Ralph Polley

Lightray

I had a feeling once about Mathematics - that I saw it all. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable - but it was after dinner and I let it go.  – Winston Churchill

I remember one afternoon, when I was about 19.  I was sitting on a couch in a small basement apartment in Albany, N.Y., trying to imagine the pathway of a ray of light.  It seemed to me that the universe, though very large (very, very large) must in some sense be finite.  I imagined that, if you chose a particular point though which the light ray had passed, that ray, if unhindered, would eventually pass through that same point again.  Like a satellite orbiting the earth passes through a point again and again.  I did not think the path was curved, but rather that straight lines in some sense eventually “come back”.  But then, as I watched the pictures in my mind, I realized the light ray would be passing through the point each time from a different direction.  And, what is more, it would pass through that point from every possible direction before once again passing through it in the direction it first did so.  I then realized that this meant that there was only one path through the universe.  Next I realized that this path must necessarily skip between points that are, from our point of view, unconnected. That is, before returning to the original point in the original direction, the unhindered ray would pass through every point in every order.  And, when it passed near to us through seemingly unconnected points, it was what we normally call very, very far away in space and time.  Somehow, all this implied to me that every point is right next to (whatever that means) every other point.  In my mind I say the universe collapse to a single point.

This was, to me, only a charming fantasy until many years later, when I read about the Michelson-Morley experiment that attempted to measure the speed of the earth with respect to the ether.  The experiment failed of course; there is no ether.  But as a result of the experiment, George FitzGerald deduced that distances were foreshortened from the point of view of an object that was moving very quickly.  That is, a distance that would appear to me to be a million miles, would look shorter to a person traveling at, say, 200,000 m/s.  If  is the distance as it appears to me and  is the foreshortened distance, then the exact formula for calculating that foreshortened distance, is.  (We can take the speed of light to be about 300,000 kilometers per second.)

To make this clearer, let’s consider two observers, Burt and Ernie.  Imagine that Burt is floating in space; there are no objects nearby and Burt feels himself to be still.  Burt sees a star that is, form his point of view, 1,000,000,000 kilometers away.  Just then, Ernie comes speeding by, traveling in the direction of the star at 299,000 kilometers per second (very fast - % of the speed of light).  Now, for the purposes of the formula above, and kilometers per second.  From Ernie’s point of view, the distance to the star is found by  kilometers.  This is less than 10% of the distance as it appears to Burt. 

Now consider what happens if Ernie speeds up to the speed of light.  The distance to the star becomes .  No matter how great a distance is considered, Ernie, from his own point of view will arrive there instantly.  For Ernie, there is no distance; the universe has no size.  Ernie has no size either.

Shortly afterward, Hendrick Lorentz discovered that the mass of an object is similarly affected.  If Ernie’s mass when floating next to Burt is 1 kilogram, then his mass, , as he flies by Burt at 299,000 kilometers per second is  kilograms.  Ernie is more than 12 times more massive. 

If Ernie increases his speed until he approaches the speed of light, the second term inside the radical in the denominator approaches one, the denominator approaches zero, and Ernie’s mass grows without limit.  If Ernie flies by at the speed of light, he (and the whole universe from his point of view) has no size and he has infinite mass. 

As in my vision, the universe has collapsed to a single point, the speeding object has merged with it and become both infinitely small and infinitely massive. 

Afterward:      I understand that I have not sufficiently studied the concepts presented here to say that this is true.  It is likely that I have badly misapplied the equations involved.  I only offer this essay as a possible example of the confluence of intuition and science.

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