On 1/12/24 8:11 PM, patdolan wrote:
On Friday, January 12, 2024 at 3:18:21 PM UTC-8, Tom Roberts wrote:
That's easy: Kepler's laws (all of them) are quite clearly not
laws of physics, they are at best approximations.
Then tell this forum, Tom Roberts, what are the laws of planetary
motion that Einstein promised us which are true in all inertial
frames of reference? Show them to us.
For the case of a very massive sun and planets of negligible mass, which
is an approximation that is very good in the solar system, then each
planet follows a geodesic in the spacetime defined by the sun:
D_v v = 0
where v is the 4-velocity of the planet (i.e. the tangent 4-vector of
its path), and D_v is the covariant derivative along v.
Note this equation is completely independent of coordinates, and can be projected onto any coordinate system you wish, not just (locally)
inertial frames.
If you want to avoid that approximation, then you must use the Einstein
field equation:
G + Λ g = T
where G is the Einstein curvature tensor, g is the metric tensor, Λ is
the cosmological constant, and T is the energy-momentum tensor for sun
and planets (units have 8 π G = c = 1, where this G is Newton's
gravitational constant). It is infeasible to solve this analytically for
even a two-body system, much less the solar system, but it can be solved numerically to essentially arbitrary precision. Like the previous
equation, this equation is completely independent of coordinates, and
can be projected onto any coordinate system you wish, not just (locally) inertial frames.
[Don't expect me to explain this further, as it is quite
clear you do not understand the requisite physics or
the underlying mathematics.]
Tom Roberts
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