Hyperphysics: Hafel-Keating experiment

[Rasalhague]

http://hyperphysics.phy-astr.gsu.edu/HBASE/relativ/airtim.html#c5

I don't understand the approximation T₀ = -T_S that they make in the final step of the section "Kinematic Time Shift Calculation". From this, and the other equations in this section, I get

$$-T_0=T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )$$

$$-1=1+\frac{R^2\omega^2}{2c^2}$$

$$c^2=\frac{R^2\omega^2}{-4}$$

$$c=\pm \frac{R\omega}{2i}$$

but how can this be when c is a constant positive real number, not dependent on the product of the rotation of the Earth with its radius? And

$$T_A=T_S-T_S\left ( \frac{2R\omega v+v^2}{2c^2} \right )$$

$$T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=-T_S\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )$$

$$T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=T_0\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )$$

$$4c^2= 2R\omega v+v^2 - R^2\omega^2$$

$$c=\frac{\sqrt{(R\omega+v)^2-2R^2\omega}}{2}$$

which can't be right, since c doesn't depend on these arbitrary variables: radius of the earth, etc.

 

[starthaus]

http://hyperphysics.phy-astr.gsu.edu/HBASE/relativ/airtim.html#c5

I don't understand the approximation T₀ = -T_S that they make in the final step of the section "Kinematic Time Shift Calculation". From this, and the other equations in this section, I get

$$-T_0=T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )$$

$$-1=1+\frac{R^2\omega^2}{2c^2}$$

$$c^2=\frac{R^2\omega^2}{-4}$$

$$c=\pm \frac{R\omega}{2i}$$

but how can this be when c is a constant positive real number, not dependent on the product of the rotation of the Earth with its radius? And

$$T_A=T_S-T_S\left ( \frac{2R\omega v+v^2}{2c^2} \right )$$

$$T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=-T_S\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )$$

$$T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=T_0\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )$$

$$4c^2= 2R\omega v+v^2 - R^2\omega^2$$

$$c=\frac{\sqrt{(R\omega+v)^2-2R^2\omega}}{2}$$

which can't be right, since c doesn't depend on these arbitrary variables: radius of the earth, etc.

HK is poorly explained using SR, a correct explanation requires GR. I am quite sure that I gave a GR-based explanation for HK somewhere in this forum. It is simply calculating the proper time $$\tau$$ by integrating the expression in coordinate time t. The expression can be derived straight from the general Schwarzschild metric setting:

$$dr=d\theta=0$$,
$$\frac{d\phi}{dt}=\omega +\frac{v_1}{R}$$
$$\frac{d\phi}{dt}=\omega -\frac{v_2}{R}$$

depending on the direction of plane motion