Break speed of light with infinite mass

Question

I've heard that a spacecraft could never exceed the speed of light because it's (relativistic) mass quickly approaches infinity and therefore there could never create a big enough rocket to propel it faster and faster.

I'm 99% sure this is a stupid question, but I'll ask it anyway :). Theoretically if a spacecraft had unlimited fuel (by travelling through a nebula picking up hydrogen/oxygen and whatever else it needed as a propellant), couldn't it actually break the speed of light?

Yes I know it gets harder and harder to accelerate the craft because it gets more and more massive (relativistically speaking), but as the craft and propellant tanks get more and more massive so does the thrust right? 1 million pounds of propellant exploding out the back gives 1 million pounds of thrust at Newtonian speeds. As the spacecraft gets heavier and heavier, the thrust would also get more and more massive would it not? In that way more massive mass is countered by more massive thrust. If not, why is the reaction mass any different to the rest of the spacecraft's mass?

Answer 1

I've heard that a spacecraft could never exceed the speed of light because it's (relativistic) mass quickly approaches infinity and therefore there could never create a big enough rocket to propel it faster and faster.

In fact, the spacecraft could never even reach, much less exceed the speed of light.

I think that you'll agree that the spacecraft, no matter what speed it may have relative to some other object, is at rest with respect to itself.

Think carefully about that! The spacecraft (or any material object) is not moving with respect to itself.

This seems so intuitive, so unquestionably true that you might think that there is no reason to even mention it.

But, according to Special Relativity, something that moves with speed c (the speed of light in vacuum) relative to some other object, moves with speed c relative to any object (c is an invariant speed).

In other words, if it were the case that the spacecraft could obtain the speed of light, it would be moving at the speed of light with respect to itself.

This is so plainly incomprehensible that it is, in fact, a relief to know that the spacecraft can never attain the speed of c.

More precisely, according to the Lorentz transformations, there is no frame of reference that moves with speed c relative to another frame of reference.

More generally, this fact "shows up" as nonsense statements like "the mass is infinite at the speed of light" or "infinite force is required to get to the speed of light".

There's no such thing as infinite mass or infinite force which is to say, you can't get to c from here.

Answer 2

The problem is- The rocket is not 'fighting' with any force-field, it is 'fighting' with the very nature of space-time. So unless we have something of zero rest mass 'things' will tend to infinity.

And yes the thrust will increase but space-time will distort( following Lorentz transformation, no GR effect here) in such a way that reaching 'c' 'tests' our infinities

P.s.- terms in ' ' are indicative(to give a feel) and not to be taken literally.

Answer 3

In simple terms: No amount of finite numbers will ever sum to infinity.

Scoop as much mass as you want, magically accelerate it so it doesn't slow you down, carry it on board. You will have a finite spaceship containing finite mass and obtaining finite mass from the ambient environment requiring infinite acceleration.

Your mass inflow rate would need to be infinite to reach $c$.

What happens instead is "asymptotic" approach - the separation between your speed and that of an object at lightspeed gets smaller and smaller and smaller, but never reaches zero.

The answers above give you the physical background that results in this "infinity", but it is worth noting the answer in the terms used in the question.

Answer 4

The speed of light is "broken" each time when a particle moves faster than the speed of light in some transparent media with $n\gt 1$. No infinite mass/energy is necessary for that. Remember Cherenkov's radiation for fast charged particles.