Everything about Lorentz Boost totally explained
In
physics, the
Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other. In classical physics (
Galilean relativity), the only conversion believed necessary was
, describing how the origin of one observer's coordinate system slides through space with respect to the other's, at speed
and along the x-axis of each frame. According to
special relativity, this is only a good approximation at much smaller speeds than the speed of light, and in general the result isn't just an offsetting of the x coordinates; lengths and times are distorted as well.
If space is
homogeneous, then the Lorentz transformation must be a
linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the
spacetime interval between any two events in
Minkowski space. The Lorentz transformations describe only the transformations in which the event at x=0, t=0 is left fixed, so they can be considered as a
rotation of
Minkowski space. The more general set of transformations that also includes translations is known as the
Poincaré group.
Henri Poincaré named the Lorentz transformations after the
Dutch physicist and
mathematician Hendrik Lorentz (
1853-
1928) in 1905. They form the mathematical basis for
Albert Einstein's theory of
special relativity. The Lorentz transformations remove contradictions between the theories of
electromagnetism and
classical mechanics. They were derived by
Joseph Larmor in 1897, and Lorentz (1899, 1904). In 1905 Einstein derived them under the assumptions of
Lorentz covariance and the constancy of the speed of light in any inertial reference frame.
Lorentz transformation for frames in standard configuration
O and
, each using their own
Cartesian coordinate system to measure space and time intervals.
O uses
and
Q uses
. Assume further that the coordinate systems are oriented so that the
x-axis and the
x' -axis overlap, the
y-axis is parallel to the
y' -axis, as are the
z-axis and the
z' -axis. The relative velocity between the two observers is
v along the common
x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate systems are said to be in
standard configuration. A
symmetric presentation
between the forward Lorentz Transformation and the inverse Lorentz Transformation
can be achieved if coordinate systems are in
symmetric configuration.
The symmetric form highlights that all physical laws should be of such a kind that
they remain unchanged under a Lorentz transformation.
The Lorentz transformation for frames in standard configuration can be shown to be:
»
Apparently
can't be negative because otherwise there would be a transformation which transforms time into spatial coordinate and vice versa. This is no good (at least in special relativity) since time can only run in the positive direction while coordinates in both. If then
it's apparently the highest achievable velocity. Theoretically it can be either infinitely large, which gives Galilean transformation and Euclidean world with absolute time, or it can be finite, which gives Lorentz transformation and Minkowski world of special relativity. The experiment tells us that it's finite,
299792458m/s.
