Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.
---- John Wheeler
Einstein Tensor
The Einstein tensor
on a semi-Riemannian manifold is defined as where is the Ricci curvature tensor and is the Ricci scalar.
Proposition
The Einstein tensor satisfies the equation
where is the Levi-Civita connection associated with the metric .
The Energy Momentum Tensor
In general relativity, continuous matter distributions and fields are described by a tensor of type
Energy-Momentum Tensor
The energy-momentum tensor is a
-tensor such that
- It is symmetric:
; - It vanishes on an open set
if and only if all the matter fields vanish on ; - Obeys the equation
.
Einstein Field Equation
The Einstein field equation is
where is the Einstein tensor, and is the energy-momentum tensor.
Remark
The Einstein field equation gives a relation between the geometry of spacetime and the matter (energy) content of the universe.
Vacuum Condition
Vacuum Space
A vacuum space is a spacetime with no matter or energy present, i.e.
.