Acceleration towards c without a reference frame and changes

[infector]

Hello everyone.
Below are two problems I have been thinking about lately.
Let’s consider two cases:

1. we have a spaceship surronded by an utter void - nothing outside which the spaceship’s pilot could refer to. The pilot (in his robotic body, allowing him to withstand enormous G-forces) turns on engines and starts accelerating at 100,000g and after few minutes reaches 270,000km/s (0.9c) - or is he? If there is no reference frame is the spaceship moving at all since speed is relative?
2. we have two spaceships and nothing else, as described above. Spaceship A starts accelerating and after some time it reaches 0.9c (acceleration phase 1), so it is moving 270,000km/s with relation to the spaceship B. Now spaceship B turns on its engines and starts accelerating at even higher rate than previously spaceship A and it is doing that as long until it catches up with the first spaceship and then it kill its thrusters; now the ships are moving next to each other with the same speed - their relative speed is equal to 0. Next, the spaceship A turns on its engines and again starts accelerating to 0.9c (acceleration phase 2) and eventually it is moving again at 270,000km/s with relation to the spaceship B.
   

The question - would the energy expenses on acceleration to the same speed be higher during one of the phases; which one? Or would they be equal? Besides, what about relativistic masses of the spaceships? Is mass of the spaceship A bigger after acceleration to 0.9c first phase than before that phase? Is it even bigger after acceleration during phase 2?

Thank you.

 

[Daniel Gallimore]

I'm going to point out a few issues the the way you posed your problem.

If there is no reference frame is the spaceship moving at all since speed is relative?

You must define a reference frame. Always. The frame of reference for which the spaceship is stationary is the frame of the spaceship itself or of a nearby observer with the same speed and acceleration moving parallel to (i.e., in the same direction as) the spaceship. In a universe that is empty of everything except the robot, the robots frame of reference would be the only reference frame. However, if the robot has a speedometer on his craft reading $0.9c$, how is that speedometer getting that number? It can't be getting its number from the first robot's reference frame. To him, the spaceship is stationary. Instead, consider a stationary second robot (with respect to our own frame of reference) that measures the speed of the spaceship to be $0.9c$ as it passes. This is the reference frame that the speedometer is getting its number from.

Let's think about this in a nonrelativistic setting. Consider a car accelerating down a highway. Now suppose there is some stationary (with respect to our own frame of reference) observer standing on the sidewalk. When the car passes the observer, the speedometer reads $10 \, m/s$. Where is the speedometer getting this number from? After all, from the driver's frame of reference, the car is stationary. The answer is that the speedometer is considering what the stationary observer sees as the car passes.

Thus, as soon as you include a speedometer on the spaceship and give the first robot a sense of speed external to what he can perceive by himself, the universe contains more than just his own reference frame. To say that there is nothing which the spaceship's pilot could refer to is a contradiction since the speedometer must be referring to some reference frame external to itself to read anything at all.

Now spaceship B turns on its engines and starts accelerating at even higher rate than previously spaceship A and it is doing that as long until it catches up with the first spaceship and then it kill its thrusters; now the ships are moving next to each other with the same speed - their relative speed is equal to 0.

This part of your problem statement is unclear. Is spaceship B accelerating till it is beside spaceship A or is spaceship B accelerating until it reaches the location where spaceship A stopped accelerating?

If the first is true, then spaceship B would be going way faster than spaceship A. This is because spaceship A has moved at a constant speed in the time it took spaceship B to catch up to it. So spaceship B has accelerated with a greater acceleration than spaceship A over a longer distance, which means its speed is larger. So their relative speed is not zero.

If the second is true, spaceship B would still be going faster than spaceship A since it accelerated with a greater acceleration than spaceship A but over the same distance. In this situation, spaceship B would eventually pass spaceship A with some constant, positive speed. So their relative speed is not zero.

If you fix your assumptions and make some clarifying adjustments to how you phrase your question, then we can talk about energy.

 

[PeroK]

For question 1, the ship is traveling at 0.9c relative to its initial reference frame, but has zero velocity in its own reference frame.

You could take from this that a period of acceleration does not cause a state of absolute motion to be reached. Rather, acceleration causes a change of inertial reference frames.

For question 2, both the first and second acceleration phases for ship A would be the same. As the answer to question 1 shows, there is essentially no difference in the two scenarios.

In reality, ship A may have lost mass in order to accelerate, so things would be different in that sense.

 

[pervect]

It seems to me this question is rather similar to asking "If a tree falls in the forest, does it make a sound?". Once you know whether a tree falling in the forest makes a sound if nobody's around to hear it, you can mostly likely, by analogy, decide if a reference frame exists if there's no material object in it.

Though I was thinking about it some more, it's probably not guaranteed to be the same answer to both questions.

But the underlying issue appears to me to be philosophy, not science, in both cases.

 

[PeroK]

It seems to me this thread is a variant of "If a tree falls in the forest, does it make a sound?". Once you know whether a tree falling in the forest makes a sound if nobody's around to hear it, you can mostly likely, by analogy, decide if a reference frame exists if there's no material object in it.

But the underlying issue appears to me to be philosophy, not science.

A reference frame can't he wished out of existence. That part of the OP makes no sense.

 

[A.T.]

reaches 270,000km/s (0.9c) - or is he? If there is no reference frame

Buy stating his velocity, you have defined a reference frame.

 

[infector]

This part of your problem statement is unclear. Is spaceship B accelerating till it is beside spaceship A or is spaceship B accelerating until it reaches the location where spaceship A stopped accelerating?

My mistake, I am sorry; I see that now when you pointed it out. What I originally wanted to mean was that second spaceship accelerates to higher velocity (let's say 0.99c) just to be able to catch up with the first spaceship, then eventually catches up with it and then deccelerates to match the velocity of that first spaceship. Truth is it would suffice to have the second ship matched ship A's velocity only, without closing in on the first ship's actual location (so they both would be separated by a vast distance) - in that case it would be enough to have the spaceship B accelerated with the same acceleration and for the same time as the first spaceship. However, I thought having both ships next to each other would make further considerations easier.
So: we have a situation where both spaceships are next to each other and traveling with exact same velocity (and then the spaceship A turns on its engines and accelerates to 0.9c - please refer to the first post).
Please tell if there is still some clarification needed; otherwise I hope you could shed more light now on my original question regarding energy expenses.

 

[Nugatory]

It is worth remembering that there are no reactionless drives, so there is no such thing as accelerating in an otherwise empty universe - you're always in motion relative to your own reaction mass.

 

[Daniel Gallimore]

My mistake, I am sorry; I see that now when you pointed it out. What I originally wanted to mean was that second spaceship accelerates to higher velocity (let's say 0.99c) just to be able to catch up with the first spaceship, then eventually catches up with it and then deccelerates to match the velocity of that first spaceship. Truth is it would suffice to have the second ship matched ship A's velocity only, without closing in on the first ship's actual location (so they both would be separated by a vast distance) - in that case it would be enough to have the spaceship B accelerated with the same acceleration and for the same time as the first spaceship. However, I thought having both ships next to each other would make further considerations easier.
So: we have a situation where both spaceships are next to each other and traveling with exact same velocity (and then the spaceship A turns on its engines and accelerates to 0.9c - please refer to the first post).
Please tell if there is still some clarification needed; otherwise I hope you could shed more light now on my original question regarding energy expenses.

Let me point our that if they are traveling side by side at the same velocity, then spaceship A is stationary relative to spaceship B and spaceship B is stationary to spaceship A, so if one accelerates, the problem is identical to if they had both started from rest.

Consider spaceship A. If it is going some constant speed and then accelerates with an acceleration $a$ for some amount of time, it takes the same amount of energy input as if spaceship A had accelerated from rest for the same amount of time. It doesn't matter if it's going slow or its going at $0.999999c$.

I would start a thread dedicated explicitly to what relativistic mass is if one does not already exist. It's a contentious topic and I'm sure you would get a plethora of interesting perspectives. Typically, however, the term "relativistic mass" is considered old fashioned.

 

[infector]

If it is going some constant speed and then accelerates with an acceleration aa for some amount of time, it takes the same amount of energy input as if spaceship A had accelerated from rest for the same amount of time. It doesn't matter if it's going slow or its going at 0.999999c0.999999c.

and

I would start a thread dedicated explicitly to what relativistic mass is if one does not already exist.

Relativistic mass was first thing which came to my mind when you mentioned about the same energy input. And I must ask: isn't it true that it takes more and more energy to increase velocity when approaching the speed of light? E.g. it would cost much more energy to increase velocity +100km/s when traveling at 0.9c than when at 0.01c, right? That seems to me to be in a contradiction with what you wrote about equal energy inputs in both cases (or probably I am getting something wrong here or maybe not taking time dilation or something else into account).
To clarify: we have two moving ships, one at 0.01c and one at 0.9c. They both start to accelerate at 10km/s² for 10 seconds. I assume we are talking here about 10 seconds of each ship's proper time, i.e. on each ship's onboard clock? Would both ships really spend same amounts of energy and increase their velocities for 100km/s? But what about what I wrote above about higher energy demands required for accelerating an object in relativistic velocities than in non-relativistic ones?

 

[Ibix]

Velocity is relative. If you must use relativistic mass (and I'd recommend you don't) then your own assessment of your relativistic mass is always equal to your rest mass because you are always at rest with respect to yourself. So if you fill your fuel tanks, deploy a marker buoy, and then expend some energy E to accelerate to v with respect to the buoy, you may consider yourself at rest and the buoy to be moving at -v. Therefore if you refill your fuel tanks, deploy another buoy, and then expend energy E you will find yourself traveling at v with respect to the second buoy. Repeat as often as you like.

After the second acceleration phase you will not be doing 2v with respect to the first marker buoy. You may interpret that as the first marker buoy seeing you as having an increased relativistic mass if you like (although I'd recommend against it). But, again, you will always see your relativistic mass equal to your rest mass.

 

[Daniel Gallimore]

andRelativistic mass was first thing which came to my mind when you mentioned about the same energy input. And I must ask: isn't it true that it takes more and more energy to increase velocity when approaching the speed of light? E.g. it would cost much more energy to increase velocity +100km/s when traveling at 0.9c than when at 0.01c, right? That seems to me to be in a contradiction with what you wrote about equal energy inputs in both cases (or probably I am getting something wrong here or maybe not taking time dilation or something else into account).
To clarify: we have two moving ships, one at 0.01c and one at 0.9c. They both start to accelerate at 10km/s² for 10 seconds. I assume we are talking here about 10 seconds of each ship's proper time, i.e. on each ship's onboard clock? Would both ships really spend same amounts of energy and increase their velocities for 100km/s? But what about what I wrote above about higher energy demands required for accelerating an object in relativistic velocities than in non-relativistic ones?

You've confirmed a hunch of mine. You don't seem to be familiar with the velocity addition equation at relative speeds.

Suppose we are observing spaceships A and B. Let spaceship B's speed relative to our frame of reference be $u$, let spaceship A's speed relative to spaceship B be $v$, and let spaceship A's speed relative to our frame of reference be $w$. You may think that $w=u+v$, but actually $$w=\frac{u+v}{1+\frac{uv}{c^2}}$$ Thus, you can't keep adding speed until you surpass the speed of light. You can continue to accelerate, and that takes energy, but you won't continue to add speed at the rate you previously were. The important thing, however, is that accelerating for a time $t$ uses some amount of energy $E$ however fast you're going, even if you gain less speed in some other observer's reference frame.

I think your confusion may be coming from the way I carelessly used the word "acceleration" without defining the reference frame. Adding energy doesn't just accelerate the spaceship, it makes it more difficult to accelerate, but only in our frame of reference. In the robot's own frame of reference frame, it will continue to feel a force due to the accelerating craft, and this force will be constant if the acceleration is constant. Since the speedometer in the spaceship is depending on what we see in our reference frame to get a speed for the craft, the speedometer tells the robot that the craft is accelerating less and less, which is very confusing to the robot who is nearly being pressed straight through his seat.

It's a little counterintuitive, and I'm sure it seems contradictory if you're used to working in a nonrelativistic setting, but there is one property of matter that ties this all together. Einstein showed that relativistic momentum is not $m_0v$ but actually $\gamma m_0v$. Doing work on the spaceship (i.e., adding energy) changes the momentum, so it not so surprising that you get diminishing returns for adding the same amount of energy. Also remember that acceleration, like speed, depends on your reference frame. As spaceship A accelerates away from spaceship B, we see it accelerate (that is, change velocity) rather slowly, and that acceleration appears to get smaller and smaller as the spaceship gets closer and closer to the speed of light, when in fact the spaceship has changed nothing about how it is accelerating.

 

[infector]

You've confirmed a hunch of mine. You don't seem to be familiar with the velocity addition equation at relative speeds.

You were right, thank you for providing the correct formula.
Your detailed answer finally made me understood the whole problem I described in the first post and for that I am very thankful.
May I ask if you are studying physics or is it just your hobby?

 

[Daniel Gallimore]

@infector It's my major, but it has all the joy of a hobby.

 

[infector]

Ok. Once again thanks for all explanations.
My thanks also go to everyone else who tried to help sharing their answers here.
All the best.

 

[Buckethead]

One thing that was not mentioned is that with a lone ship, there may not be any acceleration felt. An empty universe is an abstract idea. For example, let's say the exhaust of the ship has a velocity relative to the ship of 1000mph. You might think the ship would accelerate from 0 and after a certain time, you might think the ship will continue to accelerate past the velocity of the exhaust which would happen in real life, but the thing is, the exhaust will never be faster than 1000mph relative to the ship, so the ship can never travel faster than 1000mph relative to anything in the universe which in this case would just be the exhaust. And would you feel acceleration in such a circumstance? There is a simply way to think of this. As soon as the engine fires, the ship would be moving instantly at 1000mph relative to the rest of the universe (the exhaust) which means that after an initial shock the ship would simply be moving at a constant 1000mph and the pilot would feel no acceleration even with the engines continuing to run. So much for reaching .9c in any case.

 

[PeterDonis]

An empty universe is an abstract idea.

Not if it means Minkowski spacetime; that is a perfectly well-defined model and makes definite predictions. Which are not, btw, the ones you describe in the rest of your post, which is simply incorrect as a description of what SR predicts in this scenario.

 

[PeterDonis]

the exhaust will never be faster than 1000mph relative to the ship, so the ship can never travel faster than 1000mph relative to anything in the universe

This is incorrect, because the exhaust is emitted over time, not all at once. The exhaust emitted after the ship has gone from, say, 0 mph to 100 mph relative to the original piece of exhaust, is moving relative to the original piece of exhaust (at 100 mph, if we ignore the small relativistic correction at these low velocities). You need to take that into account in your analysis, and you haven't.

 

[Buckethead]

This is incorrect, because the exhaust is emitted over time, not all at once. The exhaust emitted after the ship has gone from, say, 0 mph to 100 mph relative to the original piece of exhaust, is moving relative to the original piece of exhaust (at 100 mph, if we ignore the small relativistic correction at these low velocities). You need to take that into account in your analysis, and you haven't.

You are making the assumption that there is a third thing involved here, the ship, the exhaust, and space. In an otherwise empty universe there is no space. The ship would always just move at 1000mph since there would never be anything it could move relative to other than the exhaust, not the original exhaust velocity as you suggest.

 

[Nugatory]

You are making the assumption that there is a third thing involved here, the ship, the exhaust, and space. In an otherwise empty universe there is no space. The ship would always just move at 1000mph since there would never be anything it could move relative to other than the exhaust, not the original exhaust velocity as you suggest.

The exhaust emitted at different times is moving at different speeds relative to the exhaust emitted at other times, so the "third thing" that you're thinking about is just the exhaust from a few moments back. The exhaust always leaves the ship with the same speed relative to the ship, so as the ship accelerates the speed of the exhaust relative to previously emitted exhaust is different.

 

[Buckethead]

The exhaust emitted at different times is moving at different speeds relative to the exhaust emitted at other times, so the "third thing" that you're thinking about is just the exhaust from a few moments back. The exhaust always leaves the ship with the same speed relative to the ship, so as the ship accelerates the speed of the exhaust relative to previously emitted exhaust is different.

This is what PeterDonis was alluding to as well. As I mentioned in my post, in our reality, yes, the ship would continue to accelerate as long as there was fuel, but not in the scenario suggested by the OP. In a purely empty universe there is no previous speed, no accumulation of speed. All that there is, is the relationship between the ship and the exhaust and it doesn't involve a previous speed, because there is no previous speed. There is always just the relationship between the ship and the exhaust which in this case is always 1000mph. I grant that this is highly speculative since it is not verifiable, but it is logical nonetheless.

 

[Ibix]

In a purely empty universe there is no previous speed, no accumulation of spee

But the universe is no longer empty after the rocket has started firing, unless you are additionally proposing that the rocket exhaust vanishes upon leaving the ship. You can always look back at your whole exhaust plume to get a history of your state of motion.

I grant that this is highly speculative since it is not verifiable, but it is logical nonetheless.

I dispute the "logical". This is a specific case of something that always seems to come up in these "empty universe except for one thing" scenarios. They always seem to me to be, on closer inspection, "empty universe except for some mechanism made of multiple parts that can be compared to one another, thus contradicting their own empty-except-for-one-thing premise" scenarios.

 

[Buckethead]

But the universe is no longer empty after the rocket has started firing, unless you are additionally proposing that the rocket exhaust vanishes upon leaving the ship. You can always look back at your whole exhaust plume to get a history of your state of motion.
I dispute the "logical". This is a specific case of something that always seems to come up in these "empty universe except for one thing" scenarios. They always seem to me to be, on closer inspection, "empty universe except for some mechanism made of multiple parts that can be compared to one another, thus contradicting their own empty-except-for-one-thing premise" scenarios.

With regard to your first sentence, think of it as a polygraph. The paper always rolls out at the same speed relative to the pen regardless of how long that goes on.
With regard to the rest of your post, yes, excellent observation. On close inspection it seems to be impossible to think of motion at all unless relative to something else. A ship can't move unless it is moving relative to the gas it spits out. This was actually a point that there is no motion or indeed acceleration unless relative to something else. Truly, acceleration and it's relative, rotational forces, are about as bizaare as you can get. When you accelerate, what are you accelerating relative to? In an otherwise empty universe, it is the exhaust gas, otherwise you are not moving. You can't just move! it's always a reactionary situation. It's worth thinking about. It's not as simple as it seems.

 

[Umaxo]

This was actually a point that there is no motion or indeed acceleration unless relative to something else. Truly, acceleration and it's relative, rotational forces, are about as bizaare as you can get. When you accelerate, what are you accelerating relative to? In an otherwise empty universe, it is the exhaust gas, otherwise you are not moving. You can't just move! it's always a reactionary situation. It's worth thinking about. It's not as simple as it seems.

Is it really neccesary to have frame of reference in order to have acceleration? Couldnt you just define acceleration by fictious forces you are experiencing?

 

[Ibix]

With regard to your first sentence, think of it as a polygraph. The paper always rolls out at the same speed relative to the pen regardless of how long that goes on.

If the pen is the rocket and the paper is the exhaust plume then your scenario does not conserve momentum.

 

[A.T.]

...there is no space. The ship would always just move ...

If there is no space, how can it move?

 

[Dale]

Couldnt you just define acceleration by fictious forces you are experiencing?

Yes, you can. @Buckethead is wrong in this, IMO

 

[Dale]

With regard to your first sentence, think of it as a polygraph. The paper always rolls out at the same speed relative to the pen regardless of how long that goes on.

This is not correct. The paper is a "rigid" object with intermolecular forces that maintain distance so that the speed of all the unrolled material is constant. The exhaust is a gas which is not rigid but has different speeds.

 

[Dale]

I grant that this is highly speculative since it is not verifiable, but it is logical nonetheless.

The known laws of physics are perfectly capable of analyzing a rocket alone in a flat spacetime. What you describe is not a logical result of applying the known laws of physics to such a universe.

It is possible that such a universe would have different laws of physics or not be a possible universe, but that is pure speculation at this point. Usually such speculation is motivated by Mach's principle, which has not been demonstrated to be a correct principle of physics.

 

[PeterDonis]

In an otherwise empty universe there is no space.

Yes, there is: there is Minkowski spacetime, that's what "an otherwise empty universe" means (assuming the rocket and its exhaust have negligible stress-energy themselves, so they don't curve spacetime). You can't make any physical predictions at all if you don't have some spacetime geometry; saying "an empty universe" without assuming some spacetime geometry is just meaningless words, not physics.

 

[PeterDonis]

In a purely empty universe there is no previous speed

Yes, there is, because you have a spacetime geometry and "speed" can be defined relative to a particular inertial frame (the one in which the rocket started out at rest) in that spacetime geometry. See my previous post.

 

[PeterDonis]

With regard to your first sentence, think of it as a polygraph.

You are getting very close to a warning for personal theory. You are not describing how motion in flat Minkowski spacetime works.

 

[Buckethead]

OK - I concede. I am in no position to argue if Minkowski spacetime remains intact or not in a universe that is void of everything except for one spaceship. It is very difficult for me to step away from the notion of Mach's principle.

 

[infector]

Thank you all for your input, very interesting discussion emerged here.
To all my (relatively little) knowledge on the topic I agree with @PeterDonis that:

"speed" can be defined relative to a particular inertial frame (the one in which the rocket started out at rest) in that spacetime geometry.

 

[PeterDonis]

I am in no position to argue if Minkowski spacetime remains intact or not in a universe that is void of everything except for one spaceship.

The question is not so much whether Minkowski spacetime "remains intact" -- Minkowski spacetime is a highly idealized model, everyone recognizes that. The question is, do we have any other model that could possibly describe "a universe that is void of everything except for one spaceship". The answer to that question is no, we don't. And in the absence of any other model, Minkowski spacetime is the best we can do, and anything else is just speculation and is off topic here.