# Can a rocket with no forces acting upon it except a single push force with constant acceleration keep accelerating forever?

I was wondering why a rocket with no opposing forces acting upon it couldn't keep accelerating given that it has the potential to release enough energy to maintain its acceleration at all costs. I have heard that any object with mass cannot reach the speed of light. Is that true? If so, why?

Assuming the rocket has some super fuel source (which we'll just take for granted and not look into further :-) then yes it can keep accelerating forever. However if you're standing on the Earth watching the rocket you'll never see it exceed the speed of light.

We need to be precise about what we mean when we are talking about relativistic speeds, so let's be clear what we mean by the rocket accelerating. If you're in an accelerating car, airplane or whatever you can tell you're accelerating because you can feel the $g$ forces, and the same is true of the rocket. If the rocket is floating in space with the engines off then the astronauts inside will be in free fall. But when they turn the engines on then they'll feel the acceleration, and they can tell how fast they are accelerating from the force they feel. For example if they feel a force of $1g$ then that means they know that they are accelerating at $9.81$ m/sec$^2$.

So when we say the rocket is accelerating at a constant acceleration $a$ what we mean is that the astronauts inside feel a constant acceleration $a$. Technically this acceleration is called the proper acceleration.

But suppose you are stood on Earth watching the rocket speeding off in outer space. You probably know that if an object is moving fast we get two relativistic effects time dilation and length contraction, and it's these two effects that mean you'll never see the rocket exceed the speed of light.

Let's see how this works and we'll take the length contraction first. The length contraction means that a distance $\ell$ on the rocket looks like a distance $\ell/\gamma$ to us on Earth, where $\gamma$ is the Lorentz factor:

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

Note that for any speed $v \gt 0$ the value of $\gamma$ is greater than one and as $v\to c$, i.e. as we approach the speed of light, the value of $\gamma$ goes to infinity. You'll see why this matters in a moment.

Now remember that acceleration has units of metres per second squared. But due to the length contraction a metre on the rocket looks like $1/\gamma$ metres to us i.e. it looks shorter than a metre. The result is that the length contraction causes the acceleration of the rocket, $a$, to be reduced by a factor of $1/\gamma$ when we measure it from Earth.

Now for time dilation. This means that one second on the rocket looks like a longer time $\gamma$ seconds to us on Earth. Again remember that acceleration has units metres per second squared, so if the length of a second is increased by a factor of $\gamma$ that means the acceleration we observe from earth is reduced by another factor of $1/\gamma^2$.

The end result of all this is that the acceleration of the rocket as we observe it from Earth, we'll call this $a'$, is reduced by a factor of $1/\gamma^3$:

$$a' = \frac{a}{\gamma^3} \tag{1}$$

And remember that I said $\gamma \to \infty$ as the speed $v \to c$? That means as we watch the rocket approach the speed of light we see $a'$ decrease and approach zero. So we end up with the rather surprising result that the astronauts on the rocket believe they are accelerating steadily, but we see their acceleration fall away to zero so they never reach the speed of light.

This is all a rather handwaving argument, but we can do the analysis rigorously, and the result is known as the relativistic rocket equations. The speed of the rocket as measured from Earth is given by:

$$v = \frac{at}{\sqrt{1 + \left(\frac{at}{c}\right)^2}} \tag{2}$$

You can confirm for yourself that no matter how large you make the time $t$ the calculated velocity $v$ never exceeds the speed of light $c$.

• This film, youtube.com/watch?v=B0BOpiMQXQA in stunning black and white shows how an accelerating force does not result in a linear increase in velocity... Dec 14, 2016 at 19:06

"Why?"

This is the question physics can't answer. Physics describes the universe, it does not explain it. Given some postulates that you are willing to accept as true without proof or explanation, one can draw logical conclusions, but they all trace back to the unexplained postulates.

From an observer in an inertial frame ("on earth", so to speak) the speed of the rocket will approach the speed of light but not exceed it.

But the observer on the rocket will always feel the acceleration. The astronaut will always feel pushed back against his seat. He will observe that he is going faster and faster forever.

It seems like a paradox. We don't know why it works that way. We can "explain" it if you are willing to accept Einstein's theory of relativity, but we can't explain why the theory is such a good "explainer" of the universe. That just seems to be the way the universe works.

Quoting wikipedia :

A half-myth: It gets harder to push a ship faster as it gets closer to the speed of light

This is a half-myth because it depends on the frame of reference. It is true for those watching from the planetary reference frame. For those experiencing the journey (in the ship's reference frame) it is not true. For both the planetary frame and the ship's reference frame, the ship will change speed in a Newtonian way—push it a little and it speeds up a little, push it a lot and it speeds up a lot. However, in the planetary frame the ship will appear to be gaining mass due to its high kinetic energy, and the mass–energy equivalence principle. Should the engines be giving a constant thrust, this will result in progressively smaller acceleration due to the higher mass it is required to accelerate.

From the ship's frame, the acceleration would continue at the same rate. However, due to Lorentz contraction, the galaxy around the ship would appear to become squashed in the direction of travel, and a destination many light years away would appear to become much closer. Traveling to this destination at subluminal speeds would become practical for the onboard travellers. Ultimately, from the ship's frame, it would be possible to reach anywhere in the visible universe, without the ship ever accelerating to light speed.

So, the speed you are going at is all a mater of reference frame. You can't go faster than light in the solar reference frame if you start your journey in the solar system.

To accelerate a body in space one need a thrust. Since nothing is faster the electromagnetic radiation all you can do is to accelerate your rocket with light. This indeed is possible, so sealing with the lights pressure from the sun is technical makeable. Using a light projector you can also accelerate a rocket.

But even with

the potential to release enough energy to maintain its acceleration

you would not be able to get a velocity greater the speed of light.