Pi and the curvature of the universe

[Cato]

The ratio of the circumference of a circle to the radius is Pi. Our value of Pi is an irrational number and is calculated assuming a flat curvature of space-time. But our universe as a whole, while very flat, probably has some small amount of curvature -- in addition to the greater local curvature due to Earth's gravity. Is there any observational evidence suggesting that the actual, measured value of Pi on Earth is different from the calculated value?

 

[Orodruin]

The number π, being a mathematical construct, is well defined and has a fixed (irrational) value. The question you might ask yourself is whether it describes the relation between the radius and circumference of a circle in our Universe. To the current experimental precision, it does, but physical measurements always have errors and you can only make statements within these errors.

 

[Cato]

The number π, being a mathematical construct, is well defined and has a fixed (irrational) value. The question you might ask yourself is whether it describes the relation between the radius and circumference of a circle in our Universe. To the current experimental precision, it does, but physical measurements always have errors and you can only make statements within these errors.

Yes, thank you. That is what I was asking -- Have we been able to make any measurements, these would probably have to astronomical in scale, which show that a measured Pi is different from the mathematical construction. So the answer if "no".

 

[Orodruin]

Yes, the answer is no (although what we are really doing is looking at the angles of really big triangles).

 

[Cato]

Yes, thank you, it makes sense that over enormous scales the angles of a triangle would not add to 180, though making such a measurement might not be possible. Also, I could imagine that on a smaller scale -- orbiting a neutron star, maybe -- the discrepancy might actually be measurable.

 

[HallsofIvy]

There are more problems here than you may think. Before we can measure a straight line or circle in "real space", we have to decide what a "straight line" is. For example, how do you imagine a line between, say, two planets? Do you think of some huge "measuring stick"? If so, the is no such thing as a "rigid" material, even theoretically, in relativity so such a "measuring stick" would bend in toward the sun- that would give "hyperbolic" geometry. Or do you think a light beam would make a better "straight line"? Then, sicd light is attracted by the sun, you would have to "aim" the light beams away from the sun so that they will curve back to your target. That would give an "elliptic" geometry.

 

[Hawkeye18]

The main problem here is that when you have non-zero curvature, the length of a circle is not proportional to the radius: for example in hyperbolic space the length grows exponentially in radius, so there is no "$\pi$" there.