Measuring pi using gravitational waves
Posted: Wed May 13, 2020 12:59 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).
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.
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.
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.
Only coincidentally. The Argand plane is a Euclidean 2-space, 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.
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 2-space, but it isn't a physical space or spacetime.