Inertial Observers: Galileo and Lorentz transformations

Basic Concepts and principles

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.

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 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
$$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$t'=t$$
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
The transformation of Lorentz
$$x'= \gamma(x -vt)$$ $$y'=y$$ $$z'=z$$ $$t'= \gamma(t -\frac{vx}{c^2})$$
$$\gamma= \sqrt{1-\frac{v^2}{c^2}}$$

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 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.

LetE, E', B, B' vectorial functions related by the formulas
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.

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