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Imagine there exists only a single object (say a 1 metre sphere). There is nothing else in all directions.

  1. Is the object moving or at rest? Is it even possible to tell, given that there is no frame of reference?

Extending the idea, suppose the sphere has some form of (rocket-like) propulsion, constantly accelerating it an arbitrary but single direction. The sphere's velocity, relative to what it was when we first imagined it, approaches the speed of light.

The propulsion then stops for a while (while we think about it). At this point we - I assume - still can't tell the difference between it being at rest or moving.

  1. The propulsion resumes. What exactly prevents the above from repeating, and the sphere continuing to accelerate to a velocity any arbitrary number of times the speed of light, relative to its velocity when we first imagined it?

(To preempt answers such as it's all irrelevant because there's no frame of reference, I forgot to tell you we then discover there is actually a second sphere, exceedingly far away, but in the direction of propulsion).

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  • $\begingroup$ do we suppose that the rocket also ejected some mass (giving yet more reference objects) or that the propulsion was unphysical? $\endgroup$ – ComptonScattering Nov 1 '16 at 22:42
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    $\begingroup$ Have you read any of Ernst Mach's papers on very closely related questions? en.wikipedia.org/wiki/Ernst_Mach $\endgroup$ – user108787 Nov 1 '16 at 22:52
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    $\begingroup$ Let's suppose we repeat this propulsion process, as you suggest. For ease, let's do it twice. After the first time, the speed of the sphere, relative to what it was when we first imagined it, is 99% the speed of light (say). Now we do it again: the speed of the sphere relative to what it was after the first propulsion is 99% light speed. Now we ask: what is the speed of the sphere relative to when we first imagined it? You would be tempted to say 99% + 99% = 198% light speed. In doing so, you assume the rule for combining relative speeds is to add them. Who said this was the right rule? $\endgroup$ – gj255 Nov 1 '16 at 23:10
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To answer your title question, there still isn't an absolute reference frame. You can pick any inertial frame moving at any constant sub-light-speed velocity with respect to the sphere (or not moving at all), and thus you can say the sphere is moving at any speed less than the speed of light. The lack of other objects doesn't mean you can't construct other reference frames.

Let's pick some frame of reference $S$ in which the sphere was initially at rest. At some event $\mathrm{A}$, the sphere begins to accelerate until, in the frame $S$, it is traveling at a speed $v_1$, where $v_1$ is almost but not quite $c$. The end of the acceleration is event $\mathrm{B}$.

From the perspective of an observer in $S$, the sphere is not at rest at event $\mathrm{B}$. Now, let's consider a frame of reference in which the sphere is at rest at $\mathrm{B}$, which we'll call $S'$. Now, the object begins accelerating at some later event $\mathrm{C}$ until it reaches a speed $v_1$ again, this time from the perspective of $S'$, at event $\mathrm{D}$. Again, $v_1$ is very close to $c$.

However, from the perspective of the observer in $\mathrm{A}$, the sphere is not moving faster than light. The relativistic velocity addition formula is not simply $$\text{speed in S at D}=\text{speed in S at B}+\text{speed in S' at D}$$ It's more complicated than that, and it effectively means that you can never observe an object traveling faster than light. It is $$\text{speed in S at D}=\frac{u+v}{1+(vu/c^2)}$$ where $$u=\text{speed in S at B},\quad v=\text{speed in S' at D}$$

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  • $\begingroup$ When the first sphere arrives at the second sphere, though, is it not travelling at many times c, relative to the second sphere? $\endgroup$ – sellotape Nov 2 '16 at 18:03
  • $\begingroup$ @sellotape No, because you can't accelerate something faster than light relative to any reference frame. You can get really, really close, but you can never get to $c$, or beyond. $\endgroup$ – HDE 226868 Nov 2 '16 at 18:04
  • $\begingroup$ So relative to the second sphere, the 2nd and subsequent accelerations make absolutely no difference, and it appears to continue to travel relative to the second sphere at just below c? $\endgroup$ – sellotape Nov 2 '16 at 18:09
  • $\begingroup$ @sellotape They do make a difference, and the speed will continue to increase, but it will do so asymptotically, never quite reaching $c$. $\endgroup$ – HDE 226868 Nov 2 '16 at 18:09
  • $\begingroup$ Do we know - aside from the almost universal acceptance that is it not possible - exactly what prevents the first sphere, in the complete absence of anything else, from continuing to accelerate (let's say linearly as in v=at) indefinitely? If that's a larger subject than a comment permits, perhaps you have a link or two to share? $\endgroup$ – sellotape Nov 2 '16 at 18:21
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I would like to make a connection between what @Bill Alsept and what @HDE 226868 said.

Note to the Fifteenth Edition [The Special and the General Theory]

In this edition I have added, as a fifth appendix, a presentation of my views on the problem of space in general and the gradual modifications of our ideas on space resulting from the influence of the relativistic view-point.
I wished to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept "empty space" loses its meaning.

June 9th, 1952, A. Einstein

  1. [Also] any mass has a gravitational field. Any change in field propagates with the speed of light. So if your object is moving, the gravitation field will vary in time at the point that you chose as reference. Also if your object is charged again the E/M field will vary at your reference point.
    Your object will no longer be "alone" if it's accelerates - it emits gravitational waves.

  2. It is in the relativistic view that no object can move faster than the speed of light. Here's way :
    To accelarate an object you need energy to give to the object.
    The total energy will become:
    E = mc2/(1 - v2/c2)
    It's clear that in order to accelerate it to a velocity close to the speed of light you will need almost infinite amount of energy. You will run out of fuel in your attempt, because if you have fuel than the fuel has a mass mfuel. So your object will have a total mass M = m + mfuel.
    The max amout of energy you can extract from your fuel is
    Efuel = mc2
    which is not enought to accelerate it to the speed of light.
    One other solution is to convert your object to light (electromagnetic waves) or gravitational waves which will have the speed of light.
    And other [hypothetical] solution would be to create a traversable wormhole and you would travel to other place in less time, but never during your "trip" will you exceed speed of light.

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A universe is made of both matter and space. The area between the matter is space. There cannot be one single object in the middle of a universe of space. If all the matter in the universe condensed down to one object there would be no space left. If that were the case then relevant to itself the object would not be moving. If there were other universes then maybe it would be moving compared to them but we do not know that.

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