What are the formula for applying Einstein's relativity in mathematics? Suppose, for example, that I wanted to resolve the twin paradox. Couldn't I do it by applying one of the formula of relativity theory and calculate how much time goes by for each twin, and therefore which one ages more? What would these formula be?

In your particular case, that would formally be something like [math] \tau = \int_{\Gamma} \sqrt{ \pm \left< \frac{dx}{d\gamma} \right| \left. \frac{dx}{d\gamma} \right> }\mathrm{d} \gamma[/math], where [math]\tau[/math] is the time measured by the respective object, [math]\Gamma[/math] is a path through spacetime, [math]\gamma[/math] a small piece of the path (in physicists' language), x a location in spacetime, and [math] \langle \cdot | \cdot \rangle [/math] the pseudo-scalar-product of relativity (which encodes the structure of spacetime). The plus of minus sign depends on the sign convention chosen for this pseudo-scalar-product (both conventions are used roughly equally often).

 

As you may have realized already, understandig physics is not only about visually seeing the formula, but also about understanding the meaning of the mathematical constructs and the physical concepts behind them. This example may not be the most central equation within relativity itself. Relativity is not inherently about twins in rockets (as well as Quantum Mechanics is not primarily about cats). But as [math]\tau[/math] is the physical time, i.e. the scale on which physical processes happen, it is rather central for relativistic physics (and then appears in much simpler forms, e.g. for [math]\Gamma[/math] being a straight line in flat Minkowski spacetime).

http://en.wikipedia.org/wiki/Time_dilation#Simple_inference_of_time_dilation_due_to_relative_velocity

http://en.wikipedia.org/wiki/Length_contraction

Thanks both.

 

Does anyone know the name of these formulas?