Non-Zero Coriolis Field in Ehlers’ Frame Theory
Ehlers’ Frame Theory is a class of geometric theories parameterized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>:</mo><mo>=</mo><mn>1</mn&g...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Galaxies |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-4434/13/2/38 |
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| Summary: | Ehlers’ Frame Theory is a class of geometric theories parameterized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>:</mo><mo>=</mo><mn>1</mn><mo>/</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> and identical to the General Theory of Relativity for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>→</mo><mn>0</mn></mrow></semantics></math></inline-formula> does not recover Newtonian gravity, as one might expect, but yields the so-called Newton–Cartan theory of gravity, which is characterized by a second gravitational field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">ω</mi></semantics></math></inline-formula>, called the Coriolis field. Such a field encodes at a non-relativistic level the dragging feature of general spacetimes, as we show explicitly for the case of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> geometries. Taking advantage of the Coriolis field, we apply Ehlers’ theory to an axially symmetric distribution of matter, mimicking, for example, a disc galaxy, and show how its dynamics might reproduce a flattish rotation curve. In the same setting, we further exploit the formal simplicity of Ehlers’ formalism in addressing non-stationary cases, which are remarkably difficult to treat with the General Theory of Relativity. We show that the time derivative of the Coriolis field gives rise to a tangential acceleration which allows for studying a possible formation in time of the rotation curve’s flattish feature. |
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| ISSN: | 2075-4434 |