Does the Einstein -Hilbert Lagrangian has a potential term?

In summary, the Lagrangian has a potential term that gives the field source. This potential term is not renormalisable.
  • #1
Karlisbad
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0
If you can derive the Eisntein equations from:

[tex] L=\int_{V} d^{4}x\sqrt (-g)R [/tex] but does L has a potential term so we can do Qm with it?..:confused: :confused:
 
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  • #2
Nope, the lagrangian is purely kinetic at first sight. However, perturbative expansion reveals an infinite series in the (self)coupling constant for the graviton, [itex]\kappa[/itex]. So you can think of the Pauli-Fierz lagrangian as the purely kinetic term and the rest of the lagrangian (depending on powers of [itex] \kappa [/itex] from 1 to infinity) as the potential one.

Daniel.
 
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  • #3
Yes you can to QM with it, Feyman derived the rules for the expansion. The problem is that it is not renormalisable

So you can think of the Pauli-Fierz lagrangian as the purely kinetic term and the rest of the lagrangian as the potential one.

This does not make sense. The 00 component of the stress tensor will give the energy density, integrted to give the energy. The expansion will give corrections to the interactions, however as said, these are not renormalisable. The first thing one is required to do is construct an interacting theory from gauge symmetry arguements, as these are renormalisable and at present is the only theoretically consistent way of non-empirically including interactions. This has not been done for gravitation.
 
  • #4
Who said anything about the stress tensor of the field? I was merely talking about a series expansion of the [itex]\sqrt{|g|} \ R [/itex] aroud a flat spacetime metric, the usual Minkowski metric.

The cubic term, the quartic term, etc. can all be viewed as self-interaction terms, hence can be considered potential energy, just like the [itex]\frac{\lambda}{4!}\varphi^{4} [/itex] can be considered that way for the scalar field theory.

Daniel.
 
  • #5
Hi, I am not sure, but your Lagrangian would give "free field", where there is no generating source for the gravitational field (you would have "R-gR/2=0") - like for gravitation waves. Actually, any field can be added as "second" term to you Lagrangian (even QM Lagrangian), which would end up on right side of Einstein equations as the "field source".
 

1. What is the Einstein-Hilbert Lagrangian?

The Einstein-Hilbert Lagrangian is a mathematical equation used in the theory of general relativity to describe the dynamics of gravity. It is a fundamental part of Einstein's field equations and is used to determine the curvature of spacetime in the presence of matter and energy.

2. Does the Einstein-Hilbert Lagrangian have a potential term?

Yes, the Einstein-Hilbert Lagrangian does have a potential term. This term is known as the cosmological constant and is represented by the Greek letter lambda (λ). It is used to account for the observed expansion of the universe and is a key component in understanding the behavior of gravity on a large scale.

3. How is the potential term in the Einstein-Hilbert Lagrangian related to dark energy?

The potential term in the Einstein-Hilbert Lagrangian is closely related to dark energy, which is a hypothetical form of energy that is believed to be responsible for the accelerating expansion of the universe. The cosmological constant, represented by the potential term in the Lagrangian, is one possible explanation for dark energy.

4. Can the potential term in the Einstein-Hilbert Lagrangian be modified?

Yes, the potential term in the Einstein-Hilbert Lagrangian can be modified to account for different theories of gravity. For example, in modified gravity theories such as f(R) gravity, the form of the potential term is altered to include higher-order terms in the curvature of spacetime.

5. What are the implications of the potential term in the Einstein-Hilbert Lagrangian?

The potential term in the Einstein-Hilbert Lagrangian has significant implications for our understanding of the universe and the behavior of gravity. It is a crucial component in the theory of general relativity and plays a key role in describing the dynamics of the universe on a large scale. The precise form and value of the potential term also have a profound impact on other areas of physics, such as cosmology and the study of black holes.

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