Measuring pi using gravitational waves
Re: Measuring pi using gravitational waves
My question is whether pi (as in circumference/diameter) is dependent on the curvature of spacetime.
My avatar was a scientific result that was later found to be 'mistaken'  I rarely claim to be 100% correct
Re: Measuring pi using gravitational waves
I think that's factored into the mumble mumble
Re: Measuring pi using gravitational waves
It is. The proper distance in GR is path dependent, so the proper distance along a circular path about a mass is not π times the proper distance along a diameter (if a diameter is even uniquely defined).
See the Black Hole Image  measuring the distance travelled by a photon across the last stable circular orbit can mean taking a path that starts off moving away from the center depending on your choice of direction.
Re: Measuring pi using gravitational waves
That's interesting, does that mean that Euler's Identity (e^{ iπ} + 1 = 0) is also dependent on the curvature of the universe?dyqik wrote: ↑Wed May 13, 2020 7:19 pmIt is. The proper distance in GR is path dependent, so the proper distance along a circular path about a mass is not π times the proper distance along a diameter (if a diameter is even uniquely defined).
See the Black Hole Image  measuring the distance travelled by a photon across the last stable circular orbit can mean taking a path that starts off moving away from the center depending on your choice of direction.
My avatar was a scientific result that was later found to be 'mistaken'  I rarely claim to be 100% correct
Re: Measuring pi using gravitational waves
I think we should differentiate between the mathematical definition of pi, and its measurement in real life.
If you want to define pi using a circle as a mathematical concept, gravity doesn't play a role. But then if you want to apply it in the real world, your circle may not really be a circle anymore.
If you want to define pi using a circle as a mathematical concept, gravity doesn't play a role. But then if you want to apply it in the real world, your circle may not really be a circle anymore.
Re: Measuring pi using gravitational waves
No, because that doesn't involve a ratio between a circumference and a diameter in 2 or 3 spacelike dimensions.Gfamily wrote: ↑Wed May 13, 2020 10:16 pmThat's interesting, does that mean that Euler's Identity (e^{ iπ} + 1 = 0) is also dependent on the curvature of the universe?dyqik wrote: ↑Wed May 13, 2020 7:19 pmIt is. The proper distance in GR is path dependent, so the proper distance along a circular path about a mass is not π times the proper distance along a diameter (if a diameter is even uniquely defined).
See the Black Hole Image  measuring the distance travelled by a photon across the last stable circular orbit can mean taking a path that starts off moving away from the center depending on your choice of direction.
Re: Measuring pi using gravitational waves
Hmm, the reason I asked is that the derivation of Euler's Identity (AFAIK) does involve the McLaurin expansion of the trigonometric functions sin and cos, which means they map to a flat geometry. So in a sense it does.dyqik wrote: ↑Thu May 14, 2020 1:30 pmNo, because that doesn't involve a ratio between a circumference and a diameter in 2 or 3 spacelike dimensions.Gfamily wrote: ↑Wed May 13, 2020 10:16 pmThat's interesting, does that mean that Euler's Identity (e^{ iπ} + 1 = 0) is also dependent on the curvature of the universe?dyqik wrote: ↑Wed May 13, 2020 7:19 pm
It is. The proper distance in GR is path dependent, so the proper distance along a circular path about a mass is not π times the proper distance along a diameter (if a diameter is even uniquely defined).
See the Black Hole Image  measuring the distance travelled by a photon across the last stable circular orbit can mean taking a path that starts off moving away from the center depending on your choice of direction.
My avatar was a scientific result that was later found to be 'mistaken'  I rarely claim to be 100% correct
Re: Measuring pi using gravitational waves
Only coincidentally. The Argand plane is a Euclidean 2space, but it isn't a physical space or spacetime.Gfamily wrote: ↑Thu May 14, 2020 1:43 pmHmm, the reason I asked is that the derivation of Euler's Identity (AFAIK) does involve the McLaurin expansion of the trigonometric functions sin and cos, which means they map to a flat geometry. So in a sense it does.

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Re: Measuring pi using gravitational waves
See this kind of stuff is why I come here, even if I only partially understand it.dyqik wrote: ↑Thu May 14, 2020 1:55 pmOnly coincidentally. The Argand plane is a Euclidean 2space, but it isn't a physical space or spacetime.