Einstein group
Encyclopedia
Albert Einstein
, in searching for the transformation group for his unified field theory
, wrote:
The Poincaré group
, the transformation group of special relativity
, being orthogonal, the inverse of a transformation equals its transpose, introducing discrete reflections. This, in turn, violates Einstein's dictum for a group "no less general than that of the continuous transformations of the four coordinates." Specifically, any pair of Euler angles θk and −θk are not independent, as are not any pair of boosts vk/c and −vk/c. Available parameters are thus reduced, from the 16 needed to express all transformations in a curved spacetime, per the general principle of relativity
, ∂xμ′/∂xν, to the 10 of the Poincaré group.
Mendel Sachs
, in the 1960s, found the transformation group that Einstein had sought, the "Einstein" group. The Einstein group can be obtained by factorizing the squared spacetime invariant interval
into a quaternion
-valued form and its conjugate, ds ds*, where
and qμ(x) is a four-vector
of Hermitian
quaternions.
Note that the Einstein group approaches—but never reaches—the Poincare group as the flat spacetime (special relativity
limit) is approached.
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, in searching for the transformation group for his unified field theory
Unified field theory
In physics, a unified field theory, occasionally referred to as a uniform field theory, is a type of field theory that allows all that is usually thought of as fundamental forces and elementary particles to be written in terms of a single field. There is no accepted unified field theory, and thus...
, wrote:
Every attempt to establish a unified field theory must start, in my opinion, from a group of transformations which is no less general than that of the continuous transformations of the four coordinates. For we should hardly be successful in looking for the subsequent enlargement of the group for a theory based on a narrower group.
The Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
, the transformation group of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, being orthogonal, the inverse of a transformation equals its transpose, introducing discrete reflections. This, in turn, violates Einstein's dictum for a group "no less general than that of the continuous transformations of the four coordinates." Specifically, any pair of Euler angles θk and −θk are not independent, as are not any pair of boosts vk/c and −vk/c. Available parameters are thus reduced, from the 16 needed to express all transformations in a curved spacetime, per the general principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
, ∂xμ′/∂xν, to the 10 of the Poincaré group.
Mendel Sachs
Mendel Sachs
Mendel Sachs is a US theoretical physicist who was Professor of Physics at the State University of New York Buffalo .- Education and career :...
, in the 1960s, found the transformation group that Einstein had sought, the "Einstein" group. The Einstein group can be obtained by factorizing the squared spacetime invariant interval
- ds2 = gμν dxμ dxν
into a quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
-valued form and its conjugate, ds ds*, where
- ds = qμ(x) dxμ
and qμ(x) is a four-vector
Four-vector
In the theory of 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...
of Hermitian
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...
quaternions.
Note that the Einstein group approaches—but never reaches—the Poincare group as the flat spacetime (special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
limit) is approached.