Imagine an astronaut in a space ship which, once it has left the earth's gravitational field, begins to accelerate continuously at 9.8 m/s2. I am imagining the ship's engine is efficient enough to accomplish this. By my calculation, he would be traveling at the speed of light, relative to his point of origin (earth), after 347.125 days. Assuming our space ship has a sufficiently sustaining life support system, our astronaut should be quiet happy and healthy, having spent all of that time in a simulated gravitational field. Ironically, he would have the sensation of being a body at rest the entire time.

 

What law of physics makes this imaginary scenario impossible?

Relativity. The amount you can change the speed is not given by naive application of the classical kinematic equation v = at; this is an approximation which begins to noticeably fail once you get to a reasonable fraction of c. You would not be able to sustain the acceleration you originally had. The kinetic energy of your ship is not given by ½ mv². This, too is an approximation that is valid at speeds below ~ 0.1c

Thus applying Relativity to the problem it turns out that the ship, accelerating at 1g as experienced by the astronaut, will reach 70% of c in 347.125 days as measured by Earth, and 75% of c in 347.125 days as measured by the astronaut.

Special relativity treats acceleration differently then classical physics does.

In special relativity, the velocity of an object after moving for a time t at an acceleration a is

[latex]v=\frac{at}{\sqrt {1+\frac{a^2t^2}{c^2}}}[/latex]

This is to ensure that the velocity never exceeds the speed of light, even in the presence of acceleration.

Also, as he is accelerating, he wouldn't have the sensation of being a body at rest. He'd have the sensation of being in an Earth-equivalent gravitational field, but those aren't exactly the same thing.

Still, it is a difficult thing for me to visualize how two objects traveling at 99% the speed of light, in opposite directions, still can't reach the speed of light relative to each other. I know that the rule is that the speed of light is constant regardless of the motion of it's source, but the relative speed of the objects themselves is where I get confused.

Still, it is a difficult thing for me to visualize how two objects traveling at 99% the speed of light, in opposite directions, still can't reach the speed of light relative to each other. I know that the rule is that the speed of light is constant regardless of the motion of it's source, but the relative speed of the objects themselves is where I get confused.

 

You and a lot of people. It's outside of our normal experience, but the theory says it's so, and it's backed up by tons of experimental evidence. It's how the universe behaves.

Most of relativity is difficult to visualize.

Still, it is a difficult thing for me to visualize how two objects traveling at 99% the speed of light, in opposite directions, still can't reach the speed of light relative to each other. I know that the rule is that the speed of light is constant regardless of the motion of it's source, but the relative speed of the objects themselves is where I get confused.

You can say that they have a separation speed greater than the speed of light, that is no problem at all and does not break any rules of relativity. However this is not the relative speed. No observer can interpret this separation speed as a relative speed. In particular it is not the relative speed that one of the objects measures the other object to have, which will always be less than the speed of light.

You can say that they have a separation speed greater than the speed of light, that is no problem at all and does not break any rules of relativity. However this is not the relative speed. No observer can interpret this separation speed as a relative speed. In particular it is not the relative speed that one of the objects measures the other object to have, which will always be less than the speed of light.

 

 

Can not a third party observer see them as separating at faster than light?

 

 

Can not a third party observer see them as separating at faster than light?

 

It's right there in the post. Separation speed is not limited to c.

 

It's right there in the post. Separation speed is not limited to c.

 

 

My mistake, i read it wrong, i see it now...

Just to be complete: Separation speed *is* limeted to 2c as long as we stay in special relativity so that space itself cannot expand.