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XI. Field Equations
In the Newton theory of gravitation, mass is the source of the
gravitational force. This suggests for GRT that mass, or rather energy
since it is a conserved quantity, is affecting geometry. However, in
relativity theory, energy is only conserved in combination with
momentum. Therefore Einstein surmised that the energy-momentum
tensor ${\mathbf{T}} = {\text{T(}}{{\mathbf{g}}_\mu })$ , with
components ${T^{\mu \nu }}$, is the frame-independent geometric
object that must act as the source of gravity.
It is a theorem of relativistic field theories that the
energy-momentum tensor is symmetric: ${T^{\mu \nu }} =
{T^{\nu \mu }}$; this makes index raising/lowering unambiguous.
Moreover, to uphold the law of conservation of energy-momentum, this
tensor must be covariantly conserved
\[{D_{_\mu }}{T^{\mu \nu }} =
0{\qquad}{D_\mu }\operatorname{T} {\text{(}}{{\mathbf{g}}^\mu }) =
0\]
In the now famous article of November 25, 1915, Albert Einstein
derived his gravitational field equation by setting the
Einstein tensor proportional to the energy-momentum tensor:
\[{G_{\mu \nu }} =
\kappa {T_{\mu \nu }}{\qquad} \operatorname{G} ({{\mathbf{g}}_\mu })
= \kappa \operatorname{T} {\text{(}}{{\mathbf{g}}_\mu })\]
The left-hand side describes the geometry of space-time; the
right-hand side the matter content of the universe. This simple
looking equation actually stands for a complicated set of second
order non-linear differential equations that relate the local
spacetime curvature, i.e. the metric tensor, to the local density
and flow of energy and momentum, quantified by the energy-momentum
tensor. Hence, one usually speaks of Einstein equations (plural).
The Einstein gravitational constant $\kappa$ is a
proportionality factor that is determined by a comparison with the
Newton theory of gravitation in the appropriate limit. This yields
$\kappa = 8\pi $, or in physical units $\kappa = 8\pi G/{c^4}$, with
Newtons gravitational constant and the speed of light, made explicit.
This constant is extremely small:
$\kappa \approx 2 \times {10^{ - 43}}{\text{ }}{{\text{N}}^{ - 1}}$.
Only in the presence of matter under extreme conditions, the effect of gravity
on space and time can be strong.