# ULTIMATE SYMMETRY:

Fractal Complex-Time and Quantum Gravity

# 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.

# Other Pages Related to Search Keywords:

• ... 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 ...

• ## ... Fundamental Theories =>:

• ... 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 ...

• ## ... Time Symmetries =>:

• ... 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 ...

• ## ... Lie Algebra =>:

• ... 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 ...

• ## ... Time Field =>:

• ... 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 ...

• ## ... Local Symmetries =>:

• ... 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 ...

• ## ... Continuous Symmetry =>:

• ... 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 ...

• ## ... Killing Vector =>:

• ... 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 ...

• ## ... Lorentz Symmetry =>:

• ... 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 ...

• ## ... Differentiable Mappings =>:

• ... 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 ...

• ## ... Isometry Group =>:

• ... 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 ...

• ## ... Local Symmetry =>:

• ... 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.
Ibn al-Arabi [The Meccan Revelations: Volume I, page 156. - Trns. Mohamed Haj Yousef]

### The Sun from the West:

Welcome to the Single Monad Model of the Cosmos and Duality of Time Theory

### Message from the Author:

I have no doubt that this is the most significant discovery in the history of mathematics, physics and philosophy, ever!

By revealing the mystery of the connection between discreteness and contintuity, this novel understanding of the complex (time-time) geometry, will cause a paradigm shift in our knowledge of the fundamental nature of the cosmos and its corporeal and incorporeal structures.