The Einstein Condition and Einstein's Equation
In local coordinates the condition that (M, g) be an Einstein manifold is simply
Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by
where n is the dimension of M.
In general relativity, Einstein's equation with a cosmological constant Λ is
written in geometrized units with G = c = 1. The stress-energy tensor Tab gives the matter and energy content of the underlying spacetime. In a vacuum (a region of spacetime with no matter) Tab = 0, and one can rewrite Einstein's equation in the form (assuming n > 2):
Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.
Read more about this topic: Einstein Manifold
Famous quotes containing the words einstein, condition and/or equation:
“God is subtle, but he is not malicious.
[Raffiniert ist der Herr Gott, aber boshaft ist er nicht.]”
—Albert Einstein (1879–1955)
“An Englishman, methinks,—not to speak of other European nations,—habitually regards himself merely as a constituent part of the English nation; he is a member of the royal regiment of Englishmen, and is proud of his company, as he has reason to be proud of it. But an American—one who has made tolerable use of his opportunities—cares, comparatively, little about such things, and is advantageously nearer to the primitive and the ultimate condition of man in these respects.”
—Henry David Thoreau (1817–1862)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)