to the velocity, $v$, via the relation $$ E=\gamma mc^2=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ As $v\to c$, $E\to\infty$, but gives an indeterminate operation at $v=c$.

Surely the ability to accelerate further cannot possible be impeded because speed is all relative, there should be no limit to acceleration? If I 'accelerate' a further 50MPH, will I get to the destination exactly 50 miles early?

(a) 50 mph is a speed, not an acceleration and (b) you can't get somewhere "50 miles early." If your trip is 300 miles, you can't get there in 250 miles. You can get there in a shorter amount of time.

From what I can gather you 'can' accelerate FTL (sort off) but instead space bends towards you so you will get to your destination 'ftl' but only due to the curvature in space? So in effect, you can go light years in seconds (lets forget the practicals for a second), but from anyone observing, this will ALWAYS take light years.

The 2nd postulate of special relativity states that $c$ is invariant of reference frame (constant for everyone), so nothing can accelerate to speeds faster than light.

Also, if for me I am going 'FTL', does outside observers see me as going light speed, or is it 99.999%.. is there a specific number?

Faster than light means that you have a speed $v>c=2.9979\times10^{10}$ cm/s. If you have done this, then kinematics suggest you have a complex velocity (as in the imaginary number complex) which is utter nonsense since velocity is a physical (real) quantity.

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You are right that there is no such thing as traveling at "99.9% the speed of light" (or for that matter any other velocity) except relative to something.

If you are traveling at 99.9% the speed of light relative to me, you will consider yourself still. In your coordinates, you can certainly accelerate to 50mph, using exactly the same amount of energy that I (in my coordinates) would need to do the same thing. But of course in my coordinates, your speed has increased only by a tiny fraction of 50mph.

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Your velocity, in these cases, is always taken relative to an outside observer, yes. As you intuited, we are already, for free, without having to work at all, moving away from a near- infinite number of photons at the speed of light.

So, in the "infinite energy" case, the implied external observer is your start point.

If you have any mass, it would take infinite energy to accelerate away from Earth to the point where you were moving away from Earth at the speed of light.

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Say we take your wifes point of view as reference and you move very fast relative to her (98% the speed of light, say). We could now change the reference frame to your point of view. Since indeed speed is relative, in this new frame you don't move at all and accelerating another 50 MPH is not a problem.

But the issue is that a coordinate transformation changes the notion of space-direction, the "now". What happens is that space and time mix. And from the frame of reference of your wife, you haven't accelerated 50 MPH. Take a look at the velocity-addition formula. If you iteratively gain speed in an ever new adjusted frame or reference, the gain of each individual boost gets smaller. Learn to read a Minkowski diagram and you see it graphically. The result is that you can't reach speed of light, as viewed from any frame of reference.

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  • $\begingroup$ So if I accelerated to what I consider to be enough to push me to 200% light based on simple calculations of acceleration, I will get to a point in space that is 2 light years in one year? (due to length contraction) $\endgroup$
    – user64272
    Dec 24, 2013 at 15:52
  • $\begingroup$ The answer to the above comment depends on what you consider to be enough energy to push you to 200% of the speed of light. Since there is no such amount of energy, your considerations are certain to be wrong, but without telling us exactly which wrong assumptions you're making, it's quite impossible to tell how much energy you're talking about, and hence quite impossible to tell you what would happen if you expended that energy. $\endgroup$
    – WillO
    Dec 24, 2013 at 17:45
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It means 99.9% of the speed of light in a vacuum.

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  • $\begingroup$ Could those who downvoted please give reasons why you downvoted? $\endgroup$
    – Geremia
    Jul 11, 2015 at 8:50
  • $\begingroup$ (I did not vote). Although your post is correct, it doesn't answer the question, I think it was voted down on this reason. $\endgroup$
    – peterh
    Oct 10, 2018 at 23:36

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