Theoretical Motivation For General Relativity - Einstein Field Equation

Einstein Field Equation

We obtain the Einstein field equation by equating the acceleration required for circular orbits with the acceleration due to gravity

.

This is the relationship between curvature of spacetime and the stress-energy tensor.

The Ricci tensor becomes

.

The trace of the Ricci tensor is

.

Comparison of the Ricci tensor with the Ricci tensor calculated from the principle of least action, Theoretical motivation for general relativity#Principle of least action in general relativity identifying the stress-energy tensor with the Hilbert stress-energy, and remembering that A+B=1 removes the ambiguity in A, B, and C.

and

.

This gives

.

The field equation can be written

where

.

This is the Einstein field equation that describes curvature of spacetime that results from stress-energy density. This equation, along with the geodesic equation have motivated by the kinetics and dynamics of a particle orbiting the earth in a circular orbit. They are true in general.

Read more about this topic:  Theoretical Motivation For General Relativity

Famous quotes containing the words einstein, field and/or equation:

    When you are courting a nice girl an hour seems like a second. When you sit on a red-hot cinder a second seems like an hour. That’s relativity.
    —Albert Einstein (1879–1955)

    Mine was, as it were, the connecting link between wild and cultivated fields; as some states are civilized, and others half-civilized, and others savage or barbarous, so my field was, though not in a bad sense, a half-cultivated field. They were beans cheerfully returning to their wild and primitive state that I cultivated, and my hoe played the Ranz des Vaches for them.
    Henry David Thoreau (1817–1862)

    A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.
    Norman Mailer (b. 1923)