Since frames of reference are arbitrary, I can define a frame of reference that moves backwards with a speed greater than $c$, then any static object in that frame of reference is already traveling forwards faster than $c$!
So what does "nothing can travel faster than light" exactly mean, and where does it apply?
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$\begingroup$ What does your frame of reference move relative to? $\endgroup$– Dmitry BrantCommented Jan 11, 2014 at 21:43
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$\begingroup$ Really I am asking, not arguing! I can't answer your question, but say it's moving relative to earth, then? $\endgroup$– MohsinCommented Jan 11, 2014 at 22:06
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$\begingroup$ Nothing can travel faster than light in the frame of reference of any particle traveling along a timelike worldline. $\endgroup$– WillOCommented Jan 11, 2014 at 22:45
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$\begingroup$ @WillO: modulo acceleration/gravity (but instantaneous proper speed will of course remain $c$) $\endgroup$– ChristophCommented Jan 11, 2014 at 22:56
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$\begingroup$ @WillO: Can you explain in simple terms what a timelike worldline is? $\endgroup$– MohsinCommented Jan 11, 2014 at 23:04
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2 Answers
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You have to measure everything from the same frame of reference. Your own frame of reference obviously has a velocity of 0, relative to you. The other object moving toward you, or away from you, will never move faster than the speed of light as seen from your frame of reference.
A third observer can see two objects, each moving at the speed of light. Toward each other, or away from each other - it does not matter.
Their distance can decrease or increase faster than the speed of light, as observed from the third observer, but no object will move faster than the speed of light with respect to the third observers frame of reference. Each of the two objects can calculate the other objects' velocity, and they will always get a number less than the speed of light. This is because time is relative, and when you measure somebody elses velocity, you use your own time.
An interesting side note:
You can travel faster than the speed of light, if you measure distance in one frame of reference, then travel in another frame of reference. Nothing else can, unless it travels with you.
The short reason for this is:
- As your velocity increases, passage of time slows down for you.
If you were traveling to for example the Alpha Centauri, about 4.5 light years away. You could perform this journey in 8 of your months.
- People on earth will have aged 5 years and 2 months.
- You will have aged 8 months.
- People on Alpha Centauri will have aged 9 years and 8 months, compared to the image you saw when you left earth.
- People on Alpha Centauri will actually have aged 5 years and 2 months, taking into account that the image from them you saw when you started traveling was 4.5 years old.
To conclude:
- Everybody else will have aged 5 years and two months, unless they also maintained a high velocity until they met you again.
- You will have aged 8 months.
- Your speed will appear to have been $2×10^9\text{ m/s}$, much faster than the speed of light.
- Others will have seen your speed as approximately $260300\text{ km/s}$.
Calculations: Can the apparent velocity of an approaching spacecraft be faster than light?
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I'm going to take a slightly different approach to explaining this, in analogy with a great answer about the ontological nature of Newton's Laws.
First, let's posit the existence of an inertial reference frame. It doesn't matter which one, but there has to be one. This is an important point, and one that's often overlooked. In it, nothing is moving faster than the speed of light.
Now given one reference frame, we can ask about all others. But what are all others? In Newtonian physics, they consist of anything we get by offsetting our coordinates and/or adding a constant velocity to everything. That is, anything reachable from our original frame by a Galilean transformation. One can easily prove that any two such frames can be transformed into one another via such transformations.
In special relativity, on the other hand, the transformation law to respect is that of Lorentz. Valid frames consist solely of those that are reached via Lorentz transformations, which are defined to have boosts of speeds less than $c$. Again, one can show that any two such frames are a single appropriate transformation apart.
You might be tempted to define "a frame moving with speed $10c$ in the $x$-direction relative to this one," but that definition isn't allowed in relativity.
In summary, nothing will ever travel faster than $c$ in any frame obtained by a valid Lorentz transformation for which this was already true.