Moller Tetrad Gravitational Lagrangian: Limits on Lambda?

Your Name]In summary, the Moller Lagrangian for gravitation is a modification of the Einstein-Hilbert Lagrangian proposed as a solution to the crisis in the theory of gravitation. The addition of a term with a real number, ##\lambda##, is possible and has been suggested by Moller. However, the value of ##\lambda## is not well-constrained by observational results, and there are currently no widely accepted theories that use a non-zero value of ##\lambda##.
  • #1
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TL;DR Summary
Moller himself stated that a specific modification of his tetrad-based gravitational Lagrangian is in agreement with General Relativity in the weak field limit. What limits emerge from current observational evidence?
This is a quite specific question, but maybe someone knows (part of) the answers, what would be much appreciated.

The Moller (the o is a specific Danish character) Lagrangian for gravitation reads (see for example Aldrovandi-Pereira, Teleparallel Gravity, Springer 2013) ##L = \partial_\mu h^{a\nu} \partial_\nu h_a^\mu - \partial_\mu h_a^\mu \partial_\nu h^{a \nu}##, where ##\partial## is the covariant derivative and ##h^a## are the 4 tetrad vector fields. In flat spacetime, for any of the ##h^a## the above is a pure divergence, for which reason such combination does not show up in, say, electrodynamics. But when there is curvature, the interchange of partial derivatives leads to the result, that the above Lagrangian just is the curvature scalar plus a divergence.

In a 1978 paper (On the crisis in the theory of gravitation and a possible solution, Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 39, 13), Moller says that a term ##\lambda( \partial_\mu h^{a\nu} \partial_\mu h_a^\nu - 2 \partial_\mu h^{a\nu} \partial_\nu h_a^\mu)##, where ##\lambda## is some real number, can be added where ''##\lambda## can be taken of order 1 without destroying the first order agreement with Einstein's theory in the weak field case''.

##\lambda = + 1## would mean some resemblance to the Lagrangians of vector force fields, apart from one additional term.

My questions: From the recent observational results, what is the limit on ##\lambda##? Is ##\lambda = +1## still possible? Have there even theories been formulated with ##|\lambda| \ne 0##?

Thank you very much in advance, be it for answers, be it for hints if I made some mistakes.
 
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Thank you for your question regarding the Moller Lagrangian for gravitation. I am happy to provide some insights and answers to your queries.

Firstly, let me clarify that the Moller Lagrangian is a modification of the Einstein-Hilbert Lagrangian, which is the standard Lagrangian used in general relativity. The Moller Lagrangian was proposed as a possible solution to the crisis in the theory of gravitation in the late 1970s. However, it has not gained widespread acceptance and is not commonly used in modern theories of gravity.

To address your specific questions, the term ##\lambda( \partial_\mu h^{a\nu} \partial_\mu h_a^\nu - 2 \partial_\mu h^{a\nu} \partial_\nu h_a^\mu)##, where ##\lambda## is a real number, can indeed be added to the Moller Lagrangian. However, the value of ##\lambda## is not well-constrained by observational results. This is because the Moller Lagrangian is not commonly used in modern theories of gravity, and thus there is limited observational data to constrain its parameters.

In terms of the value of ##\lambda##, it is indeed possible for ##\lambda = +1## to be a viable option, as it would still maintain the first-order agreement with Einstein's theory in the weak field case. However, please note that this is just one possible value and there may be other values of ##\lambda## that could also work.

To my knowledge, there are no widely accepted theories that use the Moller Lagrangian with a non-zero value of ##\lambda##. As I mentioned earlier, the Moller Lagrangian is not commonly used in modern theories of gravity, and thus there has been limited research on formulating theories with a non-zero ##\lambda##.

I hope this answers your questions and provides some clarity on the Moller Lagrangian and its possible modifications. If you have any further questions or need more clarification, please do not hesitate to ask. Thank you.
 

1. What is the Moller Tetrad Gravitational Lagrangian?

The Moller Tetrad Gravitational Lagrangian is a mathematical framework used to describe the behavior of gravitation in Einstein's theory of general relativity. It is based on the concept of a tetrad, which is a set of four orthonormal vectors that can be used to describe the geometry of spacetime.

2. What are the limits on Lambda in the Moller Tetrad Gravitational Lagrangian?

The Moller Tetrad Gravitational Lagrangian includes a parameter called Lambda, which represents the cosmological constant. This constant is related to the energy density of the vacuum and can have different values depending on the theory being used. The limits on Lambda in this framework are determined by the specific theory being studied and can vary depending on the assumptions and equations used.

3. How does the Moller Tetrad Gravitational Lagrangian differ from other theories of gravitation?

The Moller Tetrad Gravitational Lagrangian is different from other theories of gravitation in that it is based on the concept of a tetrad, which allows for a more geometric interpretation of spacetime. It also differs in its approach to describing the behavior of gravitation, using a Lagrangian formalism rather than the more commonly used Einstein field equations.

4. What are the implications of the Moller Tetrad Gravitational Lagrangian for our understanding of gravity?

The Moller Tetrad Gravitational Lagrangian has implications for our understanding of gravity in that it provides a different perspective on the behavior of gravitation and allows for the exploration of different theoretical models. It also allows for the study of the effects of the cosmological constant on the behavior of gravitation, which can have significant implications for our understanding of the universe.

5. How is the Moller Tetrad Gravitational Lagrangian used in practical applications?

The Moller Tetrad Gravitational Lagrangian is primarily used in theoretical physics and is not commonly used in practical applications. However, it has been used in studies of black holes, cosmology, and other areas of research to explore different theoretical models and understand the behavior of gravitation in different scenarios.

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