Einstein-Hilbert Action, Lagrangian Density & Vacuum Fields

In summary, the conversation discusses the use of the Ricci scalar in the action for the gravitational field, as it yields field equations in vacuum. It is suggested that this is due to the fact that the lagrangian density corresponding to vacuum is the Ricci scalar, which is the simplest possibility when constructing invariants from the Riemann tensor.
  • #1
Apashanka
429
15
From the action ∫Ldt =∫ι√|g|d4x where |g| is the determinant of the metric .and ι the lagrangian density.
For gravitational field why is this ι is replaced by the Ricci scaler R which yield field equations in vaccum.(Rij-1/2Rgij)=0
Is it that the lagrangian density corresponding to vacuum is the Ricci scaler??
 
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  • #2
The action defines the model. Your action must be of the form of an integral over spacetime of some scalar (not "scaler") function multiplied by the spacetime volume element. Having only the metric to play with, you can essentially only end up with invariants constructed from the Riemann tensor. The simplest (apart from just a constant) possibility is then the Ricci scalar. Technically you could input any scalar function constructed from the metric and Riemann tensor, but that will result in a different model and GR works very well.
 

1. What is the Einstein-Hilbert action?

The Einstein-Hilbert action is a mathematical expression used in the theory of general relativity to describe the dynamics of gravity. It is a function of the spacetime metric, which describes the curvature of spacetime, and the cosmological constant, which represents the energy density of empty space.

2. What is a Lagrangian density?

A Lagrangian density is a mathematical quantity used in the Lagrangian formulation of classical mechanics and field theory. It is a function of the fields and their derivatives, and it describes the dynamics of a physical system. In the context of general relativity, the Lagrangian density is used to describe the dynamics of the gravitational field.

3. How are the Einstein-Hilbert action and Lagrangian density related?

The Einstein-Hilbert action is a specific form of the Lagrangian density used in general relativity. It is obtained by integrating the Lagrangian density over spacetime, and it is a function of the metric and its derivatives. The Einstein-Hilbert action is used to derive the equations of motion for the gravitational field.

4. What are vacuum fields?

Vacuum fields, also known as vacuum solutions, are solutions to the equations of motion for the gravitational field in the absence of matter or energy. In other words, they describe the curvature of spacetime in the absence of any sources of gravity. These solutions are important in general relativity as they can help us understand the behavior of the gravitational field in different scenarios.

5. How does the Einstein-Hilbert action describe the behavior of vacuum fields?

The Einstein-Hilbert action includes a term that represents the energy density of empty space, known as the cosmological constant. This term affects the behavior of the gravitational field in the absence of matter or energy, and it can lead to the expansion or contraction of spacetime. The equations of motion derived from the Einstein-Hilbert action can be used to study the evolution of vacuum fields and their impact on the overall curvature of spacetime.

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