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micomaco86572
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Can any continuous coordinate transformation on a differential manifold be viewed as a poincare transformation locally in every tangent space of this manifold?
Thx!
Thx!
arkajad said:This is what you do in gauge theories of the Poincare group.
http://www.springerlink.com/content/qv46n02uq4301315/" by K. Pilch.
micomaco86572 said:Yeah，so I am thinking about whether the local poincare transformation includes all the continuous coordinate transformations except the scale transformation mentioned above.
arkajad said:Infinitesimally, we can expand a vector field (infinitesimal diffeomorphism) as:
[tex]\xi(x+h)^\mu =\xi^\mu (x)+a^\mu_\nu h^\nu +...[/tex]
To kill the unwanted degrees of freedom we impose the condition of "no scalings" which amounts to assuming that [tex]a^\mu_\nu[/tex] is a Lorentz matrix.
bcrowell said:Are you claiming that this works on a one-dimensional manifold?
The Local Poincare Transformation is a mathematical concept used in physics to describe the behavior of physical systems under changes in reference frames. It is based on the principles of relativity and is used to understand the effects of motion on space and time.
The Global Poincare Transformation describes the transformation of space and time between two inertial reference frames, while the Local Poincare Transformation takes into account the changes in space and time at each point in a non-inertial frame. In other words, the Local Poincare Transformation is more general and can be applied to accelerated frames of reference.
The Local Poincare Transformation is a fundamental concept in physics, as it allows us to understand the behavior of physical systems in different reference frames. It is essential in the development of theories such as relativity and quantum mechanics, and is used in a wide range of fields including astrophysics, particle physics, and cosmology.
In experiments, the Local Poincare Transformation is used to correct for the effects of motion and acceleration on measurements. This is crucial in high-precision experiments, where even small changes in reference frames can have significant impacts on the results. By applying the Local Poincare Transformation, scientists can account for these effects and make more accurate measurements.
Like any mathematical concept, the Local Poincare Transformation has its limitations. It is based on the assumptions of special relativity and may not accurately describe the behavior of physical systems in extreme conditions, such as near black holes or at the quantum scale. Additionally, the Local Poincare Transformation does not take into account the effects of gravity, which requires the use of more complex mathematical frameworks.