# Is this evidence for a space-time continuum?

I have always wondered if spacetime is actually smooth, or if it is granular such as a quantized spacetime.

I remember learning that if you draw the smallest possible square around a circle, you will, of course, get a perimeter, 8 times the radius of the circle. Then if you draw in the largest possible squares in each of the four corners between the inside of the large square and the outside of the circle, you will get closer to the area of the circle. But you won't get closer to the circumference.

If you kept on cutting out the largest possible square in the spaces between the outer edge of the circle and the inside of the original square, you will approach the circle's area, but the perimeter stays constant at 8r.

Now here's the point. Let's say that the length of a quantum of spacetime is d (let's just forget about Planck lengths for now). No matter what shape the quantum of this spacetime is in, there will still have to be an n number of them in the x direction and the same number in the y direction in getting around a circle. This would mean that any circle that we think is a circle actually has a circumference 8r.

But, I did a simple experiment. I took a piece of measuring paper and wrapped it around a disk. When I took the same piece of measuring paper and measured the diameter of the disk, as you probably have guessed, the length of the piece of paper was about 6.24r.

I even laid down the measuring paper and rolled the disk over the length to make sure that the paper didn't somehow lose length when curling around the disk. The results were the same.

How can such a difference between 8r and 6.24r be explained?

Is this evidence of infinitesimals of spacetime?

Nugatory
Mentor
How can such a difference between 8r and 6.24r be explained?

Is this evidence of infinitesimals of spacetime?

No. The easiest way to see this is to try using a different procedure to approximate the circumference of the circle: the four-sided square you started with, then a five-sided pentagon, the a six-sided hexagon, and so forth. As the number of sides increases, the perimeter of the polygon gets closer to ##2\pi{r}## and they become equal in the limit when the number of sides approaches infinity.

The fact that the two approximation procedures give different results is a very strong hint that they aren't both right. Your experiment with the rolled paper suggests that it's the ever-smaller-squares one that is wrong.

You will learn why it doesn't work (in other words, why the result of your experiment was a measurement of 6.24r instead of 8r) in a course on integral calculus.

phinds
Gold Member
2021 Award
I have always wondered if spacetime is actually smooth, or if it is granular such as a quantized spacetime.
That question is of great interest to lots of folks and there have been probably hundreds of threads on it here. I suggest a forum search.

No. The easiest way to see this is to try using a different procedure to approximate the circumference of the circle: the four-sided square you started with, then a five-sided pentagon, the a six-sided hexagon, and so forth. As the number of sides increases, the perimeter of the polygon gets closer to ##2\pi{r}## and they become equal in the limit when the number of sides approaches infinity.

But that's my whole point. Since the measurement is roughly 2pi and not 8r, then we know that there is a true curvature. If the smallest possible space has real numbers for its height, width and depth, then at any rotation, the circle would have to have a finite number of heights and a finite number of widths. Then the circle would have to be 8r for reasons that I gave above.

The fact that the two approximation procedures give different results is a very strong hint that they aren't both right. Your experiment with the rolled paper suggests that it's the ever-smaller-squares one that is wrong.

The point that I was trying to make is that if it is wrong, then how could spacetime consist of real heights, widths and depths?

You will learn why it doesn't work (in other words, why the result of your experiment was a measurement of 6.24r instead of 8r) in a course on integral calculus.

I have taken integral calculus, but we did not get into analysis.

Nugatory
Mentor
The point that I was trying to make is that if it is wrong, then how could spacetime consist of real heights, widths and depths?
Just because you're thinking in terms of heights and widths (leave depth out for now, because you started with a two-dimensional example), that doesn't mean that you're limited to approximating shapes as squares - triangles have widths and heights too.

If you were to use polar coordinates instead of cartesian coordinates you would find the n-sided polygon approach to be more natural, but the geometry of the two-dimensional surface of the sheet of paper on which you drew your circle is the same either way.

I have taken integral calculus, but we did not get into analysis.
The important point here is the way that we prove that the method of approximating the area under the curve with more and narrower rectangles converges on the precise value of the area under the curve. We can draw the little rectangles with their top left corner or their top right corner touching the curve. One way gives us an overestimate and one way gives us an underestimate, and we can prove that we can make the two values arbitrarily close to one another so that they squeeze down to the right value. You cannot do anything similar if you try approximating the circumference of a circle with your stairstep pattern of cubes, and that's why it doesn't work.

Depending on your calculus course, you may not have gone very far into these proofs as some courses (reasonably) focus more on what works than why it works.

Just because you're thinking in terms of heights and widths (leave depth out for now, because you started with a two-dimensional example), that doesn't mean that you're limited to approximating shapes as squares - triangles have widths and heights too.

I think that any shape with a real area must ultimately have a real width and a real length . So if we follow these real lengths and real widths around the circle, they should do no better that the squares.

The important point here is the way that we prove that the method of approximating the area under the curve with more and narrower rectangles converges on the precise value of the area under the curve. We can draw the little rectangles with their top left corner or their top right corner touching the curve. One way gives us an overestimate and one way gives us an underestimate, and we can prove that we can make the two values arbitrarily close to one another so that they squeeze down to the right value. You cannot do anything similar if you try approximating the circumference of a circle with your stairstep pattern of cubes, and that's why it doesn't work.

But that's for area; I agree that the sum of the areas of the squares would approach the area of a circle as the squares get smaller and more squares are added.

Depending on your calculus course, you may not have gone very far into these proofs as some courses (reasonably) focus more on what works than why it works.

In physics, I actually learnt a little more about how the shape of infinitesimals actually become important. But I only took the usual first-year calculus course in university.

jbriggs444
Homework Helper
I think that any shape with a real area must ultimately have a real width and a real length.
It is easy to come up with examples of unbounded shapes with finite areas. So what are you trying to claim here?

So if we follow these real lengths and real widths around the circle, they should do no better that the squares.

If you decide that the taxicab metric is the "true" metric of the universe, how do you account for the fact that a 12 inch ruler (as measured east/west) will cover more than half of the diagonal of a 12 inch by 12 inch square?

If rulers change their length when they change their angle, how do you propose to physically measure the "true" circumference of a circle?

If this "true" metric does not work very well, why use it in the first place?

It is easy to come up with examples of unbounded shapes with finite areas. So what are you trying to claim here?

If you decide that the taxicab metric is the "true" metric of the universe, how do you account for the fact that a 12 inch ruler (as measured east/west) will cover more than half of the diagonal of a 12 inch by 12 inch square?

The point is that reality does not seem to follow the taxicab rules. This seems to suggest that spacetime is smooth and curved like Einstein claimed. But there is great interest and hope in quantifying spacetime.

If rulers change their length when they change their angle, how do you propose to physically measure the "true" circumference of a circle?

They obviously don't change at an angle. An obvious reason would be that spokes wouldn't fit when bicycle rims rotate.

If this "true" metric does not work very well, why use it in the first place?

Because, a quantized spacetime would seem to imply wrong measurements like a circle having a circumference of 8r.