Speed of light problem

We all know about time dilation as you get closer and closer to C.
for instance, at 0.9C, the dilation is about 2.3:1. But at 0.1C the time dilation is only about 1:1.0101, a shortening of the journey by about 1 percent. So really this time dilation thing has two parts. One part is the actual velocity, and the other the time dilation ratio.
So at 0.9C, using 186,282 MPS as C, then 0.9C is 167653.8 MPS
Consider the distance to Alpha Centauri, the closest known star, at least one of the triple anyway at 4.3 odd LY. If time dilation were not a factor, if you went at C, it would take you 4.3 years, and of course if you were going at C, people on earth would still think it took you 4.3 years but it would be near instantaneous. Impossible of course.
But look at lower numbers.
If you go 0.1 C, ignoring time dilation, it would take 43 years to get to AC.

With time dilation taken into account, your time is about 1 percent less, or about 42.57 years. On board the craft, you know time went by a bit faster and you shaved 6 months off your journey, but you know it was not even close to the speed of light so you turn around and go back home, you have aged one year less, but your clocks are only out by that much. So you get closer to C. Suppose you go 0.5 C.
93,141 MPS. Time dilation now makes your clocks run about 15 percent faster, but you are still going only 0.5C. so now that same journey not counting time dilation, takes 8.6 years, but you think its 15% faster or about 7.3 years. Not quite up to light speed yet.
But you notice, at some point you getting closer to C, there is a velocity you travel that has you going at the speed of light, that is to say, it takes 4.3 years according to your clock so you THINK you are going at the speed of light but sadly, you get home and somewhat more time has elaped on Earth. What is that velocity? If you go at 0.9 C, the journey would take 1.1111 times 4.3 years or about 4.77 years
but the time dilation ratio at that velocity is about 2.3:1, so your clock says it took 2.07 years so you think you went over TWICE the speed of light. Of couse earth clocks disagree, but what is the velocity where you think you went exactly at C?

Three equations, three unknowns:

{x} = the square root of x

t' = t/{1-(v/c)^2}

c = d/t'
v = d/t

=> c = d{1-(v/c)^2}/t
=> t = d{1-(v/c)^2}/c

=> v = c/{1-(v/c)^2}
=> v^2 (1-(v/c)^2) = c^2
=> v^2 - v^4/c^2 = c^2
=> v^2 c^2 - v^4 = c^4
=> v^4 - v^2 c^2 + c^4 = 0
=> v^2 = (c^2 -/+ {c^4 - 4c^4})/2
=> (v/c)^2 = (1 -/+ i{3})/2
=> v/c = (1 +/- i{3})/2
=> v = (1/2 +/- i{3}/2) c

I'm getting an unreal answer.

You are getting i*1.366 and i*0.366 C, neither of which helps even if you ignore i. 1.732/2= 0.866 +/- 0.5
Even 0.866 by itself is no help.
.86C looks like sqrt 3 C to the people on the ship, 1.732C. Back to the drawing board.

can you please reformulate your questions? they make no sense!

Originally posted by 3v1l5w1n
can you please reformulate your questions? they make no sense!

If you are in a spaceship, and go say, half the speed of light, there is a certain time dilation, given by the famous T= T1/Sqrt (1-V^2/C^2)
At 0.5C, that # is 1.1547, so your clock and your atoms and all else is going about 15% faster. Meanwhile, you are going at 0.5C.
without time dilation, going say, to Alpha Centauri, you would take twice as long as a light beam, or about 8.6 years, ok?
But with time dilation, you take the inverse of 1.1547, or 0.866 times that 8.6 year figure, or about 7.5 years. So you think you are going not 0.5C but 4.3/7.5 or about 0.57C. You have gained a bit on light because of the time dilation.
So if you ramp up the speed, now you are going at 0.9 C, Time dilation works out to about 2.2C, times 0.9, your actual speed, and you think you are going about 2C. So what would your velocity have to be to make you think you are going right at the speed of light?

Originally posted by sonhouse
If you are in a spaceship, and go say, half the speed of light, there is a certain time dilation, given by the famous T= T1/Sqrt (1-V^2/C^2)
At 0.5C, that # is 1.1547, so your clock and your atoms and all else is going about 15% faster. Meanwhile, you are going at 0.5C.
without time dilation, going say, to Alpha Centauri, you would take twice as long as a ...[text shortened]... what would your velocity have to be to make you think you are going right at the speed of light?

Things don't work like that. Time dilation and space contraction happens at the same time. So whatever speed you have both effects account for you to think that v is always less than c. Things a little bit trickier than this but this gives you the rough idea.

Originally posted by adam warlock
Things don't work like that. Time dilation and space contraction happens at the same time. So whatever speed you have both effects account for you to think that v is always less than c. Things a little bit trickier than this but this gives you the rough idea.

exactly, i guess that's why i found your questions confusing ...

First of all, when you travel at 0.5c "your clock and your atoms and all else is going about 15%" SLOWER and not faster ... and it is all relative.

For v = 0.5c the lorentz factor gamma is 2/sqrt(3) ~= 1.15 hence the distance you cover from your point of view is only L' = L / gamma ~= 3.72ly. From your point of view, the distance is not 4.3ly but 3.77ly. And you'll measure that it took you 7.44 years to get there and not 8.6

Your problem is that you tried to use classical distance formula with speeds relatively close to c. When you move at 0.5c it is no longer true that the time it takes to cover 4.3ly is 4.3/0.5 = 8.6y. This formula works only for non-relativistic speeds.

Finally, according to relativity, you can't appear to be moving at 1.0c in only one reference frame. That's what it means that the speed of light is absolute. If you move at 1.0c and you shoot from your laser gun, the light beam will still travel at 1.0c from your perspective as well from the perspective of somone who's standing still. Similarly, if you have two spaceships travelling at 1.0c each in a head-on collision course, each one will determine that the other is moving at 1.0c and a starionary 3rd party will determine that each is moving at 1.0c and that they are approaching each other at 1.0c as well ( that's what it means that you can't move faster than light ).

Originally posted by 3v1l5w1n
exactly, i guess that's why i found your questions confusing ...

First of all, when you travel at 0.5c "your clock and your atoms and all else is going about 15%" SLOWER and not faster ... and it is all relative.

For v = 0.5c the lorentz factor gamma is 2/sqrt(3) ~= 1.15 hence the distance you cover from your point of view is only L' = L / gamma ~= 3 ...[text shortened]... other at 1.0c as well ( that's what it means that you can't move faster than light ).

As you get closer to C, from your viewpoint in the spacecraft, you think you are going faster than C at some point. At 0.5 C without relativistic effects, you take 8.6 years to get to AC, 4.3 LY away. But if you get close enough to C, from your viewpoint your clocks tell you it takes, say, one year to get to AC. Of course 4 years still goes by on earth. but your clocks tell you you are now going about 4 times the speed of light. So from your viewpoint, there is a velocity where YOU think you are going exactly at the speed of light. Its not quite as tricky as you try to make it out. If you go 0.1C, you know full well and good you are not going anywhere near C, but at 0.99C your clocks tell you you are going 7 times C. So somewhere between those two numbers your clocks tell you you are going at the speed of light. What is so hard to understand about that?

As you get closer to C, from your viewpoint in the spacecraft, you think you are going faster than C at some point. At 0.5 C without relativistic effects, you take 8.6 years to get to AC, 4.3 LY away.

No, you can't do that. When you move at 0.5c you have to take the relativistic effects into account. If you don't the classical distance-velocity-time formulae are simply no longer correct! It will not take you 8.6 years to get to AC, 4.3 ly away. It just won't!

But if you get close enough to C, from your viewpoint your clocks tell you it takes, say, one year to get to AC. Of course 4 years still goes by on earth. but your clocks tell you you are now going about 4 times the speed of light.

No! It seems you just can't wrap your head around it ... if you're going at
0.99c your clocks nor any other clock you can think of, will ever tell you you're going faster than 1.0c. It just doesn't work that way. If it took you a year to get there at 0.99c from your point of reference you will think that AC is only 0.99ly away and not 4.3 (because of the length contraction).

Basically, the problem is that you assume the distance 4.3ly to AC to be absolute - but that is not true. The distance 4.3ly is correct only from our point of view (as beings living in an environment moving way below relativistic speeds). If we existed as highly energetic particles living in an environment where things move at 0.99c ( which gives gamma equal roughly 7) we'd think that AC is only 4.3/7 = 0.61ly away. And the important thing is that in both cases we are all correct in our perception of the distance. That is because the distances are linked with our reference frame (that is why it is called the theory of relativity).

... which basically means, if your friends tells you that the Sun is 1AU awayfrom the Earth and you say that is it only 0.14AU you can both correct (assuming you took your measurements while moving at 0.99c)

Originally posted by sonhouse
As you get closer to C, from your viewpoint in the spacecraft, you think you are going faster than C at some point.

No you don't. This shows that you lack a fundamental understanding of SR. I don't mean this in a mean way but that's exactly what you are showing. Just read carefuly what 3v1l5w1n is saying and you'll see that he's right.

Originally posted by adam warlock
No you don't. This shows that you lack a fundamental understanding of SR. I don't mean this in a mean way but that's exactly what you are showing. Just read carefuly what 3v1l5w1n is saying and you'll see that he's right.

Well, its a valuable discussion anyway! But when you get to AC and you get back home and you find you aged less than earthers, you would have to say, I must have been going faster than light. If the trip length was the only thing changing, say earth says its 100 LY to a certain star and I go close enough to C to make me think its only 1 LY away, I am also aging less so it has to seem to me l went faster than C. Of course you didn't but I am still alive at the end of the journey, and at slower speeds would not be. If I went at a velocity where the distance seemed like 75 LY instead, I would be dead when I get there most likely. That alone would have to be saying something about my apparent velocity. Its a trip into the future, timewise. I guess maybe thats all it is in the end, a trip to your own future.

I knew about the foreshortening of a spacecraft as you approach C but I thought that was limited to just the spacecraft. But you are saying your entire universe is shortening up as you approach C, another ball of wax entirely. So from your viewpoint, the entire UNIVERSE shrinks as you get closer to C? I guess it only shrinks in the direction of travel so only one dimension shrinks, the universe turns into a pancake. I had thought of only the ship turning into a pancake. So its kind of like as you approach C, you could think of the universe as billions of sheets (space) and as you get closer to C, those sheets get closer together, the sheets being maybe plank units apart but the spaceship causes them to stack up one on the other because of its velocity, squashing space and time together.

Originally posted by sonhouse
I knew about the foreshortening of a spacecraft as you approach C but I thought that was limited to just the spacecraft. But you are saying your entire universe is shortening up as you approach C, another ball of wax entirely. So from your viewpoint, the entire UNIVERSE shrinks as you get closer to C? I guess it only shrinks in the direction of travel so on ...[text shortened]... es them to stack up one on the other because of its velocity, squashing space and time together.

It is limited to everything. 🙂

The point is that if you want to know a distance you have to have some method to measure it. Either you are measuring a spaceship or a distance between galaxies you are stil measuring so the result has to be the same. I think that your problem in here is that you are confusig both frames. The one that is fixed on earth and has a distance of 10Ly to whatever place you wanna go or the frame on the space ship that goes at a speed near c and sees the distance as being just 0.1Ly. This can't be done. If you are on the space ship and you'd get some way to measure the distance to your destination while you're travelling you will always measure a very smaller distance to when compared to the distance people that stayed on earth measured. And since speed is distance divided by time you'd always get something smaller than c.

One other thing though is to imagine that you have no knowledge of special relativity and would embark on such a voyage. If you wouldn't measure the distance to your destination while you're travelling and just took notice of the time elapsed you'd think you'd be travelling faster than light. But then of course on Newtonian dynamics there's nothing wrong with that and all the shock wouldn't be a shock at all.

And how do you measure distances to objects while you're travelling? Well you can do it in principle by using light signs that you emit to your destination and are reflected by it. Of course this is easier said than done but if you just go by that reasoning and mess around a little bit with SR formulas you'll see the result cristal clear.

Just one last thing though. I think this method of emitting radiation to an object and getting it's reflection to determine it's distance to us was done already. And it was done to the Moon.

Well the Universe does shrinks to a pancake in one direction while you are approahing c but fortunately for all the rest of life on the Universe it only does so to you. 😛 As on the rest to be quite honest with you I couldn't understand what are you trying to convey.

Originally posted by adam warlock
It is limited to everything. 🙂

The point is that if you want to know a distance you have to have some method to measure it. Either you are measuring a spaceship or a distance between galaxies you are stil measuring so the result has to be the same. I think that your problem in here is that you are confusig both frames. The one that is fixed on earth ...[text shortened]... to be quite honest with you I couldn't understand what are you trying to convey.

That last was just an attempt at analogy, a way of maybe visualizing what is happening. Like squashing the 'distance' between world frames. The assumption is spacetime is quantized and on a small enough scale there would be gaps between space and time and those gaps get closer together. Not saying that's real, just a way of looking at it.

Originally posted by sonhouse
That last was just an attempt at analogy, a way of maybe visualizing what is happening. Like squashing the 'distance' between world frames. The assumption is spacetime is quantized and on a small enough scale there would be gaps between space and time and those gaps get closer together. Not saying that's real, just a way of looking at it.

And I wasn't saying you were wrong I just didn't get what you meant. My first language isn't english and even though I normally understand things at first this was one of those times I didn't.

Ignoring that we need to rethink SR when taking into account QM I'd say that the gaps don't become any closer. Remember that you'd still have to measure the distance between gaps and that distance would shorten in the same proportion, so everything would look the same to you.

Edit: Ignore my last paragraph because it is wrong.