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Doesn't really work this way. A lot of the wonkiness in SR is tied to the fact that the speed of light is the same, measured in _any_ reference frame.

So, say you're on earth and you measure the speed of light... you find that it's c (~3x10^8 m/s).

Now you get on a spaceship and accelerate to 0.5c with respect to earth, and you measure the speed of light relative to your spaceship... still c!

In this way, you can't really define a reference frame with a speed "the same as the speed of light". And if you try, you'll run into nasty infinities in all your equations that will cause them to blow up and stop being useful.

2 comments

So depending on how you measure, you’re always stationary or moving near light speed, or somewhere in between, depending on your measurement reference (the thing you’re moving relative to)?

How is there a speed limit at all, if that’s the case? You can accelerate to 0.5c and then toss an apple out the window and say you’re moving at the speed of an apple tossed out of the window, relative to the apple. You have all of c available as headroom again? You can accelerate up to 0.5c again, relative to the apple you tossed out the window?

I am imagining you will say that it will seem like this is what is happening to folks in the spaceship, but what’s really happening is that time is slowing for the spaceship and it’s passengers, and that they still can’t reach c. Fine. But c relative to what? There is no absolute c because there are no truly fixed points, so c relative to what?

There is no underlying reference frame. All motion is relative. Everyone, no matter how fast they are already going, will measure the speed of light as c. Accelerate to .99c and shine a flashlight in front of you. That light is moving ahead of you at the speed of c. Because to you, you are not moving.

That's true for the laws of physics, yes, but our universe does have a 'natural' frame of reference.

> A comoving observer is the only observer who will perceive the universe, including the cosmic microwave background radiation, to be isotropic. Non-comoving observers will see regions of the sky systematically blue-shifted or red-shifted. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local frame of reference called the comoving frame. The velocity of an observer relative to the local comoving frame is called the peculiar velocity of the observer.

From https://en.wikipedia.org/wiki/Comoving_and_proper_distances#...

> You can accelerate to 0.5c and then toss an apple out the window and say you’re moving at the speed of an apple tossed out of the window, relative to the apple. You have all of c available as headroom again? You can accelerate up to 0.5c again, relative to the apple you tossed out the window?

Yes you can. You can even do it with 0.6c for both those speeds.

But critically, having done that, you still won't be going >=1.0C relative to the road.

Yep, exactly.

> so c relative to what?

it's either "relative to any observer." or "relative to any inertial reference frame". no matter where you go (on the ship, on a planet you pass by, on another ship) you will never see the apple travel as fast as the photons coming out of your flashlight. Depending on where the observer is, they will see the apple accelerate to 0.5c (if they are aboard the ship) or they will see it gain mass (or rather, see you throw it more slowly as if it had gained mass), contract in the direction it's thrown, and slow down (due to time dilation...relative to the moving frame).

The case I don't know how to answer is two apples thrown at each other, each with a speed greater than 0.5c.

If you want to explore/understand the velocities of these relativistic apples, look into the Velocity-addition formula[1].

1: https://en.wikipedia.org/wiki/Velocity-addition_formula

I was thinking of this differently.

Suppose there is a starting point A from which your ship is moving away from. At the same time, a photon is shot out from A. You can take the distance traversed between A and the ship as D1, and the distance traversed by the photon as D2. Then your "percentage of C" is D1/D2.

How can you get the distance D2? I'm not sure. I guess we have to pretend it's also a ship that is traveling at C that can emit information to us (also at C :p )