*What are straight lines, to begin with?
*

We are taught in schools that light travels in straight lines through air or vacuum, and we also have “experiments” to “prove” this. One of them that was in my books was to arrange three cards with a small hole in each them so that the holes align in a line. Then we put a candle on one side so that its flame is also aligned with the holes, and then look at it from the other side. To our delight, we see the flame. Now move the middle card so that the holes are no more aligned, and what a surprise, we don’t see the flame through the holes any more!

There are other “proofs” and “experiments”, some of which you can get with a google search on a string such as *light travels in straight line*. As of now, this is what a few top results talk about in our simple context (i. e. apart from those talking about changing refractive index or talking about General Theory of Relativity): A few of them replace our rudimentary candle with a flashlight. A few others use a pinhole camera as an example and draw lines from the object to the camera screen. Then there are arguments based on sharpness and position of shadows. Yet others show the magnificent photographs of the shadow lines that are formed by clouds on the other side of the Sun when they block the sunlight.

All of them, however, take it for granted that the concept of a straight line is well-defined, *a priori*, independently. In pure (Euclidean) geometry it may be, but we need to examine how do we determine whether some *physical* shape is a straight line. Again let us go back to the three cards experiment. How do we check whether the card holes are aligned in a straight line?

If we decide that by looking through the holes to make sure we can see through, then clearly our reasoning is circular! All that we can conclude when we place the candle on the other side and are able to see the flame, is that *light follows the path that light follows*. So let us try to be sophisticated and see if we can confirm the collinear alignment without looking through the holes. One way is to make the three cards of exactly the same size and make a hole in them in exactly the same position. Then we can align the cards and the holes will be aligned with them. But how do we align the cards? If we check their alignment by trying to see if their corresponding corners and edges visually coincide from a particular position of the eye, then we are doing the same thing in essence: trying to align them *in a path that light follows*, which is again circular. What if we push them all together by a straight ruler so that they are aligned with the ruler? Sounds good, except that how do we know that the ruler’s edge is a straight line? By looking at it from the side so that all the edge of the ruler seems like a point? Circular again.

In short, one of our main methods to check whether something is straight, involves trying to put our eye in the line and see if it is all one behind the other. One of our practical definitions of a straight line in the physical world is the path followed by light. It is almost our natural instinct, so much, that we even use it while talking about path of light itself! Obviously in this case we are being circular in our reasoning.

If we want to really verify that light travels in a straight line, we must base our method of verifying the straight line on some other definition. One such definition that I can think of, in the context of Euclidean geometry, is that of the shortest path between two points. How do we *physically* find out the shortest path? One way is to take a very light, thin, unstretchable but otherwise flexible string, and hold it at the two given points so that it is fully stretched. Now we can convince ourselves that the string indeed occupies the shortest path between the two points. (If it did not, then we could stretch it further until it did.) Now if light follows this path, then it can be indeed said that light has followed a straight line, a straight line confirmed by other means than light itself.

So if we insist on the candle-behind-cards experiment again, the holes can be aligned by passing this string through them and stretching it so that the cards move and holes align, fixing the cards, and then removing the string. Now if we see the flame through the holes, or even if we could just see through the holes, we are done. But why do we require the cards and the candle? Just stretch the string, put our eye in the line of the string on one end, and indeed we will see that all the string looks like a point. So the path of light is along the string, which is a straight line.

Of course I am not considering change in medium (varying refractive index), or effects of gravity (via General Theory of Relativity). But it is interesting to see that even in a very elementary setting such as the ordinary three-dimensional Euclidean geometry, and only air or vacuum as the medium for light, we can commit blunders of reasoning.

I am interested to know any other independent definitions of a straight line that can be physically used in this context. Please do comment if you have any.

Hey!! that was a good reasoning. This question i had gave up thinking about a long time ago!!

Both your examples (light & string) assume that you have already “defined/assumed/implicitly-know” the concept of distance (length) which is a “higher structure” definable only after defining a metric..

One could go via defining the “least” action and show that for “straight lines” the least action path is same as least time path..

also, in your examples, a child could ask: how do you know that these are really the smallest lenght paths that you say as straight lines?? similar to “sum over all paths” idea.. i.e. how do you “proove” that a “hidden secret” path that is shorter than the one you see does not exist?? can you have one such if you have higher dimensions??

and what means “keep the stings straight?” what is straight? would you use a laser to verify if strings are straight? tautology??

e.g. one could (gedanken experiment) think of making a glass slab with weird refractive index so that a laser beam moves in a crooked zig-zag manner (as seen from air), or gets wound up inside a fibre optic cable..

In both cases it is moving in straight line in respective circumstances. so try proving this to a layman..

another way would also be to think of straight line as line with largest gradient (of something, similar to action stuff).. guess gradients are easier to imagine for people than vaguer “distances, times, etc”

Thanks for the detailed comment! I hope to answer your concerns here:

I think I can do that, given the simple context of vacuum and flat spacetime.

Yes, they can be clever. Any hints?

Perhaps you misread this. It’s

stretched, notstraight.Again, this is outside the context.

Just to repeat, the point of this post was that even in such a simple context, we can commit blunders of reasoning.