Physical law from experimental data

[bernhard.rothenstein]

Consider that high quality experiments furnish data that relate proper time intervals to coordinate time intervals for different values of the relative velocity (life time dilationf moving mesons). An inteligent computer program could find out the best function of the relative speed which relates all the experimental results, i.e. the time dilation formula. Could we use it for further relativistic derivations or it is necessary to perform other experiments (light clock) in order to derive it?
We find a simillar situation in the case of the formula which accounts for the mass dilation (Bucherer's experiment).

 

[dx]

With that experimental data, together with the law of inertia and the uniformity of clock rates on any single world line, one can deduce Lorentz invariance, and all other consequences of relativity.

 

[bernhard.rothenstein]

With that experimental data, together with the law of inertia and the uniformity of clock rates on any single world line, one can deduce Lorentz invariance, and all other consequences of relativity.

Thanks. Please be more specific concerning the law of inertia. What does it introduce in the derivation?

 

[dx]

The law of inertia is just to determine what world lines look like in inertial reference frames. If we send particles with different velocities in different directions, and then record the event where their measure a time 1, we get the hyperbola. In light-signal coordinates, this hyperbola is $$\tau_1 \tau_2 = 1$$. By a simple argument, we can deduce that the light-signal coordinates of an event $$(\tau_1, \tau_2)$$ in a frame moving with velocity v relative to the original frame are

$$\tau_1 ' = \sqrt{\frac{1+v}{1-v}} \tau_1$$
$$\tau_2 ' = \sqrt{\frac{1-v}{1+v}} \tau_2$$

Thus we get lorentz invariance: $$\tau_1 \tau_2 = \tau_1 ' \tau_2 '$$.