Understanding Time Dilation and the Speed of Light in Particle Colliders

[mcjosep]

If someone were traveling at let's say 99.9999% the speed of light time would go slower for them but their speed would stay the same.

So let's say for every ten second we would experience here on Earth they would experience they experience 1 second.

so they would travel (10 seconds X 300,000 km )=3,000,000 km in 10 seconds

To them though they would travel 3,000,000 km in one second (a factor of ten times the speed of light) but supposedly they shouldn't be able to travel faster than light. to them though they are. I do not understand this, can someone explain?

Definition of speed of light is roughly 300,000 km a second not 3,000,000 km a second.

 

[yuiop]

Time measured in Earth frame = 10 seconds.
Distance measured in Earth frame = 3,000,000 km
Velocity measured in Earth frame = 300,000 km/s.

Time measured in traveller's frame = 1 second.
Distance measured in traveller's frame = 300,000 km
Velocity measured in traveller's frame = 300,000 km/s.

You are forgetting that the distance measured by the traveller will be length contracted by a factor of 10 also, or you are calculating velocity by using measurements of time in the travellers frame and distances in the Earth frame. In the latter case this is called celerity or rapidity, but it is not velocity, which is always less than the speed of light if you stick to measurements made by observers, clocks and rulers in one reference frame.

 

[turin]

If someone were traveling at let's say 99.9999% the speed of light ...
...
... for every ten second we would experience here on Earth they would experience they experience 1 second.

Well, actually, the ratio is 1000:1. You can calculate this. It is the gamma factor. In order for the ratio to be 10:1, their speed would need to be 99% the speed of light, without all of the extra decimal places. This is one of the curious features of relativistic effects. A small difference in one thing can mean a huge difference in another, and vice versa. This is important even in "practical" applications where things like timing and energy/frequency both need to be considered.

Anyway, in order to save confusion, I will from here on assume v = 99% c (instead of 99.9999%), thus a gamma factor of 10:1, to remain consistent with the rest of your post.

... their speed would stay the same.

The same as what?

so they would travel ... 3,000,000 km in 10 seconds

It is important to understand that those seconds and km's are measured by the Earth observer. In relativity, the observer who makes the measurement can be an important consideration.

To them though they would travel 3,000,000 km in one second (a factor of ten times the speed of light) but supposedly they shouldn't be able to travel faster than light. to them though they are. I do not understand this, can someone explain?

No. To them, they are at rest, and the Earth is flying by. As you have mentioned, there is some time dilation. That is, an Earth observer would say that the traveler's time is 10 x slower than normal. The other piece of the puzzle that you're missing is length contraction. To the traveler, everything is squished into shorter distances due to its motion past the traveler.

However, there is the notion of proper velocity, which can exceed the speed of light, and this is basically what you're talking about. You are considering two kinds of velocity: coordinate (what the Earth observer measures) and proper (a useful, though probably confusing, notion). They are related through the gamma factor. That is, the proper velocity is gamma x the coordinate velocity. The proper velocity measures how much Earth frame distance is covered in a given amount of traveler time.

(Oh, kev already said this stuff about length contraction and proper velocity (celerity).)

 

[yuiop]

If you are going to use the "pick and mix" method of using measurements from different frames to determine velocity, you could also say that because the Earth measures the elapsed time as 10 seconds and the traveller measures the distance as 300,000 km, the "velocity" of the traveller is 30,000 km/s or one tenth the speed of light.

 

[turin]

If you are going to use the "pick and mix" method of using measurements from different frames to determine velocity, you could also say that because the Earth measures the elapsed time as 10 seconds and the traveller measures the distance as 300,000 km, the "velocity" of the traveller is 30,000 km/s or one tenth the speed of light.

True. However, there is a practical reason for picking length from the Earth frame and time from the traveling frame, namely, for considerations in designing particle colliders.