Derivation of the two criteria for the inertial frame of reference.

Rob Kamp, Netherlands

Version 1.0.0: April, 2005.

An inertial frame of reference is a celestial object (e.g. a planet or a space craft) such that Newton's laws hold in any coordinate system that is attached to it. In particular Newton's second law applies: F = m x a. It is shown that there are 2 sufficient and necesary conditions for a reference frame to be inertial.

Criterium 1: The total force on the frame is zero.

On a small mass m in the vicinity of a planet-frame with mass M, there is exerted no external electromagnetic forces (that is given). Thus the only force is the gravitational force due to the presence of the planet. If the frame is inertial, then by definition: F_m = F_m,M + SIGMA_(i=1;N){F_m,i(x)} = F_m,M, in which i is the index of all distant (fixed) stars in the Universe. Thus:

(1)   SIGMA_(i=1..N){F_m,i(x)} = O.

According to Newton's fourth (gravitational) law:

(2)   F_m,i(x) = (M_i x m)/(|DR_i|²)r_i

In the above equation, r_i is the unit vector pointing in the direction of star i and DR_i = R_i - R_m.

Three equations related to |DR_i|²:

(3)   |DR_i|² = |R_i|² - 2r_m.R_i + |r_m|²

(4)   -2.r_m.R_i =< 2.|r_m|.|R_i| << |R_i|²

(5)   |r_m|² << |R_i|²

From (3), (4) and (5) it follows:

(6)   |DR_i|² *= |R_i|² ("*=" means: almost equal)

Combination of (1), (2) and (6):

(7)   (m/M) x SIGMA_(i=1..N){(M_iM)/(|R_i|²)r_i)} *= O

Equation (7) implies:

(8)   SIGMA_(i=1..N){F_M,i} *= O.

Equation (8) means that the total gravitational force of the distant stars on the frame of reference is (virtually) zero. This is the first criterium.

Criterium 2: The frame does not rotate.

The average force of all stars in the universe is in cylindrical coordinates:

(9)   <F_i> = (<M_ir_i''> - <M_ir_iθ_i^{' 2}>) r + (<2M_ir_i'θ_i'> + <M_ir_iθ_i''>) θ + <M_iz_i''> z

According to Newton's third law the average internal force between the stars in any inertial frame is zero:

(10)   <F ^Int_i> = (1/N).SIGMA_(i=1..N){F_i} = O

There are no external forces on the universe. So:

(11)   <F_i> = <F ^Int_i> = O

The average angular velocity in the z-direction is:

(12)   ω = (1/N).SIGMA_(i=1..N){ω + δω_i} = (1/N).SIGMA_(i=1..N){θ'_i}.

It follows that:

(13)   SIGMA_(i=1..N){δω_i} = 0.

The δω_i are not all zero. It follows from (9) that if ω = 0 (in this case θ'_i = δω_i )

(14)   <M_ir_i''> = <M_ir_iθ_i^{' 2}> <> 0.

But suppose ω <> 0. So the inertial frame would rotate in the universe. This implies that relative the inertial frame, the universe would rotate around the z-axis of the coordinate frame of the reference frame (e.g. planet). Centripetal forces would have to keep the stars moving on average in circular orbit: such that <r_i'> does not change. The latter is implied by the fact that the motion of the stars are not affected by an inertial frame that starts rotating. But the gravitational forces can not give rise to such centripetal forces. So a rotating inertial frame is impossible. This is derived below. (This means that a planet is not an exact inertial frame. Only approximately.)

If ω <> 0, then

(15)   <M_ir_i''> < <M_ir_iθ_i^{' 2}>

Proof of (15):

(16)   <M_ir_iθ_i^{' 2}> = <M_ir_i.(ω + δω_i) ²> = <M_ir_i.(ω² + 2.ωδω_i + δω_i²)> = <M_ir_i.ω²> + 2.ω<M_ir_iδω_i> + <M_ir_iδω_i²> = <M_ir_i.ω²> + <M_ir_iδω_i²>

The term 2.ω<M_ir_iδω_i> in (16) is postulated to be (virtually) zero; for all i with fixed values for M_ir_i, there are equally many positive as negative values of δω_i (Symmetric Gaussian distribution).

Substitution of (16) in the r-component of (9):

(17)   <M_ir_i''> - (<M_ir_i.ω²> + <M_ir_iδω_i²>) = (<M_ir_i''> - <M_ir_iδω_i²>) - <M_ir_i.ω²> = 0 - <M_ir_i.ω²> = - <M_ir_i>.ω² Q.E.D.

The last step in (17) follows from (14). (δω_i = θ_i' in case ω = 0).

According to (11) though the following should hold in an inertial frame: <F_i> = <F ^Int_i> = O. This implies (14). But (14) contradicts (15),
which holds when ω <> 0, i.e. when the inertial frame starts to rotate. Conclusion: ω = 0; there are no rotating inertial frames. Stated differently:
<F ^Int_i> = O <> - <M_ir_i>.ω² r, except when ω = 0. This is the second criterium.