ULTIMATE SYMMETRY:
Fractal Complex-Time and Quantum Gravity
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III.1.4 Space-Time Symmetries and
Diffeomorphisms
In General Relativity, space-time symmetries are used in
studying exact solutions of Einstein s field equations, and they are different
from internal symmetries. For example, the role of spherical symmetry is
important in deriving the Schwarzschild solution and studying its physical
consequences, such as the nonexistence of gravitational radiation in a
spherically pulsating star. In the cosmological principle, symmetry restricts
the type of universes that are consistent with large-scale observations, such
as the Friedmann-Lema tre-Robertson-Walker metric. There various corresponding
preserved properties, including: geodesics of the space-time, the metric tensor
and the curvature tensor.
A rigorous definition of symmetries in General Relativity
has been given by Hall in 2004, based on the idea of using smooth vector fields
whose local flow diffeomorphisms preserve some property of the space-time.
Because of this restricting diffeomorphism, the behavior of objects may not be
as manifestly symmetric on large scales. A symmetry on the space-time is a
smooth vector field whose local flow diffeomorphisms preserve some geometrical
feature of the space-time, such as the metric, the energy-momentum tensor, or
other aspects of the space-time, including its geodesic structure. The vector
fields are called collineations or symmetries, and the set of all symmetry
vector fields forms a Lie algebra under the Lie bracket operation, if the
smoothness condition is fulfilled.
A Killing vector field is one of the most important types
of symmetries and is defined to be a smooth vector field that preserves the
metric tensor. Killing vector fields are related to conservation laws, and they
have extensive applications, also in classical mechanics. An affine vector
field preserves geodesics and preserves the affine parameter, while the
homothetic vector fields are important in studying singularities. Other kinds
of symmetry preserve the energy-momentum tensor, which are referred to as
matter collineations or matter symmetries. It may be also shown that every
Killing vector field is a matter collineation by the Einstein field equations.
Therefore, a vector field that preserves the metric necessarily preserves the
corresponding energy-momentum tensor. When the energy-momentum tensor
represents a perfect fluid, every Killing vector field preserves the energy
density, pressure and the fluid flow vector field. When the energy-momentum
tensor represents an electromagnetic field, a Killing vector field does not
necessarily preserve the electric and magnetic fields.
... 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 ...
... are used in studying exact solutions of Einstein s field equations, and they are different from internal symmetries. For example, the role of spherical symmetry is important in deriving the Schwarzschild solution and studying its physical consequences, such as the nonexistence of gravitati ...
... -momentum tensor represents a perfect fluid, every Killing vector field preserves the energy density, pressure and the fluid flow vector field. When the energy-momentum tensor represents an ELECTROMAGNETIC FIELD , a Killing vector field does not necessarily preserve the electric and magneti ...
... metry is important in deriving the Schwarzschild solution and studying its physical consequences, such as the nonexistence of gravitational radiation in a spherically pulsating star. In the COSMOLOGICAL PRINCIPLE , symmetry restricts the type of universes that are consistent with large-scal ...
... r fields whose local flow diffeomorphisms preserve some property of the space-time. Because of this restricting diffeomorphism, the behavior of objects may not be as manifestly symmetric on LARGE SCALE s. A symmetry on the space-time is a smooth vector field whose local flow diffeomorphisms ...
... preserve the energy-momentum tensor, which are referred to as matter collineations or matter symmetries. It may be also shown that every Killing vector field is a matter collineation by the Einstein field equations. Therefore, a vector field that preserves the metric necessarily preserves ...
... density, pressure and the fluid flow vector field. When the energy-momentum tensor represents an electromagnetic field, a Killing vector field does not necessarily preserve the electric and MAGNETIC FIELDS . Read Other Books: The Single Monad Model of the Cosmos: Ibn Ar ...
... tum tensor, or other aspects of the space-time, including its geodesic structure. The vector fields are called collineations or symmetries, and the set of all symmetry vector fields forms a Lie algebra under the Lie bracket operation, if the smoothness condition is fulfilled. A Killing vec ...
... ex-Time and Quantum Gravity by Mohamed Haj Yousef Search Inside this Book III.1.4 Space-Time Symmetries and Diffeomorphisms In General Relativity, space-time symmetries are used in studying EXACT SOLUTION s of Einstein s field equations, and they are different from internal symmetries. For ...
... ed. A Killing vector field is one of the most important types of symmetries and is defined to be a smooth vector field that preserves the metric tensor. Killing vector fields are related to CONSERVATION LAWS , and they have extensive applications, also in classical mechanics. An affine vect ...
... ed. A Killing vector field is one of the most important types of symmetries and is defined to be a smooth vector field that preserves the metric tensor. Killing vector fields are related to CONSERVATION LAW s, and they have extensive applications, also in classical mechanics. An affine vect ...
... Einstein s field equations, and they are different from internal symmetries. For example, the role of spherical symmetry is important in deriving the Schwarzschild solution and studying its PHYSICAL CONSEQUENCES , such as the nonexistence of gravitational radiation in a spherically pulsatin ...
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.
