# Breaking the speed of light limit.

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1. Jan 3, 2014

### dratone

"breaking" the speed of light limit.

I couldn't quite figure out how to describe it shortly hence the weird title.

I was wondering what it would "look" like if the following situation happened.

Lets say you have a space craft. It is going at a speed of 0.7 c. in relation to star X.

Accoording to special relativity (or was it general relativity?) you could also say your spacecraft is not moving and star X is heading in the opposite direction at a speed of 0.7c.. Correct?

But, in "reality" you are the one moving.. Now..

There is another spacecraft heading in the opposite direction at a speed of 0.7c in relation to the same star. The spacecraft meet... So.. you get the folowing situation..
a = Spacecraft A
b = Spacecraft B
c = stationary star.

a ====> c <=== b

The velocity difference is 1.4c.. more then the speed of light..
Now I understand that none of the spacecraft is actually moving faster then the speed of light, but from what I understand it is fair to say that spacecraft A has velocity 0c and B has velocity 1.4c.

What would that look like from the perspective of spacecraft A? How could that possibly be turned that B has a velocity of <1c?

I hope I'm making myself clear. I'm heaving a bit of difficulty trying to formulate my question.

2. Jan 3, 2014

### ZapperZ

Staff Emeritus
3. Jan 3, 2014

### dratone

Thanks. I knew I was missing something but couldn't quite figure out how to put it.

The only question I have then is I understand that you need to add them that way... But why?
I couldn't quite find the reasoning behind it, just that you need to do it.

4. Jan 3, 2014

### Staff: Mentor

From the frame of reference of the star, both spacecraft are moving towards it at 0.7c. From the frame of either spacecraft, the other spacecraft is approaching at 0.94c. From the frame of the star, the distance between both spacecraft is closing at 1.4c, but that's okay since neither spacecraft is traveling FTL.

As for why, that has to do with how the speed of light is invariant in all inertial frames and several other concepts in relativity.

5. Jan 3, 2014

### ZapperZ

Staff Emeritus
When you ask why, are you asking for the derivation of relativistic velocity addition?

http://www.phys.unsw.edu.au/einsteinlight/jw/module4_Lorentz_transforms.htm

If you asking why it has that mathematical form, then you need to go back to the postulates of Special Relativity and the idea behind the Lorentz transformation.

Zz.

6. Jan 3, 2014

### ghwellsjr

You've done an excellent job and welcome to the Physics Forums.

Correct. According to Special Relativity, you need to specify the speed of each object with respect to an Inertial Reference Frame (IRF). If you specify the speed of the spacecraft with respect to star X, what you are really saying is that the spacecraft is traveling in the IRF in which star X is at rest. Once you specify a scenario according to one IRF, you shouldn't just try to guess what it looks like in another IRF. Instead, you should use the Lorentz Transformation process to calculate what it looks like in another IRF. And it will turn out that if you transform to the IRF in which the spacecraft is at rest, star X will be traveling at the same speed in the opposite direction.

No, you can't say that one IRF is closer to "reality" than any other IRF. They are all equally valid and none is preferred, not even the rest frames of any particular objects. We may prefer to describe a scenario according to one IRF because it is easier to understand but that doesn't make it "reality".

This is exactly why I said you need to use the Lorentz Transformation to see what it looks like in the rest frame of the spacecraft and it helps to draw a spacetime diagram for each IRF.

Like I said, you've done an excellent job.

We start with what you started with, the rest IRF of star X which is shown as the thick red line. The first spacecraft is shown in blue going to the right and the second spacecraft is green going to the left. You should be able to verify that their speeds are 0.7c because in 10 years they progress through 7 light-years. The dots on each worldline mark off one-year increments of each object's time. The faster they go, the farther apart the dots are compared to the Coordinate Time:

Now I want to show you how the pilot of the first spacecraft can measure the speed of star X relative to himself and then how he can measure the speed of the other spacecraft. What he has to do is measure how far away each object is at some particular time. Then he has to do it again at some later time. Then he can calculate the speed of the object by dividing the difference in the distance measurements by the difference in the time measurements.

So how does he make the required measurements? He uses radar signals which bounce off the object of interest. He keeps track of his time when the radar signal was sent and his time of when the reflection was received. By dividing the difference in the times by two, he assumes that the signal took the same amount of time to get to the object as it took for the reflection to get back (Einstein's convention) and assumes that the distance is how far the signal would progress at the speed of light.

Then for the time at which the distance measurement is made, he simply takes the average of the two times.

He could repeat the process again but since the spacecraft eventually will arrive at star X, we can use the time at which the distance is zero as the second point.

So let's see what the pilot gets. It is customary to synchronize all the clocks so that they are at zero when they get to the origin of the IRF so I have marked in the important times for the pilot. We'll focus first on the measurement for star X. I think you can see that the radar signal was emitted at the pilot's time of -10 years and he receives the reflection at his time of -1.8 years. The average is -5.9 years. The difference is 8.2 years and half that is 4.2 years so he establishes that star X was 4.2 light-years away at his time of -5.9 years. Since its distance at time zero is zero, he establishes the speed to be 4.2 light-years divided by 5.9 years or 0.71c, close enough for eyeballing the times off the diagram.

Now for his calculation of the speed of the other spacecraft, we use the same radar sent time of -10 years but the reflection received time of -0.3 years. The average is -5.15 years and the distance measurement is 9.7/2 = 4.85 light-years so the speed is 0.94c. This is the same value that Drakkith reported in post #4.

But we can also use the Lorentz Transformation to determine the speeds as shown in this diagram for the rest IRF of the first spacecraft:

Now, because the spacecraft is at rest in this IRF, the Coordinate Distances match the measurements that the pilot made.

Does this make perfect sense to you? Any questions?

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