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There is a spaceship, moving at a speed close to the speed of light.

The control room of the spaceship is in the front and the sleeping quarters are at the back.

One of the astronauts, Joe, decides to take a nap and floats back to his sleeping bag.

Will he be able to return to his post in the control room?

When he is sleeping, he is at rest with respect to the spaceship. However, to move to the control room, he must accelerate first, heading to the front, adding to his huge speed with respect to a "stationary" observer. But the spaceship is already going close to the speed of light, and there is this notion from a "stationary" observer's point of view that you must exert ever bigger forces to accelerate, the closer you already have got to the speed of light.

According to a "stationary" observer, Joe is close to the speed of light, and still accelerating by pushing the spaceship backwards. For the purpose of acceleration, he weights millions of billions of tons. Can his muscles keep up?

I understand that with respect to the spaceship, nothing strange happens, they are not accelerating, it is just like being on ISS. I don't understand, how to match this up with what an external observer moving slower can see?

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    $\begingroup$ Have you tried applying the Lorentz equations? $\endgroup$ – D. Halsey Jul 10 at 16:16
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    $\begingroup$ According to that astronaut, your house is currently moving close to the speed of light. When you wake up in the morning, do you have any trouble making it back to the living room? $\endgroup$ – WillO Jul 10 at 17:50
  • $\begingroup$ @WillO Well at least you just gave me a reason to justify how hard it can be to get up in the mornings. "Relative to some observer, this takes a lot of effort." $\endgroup$ – JMac Jul 10 at 18:39
  • $\begingroup$ Joe's acceleration is small in the stationary frame, so the force is the same. The gamma factors in the relativistic mass and speed cancel out. $\endgroup$ – safesphere Jul 10 at 20:20
  • $\begingroup$ WillO: yes, that is the solution, but I am lacking at the explanation department... $\endgroup$ – Zoltan K. Jul 11 at 18:02
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There is a spaceship, moving at a speed close to the speed of light.

Uniform motion is relative so, to help yourself think more clearly about this, try writing something like this instead:

Alice observes a spaceship moving with speed close to the speed of light.

While this may not seem like a big difference, it helps to note that we can easily add something like

Bob observes the same spaceship moving with a speed much less than the speed of light.

without changing the problem at all from Alice's perspective. But now your question nearly answers itself: of course Joe is able to return to his post in the spacecraft because there are an infinity of 'Bobs' (observers) that observe the spacecraft (and Joe) to have speed much less than the speed of light (essentially no relativistic effects).

Now, due to the relativity of (uniform) motion, it follows that Joe, at rest with respect to, and in the back of the spaceship, observes that Alice is moving with speed close to the speed of light and that Bob is moving with speed much less than the speed of light.

For the purpose of acceleration, he weights millions of billions of tons. Can his muscles keep up?

It isn't clear what your reasoning here is. The spacecraft is in free-fall, moving uniformly through space, i.e., Joe is weightless. His muscles impart a force that accelerates him towards his post. All observers agree that Joe accelerates but they don't generally agree on the magnitude of Joe's acceleration.

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  • $\begingroup$ So basically, you say Alice sees the time on the spaceship has slown down significantly. Because of this, when Joe returns to his post, according to Alice, Joe accelerates very_very_very slowly. Which is consistent with him gaining a lot of weight, so his muscles can propel him to the front only so fast (in this case, so slow). $\endgroup$ – Zoltan K. Jul 11 at 18:06
  • $\begingroup$ "It isn't clear what your reasoning here is." I was thinking that if a spaceship has a constant trust X, according to an external, "stationary" observer (Alice), it gains less and less momentum from the same trust, as if its weight was growing. When Joe moves forward in the spaceship, it is not different from a spaceship gaining momentum by throwing stuff backwards. So Alice must see that Joe has gained weight, as if he was a spaceship himself, for the purpose of accelerating forward. $\endgroup$ – Zoltan K. Jul 11 at 18:10
  • $\begingroup$ However, even though Joe gains weight, according to Alice, general relativity says nothing about having stronger muscles. The sentence "All observers agree that Joe accelerates but they don't generally agree on the magnitude of Joe's acceleration." put me in the right track, to understand why - from the viewpoint of the "stationary" Alice everything seems normal. $\endgroup$ – Zoltan K. Jul 11 at 18:12
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    $\begingroup$ @ZoltanK. wrote "However, even though Joe gains weight, according to Alice..." - Joe doesn't gain weight (or mass) according to Alice and that's all I have to say about that. You may want to search this site and elsewhere for the term invariant mass. For example, see Ben Crowell's answer here $\endgroup$ – Alfred Centauri Jul 12 at 0:48
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As you stated, in the spaceship inertial reference frame nothing strange happens.

Instead, according to special relativity, the energy required to accelerate an object closer and closer to the speed of light, with respect to an external observer, becomes progressively bigger and bigger and eventually infinite. Using the out-of-date concept of relativistic mass, we say that the relativistic mass becomes infinite when we approach the speed of light. That means that the same amount of energy provided to the object allows a smaller and smaller increase in speed.

This reasoning applies both to the spaceship boosted by the engines and the astronaut pushing backward the spaceship craft. In the latter I consider the astronaut as the object measured by the external observer. The astronaut gains at every push back a small amount of energy which thrusts him/her closer and closer to the speed of light, but without reaching it.

Do not be misled by the concept of relativistic mass, as it is an obsolete way to describe the special relativity events. The relativistic energy $E$ of an object with proper mass $m$ moving at velocity $v$ in an inertial reference frame is $E = \gamma m c^2$. The Lorentz factor $\gamma$ diverges when $v \to c$ and the energy is consequent. To attach the Lorentz factor to the mass is not physically meaningful. That is the modern reading.

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  • $\begingroup$ Yes, this is exactly what I don't understand! What is the difference between the spaceship burning fuel, and expel it to the back to gain more momentum vs. an astronaut pushing a spaceship backwards to gain more momentum, from an external, "stationary" observer's point of view? Why is the astronaut's way of propulsion so special? This would mean, we could build a matrioshka spaceship that can easily accelerate near the speed of light, by the internal ships pushing against the external ships. $\endgroup$ – Zoltan K. Jul 11 at 17:59
  • $\begingroup$ @Zoltan K. I edited my post. Please refer to it. $\endgroup$ – Michele Grosso Jul 12 at 13:04
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It's all a matter of reference frames. To anyone in the spaceship everything would seem normal, no length contraction, no time dilation, no mass increase for anything aboard the spaceship. From the reference frame of an observer on Earth, however, all these relativistic effects would manifest themselves when he viewed the spaceship, and vice versa. If the crew of the spaceship could communicate with Earth, from their point of view it is Earth and its inhabitants which would experience length contraction, time dilation and mass increase, but we ourselves would notice no change. If, however ,we were supplying the spaceship with power from a laser beam, we would notice that as the ship acquired relativistic mass increase, it became harder and harder to accelerate it, until eventually there would be no detectable increase in velocity but a very gradual increase in (inertial) mass. I wish you would tell us what fuel the spaceship uses to reach near light velocity, because scientists would dearly like to know. There is no known fuel capable of producing such energy, especially as the spaceship would have to carry heavy radiation shielding. The laser beam from Earth would be totally inadequate.

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See the first table here:

http://www.math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

Note that in about 2000 years gamma becomes about 2000, in about 100000 years gamma becomes about 100000, and so on. And note that transverse relativistic mass is gamma times rest mass.

So if Joe was onboard on that rocket for million years, his mass would become million rest masses. And each year his mass would increase by one rest mass.

So let us say that at the launchpad Joe's rest mass is 100 kg. After the rocket motors have been pushing Joe at constant 1000 N force for million years, Joe's mass is million times his rest mass.

Now the rocket has stopped accelerating and Joe starts accelerating using his legs that generate a 1000 N force for a year. That causes Joe's mass to increase by 100 kg. Joe's mass was 100 million kg, it increased by 100 kg. Joe does not notice such small change. "My legs did not accelerate me to relativistic speed", Joe says.

Or maybe I should say that Joe's legs do not notice the small mass increase of Joe's upper body.

In that above text "mass" means transverse relativistic mass, except when I say "rest mass". And "year" means year in the launchpad frame.

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  • $\begingroup$ As I understand your argument, since Joe did not notice his weight going from 100kg to 100million kg, he surely won't notice it going up another 100kg. Now the obvious question is, why did he not notice the first weight change - from the viewpoint of a "stationary" observer, left on the launchpad? $\endgroup$ – Zoltan K. Jul 12 at 17:29
  • $\begingroup$ Or is this an argument about Joe's upper body being accelerating even a little bit, since he is already close to the speed of light, the upper body will be compressed, somehow giving more room for the legs to stretch, without doing much work? $\endgroup$ – Zoltan K. Jul 12 at 17:32
  • $\begingroup$ @ZoltanK. If Joe increased his coordinate speed by a little bit, then Joe's upper body's might compress by 10 %, and its mass might increase by 10%, which two things Joe's legs might notice. But it's extremely hard for Joe to increase his coordinate speed a little bit, because of his large longitudinal relativistic mass, which in my example was million to third power times 100kg. I assumed that Joe pounced to the other end of the spaceship as fast as possible using legs that were able to generate a 1000N thrust. The force Joe felt was 1000N. $\endgroup$ – stuffu Jul 13 at 2:37
  • $\begingroup$ So, because Joe can only ever produce 1000N of trust, but his weight is now 10^18 * 100kg, his apparent movement speed is really slow, "as if time was slowing down on the spaceship". $\endgroup$ – Zoltan K. Jul 13 at 13:41

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