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From a traveler's point of view, as he is accelerating with $1g$, in under one year he would reach the speed of light. Note that from his point of view, everything looks normal so he could keep accelerating, except of course other than all distances become zero and time looks frozen, and probably eventually points ahead of him would become his behind, and time started moving backward.

So, what prevents him from reaching speed faster than light?

tpg2114
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Ari Royce Hidayat
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    Related: https://physics.stackexchange.com/q/517015/ – Mohammad Javanshiry Dec 28 '19 at 07:19
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    Thanks, read that. But how if there is no object outside the spaceship? – Ari Royce Hidayat Dec 28 '19 at 07:33
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    The question isn't what stops him from traveling faster than light; it's what stops everything else from traveling faster than light. – WillO Dec 28 '19 at 07:43
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    @WillO Actually I think that would almost be usefully posted as an answer. It wouldn't take much more to turn it into a good answer. – David Z Dec 28 '19 at 07:58
  • @AriRoyceHidayat Then, there is nothing outside the spaceship to which the observer can attribute his velocity. To him, solely an acceleration is meaningful unless we want to consider Mach's principles according to which there is even no acceleration induced in the spaceship in the absence of outer objects. – Mohammad Javanshiry Dec 28 '19 at 08:00
  • @MohammadJavanshiry I see in the answer referenced where it compared the rate of energy used per unit time. However I don't see where it is relevant, as e.g. when the traveler almost reaches the speed of light, and then he looks outside, he would see the entire universe is frozen which is of course he would see almost zero energy spent in the entire universe, thus a lower energy consumption, but how could it affect his own energy spent to keep accelerating? – Ari Royce Hidayat Dec 28 '19 at 08:07
  • @PM2Ring Yes, but it's from point of view of an observer staying on earth, but it's not so from traveler's point of view. – Ari Royce Hidayat Dec 28 '19 at 08:08
  • @PM2Ring But it's not. When we are sitting on earth with 1g, our spaceship is the entire universe. When we accelerate from earth with 1g, we left the entire universe with 1g in acceleration, and that's the case, what prevents it from reaching the speed of light? – Ari Royce Hidayat Dec 28 '19 at 08:24
  • Let us continue this discussion in chat. – PM 2Ring Dec 28 '19 at 08:35
  • @AriRoyceHidayat The fuel consumption is not lowered from the viewpoint of the observer in the spaceship frame and it remains the same all times, but rather he claims that the spaceship consumes much fuel than before for getting a few distances away from, say, the earth as the spaceship reaches close to the speed of light from the viewpoint of the observer on the earth. I used much fuel figuratively ... – Mohammad Javanshiry Dec 28 '19 at 08:37
  • Indeed, I mean, when the spaceship is far away from the earth, its engines should be "on" for a long time to make a very short displacement compared to when the spaceship was close to the earth having small velocities. – Mohammad Javanshiry Dec 28 '19 at 08:38
  • @MohammadJavanshiry But a distance from traveler's viewpoint, would still look the same, and consumes the same amount of energy to travel it faster. – Ari Royce Hidayat Dec 28 '19 at 08:41
  • @AriRoyceHidayat Nope. Remember that if the observer in the spaceships undergoes a constant acceleration, the spaceship's acceleration is no longer constant from the viewpoint of the observer on the earth. And vice versa, if the spaceship acceleration is measured constantly by the observer on the earth, it is no longer measured constantly by the observer in the spaceship. – Mohammad Javanshiry Dec 28 '19 at 08:51
  • @AriRoyce Hey! it's generally look bad to edit out question's like that because puts answers like the ones posted by Cleonis out of context. Hence, I suggest to revert the edit :^) – tryst with freedom Jan 24 '21 at 07:35
  • @Buraian Ok noted, how do I revert it? Thanks. – Ari Royce Hidayat Jan 24 '21 at 07:37
  • I didn't know, but thankfully tpg has edited it now :^) – tryst with freedom Jan 24 '21 at 07:51
  • @Buraian Ok thanks – Ari Royce Hidayat Jan 24 '21 at 08:34

1 Answers1

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This is the case of hyperbolic motion.

Let me shift the thought experiment to a related case.

In a linear accelerator a particle is accelerated by an electric field. You can accumulate kinetic energy as follows:

You create an electric potential that extends from the current position of the particle to some distance ahead of it. By the time the particle has traveled the extent of the potential you switch it off, and you set up a new potential, once again ahead of the traveling particle. You keep repeating that pattern.

Since this is a thought experiment there is no upper limit to the length of this linear accelerator.

So we have a particle traveling along the length of this linear accelerator, and with every stage of acceleration the particle accumulates more kinetic energy.

There is no upper limit to the amount of kinetic energy of the particle.

At each acceleration stage you can describe the physics in terms of some instantaneously co-moving frame, and at each acceleration stage the kinetic energy is increased.

Quantity of motion

The kinetic energy is a valid measure for what we can refer to as quantity of motion.

There is no upper limit to the quantity of motion, since there is no upper limit to the amount of kinetic energy that matter can accumulate.

I'm in favor of shifting to thinking of kinetic energy as the quantity of motion: no upper limit.

One more remark: kinetic energy isn't an intrinsic property. Kinetic energy is meaningful only in terms of relative motion. It's when an object impacts another object that kinetic energy comes in. The larger the relative velocity the larger the amount of kinetic energy that comes in.

Geometry of spacetime

According to Special Relativity the upper limit of velocity should not be seen as a property of matter, it should be seen as a geometric property of spacetime itself.

diagram, minkowski spacetime, hyperbolic motion

Image source: wikipedia hyperbolic motion image file

The image illustrates the geometric property.
The blue line represents the worldline of a point that is subject to uniform acceleration.

The vertical axis is labeled 'T' for time, the horizontal axis is labeled 'x' for position.

The blue line is the graph of the following expression:

$$ x^2 - T^2 = 1 $$

To ask: "In Minkowski spacetime, what prevents the particle from exceeding the speed of light?" is like asking: "Given the expression $ x^2 - T^2 = 1 $, what prevents the blue graph to cross over the black diagonal?

Cleonis
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  • But it is still from the point of view of standing observer, surely there is an upper limit be it geometric reason or simply infinite mass etc. From traveler's point of view, he could keep accelerating and insists his speed is $s = gt$, and when $t$ is almost one year, $s > c$, light would still be seen travelling at speed $c$, but he could not determine its absolute speed rather than relative speed from when he started the journey, etc, i.e. principle of relativity is not broken. What prevents him from having $s \geq c$? – Ari Royce Hidayat Dec 28 '19 at 14:00
  • If the acceleration consists of many identical "burns", then doing a burn changes the previous burns by amount depending on the change of the gamma factor. Durations of burns increase. Thrusts do not change. Inertias that resisted the thrusts increase a lot. In the new frame those old burns look different. – stuffu Dec 28 '19 at 16:16
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    Since the traveler pushes into the realm of relativistic velocity the expression $s = gt$ is not applicable. Instead one must use the rules of relativistic velocity addition. When the rules of relativistic velocity addition are applied the outcome is that the velocity with respect to the point of departure does not exceed the speed of light. So unfortunately, while you assert that you endorse the principles of relativity, you are not actually applying the principles of relativity. – Cleonis Dec 28 '19 at 16:50
  • @Cleonis Relativistic velocity addition is when we want to calculate the total speed from point of view of a standing observer of an object moving relative to another moving object. It's still not from the point of view of the traveler. – Ari Royce Hidayat Dec 28 '19 at 17:45
  • @AriRoyceHidayat Relativistic velocity addition works both ways, that's the whole point of it. You can take any instantaneously co-moving frame (along the traveler's worldline) and then represent all of the motion taking that instanteneously co-moving frame as a point of zero velocity. The representation of the point of departure then accelerates away from the traveler. The relative velocity is the same one way or the other. You seem trapped in a belief system in which special relativity is valid for standing observers, but not for travelers. – Cleonis Dec 29 '19 at 10:16
  • @Cleonis Sorry, wrong comment above, and yes it works both way. So of course if he looks outside the situation would be just like experienced by the standing observer, that in this case no force could accelerate the rest of the universe to speed faster than light. I'm working on something, and curious for what it is like if the traveler could insist his own "absolute" speed because he feels the acceleration, while others don't, and think there should be some continuity for distances and time from almost zero to negative. But never mind, and thanks anw :D) – Ari Royce Hidayat Dec 29 '19 at 12:37