The Speed of Light - Nothing can go faster?

[Passionflower]

Here is a graph of a rocket traveling with a proper acceleration of 1 showing velocity, celerity and rapidity and the diminishing coordinate acceleration.

The area bounded by the Coordinate Acceleration is the velocity while the area bounded by the velocity is the total distance traveled. The rapidity (because the proper acceleration is 1) also shows the proper time for an observer in the rocket.

 

[kamenjar]

A graph to show velocity, celerity and rapidity and to show the diminished coordinate acceleration while the proper acceleration remains constant:

View attachment 28923

So then celerity is perceived (proper?) speed/acceleration from the moving reference frame of the object that is traveling, considering that object's clock?

 

[Passionflower]

So then celerity is perceived (proper?) speed/acceleration from the moving reference frame of the object that is traveling, considering that object's clock?

Celerity is basically chart velocity. Ignoring acceleration to keep it simple, suppose you want to go to a far a way planet. You check the latest edition of Google SpaceCharts and see it is 1 light year away. Say you want to get there in half a year, then the rocket must travel with a celerity of 1/0.5= 2

Then with a celerity w=2 the rocket's velocity would be:

$${w \over \sqrt{1+w^2}} = 0.8944271908$$

The rapidity would be:

$$ln(w+ \sqrt{1+w^2}) = 1.443635475$$

On arrival the clock at home would read:

$${1 \over v} = 1.118033989$$

An observer on Earth watching through a telescope would see your clock during travel slowed down by a factor of:

$${w + \sqrt{1+w^2}} = 4.236067977$$

While, if he takes into account the finite speed of light, he would conclude your clock is slower by a factor:

$$\sqrt{1+w^2} = 2.236067976$$

As you can see simply the Doppler factor minus the celerity.

As you can see, starting from celerity the formulas as easy to remember the recurring part is

$$\gamma = \sqrt{1+w^2}$$

Which is sometimes called the Lorentz factor.

How do you add two celerities together? Just multiply the Doppler factors!

Now of course it is simple if all the motion is one dimensional, when we get tree dimensional motion it becomes far more complicated, unfortunately I cannot find anything in the literature giving a comprehensive description of celerity in three dimensions and all the relevant formulas. If someone knows, please let me know where we can find this.

 

[TheTechNoir]

Somewhere while browsing around this article, I found the answer, I guess...:

A controller based on Earth is monitoring a spaceship moving away at a speed 0.8c. According to the theory of relativity, he will observe a time dilation that slows the ship's clocks by a factor of 5/3, even after he has taken into account the Doppler shift of signals coming from the space ship. If he works out the distance moved by the ship divided by the time elapsed as measured by the onboard clocks, he will get an answer of 4/3 c. He infers from this that the ship's occupants determine themselves to be traversing the distances between stars at speeds greater than the speed of light when measured with their clocks. From the point of view of the occupants their clocks undergo no slowing; rather, they maintain that it is the distance between the stars which has contracted by a factor of 5/3. So they also agree that they are covering the known distances between stars at 4/3 c.

Which to them appears as speed that is faster than however long the light would take in "rest" reference frame to reach. I didn't realize that you perceive this speed whenever you reach some significant relativistic speeds (there is some break-off). And of course, that doesn't mean that they would reach the star before the light would and see themselves taking off from earth, or that would pass the light from my flashlight inside the ship. It still though appears that there is still no relativistic barrier to you accelerating and continuing to accelerate at a rate that appears constant to you and "gaining speed continuously" with respect to distances measured in the Earth reference frame.

May of you ask "what you mean by speed?" or "speed of light from where?". I guess I wasn't clear - the speed would be "static reference frame distance, over dilated time", which is in a way a wrong way to measure things, but still I guess a meaningful way.

You can just keep increasing that speed until you go nuts (if you could fuel) at what appears to you as a CONSTANT rate. And I was wrong to think that something significant happens at some break-off point, but there isn't one.

The reason I thought about it was the case when you leap towards a black hole. It looks similar to constantly accelerating ship. To an outside observer, you never reach the black hole. One would think that in your time frame you would simply fall into the event horizon, but even in this case you don't! The black hole evaporates before you reach it

In other words, each of you is in a way right, but you guys get too bogged down in perceiving speed from a "earth reference frame".

I believe I am completely understanding your thought process behind this perspective.

You are thinking if you hypothetically continue accelerating to arbitrary velocities, then while relativistic effects prevent you from surpassing the speed of light surely in your OWN frame of reference if you hypothetically continued accelerating for an infinite amount of time let's say, then it would be inevitable that from your own frame of reference you have indeed broken the light barrier. You are also acknowledging and understanding that if you shine a flashlight even after you have passed the light barrier it will still shine ahead of you at the speed of light. So while you are not technically past the speed of light in your frame of reference, you have in some other way still actually achieved light speed as you have been accelerating consistently for up to an infinite amount of time. No?

The problem with this mentality though is that if you were to hypothetically continue said acceleration you still are always only accelerating in reference to something else, again there is no fixed space in which you are moving across faster and faster. So for you, you will never even approach light speed because light will never slow down for you. So from your perspective this acceleration will have relativistic effects on external things you observe but never directly effect you personally. Whereas from any other observers perspective, as you continue accelerating you get closer and closer but you never reach light speed as the acceleration slows more and more. This is all that matters, what you experience (no light speed) and what others experience (they see you traveling at less than c) there is no being immune to relativity that can observe you accelerating at a defined/fixed/true consistent speed until you eventually surpass c.