Relativistic kinetic energy formula
mlut...@wanadoo.fr
http://groups.google.fr/group/sci.physics.research/browse_frm/thread/b7e070676c691cf6/a47afcbbceff2a71?hl=en&lnk=st&q=relativistic+escape+velocity#a47afcbbceff2a71
When the relativistic kinetic energy formula
K = mc^2(1/(1-sqrt(v^2/c^2)) - 1) is used
instead of K = 1/2 mv^2, one gets with the
Newtonian gravitational potential energy
sqrt(1-v^2/c^2) = 1/(GM/(c^2*d) + 1,
and the redshift observed at d1 of light emitted
at d2 is (GM/(c^2*d1)+1) / (GM/(c^2*d2)+1) - 1,
where d2 and d1 are distances from the center of
mass of an object of mass M.
If light is emitted at a distance d1 from an object of
mass M situated at a distance d2 from an observer
on Earth, the shift formulas are
Shift ('new' GR formula) =
(GM/(c^2*d2)+1) / (GM/(c^2*d1)+1) - 1, against
Shift (classical GR formula) =
sqrt(1-2GM/c^2d2) / sqrt(1-2GM/c^2d1) - 1.
For instance,
1) shift observed on Earth of light emitted
by the Sun (d1 = Sun's radius, d2 =
distance Sun-Earth)
Shift ('new' GR formula) = -2.11237 * 10^-6
Shift (classical GR formula) =
-2.11236 * 10^-6
2) shift observed on Earth of light emitted
by a neutron star of solar mass and
Earth radius (= d1), whose distance from
Earth d2 = 1000000 * distance Sun-Earth:
Shift ('new' GR formula) = -2.31787 * 10^-4
Shift (classical GR formula) =
-2.31867 * 10^-4
The 2 formulas give almost identical shifts for
"normal" objects.
However, the calculated shifts will be very
different for hypothetical objects like black
holes.
My question is:
Why give the 2 different formulas generally give
so close results?
Marcel Luttgens