# A Roller Coaster Ride through Relativity/The Bending of Light

## The Bending of Light

This is easy to understand. Imagine that you are in a rocket playing with a laser pen, pointing at a picture of Scooby-doo on the opposite wall. Suddenly, you see the laser beam bend downwards towards the back of the rocket. At the very same instant, the copy of the Hitch-hikers Guide to the Galaxy which had been quietly floating weightless in front of you careers off in the same direction. Of course the reason is obvious. The rocket is accelerating so that the book is, as it were, left behind. In the same way, in the time it took for the light to cross the width of the rocket, the rocket had moved an extra distance forwards causing the spot to hit the wall of the rocket further behind its original position.

The bending of light in an accelerating rocket

If the rocket has a width of l and is accelerating with an acceleration of a then in the time it takes for light to cross the width of the rocket t (where t = l/c) the rocket's speed will have increased by dv = at = al/c. (It is here assumed that this increase is much smaller than the speed of light so we don't have to take any special relativity considerations into account)

The angle of deflection of the beam when it hits the wall (in radians) will therefore be approximately al/c2.

Now from the Fundamental Principle, what is true in an accelerating system is true in a gravitational field so it follows that when a beam of light crosses a gravitational field of strength g (= a), it will bend through an angle ${\displaystyle \alpha }$ where:

${\displaystyle \alpha ={lg \over c^{2}}}$

It is also quite easy to show that the distance it will fall s is given by

${\displaystyle s={\tfrac {1}{2}}{l^{2}g \over c^{2}}}$

Not surprisingly, this turns out to be rather small where the Earth's gravity is concerned. A laser beam shining horizontally over a distance of 1 km in the Earth's gravitational field falls by only 5 x 10−11 m. That is about a quarter of the diameter of an atom!

On the other hand, when light from a distant star passes close to the surface of the sun, it is significantly deflected and, what is more, this deflection can fairly easily be measured. It causes the stars immediately behind the Sun to appear further away from the Sun than they really are.

The bending of starlight round the Sun

The idea that gravity could bend light is not new, however. It would not have surprised Newton in the least who always thought light was a stream of particles which were influenced by gravity. On the other hand, the proponents of the wave theory of light, Huygens and, later, Maxwell, would have hotly denied the possibility. After all, gravity acts on mass, how could it possibly affect an electromagnetic wave? And yet it does.

[I have to add a cautionary note here: in fact, gravity bends light by more than the amount calculated above. The flaw in our argument lies not in the approximation that the speed of the rocket is much less than the speed of light, but in the assumption that space and time are unaffected by the acceleration of the rocket (or, equivalently, the gravitational field). When the effect of the distortion of space and time is taken into account as well, the full General Theory predicts an angle of deflection of exactly twice the amount calculated above and the predictions of the full theory have been verified experimentally many times since Einstein first calculated the size of the effect.]

## Down we go again

Well except for that bit about the bending being twice what it should be which I didn't understand, that was pretty easy. Doesn't this roller coaster have anything a bit more exciting to offer?

Well yes, it does - but first we have another rather familiar hill to descend. Remember - just hang on to that Principle!

As we cruise over the crest of the hill the roller coaster tilts over sideways and we find ourselves looking down a deep shaft. At the bottom of the shaft there are floodlights and workmen with clocks and measuring sticks - but they don't seem to be working very quickly - and their measuring sticks look very short - and, for some reason, they seem to be using red floodlights.

But all too soon the roller coaster rights itself and begins to hurtle down the next slope towards:

Bizarre consequence number 18
Clocks go slow and rulers shrink at the bottom of a well.

Uh?