if no preferred location in space

[billwald]

If there is no preferred location and no preferred dirrection in space then consider a spaceship in an empty part of space with no charts. The ship can accelerate to 80%C. The crew is all asleep.

The mate wakes, looks around, sees nothing of interest, puts out a marker buoy and accelerates to top speed. After awhile he goes back to his bunk. He didn't log anything.

Some hours later the skipper wakes, looks around, sees nothing, puts out a marker buoy and accelerates to 80% C with respect to the 2nd marker. What is the ship's speed with respect to the first marker and why?

 

[Avatar]

Your original assertion is incorrect if the ship can continue to accelerate after reaching .8c.

 

[Paul365]

A ship that can accelerate to 80%c would need a mechanism to measure its speed and shut the engine off when 80%c are reached. This requires a marker for a speed reference.

The speed relative to the first marker is v' = (v1+v2)/(1 + v1v2/c^2) = 1.6c/1.64 = 0.976c

 

[Tinker Grey]

There are a couple of problems with the formulation of the question. This causes us to make assumptions as to what is meant.

Going on a hunch, here is my first guess:
The OP says the ship can go 0.8C. The OP didn't really mean accelerate. If this is correct, then the OP thinks that because there are no referrents, there is nothing to stop the ship from reaching 80% of the way to C from it's current speed. But, the flaw here is the assumption that speed depends on referrents. Our ability to detect speed depends on referrents. If the top speed is 80%C, period, then the second buoy would be stationary.

If, in fact, the OP means accelerate, there is a different problem. The is only ONE speed limit in space (assuming a perfect vacuum) -- and that is C. There is no such thing as being limited to 0.8C. You may accelerate until you run out of fuel to expel. In the case of solar panels that various scientists and sci-fi writers propose, you can accelerate until the source of energy is too far away. However, since the OP specifies 'no referrents', there is no external source from which to absorb energy. Because of pratical issues, an ordinary ship requires an infinite amount of mass to expel to reach C.

I think this second option suggests that perhaps the OP wants to accelerate to 0.8C, then 80% of the distance to C, then 80% of the remaining distance, and so on. Again, this would take infinite mass to expel. Assuming some ideal ship that could carry a nearly-infinite mass, you could continue to close the gap on C. However, under those conditions, talk of 80% is meaningless.

If the question is: Could we go faster than the speed of light if we didn't know how fast we were going? Then, the answer is no. Again, your speed doesn't depend on referrents. Your ability to detect speed depends on referrents.

 

[shernren]

In any relativistic universe (and remember that the first relativity was Galilean, not Einsteinian - so this applies even classically) velocity is not absolute. The easiest way to interpret a "top speed of .8c", then, is that the ship can reach a maximum speed of .8c, on any particular trip, relative to an observer who remains stationary at the origin point of that particular trip.

This is true even classically. Consider a road with a speed limit of 80km/h. Suppose a car is going down the road with a speed of 60km/h. A cop, driving the other way on the opposite lane with a speed of 40km/h, measures the first car's speed as 100km/h. Should the driver be fined? Obviously not.

(The more mathematically tangible way to describe this is that velocity is defined as the rate of change of displacement. But displacement has to be measured from an arbitrary origin, so that effectively we are back to measuring velocity from the PoV of a stationary observer at the original position.)

Now let us rephrase the example in a classical situation. I am driving a car on a road that has a speed limit of 80km/h. I'm parked on the roadside and leave a marker there. Marker A remains stationary. I then accelerate to 80km/h - relative to Marker A, by the intuitive definition of velocity - and throw out a Marker B.

What is my speed relative to Marker B? That depends on the inertial nature of Marker B. For suppose Marker B is a kite that can fly frictionlessly in the air - therefore, it continues to move at 80km/h alongside me. If I now accelerate to 80km/h relative to Marker B, my velocity relative to either Marker A or a stationary cop by the roadside is 160km/h, and I have broken the speed limit.

On the other hand, suppose Marker B is a heavy lead weight - as soon as it hits the road it stops. Then I am already at 80km/h relative to Marker B and relative to a stationary cop by the roadside and to Marker A, and I have not broken the speed limit - but I cannot accelerate any more without breaking it. Thus if both A and B are stationary (as the original problem implies) I cannot both accelerate, and maintain a speed of 80km/h relative to both A and B. If I accelerate, my speed relative to A will be different from my speed relative to B.

Thus, there is no paradox, whether in a classical or a relativistic situation. The only difference in relativity is that if I miraculously accelerated to the speed of light, I would indeed be traveling at the speed of light relative to both A and B - and that makes all the difference.

 

[billwald]

Thanks for the responses. This sort of thing drives me nuts. I was thinking of Shroeder's (sp?) cat problem that observation creates reality?

 

[Paul365]

Thanks for the responses. This sort of thing drives me nuts. I was thinking of Shroeder's (sp?) cat problem that observation creates reality?

No, this effect is unrelated to Schrödinger's cat. It's simple relativistic velocity addition.

Normally you add velocities like v = v1 + v2 (such as when you accelerate a rocket).

This equation becomes invalid on high speeds. You then have to take into consideration the relativistic length contraction (the distance to travel becomes shorter) and time dilatation (the time in the rocket ticks away slower). This leads to the relativistic velocity addition as in my above response, and results in that you can never reach the speed of light relative to any marker buoy.