If an infinite universe containing only a single object, is the object at rest?

Question

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.

2. 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).

Answer 1

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 $$u=\text{speed in S at B},\quad v=\text{speed in S' at D}$$

Answer 2

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 m_fuel. So your object will have a total mass M = m + m_fuel.
   The max amout of energy you can extract from your fuel is
   E_fuel = 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.