Tuesday, December 21, 2004

Inertial Forces Newton -> Einstein -> Shipov
Background to "inertial forces" in Einstein's GR ~ (LC) connection field and in Gennady Shipov's extension of GR to include possibly torsion fields:

Review of the concept of "inertial forces" in Newton's kinematics of non-inertial frames of reference in Galilean relativity of absolute simultaneity when v/c << 1.

Let the rotating non-inertial frame S' have pseudo-vector W instant angular velocity

The rate of change in the rotating frame S' is the convective derivative linear operator on 3-vectors

...,t' = ...,t + Wx...

where ,t is the rate of change in the inertial frame S.
Let r be the position vector of a point test particle.

r,t' = r,t + Wxr = instant velocity

r,t',t' = r,t,t + Wxr,t + W,txr + Wxr,t + WxWxr = r,t,t + 2Wxr,t + WxWxr + W,txr

2Wxr,t = Coriolis inertial force per unit test mass in the non-inertial rotating frame

WxWxr = Centrifugal inertial force per unit test mass in the non-inertial rotating frame

W,txr is the inertial torque per unit test mass

Or in terms of Newton's F = ma

unprimed quantities are in the inertial frame, primed quantities are in the rotating non-inertial frame sharing common origin.

v' = v + Wxr

a' = a + 2Wxv + WxWxr + W,txr

Note that the Coriolis inertial force in the rotating frame is a bit like the Lorentz magnetic force on a charge e with W like magnetic field B but here we have the equivalence principle i.e. m cancels out of the problem for all inertial forces just like with gravity.

SPECIAL CASE of above

If the rotating frame is also the instant rest frame of the point test particle then

v' = 0

Therefore v = -Wxr

a' = 0

Assume also no torque so that W,t = 0

Therefore

a + Wxv = 0

Or

a - WxWxr = 0

That is inertial force compensation in the REST LNIF of the test particle where the effective acceleration of "artificial" gravity is

g = a = WxWxr

Of course, the inertial geodesic motion is a straight Euclidean line with the test particle at constant speed.

In Newton's theory there is actually an external gravity force and you can have, for example

GM/r^2 = |WxWxr|

GM = W^2r^3 = r^3/T^2

T = period of orbit. This is Kepler's law of planetary motion, M is mass of Sun.

In Newton's "Force" theory the motion of the planets is NON-GEODESIC in flat 3D space.

This world picture "paradigm" changes completely in Einstein's "force without Force" theory AKA "geometrodynamics" where now the motion of the planets is "inertial" i.e. "geodesic" in 4D space-time i.e. straightest 4D world line possible in curved spacetime.

The space-time curvatures in the above example of Kepler's law are made from

1/r*^2 = GM/c^2r^3

where r* is the scale of local radii of curvature of space-time at "distance" r in weak field limit GM/c^2r << 1.