 # Special Relativity and Minkowski's space

### Basic concepts and principles

We have seen that there were discrepancies between the cinematic classic (Galileo) theory and electromagnetic phenomena. We saw that Lorentz Transformation makes these discrepancies disappear.

In fact, many of relativistic ideas and formulas had already appeared before the famous article by Einstein in 1905, but was him 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 t2 - t1, the spacecraft will travel from x1=vt1 to x2=vt2

Now suppose an observer inside the spacecraft, he notes that

t1' = γ(t1 - v2t1/c2)
t2' = γ(t2 - v2t2/c2)

So

γ(t2' - t1') = t2 - t1, with γ > 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 x1 ', x2' for an observer traveling inside the ship, while for an observer at rest on earth these values are x1, x2. For the Lorentz transformation

x1' = γ(x1 - vt)
x2' = γ(x2 - vt)

So

x2' - x1' = γ(x2 - x1), again with γ > 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

t2 - t1 = 0, but t2' - t1' =γ(t2 - v2x2/c2) - γ(t1 - v2x1/c2) = γv(x1 - x2)/c2 ≠ 0, si x1 ≠ x2

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

5) Addiction for velocities rule: Again the same example, suppose that observer on the spacecraft observes an object moving at speed v2=dx'/dt'.

The observer at rest in this movement is seeing speed v1=dx/dt. Then

v2 = dx'/dt' = dx'dt/dt'dt = [γ(dx/dt -v)]/[γ(1 - dx/dt v/c2)] = (v1 - v)/(1 - vv1/c2)

This is called the adding velocities rule in special relativity. Note thar v1 = v2 + v for small velocities we humans, use.

On other way if v2=c then v1=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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