Grupo de Lorentz

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In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis. The components transform between these bases as the space and time coordinate differences, (cΔt,Δx,Δy,Δz) under spatial translations, rotations, and boosts (a change by a constant velocity to another inertial reference frame). The set of all such translations, rotations, and boosts (called Poincaré transformations) forms the Poincaré group. The set of rotations and boosts (Lorentz transformations, described by 4×4 matrices) forms the Lorentz group.
In physics (and mathematics), the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime
The Lorentz transformation for frames in standard configuration can be shown to be:
where is called the Lorentz factor.
Matrix form
This Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as
More generally for a boost in an arbitrary direction (βx,βy,βz),
where and .
Note that this is only the "boost", i.e. a transformation between two frames in relative motion. But the most general proper Lorentz transformation also contains a rotation of the three axes. This boost alone is given by a symmetric matrix. But the general Lorentz transformation matrix is not symmetric.
Rapidity
The Lorentz transformation can be cast into another useful form by introducing a parameter φ called the rapidity (an instance of hyperbolic angle) through the equation:
Equivalently:
Then the Lorentz transformation in standard configuration is:
Hyperbolic trigonometric expressions
It can also be shown that:
and therefore,
Hyperbolic rotation of coordinates
Substituting these expressions into the matrix form of the transformation, we have:
Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity φ represents the hyperbolic angle of rotation.
General boosts
For a boost in an arbitrary direction with velocity , it is convenient to decompose the spatial vector into components perpendicular and parallel to the velocity : . Then only the component in the direction of is 'warped' by the gamma factor:
where now . The second of these can be written as:
These equations can be expressed in matrix form as
where I is the identity matrix, v is velocity written as a column vector and vT is its transpose (a row vector).
Spacetime interval
In a given coordinate system (xμ), if two events A and B are separated by
the spacetime interval between them is given by
This can be written in another form using the Minkowski metric. In this coordinate system,
Then, we can write
or, using the Einstein summation convention,
Now suppose that we make a coordinate transformation . Then, the interval in this coordinate system is given by
or
It is a result of special relativity that the interval is an invariant. That is, . It can be shown[4] that this requires the coordinate transformation to be of the form
Here, is a constant vector and a constant matrix, where we require that
Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation.[5] The Ca represents a space-time translation. When , the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.
Taking the determinant of gives us
Lorentz transformations with are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with are called improper Lorentz transformations and consist of (discrete) space and time reflections combined with spatial rotations and boosts. They don't form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.
The composition of two Poincaré transformations is a Poincaré transformation and the set of all Poincaré transformations with the operation of composition forms a group called the Poincaré group. Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the Lorentz group.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
Special relativity
One of the most astounding predictions of special relativity was the idea that time is relative. In essence, each observer's frame of reference is associated with a unique clock, the result being that time passes at different rates for different observers. This was a direct prediction from the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz transformations that the concept of simultaneity varies between reference frames. Another startling result is length contraction.
Lorentz transformations can also be used to prove that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force. If we have one charge or a collection of charges which are all stationary with respect to each other, we can observe the system in a frame in which there is no motion of the charges. In this frame, there is only an electric field. If we switch to a moving frame, the Lorentz transformation will give rise to a magnetic field. These two fields are unified in the concept of the electromagnetic field.