Two Particles Moving Apart Each at the Speed of Light Move Apart at the Speed of Light

Einstein’s theory special relativity holds that no physical entity can exceed the speed of light, C.

This produces a counterintuitive result when one considers the case of two particles moving apart from one another in opposite directions, each with a velocity of C.

Common sense seems to suggest that the particles would be separating at a velocity of 2C.

This is because the ordinary formula for calculating the relative velocity of two objects moving apart, first given as an equation by Galileo (who was a great physicist, but only a mediocre astronomer), is

s = v + u

s for “speed” or “net relative velocity” and since u and v are the same letter, originally, they stand for the two velocities of the things moving apart.

So, by the common-sensical Galileo formula, two trains moving apart from each other, each at 5 meters per second, move apart at 10 m/s.  Human experience and careful measurement seem to confirm this.

And by the same common-sensical Galilean equation, two particles moving apart from one another each at C should move apart at 2C,  since obviously C + C = 2C.

But Einstein’s theory predicts they will move apart at C, contrary to the common-sensical Galilean equation.

What happened?

Very simply, Galileo’s equation is incorrect.  The actual equation is this:

Screen Shot 2017-12-14 at 10.40.22 PM

As we can easily calculate, where v and u both equal C, s = (C + C)/(1 + (CC/C²) = 2C/1+(C²/C²) = 2C/(1+1) = 2C/2 = C.

What about our two trains moving apart each at 5 m/s?

The answer is that they do not move apart at 10 m/s, but at 9.999999999999996 m/s.

And the difference between 9.999999999999996 m/s and 10 m/s is so small (in human terms) that it is impossible to directly measure in almost all cases, and makes no practical difference, let us say, 99.99999999999996% of the time.

In other words, the Galilean equation is slightly inaccurate but produces error so slight as to make no practical difference and thus is completely reasonable to use in almost every calculation human beings make—except of course for those human beings that are theoretical physicists who are dealing with things moving at a significant fraction of the speed of light.  When we are dealing in meters and seconds, or kilometers and hours, angstroms and picoseconds don’t make any real difference.  As Aristotle correctly point out 2400 years ago, it is as unreasonable to demand far too much precision as to demand far too little precision; what is reasonable is to be as precise as the subject matter requires.  Building a house does not require us to make measurements at the subatomic level, and to demand this of the house builder is unreasonable: it’s outside his power, and it would result in no houses being built.

I wrote this just in case anyone wondered about how it is possible for two particles moving apart each at C to move at C.  I wondered about this, a long time ago, and took the trouble to find out, so I thought I’d share.

Our common-sensical intuitions about things are not always correct, although as in this case, they are often “in the neighborhood of truth.”

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