Two observers B and C are said to be inertial if they use inertial coordinates systems, ie between these systems, there are no movement or any relative moviment between them is rectilinear and uniform .

For example, if observers B and C are at different points on our Earth, for example B in New York and C located
somewhere in Australial traveling by plane to 900km/h along the shortest path.

Could be considered that they are Inertial observers? The answer to this question
depends on the phenomenon we are considering.

In order to measure displacement of a comet orbiting around the sun, either different
situation has no effect on their measurements, both will measure the same
speed, same period in translation around the sun, etc ... In this case they are
both inertial observers.

However the situation is different if we consider a satellite orbiting the earth in
the same direction and with the same orbital period of the Earth day: 24 hours.
In this situation, from the viewpoint of B, he will say that the satellite is at
rest in the sky which does not have any movement.

From the viewpoint of C the situation is that the satellite is shifting slightly
to the west.

However, if we consider another observer D on the surface of the moon, he will say
that the satellite has a translational motion around the earth takes 24 hours and
that both B and C are moving with it (really C has more moviment).

Which of the observer is correct? The answer is that everyone, each using a
different reference system: B uses a reference system centered in New York, C uses a reference system
in uniform rectilinear motion with respect to B, and D using a
reference system with origin its location on the surface of the moon according
to which observers B and C have accelerated movements.

The conclusion to this example is that the laws of physics do not change for
different observers, provided they use reference systems at relative rest beetween them (this "beetween them" is important) or
at relative uniform rectilinear motion (one more time, beetween them).
In other cases the situations may be different for one or other observer

In classical Newtonian mechanics is considered the existence of an absolute time. This absolute time
is the same for all inertial observers. One of most revolutionary Special
Theory of Relativity idea is that: time is not absolute, depends on the reference system which is measuring.

# Inertial Observers: Galileo and Lorentz transformations

### Basic Concepts and principles

### Extended Theory

Suposse two intertial observers A and B who use (x, y, x, t) y (x', y', z', t') coordinates respectively. Suposse B moving by the x-axis of A at velocity constant v, ie for A

Both systems are related by the Galilean Transformation

$$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$t'=t$$
We saw at section Maxwell equations
that such equations may not hold for B especially if v is next to the value c = 300.000 km/s (aprox.).

Lorentz idea was study transformations that leave invariant Maxwell's equations, and concluded that the relation between both coordinates systems would be

$$x'= \gamma(x -vt)$$ $$y'=y$$ $$z'=z$$ $$t'= \gamma(t -\frac{vx}{c^2})$$

Where

$$\gamma= \sqrt{1-\frac{v^2}{c^2}}$$

This is the Lorentz Transformation, it is essential to understand special relativity theory and as we will see, from this transformation emerge all relativity revolutionary consequences. Note that for speeds which we are accustomed to, the value γ is nearly 1 and the Lorentz transformation is nearly to be identical to the Galilean transformation.

A first, important observation is that to pass from one reference system to another, we must also transform time coordinate. Ie time is relative and depends on the reference system the observer were using to measure.

We will see finally a theorem that explains how to transform electromagnetic fields for inertial observers, such as for them all, Maxwell's equations holdss:

**v**= (v, 0, 0).B reference system is moving in growing X-axis direction of A reference System. |

Both systems are related by the Galilean Transformation

**The transformation of Galileo**

$$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$t'=t$$

Lorentz idea was study transformations that leave invariant Maxwell's equations, and concluded that the relation between both coordinates systems would be

**The transformation of Lorentz**

$$x'= \gamma(x -vt)$$ $$y'=y$$ $$z'=z$$ $$t'= \gamma(t -\frac{vx}{c^2})$$

Where

$$\gamma= \sqrt{1-\frac{v^2}{c^2}}$$

This is the Lorentz Transformation, it is essential to understand special relativity theory and as we will see, from this transformation emerge all relativity revolutionary consequences. Note that for speeds which we are accustomed to, the value γ is nearly 1 and the Lorentz transformation is nearly to be identical to the Galilean transformation.

A first, important observation is that to pass from one reference system to another, we must also transform time coordinate. Ie time is relative and depends on the reference system the observer were using to measure.

We will see finally a theorem that explains how to transform electromagnetic fields for inertial observers, such as for them all, Maxwell's equations holdss:

Consider two inertial observers A and B as before A moves to B with velocity

suposse also, that both coordinate systems are related by the Lorentz transformation we have seen before then.

Let

E'1 = E1

E'2 = γ(E2 - vB3)

E'3 = γ(E3 + vB2)

B'1 = B1

B'2 = γ(B2 + vE3/c2)

B'3 = γ(B3 - vE2/c2)

Then if E and B hods the Maxwell's equations with the coordinates system of A, then E' and B' hods Maxwell's equations with the system used by B

This theorem shows that if you use the Lorentz transformation to change coordinates between inertial reference systems, Maxwell's equations are true for all them.
**v**= (v, 0, 0). Suppose A uses the coordinate system (x, y, x, t) and B uses (x ', y', z ', t').suposse also, that both coordinate systems are related by the Lorentz transformation we have seen before then.

Let

**E**,**E'**,**B**,**B'**vectorial functions related by the formulasE'1 = E1

E'2 = γ(E2 - vB3)

E'3 = γ(E3 + vB2)

B'1 = B1

B'2 = γ(B2 + vE3/c2)

B'3 = γ(B3 - vE2/c2)

Then if E and B hods the Maxwell's equations with the coordinates system of A, then E' and B' hods Maxwell's equations with the system used by B

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