Einstein-Hilbert Action, Lagrangian Density & Vacuum Fields

Click For Summary
SUMMARY

The discussion centers on the Einstein-Hilbert action, specifically the relationship between the Lagrangian density (ι) and the Ricci scalar (R) in the context of vacuum gravitational fields. It is established that the Lagrangian density for vacuum is effectively represented by the Ricci scalar, leading to the field equations in vacuum: (Rij - 1/2Rgij) = 0. The action is defined as an integral over spacetime of a scalar function multiplied by the spacetime volume element, with the Ricci scalar being the simplest invariant derived from the Riemann tensor. This formulation is crucial for the consistency and effectiveness of General Relativity (GR).

PREREQUISITES
  • Understanding of General Relativity (GR) principles
  • Familiarity with the concept of the Ricci scalar (R)
  • Knowledge of the Riemann tensor and its properties
  • Basic grasp of variational principles in physics
NEXT STEPS
  • Study the derivation of the Einstein-Hilbert action in detail
  • Explore the implications of the Ricci scalar in different gravitational contexts
  • Learn about the role of the Riemann tensor in General Relativity
  • Investigate alternative formulations of gravitational theories beyond GR
USEFUL FOR

The discussion is beneficial for theoretical physicists, cosmologists, and advanced students of physics who are delving into the foundations of General Relativity and the mathematical structures underlying gravitational theories.

Apashanka
Messages
427
Reaction score
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??
 
Physics news on Phys.org
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.
 

Similar threads

Undergrad Einstein Hilbert action
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Boundary conditions in ##\delta I=0## to derive Einstein's equations
  • · Replies 8 ·
Replies
8
Views
1K
Graduate Why is 1/G Used in GR Lagrangian?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Differences between Actions in Curved Spacetime
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Einstein Hilbert action integral
  • · Replies 32 ·
2
Replies
32
Views
6K
Undergrad Lagrangian density of a linearly elastic string under gravity
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Why Do the Terms in the Lagrangian Action Have the Same Dimension?
  • · Replies 5 ·
Replies
5
Views
6K
Graduate Hamiltonian of General Relativity
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Is the Vacuum Energy Density Sign Correct for Bosons?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad A "continous" gravitational field formula r=0 to infinity
  • · Replies 16 ·
Replies
16
Views
1K