ULTIMATE SYMMETRY:
Fractal Complex-Time and Quantum Gravity
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III.1.3 The Principle of General
Covariance
Because they are based on tensors, the equations of
motion in General Relativity are not related to the coordinate system. However,
the components of the tensors in some basis are treated as space-time fields
which are dependent on the chosen coordinate system, and because the action is
scalar it is invariant under coordinate transformations. Since all coordinates
systems are artifacts of the human mind, the laws of physics must have the same
form in any reference frame. This is called general, or diffeomorphism,
covariance, or invariance.
This General Covariance Principle was assumed by Einstein
as an extension of the Principle of Relativity. However, in contrast with the
global space-time symmetries of Special Relativity, the invariance of the form
of the laws, under transformations of the coordinates depending smoothly on
arbitrary functions of space and time, is a local symmetry with diffeomorphism
group. This extension of the concept of continuous symmetry from global to
local symmetries led to Noether s theorem which connects between global
symmetries and conservation laws.
In Special Relativity, the Minkowski metric possesses a
global Lorentz symmetry, and the full isometry group is the Poincar group
which is an extension of the Lorentz group that also includes translations.
Similarly, the theory of isometries in General Relativity on fixed backgrounds
is described by Killing vector fields which form a Lie algebra that generates
the group of isometries, but this group is non-trivial only in special cases.
The full symmetry of General Relativity is the diffeomorphism group of the
manifold which is the group of all continuous and differentiable mappings of
the manifold to itself, but this group is different for manifolds of different
topologies. For example, for the Schwarzschild metric, the corresponding
Killing group is a subgroup of the Poincar group. Nevertheless, if one
searches for the symmetry group that leaves a general metric of the Einstein
equation invariant one finds that this group contains only the identity
transformation. In contrast to all other fundamental theories of physics,
General Relativity is not based on some kind of symmetry, but it is a result of
the Equivalence Principle.
... Space Transcendence Read this short concise exploration of the Duality of Time Postulate: DoT: The Duality of Time Postulate and Its Consequences on General Relativity and Quantum Mechanics ...
... searches for the symmetry group that leaves a general metric of the Einstein equation invariant one finds that this group contains only the identity transformation. In contrast to all other FUNDAMENTAL THEORIES of physics, General Relativity is not based on some kind of symmetry, but it is ...
... ffeomorphism, covariance, or invariance. This General Covariance Principle was assumed by Einstein as an extension of the Principle of Relativity. However, in contrast with the global space- TIME SYMMETRIES of Special Relativity, the invariance of the form of the laws, under transformations ...
... nsion of the Lorentz group that also includes translations. Similarly, the theory of isometries in General Relativity on fixed backgrounds is described by Killing vector fields which form a Lie algebra that generates the group of isometries, but this group is non-trivial only in special ca ...
... they are based on tensors, the equations of motion in General Relativity are not related to the coordinate system. However, the components of the tensors in some basis are treated as space- TIME FIELD s which are dependent on the chosen coordinate system, and because the action is scalar it ...
... e coordinates depending smoothly on arbitrary functions of space and time, is a local symmetry with diffeomorphism group. This extension of the concept of continuous symmetry from global to LOCAL SYMMETRIES led to Noether s theorem which connects between global symmetries and conservation ...
... e laws, under transformations of the coordinates depending smoothly on arbitrary functions of space and time, is a local symmetry with diffeomorphism group. This extension of the concept of CONTINUOUS SYMMETRY from global to local symmetries led to Noether s theorem which connects between ...
... the Poincar group which is an extension of the Lorentz group that also includes translations. Similarly, the theory of isometries in General Relativity on fixed backgrounds is described by Killing vector fields which form a Lie algebra that generates the group of isometries, but this grou ...
... metry from global to local symmetries led to Noether s theorem which connects between global symmetries and conservation laws. In Special Relativity, the Minkowski metric possesses a global Lorentz symmetry, and the full isometry group is the Poincar group which is an extension of the Lore ...
... isometries, but this group is non-trivial only in special cases. The full symmetry of General Relativity is the diffeomorphism group of the manifold which is the group of all continuous and DIFFERENTIABLE MAPPINGS of the manifold to itself, but this group is different for manifolds of diff ...
... etries led to Noether s theorem which connects between global symmetries and conservation laws. In Special Relativity, the Minkowski metric possesses a global Lorentz symmetry, and the full ISOMETRY GROUP is the Poincar group which is an extension of the Lorentz group that also includes tr ...
... ace-time symmetries of Special Relativity, the invariance of the form of the laws, under transformations of the coordinates depending smoothly on arbitrary functions of space and time, is a LOCAL SYMMETRY with diffeomorphism group. This extension of the concept of continuous symmetry from ...
The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
