Re: Challenge for Paul; Probe that with Mercury ds^2>0 and the solution is spacelike

Den 17.02.2025 22:29, skrev rhertz:

Line element ds in the Schwarzschild metric(describing spacetime around
a massive object like the Sun):
>
ds² = -(1 - 2GM/c² r) c² dt² + 1/(1 - 2GM/c² r) dr² + r² dɸ²
>

 So you have realised that it was a blunder to think you could
use the metric for flat spacetime in an environment where geodesics
are ellipses.
 ----------------------
 It is very obvious that you don't know what a metric is, so
I will give a short lesson about the most elementary concepts
in spacetime geometry:
 In physics, an "event" is a point in space at a time,
or a point in spacetime.
 The metric can be used to find the spacetime interval between
two events, or the spacetime interval along a path between two events.
 It is quite common to use s² as the interval, but it is more 'natural' to call the interval s, so that's what I will do.
 's' consists of two components, a temporal and a spatial.
If we call the temporal component cT and the spatial component D,
we have: s² = −c²T² + D²
 If D > cT then S is spacelike  (s² > 0)  D/T > c
If D = cT then S is lightlike  (s² = 0)  D/T = c
If D < cT then S is timelike   (s² < 0)  D/T < c
 Two events on the worldline of a massive object will always be
separated by a timelike interval, because the object's speed D/T
is always less than c, and D < cT.
 In the latter case it is common to set s = -cτ, and
the Schwarzschild metric becomes:
 c²dτ² = (1 - 2GM/c²r)c²dt² - 1/(1 - 2GM/c²r)dr² - r² dɸ²
 You can see this metric applied on satellites here:
https://paulba.no/pdf/Clock_rate.pdf
 (I know I am an idiot who bother to try to teach you
  what you never will learn.)