Special Relativity and Minkowski's space

Basic concepts and principles

We have already seen that there are discrepancies between classical (Galileo's) kinematics and electromagnetic phenomena. Lorentz Transformation makes these discrepancies disappear, the price to pay for it is a relative complexity added to the formulas.

In fact, many of relativistic ideas and formulas had already appeared before the famous article by Einstein in 1905, but was he who created the comprehensive and robust framework in which these ideas had their true meaning and from it, he deduced too (as everybody knows) the equivalence of mass and energy as the most famous formula of science:

E = mc2

Special Relativity postulations

1) Laws governing physical systems are the same for all inertial observers, or in other words, physical phenomena do not change if these changes are expressed in relative, uniform rectilinear motion systems.

2) Speed of light is the same for all inertial observers.

Extetended Theory

Minkowsky Space is the space environment that will serve to develop the whole theory, we have the Lorentz Transformation , and we have defined relativistic units and show Einstein's postulates and their consequences at this section

Special relativity Consequences



1) Speed of light is maximum : For the Lorentz transformation of γ to be meaningful we need |v| < c

2) Time dilation : If an object moving at velocity v, suposse a spaceship. Suppose an observer too on the earth at rest, this observer will perceive the interval time \(t_2-t_1\), the spacecraft will travel from \(x_1=vt_1\) to \(x_2=vt_2\).

Now suppose an observer inside the spacecraft, he notes that

\( t_1'=\gamma(t_1-\frac{v^ 2t_1}{c^2}) \)

\( t_2'=\gamma(t_2-\frac{v^2t_2}{c^2}) \)

So

\( \gamma(t_2'-t_1') = t_2-t_1 \)

This implies that the time interval is longer in viewer observer inside the ship as each second must be multiplied by γ

3) Contraction of space : We take the example above, the spacecraft is travelling with velocity v. It has extremes \(x_1', x_2'\) for an observer traveling inside the ship, while for an observer on earth these values are \(x_1, x_2\) . For the Lorentz transformation

\(x_1'=\gamma (x_1 - vt) \)

\(x_2'=\gamma (x_2 - vt) \)

So

\(x_2' - x_1' = \gamma (x_2 - x_1) \)

Thus the observer who travels into the spacecraft see it larger, and a observer outside see the spacecraft shorter.

4) Relative simultaneity Again the example above, if for the observer at rest in the earth an event occurs at time t1 and the point x1 and other event occurs at time t2 and a point x2.

For an observer inside the ship, these phenomenon occurs at x1' and the instant t1' and x2' and the instant t2' respectively. For the Lorentz transformation, these values are related by

if \(t_2 - t_1 = 0 \)

\( t_2' - t_1' = \gamma (t_2 - \frac{v^2x_2^2}{c^2}) - \gamma (t_1 - \frac{v^2x_1^2}{c^2}) = \gamma (x_1-x_2)\frac{v^2}{c^2} \neq 0\Leftrightarrow x_1 \neq x_2 \)

Therefore, two events at different points are simultaneous for observer at rest are but not for the moving observer into spacecraft.

5) New addition for velocities rule: Again the same example, suppose that observer on the spacecraft observes an object moving at speed

\( v_2= \frac{\mathrm{dx'} }{\mathrm{d} t'} \).

The observer at rest on earth in this movement is seeing speed \( v_1= \frac{\mathrm{dx} }{\mathrm{d} t} \). Then

\( v_2= \frac{\mathrm{dx'} }{\mathrm{d} t'} = \frac{\mathrm{dx' dt} }{\mathrm{d} t' \mathrm{d} t}=\frac{\gamma (\frac{dx}{dt}-v)}{\gamma (1-\frac{vdx}{c^2dt})} \)

\( \displaystyle{ \Rightarrow v_2 = \frac{v_1-v}{1-\frac{vv_1}{c^2}} }\)

\( \displaystyle{\Rightarrow v_1 = \frac{v+v_2}{1+\frac{vv_2}{c^2}} } \).

This is called the adding velocities rule in special relativity. Note thar \(v_1 = v_2 + v \) for small velocities we humans, are accustomed to use.

On other way if \( v_2=c \Rightarrow v_1=c \), which is consistent with the first consequence: light speed is maximum

6) Equivalence of mass and energy : We will see it in the E=MC2 section .





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