The astronaut accelerated, the people on earth did not.

They accelerated away from him though. If there's no absolute frame of reference how can you say who accelerated?

Velocity is relative, but acceleration is not. The earth-dwellers and the astronaut will both agree that the astronaut accelerated and the Earth didn't.

Similarly, rotation rate isn't relative. Acceleration and rotation rate can both be measured against absolute states of no acceleration and no rotation (respectively). This is why our phones have accelerometers and gyroscopes which are entirely internal, but determining velocity, position, and direction require the external references provided by GPS and compass.

How can acceleration not be relative if velocity is? Isn't it just the derivative of velocity? How do you escape the relativeness? Or is that what all those theories of relativity are about?

It's possible to absolutely measure the change in a quantity without knowing the absolute value of the quantity itself. Think of a hole in the ground of unknown depth but known radius, filled with water. If the water level changes you can calculate the absolute change in water volume without knowing the total volume of water in the hole. Similarly, you can measure an absolute value for acceleration (change in velocity) even though an absolute value of velocity doesn't exist.

If you have a fancy sports car, you can feel acceleration. It slams you back in your seat. (check out this: https://secure.wikimedia.org/wikipedia/en/wiki/Equivalence_p...) You cannot however feel speed.

Therefore, you can measure acceleration with nothing but equipment (a simple spring-scale will suffice) in your car (reference frame), but you can only measure your speed relative to something outside of your car (traditionally, the road).

That might be a deeper question about the nature of the Universe.

Newton's first law (inertia): an object in motion tends to stay in motion. This gives rise to the idea of an "inertial reference frame" which is the idealized reference of a hypothetical object in motion. In simplistic terms, if you pretend that the Universe is merely Newtonian in nature and contains a preferred reference frame then you could imagine that an inertial reference frame can be described by a velocity vector (a speed and a direction). If an object has zero speed relative to the inertial reference frame that will tend to continue to be true so long as no forces intervene.

If you induce an acceleration on an object you will change it's velocity. In it's initial inertial reference frame it will appear to be accelerating. In any other inertial reference frame it will also appear to be accelerating.

There is no such thing as an accelerating inertial reference frame precisely because such a frame is not inertial, it would imply that forces are being applied to the objects involved. The laws of physics make sense if you apply them within any inertial reference frame, but they don't make sense in accelerating reference frames (mass, energy, and forces become inconsistent and confused).

Does that make sense?

I don't think acceleration is a torsor, quite, but if you understand a torsor, you understand how acceleration can be absolute even if velocity is not: http://math.ucr.edu/home/baez/torsors.html

In this example, the earth isn't in an accelerating reference frame, only the astronaut is. There isn't a "master reference frame", but that doesn't mean that acceleration is relative.

Despite the upvotes on reddit, this really is a topic that can only be done justice by good book or instructor. Any self-respecting library should have a copy of Relativity: The Special and General Theory by Einstein, it's written specifically for the layman. Come to think of it,... I should probably pick it up again myself.

Because unlike speed acceleration is not relative.

Acceleration can be measured without reference to anything else. If you are in a box and accelerating you can measure it - it feels exactly like gravity.

He feels that gravity, they don't.