# How are light and time related?

So, from what I understand:

Special Relativity says that light is always observed moving at the speed of light (c). If some object had a velocity of (3/4)c, and the object had some sort of clock attached to it, it would measure differently from a still clock.

1) What would a clock moving the speed of light measure compared to a still clock?

2) Why does it seem that the speed of light is a focal point of time?

EDIT: I may need to rephrase this slightly. What I want to know is, why is it that observed time changes at high velocities?

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Light and time are not related, because the "speed of light" is just an accidental property of light. It is not really anything to do with light in particular, it is a geometrical quantity that tells you how to change units of space and time. –  Ron Maimon Aug 31 '11 at 3:10

1) It is impossible for matter to go at the speed of light, as it takes an amount of energy equal to $mc^{2}\left(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}-1\right)$ to accelerate it to a speed $v$, and this energy becomes infinite as $v\rightarrow c$. In another sense, reference frames moving at the speed of light (or faster) are excluded from Einstein's principle of equivalence.

2) And this gets us to light. To the best of our knowledge, light is massless, but still carries energy. If you look at the equation above, you see that if you allow the mass of the object to go to zero, and still expect the particle to carry energy, then it is necessary that the speed of the object approach $c$, so that you get a $\frac{0}{0}$ indeterminate answer for the energy carried. The speed of light is the speed at which all massless particles travel. It is only happenstance that light was the first massless particle that we discovered, which caused $c$ to be named after light, a name which has stuck.

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Thanks for the response! I now see this is why light travels the speed it does and how light and velocity are related; I see no mention of time in your response though. How does velocity relate to observed time? –  BKaylor Mar 14 '11 at 3:04

This is because time is like space, and moving objects are tilted in time. It is no more mysterious than the statement that two rulers, one vertical and one tilted, have different heights. Not only that, but if you tilt your floor so that it is perpendicular to the tilted ruler, now the ruler that was previously vertical and taller is now tilted and shorter! There is no more mystery here than geometry, except that the geometry is slightly different from Euclid's in that the pythagorean theorem has a minus sign.

When you are standing still, your path in space-time is parallel to the time axis. When you are moving, your path in space-time is tilted relative to the time axis. If you go somewhere and come back, you make a triangle in space time, and the time you measure is the sum of the lengths of the two legs of the triangle that represent your path, and the time measured by someone who doesn't move is the third leg.

In ordinary geometry, the sum of two legs of a triangle are always more than the third, in relativity, it is always less (under the condition that an observer can travel on each leg). The reason for this difference is explained in this answer: Einstein's postulates <==> Minkowski space. (In layman's terms)

Because of the minus sign in the pythagorean theorem, there are special right triangles whose hypotenuse is of zero length. These triangles have a vertical side which is always proportional to the horizontal side, in other words, their hypotenuse represent an object travelling with constant speed. This speed is the speed of light.

It is just an accident of our universe that light travels at this speed. If light had a small mass, it wouldn't quite travel at c, but everything else would work out fine. But we would still have relativity exactly the same as before, and the parameter "c" that we call the speed of light would still be exactly the same. It would just get a new name, perhaps "the speed of gravity".

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