An **inertial fame**
is a reference frame in which all relative accelerations due to external forces
are eliminated.

I. The laws of nature are the same in all inertial reference frames.

II. In vacuum, light propagates with respect to any inertial frame and in all
directions with the universal speed c. This speed is a constant of nature.

Consider two reference frames K and K’.

Assume that the coordinate axes in
the two frames are parallel and that the origins of the coordinates coincide at
t = t’ = 0.

Assume that K’ is moving with velocity v**i** with respect to
K.

The Lorentz transformation gives the coordinates of a
**space-time** point (x_{0},x_{1},x_{2},x_{3})
= (ct,x,y,z) in K in terms of its coordinates (x'_{0},x'_{1},x'_{2},x'_{3})
= (ct',x',y', z') in K’ and vice versa.

β = v/c,
**β**
= **v**/c , γ
= (1 - β^{2})^{-½}.

Since 0 ≤ β ≤ 1, we may write β =
tanh(B), where B is the **boost parameter** or the
**rapidity**.

[tanh(u) = (e^{u} - e^{-u})/(e^{u} + e^{-u})].

Then γ = (1 - tanh^{2}(B))^{-½}
= cosh(B), γβ = tanh(B)cosh(B) = sinh(B), and

reminiscent of a rotation.

We define as a **4-vector**
any set of 4 quantities which transform like (x_{0},x_{1},x_{2},x_{3})
under a Lorentz transformation; (a_{0},a_{1},a_{2},a_{3})
is a 4-vector if

or

a_{0}' = γ(a_{0} – **β∙a**), **a**'_{||}
= γ(** a**_{||} – **β**a_{0}), **a**'_{⊥} =
**a**_{⊥}.

The **"dot product"
**
(a

is invariant under a Lorentz transformation.

The 4-vector (**x**_{0},**r**) defines an **event**. The **
space-time interval** between two events is ds.

ds^{2} = c^{2}dt^{2} - |d**r**|^{2}.

In a reference frame in which two events have the same space coordinates d**r
**= 0 and ds^{2} = c^{2}dτ^{2},

where dτ
= ds/c is the **proper time interval**. It is a Lorentz invariant
quantity.

dτ
= dt/γ.

**4-velocity**:
(u_{0},**u**) = (γc,γd**r**/dt) =
(dx_{0}/dτ,d**r**/dτ).

(u_{0},**u**)∙(u_{0},**u**)
= c^{2} is a Lorentz invariant scalar.

**4-vector momentum**:
(p_{0},**p**) = (mu_{0},**mu**)
= (E/c,**p**) = (γmc,γmd**r**/dt).

(p_{0},**p**)∙(p_{0},**p**)
= mc^{2} --> E^{2} = m^{2}c^{4} + p^{2}c^{2}.

A particle moves in K with velocity **u** = d**r**/dt. K' moves
with respect to K with velocity **v**. The particle's velocity in K’,
**u**' = d**r**'/dt', is given by

u'_{||} = (u_{||} - v)/(1 -** v∙u**/c^{2}),

u'_{⊥}
= u_{⊥}/(γ(1
-** v∙u**/c^{2})).

where parallel and perpendicular refer to the direction of the relative
velocity **v**.

It is impossible to obtain speeds greater than c.

In every reference frame energy and momentum are
conserved in collisions between free particles.

For each component p_{μ} of the
4-vector (p_{0}, p_{1}, p_{2}, p_{3})
we have

∑_{particles_in} p_{μ} = ∑_{particles_out} p_{μ},
or ∑_{i} (p_{i})_{μ} = ∑_{j} (p_{j})_{μ},

where i denotes the particles going into the collision and j
denotes the particles emerging from the collision.

For transformations between reference frames we have

(P_{0},**P**)∙(P_{0},**P**)
= (P_{0}',**P**')∙(P_{0}',**P**').

Here
P_{0} = ∑_{particles} p_{0} and **P**
= ∑_{particles} **p**.

This is a consequence of the invariance of the dot product under a Lorentz
transformation.

Assume we have a collection of initially free particles. They interact
with each other and possibly change into different particles. After the
interaction the new particles are free again. The following rules always apply:

(a) The "length" of the 4-vector (P_{0},**P**) is
invariant when transforming
from one one reference frame to another.

(b) P_{0} and** P** are conserved in any reference frame.

The angular frequency ω of a sinusoidal
electromagnetic wave with wave vector **k** (k = 2π/λ
= ω/c) in a reference frame K is measured as
ω' in a reference frame K' moving with uniform
velocity **v** with respect to K.

ω' = γω(1 - (v/c)cosθ),
where θ is the angle between the directions of **k**
and **v**.

ω' = ω[(1 - v/c)/(1 +
v/c)]^{½} if **k** and **v** are parallel to each other.

ω' = ω[(1 + v/c)/(1 -
v/c)]^{½} if **k** and **v** are anti-parallel to each other.