Is this evidence for a space-time continuum?

• student34
In summary, the conversation discusses the question of whether spacetime is smooth or granular, and the use of different methods to approximate the circumference of a circle. It is suggested that the difference between the two methods is a strong indication that one of them is incorrect. The conversation also mentions the use of integral calculus to explain this discrepancy.
student34
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?

student34 said:
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.

student34 said:
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.

Nugatory said:
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.

student34 said:
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.

Nugatory said:
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 learned a little more about how the shape of infinitesimals actually become important. But I only took the usual first-year calculus course in university.

student34 said:
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?

jbriggs444 said:
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.

1. What is the space-time continuum?

The space-time continuum is a conceptual framework used to describe the interwoven nature of space and time in the universe. It combines the three dimensions of space (length, width, and height) with the fourth dimension of time. According to this theory, space and time are not separate entities but are connected and influenced by each other.

2. How is the space-time continuum related to Einstein's theory of relativity?

Einstein's theory of relativity states that the laws of physics are the same for all observers in uniform motion. This theory also introduced the concept of a four-dimensional space-time continuum, where space and time are not absolute, but are relative to the observer's frame of reference. This idea revolutionized our understanding of the universe and helped explain the relationship between space and time.

3. Is there any evidence for the existence of the space-time continuum?

Yes, there is overwhelming evidence for the space-time continuum. One of the key pieces of evidence is the observation of gravitational lensing, where the light from distant objects is bent by the curvature of space-time caused by massive objects. This phenomenon is predicted by Einstein's theory of relativity and is observed in various astronomical observations.

4. Can the space-time continuum be observed directly?

No, the space-time continuum cannot be observed directly as it is a 4-dimensional concept. However, its effects can be observed and studied through various scientific experiments and observations, such as gravitational lensing, time dilation, and the behavior of objects in the presence of strong gravitational fields.

5. How does the concept of the space-time continuum impact our understanding of the universe?

The concept of the space-time continuum has greatly impacted our understanding of the universe. It has helped us explain various phenomena, such as gravity, the behavior of light, and the expansion of the universe. This theory has also led to the development of technologies, such as GPS, that rely on the principles of the space-time continuum. It continues to be a fundamental concept in physics and has greatly advanced our understanding of the universe.

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